California Maths Grade 7

interactive student edition Authors Day • Frey • Howard • Hutchens Luchin • McClain • Molix-Bailey Ott • Pelfrey • Price Vielhaber • Willard (t)Created by Michael Trott with Mathematica. From Graphica 1, Copyright ©1999 Wolfram Media, Inc., (b)Richard Cummins/SuperStock About the Cover The sailboard was invented by Newman Drake in the 1950s, and the sport was popularized by Californians Jim Drake and Hoyle Schweitzer in the 1970s. The sport spread rapidly from California, throughout the United States and around the world. By standing on the rudderless board and maneuvering the sail to glide along the water’s surface, a sailboarder can reach speeds of up to 45 miles per hour. In Chapter 4, you will learn to solve problems involving rate, speed, and distance. About the Graphics Twisted torus. Created with Mathematica. A torus with rose-shaped cross section is constructed. Then the cross section is rotated around its center as it moves along a circle to form a twisted torus. For more information, and for programs to construct such graphics, see: www.wolfram.com/r/textbook. Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-877850-6 MHID: 0-07-877850-6 Printed in the United States of America. 3 4 5 6 7 8 9 10 043/079 16 15 14 13 12 11 10 09 08 07 Start Smart: Be a Better Problem Solver Unit 1 Number and Operations: Rational and Real Numbers 1 Algebra: Integers 2 Algebra: Rational Numbers 3 Real Numbers and the Pythagorean Theorem Unit 2 Patterns, Relationships, and Algebraic Thinking 4 Proportions and Similarity 5 Percent Unit 3 Geometry and Measurement 6 Geometry and Spatial Reasoning 7 Measurement: Area and Volume Unit 4 Algebraic Thinking: Linear and Nonlinear Functions 8 Algebra: More Equations and Inequalities 9 Algebra: Linear Functions 10 Algebra: Nonlinear Functions and Polynomials Unit 5 Statistics, Data Analysis, and Probability 11 Statistics Standards Review 12 Looking Ahead to Grade 8: Probability iii Authors Rhonda J. Molix-Bailey Mathematics Consultant Mathematics by Design DeSoto, Texas Roger Day, Ph.D. Mathematics Department Chair Pontiac Township High School Pontiac, Illinois Patricia Frey, Ed.D. Math Coordinator at Westminster Community Charter School Buffalo, New York Arthur C. Howard Mathematics Teacher Houston Christian High School Houston, Texas Deborah A. Hutchens, Ed.D. Principal Chesapeake, Virginia Beatrice Luchin Mathematics Consultant League City, Texas Contributing Author Viken Hovsepian Professor of Mathematics Rio Hondo College Whittier, California iv Aaron Haupt Meet the Authors at ca.gr7math.com Kay McClain, Ed.D. Assistant Professor Vanderbilt University Nashville, Tennessee Jack M. Ott, Ph.D. Distinguished Professor of Secondary Education Emeritus University of South Carolina Columbia, South Carolina Ronald Pelfrey, Ed.D. Mathematics Specialist Appalachian Rural Systemic Initiative and Mathematics Consultant Lexington, Kentucky Jack Price, Ed.D. Professor Emeritus California State Polytechnic University Pomona, California Kathleen Vielhaber Mathematics Consultant St. Louis, Missouri Teri Willard, Ed.D. Assistant Professor Department of Mathematics Central Washington University Ellensburg, Washington Contributing Author Dinah Zike Educational Consultant Dinah-Might Activities, Inc. San Antonio, Texas v Aaron Haupt California Mathematics Advisory Board Glencoe wishes to thank the following professionals for their invaluable feedback during the development of the program. They reviewed the table of contents, the prototype of the Teacher Wraparound Edition, and the California Standards Review chapter. Cheryl L. Avalos Mathematics Consultant Retired Teacher Hacienda Heights, California William M. Bokesch Rancho Bernardo High School San Diego, California Patty Brown Teacher John Muir Elementary Fresno, California David J. Chamberlain Secondary Mathematics Resource Teacher Capistrano Unified School District San Juan Capistrano, California Eppie Chung K-6 Teacher Modesto City Schools Modesto, California Lisa Marie Cirrincione Middle School Teacher Lincoln Middle School Oceanside, California Carol Cronk Mathematics Program Specialist San Bernardino City Unified School District San Bernardino, California Ilene Foster Teacher Specialist– Mathematics Pomona Unified School District Pomona, California Grant A. Fraser, Ph. D. Professor of Mathematics California State University, Los Angeles Los Angeles, California vi Suzanne Bocskai Freire Teacher Kingswood Elementary Citrus Heights, California Beth Holguin Teacher Graystone Elementary San Jose, California Donna M. Kopenski, Ed. D. Mathematics Coordinator K-5 City Heights Educational Collaborative San Diego, California Kelly Mack 6th Grade Teacher Captain Jason Dahl Elementary San Jose, California Juvenal Martinez Dual Immersion/ESL Instructor Aeolian Elementary Whittier, California John McGuire Associate Principal Pacific Union School Arcata, California Dr. Donald R. Price Teacher, Adjunct Professor Rowland Unified School District Rowland Heights, California Kasey St. James Mathematics Teacher Sunny Hills High School Fullerton, California Arthur K. Wayman, Ph. D. Professor of Mathematics Emeritus California State University, Long Beach Long Beach, California Beverly Wells First Grade Teacher Mineral King Elementary School Visalia, California Frances Basich Whitney Project Director, Mathematics K-12 Santa Cruz County Office of Education Capitola, California vii Consultants Glencoe/McGraw-Hill wishes to thank the following professionals for their feedback. They were instrumental in providing valuable input toward the development of this program in these specific areas. Mathematical Content Graphing Calculator Viken Hovsepian Professor of Mathematics Rio Hondo College Whittier, California Ruth M. Casey Mathematics Teacher Department Chair Anderson County High School Lawrenceburg, Kentucky Grant A. Fraser, Ph. D. Professor of Mathematics California State University, Los Angeles Los Angeles, California Arthur K. Wayman, Ph. D. Professor of Mathematics Emeritus California State University, Long Beach Long Beach, California Differentiated Instruction Nancy Frey, Ph. D. Associate Professor of Literacy San Diego State University San Diego, California English Language Learners Mary Avalos, Ph. D. Assistant Chair, Teaching and Learning Assistant Research Professor University of Miami, School of Education Coral Gables, Florida Jana Echevarria, Ph. D. Professor, College of Education California State University, Long Beach Long Beach, California Josefina V. Tinajero, Ph. D. Dean, College of Education The University of Texas at El Paso El Paso, Texas Gifted and Talented Ed Zaccaro Author Mathematics and science books for gifted children Bellevue, Iowa viii Jerry Cummins Former President National Council of Supervisors of Mathematics Western Springs, Illinois Learning Disabilities Kate Garnett, Ph. D. Chairperson, Coordinator Learning Disabilities School of Education Department of Special Education Hunter College, CUNY New York, New York Mathematical Fluency Jason Mutford Mathematics Instructor Coxsackie-Athens Central School District Coxsackie, New York Pre-AP Dixie Ross AP Calculus Teacher Pflugerville High School Pflugerville, Texas Reading and Vocabulary Douglas Fisher, Ph. D. Director of Professional Development and Professor City Heights Educational Collaborative San Diego State University San Diego, California Lynn T. Havens Director of Project CRISS Kalispell School District Kalispell, Montana California Reviewers Each California Reviewer reviewed at least two chapters of the Student Edition, giving feedback and suggestions for improving the effectiveness of the mathematics instruction. Mariana Alwell Teacher & Mathematics Coach Garden Gate Elementary Cupertino, California Derrick Chun Kei Hui Certified BCLAD Mathematics Teacher Natomas Middle School Sacramento, California Rudy C. Sass Mathematics Chair Orangeview Junior High School Anaheim, California Cheryl Anderson District Mathematics Resource Teacher Cupertino School District Cupertino, California Robin Ingram Mathematics Instructor/Department Chair Alta Sierra Intermediate School Clovis, California David Schick Mathematics Teacher Wangenheim Middle School San Diego, California Aimey Balderman Mathematics Teacher Tommie Kunst Junior High School Santa Maria, California Debra C. Lonso Mathematics Teacher Dover Middle School Fairfield, California Kristine A. Banfe Mathematics Teacher Hyde Middle School Cupertino, California Roxanne Mancha Mathematics Department Chair Crystal Middle School Suisun City, California James Douglas Sherman Pre-Algebra/Algebra Instructor, 7th/8th grade Miller Middle School San Jose, California Dianne Chrisman Mathematics Teacher Coronado High School Coronado, California Mary Beth Moon Mathematics Teacher/Consulting Teacher Earl Warren Jr. High School Bakersfield, California Patricia Elmore 6th Grade Mathematics Teacher Heritage Intermediate School Fontana, California Grainne O’Malley Middle School Mathematics Coordinator Crossroad Middle School Santa Monica, California Jill Fetters Mathematics Teacher Tevis Jr. High Bakersfield, California Candice Richards Mathematics Teacher Newport Mesa Unified School District Costa Mesa, California Rosalee Hrubic Staff Development Specialist, Secondary Mathematics Riverside Unified School District Riverside, California Steven Robitaille Mathematics Instructor Trabuco Hills High School Mission Viejo, California Charles P. Toots Mathematics Department Chairsperson Le Conte Middle School Los Angeles, California Judith Vincent Teacher Cavitt Junior High School Granite Bay, California Carrie M. Wong 6th/7th Mathematics Teacher Taylor Middle School Millbrae, California ix Be a Better Problem Solver A Plan for Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problem-Solving Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 4 Multi-Step Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Tools for Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Techniques for Problem Solving . . . . . . . . . . . . . . . . . . . . 10 How to Use Your Math Book Why do I Need my Math Book? . . . . . . . . . . . . . . . . . . . . 12 Doing Your Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Studying for a Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Scavenger Hunt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 California Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 x Gabe Palmer/CORBIS CH APTER 1 Algebra: Integers A Plan for Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . 24 1-2 Variables, Expressions, and Properties . . . . . . . . . . . . . . . 29 1-3 Integers and Absolute Value . . . . . . . . . . . . . . . . . . . . . . . 35 Extend 1-3 Algebra Lab: Graphing Data . . . . . . . . . . . . . . . . . 40 1-4 Adding Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1-5 Subtracting Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1-6 Multiplying and Dividing Integers . . . . . . . . . . . . . . . . . . . 51 1-7 Writing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1-8 Problem-Solving Investigation: 1-9 Table of Contents 1-1 Prerequisite Skills • Get Ready for Chapter 1 23 • Get Ready for the Next Lesson 28, 34, 39, 45, 49, 56, 61, 69 Work Backward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Reading and Writing Mathematics Reading Word Problems: Simplify the Problem. . . . . . . 64 • Reading in the Content Area 29 • Reading Math 26, 35, 53, 71 • Reading Word Problems 64 • Vocabulary Link 31 • Writing in Math 28, 34, 39, 45, 49, 56, 61, 69, 73 Solving Addition and Subtraction Equations . . . . . . . . . . 65 1-10 Solving Multiplication and Division Equations . . . . . . . . . 70 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . . 74 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 California Standards Practice . . . . . . . . . . . . . . . . . . . 80 California Standards Practice • Multiple Choice 28, 34, 39, 45, 49, 56, 59, 61, 69, 73 • Worked Out Example 58 H.O.T. Problems Higher Order Thinking • Challenge 28, 33, 39, 45, 49, 56, 60, 69, 73 • Find the Error 34, 49, 61 • Number Sense 56, 73 • Open Ended 28, 33, 45, 49, 56, 69, 73 • Select a Technique 28 • Which One Doesn’t Belong? 39, 69 xi CH APTER 2 Algebra: Rational Numbers 2-1 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Reading Word Problems: New Vocabulary . . . . . . . . . . 90 2-2 Comparing and Ordering Rational Numbers . . . . . . . . . . 91 2-3 Multiplying Positive and Negative Fractions . . . . . . . . . . . 96 2-4 Dividing Positive and Negative Fractions . . . . . . . . . . . . 102 2-5 Adding and Subtracting Like Fractions . . . . . . . . . . . . . . 108 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2-6 Adding and Subtracting Unlike Fractions . . . . . . . . . . . . 114 2-7 Solving Equations with Rational Numbers . . . . . . . . . . . 119 2-8 Problem-Solving Investigation: Look for a Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2-9 Powers and Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 2-10 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Prerequisite Skills • Get Ready for Chapter 2 83 • Get Ready for the Next Lesson 89, 95, 101, 107, 112, 118, 123, 129 Reading and Writing Mathematics • Reading in the Content Area 84 • Reading Word Problems 90 • Vocabulary Link 85 • Writing in Math 89, 95, 101, 107, 111, 117, 122, 124, 129, 133 California Standards Practice • Multiple Choice 89, 95, 101, 107, 112, 116, 118, 123, 129, 133 • Worked Out Example 115 H.O.T. Problems Higher Order Thinking • Challenge 89, 95, 101, 106, 111, 117, 122, 129, 133 • Find the Error 100, 111 • Number Sense 95, 107, 117, 129, 133 • Open Ended 88, 95, 101, 106, 111, 117, 122, 129 • Which One Doesn’t Belong? 88, 122 xii Tom Brakefield/CORBIS ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . 134 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 California Standards Practice . . . . . . . . . . . . . . . . . . 140 CH APTER 3 Real Numbers and the Pythagorean Theorem 3-1 Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3-2 Estimating Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . 148 3-3 Problem–Solving Investigation: Use a Venn Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Reading Word Problems: The Language of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3-4 The Real Number System . . . . . . . . . . . . . . . . . . . . . . . . 155 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Geometry Lab: The Pythagorean Theorem . . . . 161 3-5 The Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . 162 3-6 Using the Pythagorean Theorem . . . . . . . . . . . . . . . . . . 167 Extend 3-6 3-7 Table of Contents Explore 3-5 Geometry Lab: Graphing Irrational Numbers . . 172 Geometry: Distance on the Coordinate Plane . . . . . . . . 173 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . 179 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 California Standards Practice . . . . . . . . . . . . . . . . . . 184 Prerequisite Skills • Get Ready for Chapter 3 143 • Get Ready for the Next Lesson 147, 151, 159, 166, 171 Reading and Writing Mathematics • Reading in the Content Area 144 • Reading Math 148 • Reading Word Problems 154 • Writing in Math 147, 151, 152, 159, 166, 171, 177 California Standards Practice • Multiple Choice 147, 151, 159, 166, 169, 171, 178 • Worked Out Example 168 H.O.T. Problems Higher Order Thinking • Challenge 147, 151, 159, 166, 171, 177 • Find the Error 151, 166 • Number Sense 147, 150 • Open Ended 147, 151, 159, 166, 170 • Select a Tool 177 • Which One Doesn’t Belong? 170 xiii CH APTER 4 Proportions and Similarity 4-1 Ratios and Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4-2 Proportional and Nonproportional Relationships . . . . . 194 4-3 Solving Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Extend 4-3 4-4 Geometry Lab: The Golden Rectangle . . . . . . . .203 Problem–Solving Investigation: Draw a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204 4-5 Similar Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Prerequisite Skills • Get Ready for Chapter 4 189 • Get Ready for the Next Lesson 193, 197, 202, 211, 218, 223, 229, 235 4-6 Extend 4-6 California Standards Practice • Multiple Choice 193, 197, 202, 209, 210, 211, 218, 223, 229, 235, 241 • Worked Out Example 209 H.O.T. Problems Higher Order Thinking • Challenge 193, 197, 202, 211, 218, 223, 229, 235, 241 • Find the Error 218, 228 • Number Sense 235 • Open Ended 197, 202, 228, 235, 241 • Which One Doesn’t Belong? 193, 223 xiv Steve Vidler/SuperStock Spreadsheet Lab: Converting Measures . . . . . . 219 4-7 Measurement: Converting Square Units and Cubic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4-8 Scale Drawings and Models . . . . . . . . . . . . . . . . . . . . . . 224 4-9 Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Reading and Writing Mathematics • Reading in the Content Area 195 • Reading Math 190, 191, 207, 220 • Writing in Math 193, 197, 202, 204, 211, 218, 223, 229, 235, 241 Measurement: Converting Length, Weight/Mass, Capacity, and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 4-10 Constant Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . 236 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . .242 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 California Standards Practice . . . . . . . . . . . . . . . . . .248 CH APTER 5 Percent 5-1 Ratios and Percents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 5-2 Comparing Fractions, Decimals, and Percents . . . . . . . . 256 Reading Word Problems: Comparing Data . . . . . . . .262 5-3 Algebra: The Percent Proportion . . . . . . . . . . . . . . . . . . .263 5-4 Finding Percents Mentally . . . . . . . . . . . . . . . . . . . . . . . .268 5-5 Problem-Solving Investigation: Reasonable Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Percent and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 275 5-7 Algebra: The Percent Equation . . . . . . . . . . . . . . . . . . . . 279 5-8 Percent of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284 5-9 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .290 Table of Contents 5-6 Extend 5-9 Spreadsheet Lab: Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . .295 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .299 Prerequisite Skills • Get Ready for Chapter 5 251 • Get Ready for the Next Lesson 255, 261, 267, 271, 278, 283, 289 California Standards Practice . . . . . . . . . . . . . . . . . .300 Reading and Writing Mathematics • Reading in the Content Area 263 • Reading Math 264, 275 • Reading Word Problems 262 • Writing in Math 255, 261, 262, 267, 271, 272, 278, 283, 289, 293 California Standards Practice • Multiple Choice 255, 261, 267, 271, 278, 283, 289, 292, 293 • Worked Out Example 291 H.O.T. Problems Higher Order Thinking • Challenge 255, 261, 267, 271, 278, 283, 289, 293 • Find the Error 260, 271, 289 • Number Sense 278 • Open Ended 255, 261, 271, 293 • Which One Doesn’t Belong? 255 xv CH APTER 6 Geometry and Spatial Reasoning 6-1 Line and Angle Relationships . . . . . . . . . . . . . . . . . . . . .306 Extend 6-1 6-2 Geometry Lab: Constructions . . . . . . . . . . . . . . . 311 Problem-Solving Investigation: Use Logical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 6-3 Polygons and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 6-4 Congruent Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 Extend 6-4 Geometry Lab: Investigating Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 Prerequisite Skills • Get Ready for Chapter 6 305 • Get Ready for the Next Lesson 310, 319, 323, 331, 336 6-5 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 6-6 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 6-7 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 ASSESSMENT Reading and Writing Mathematics • Reading in the Content Area 306 • Reading Math 307, 308, 311, 313, 321, 332 • Writing in Math 310, 314, 318, 323, 331, 336, 341 California Standards Practice • Multiple Choice 310, 319, 323, 331, 336, 339, 341 • Worked Out Example 338 H.O.T. Problems Higher Order Thinking • Challenge 318, 323, 331, 336, 341 • Open Ended 310, 336 • Reasoning 310, 341 xvi Jon Hicks/CORBIS Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . .342 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 California Standards Practice . . . . . . . . . . . . . . . . . .348 CH APTER E.S. Ross/Visuals Unlimited 7 7-1 Circumference and Area of Circles . . . . . . . . . . . . . . . . . 352 Extend 7-1 7-2 Measurement: Area and Volume Geometry Lab: Investigating Arcs and Angles . . 358 Problem-Solving Investigation: Solve a Simpler Problem . . . . . . . . . . . . . . . . . . . . . . . . .360 Explore 7-3 Measurement Lab: Area of Irregular Figures . .362 7-3 Area of Complex Figures . . . . . . . . . . . . . . . . . . . . . . . . .363 7-4 Three-Dimensional Figures . . . . . . . . . . . . . . . . . . . . . . .368 7-5 Volume of Prisms and Cylinders . . . . . . . . . . . . . . . . . . . 373 Table of Contents Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 7-6 Volume of Pyramids and Cones . . . . . . . . . . . . . . . . . . .380 Explore 7-7 Measurement Lab: Surface Area of Cylinders . . . . . . . . . . . . . . . . . . . . . . . . .385 7-7 Surface Area of Prisms and Cylinders . . . . . . . . . . . . . . .386 Extend 7-7 Measurement Lab: Net of a Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .392 7-8 Surface Area of Pyramids . . . . . . . . . . . . . . . . . . . . . . . .393 Explore 7-9 7-9 Spreadsheet Lab: Similar Solids . . . . . . . . . . . . . 397 Prerequisite Skills • Get Ready for Chapter 7 351 • Get Ready for the Next Lesson 357, 367, 372, 378, 384, 391, 396 Similar Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .399 Reading and Writing Mathematics ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . .405 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . .409 California Standards Practice . . . . . . . . 410 • Reading in the Content Area 363 • Vocabulary Link 368, 386 • Writing in Math 357, 360, 367, 372, 378, 384, 391, 396, 403 California Standards Practice • Multiple Choice 357, 367, 372, 378, 384, 391, 396, 402, 404 • Worked Out Example 401 H.O.T. Problems Higher Order Thinking • Challenge 357, 367, 372, 377, 384, 391, 396, 403 • Find the Error 377 • Number Sense 356, 384, 391 • Open Ended 356, 372, 377, 384, 396, 403 • Reasoning 391, 403 • Select a Tool 378 • Select a Technique 403 xvii CH APTER 8 Algebra: More Equations and Inequalities 8-1 Simplifying Algebraic Expressions . . . . . . . . . . . . . . . . . . 416 8-2 Solving Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . .422 8-3 Writing Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . .427 Explore 8-4 Algebra Lab: Equations with Variables on Each Side . . . . . . . . . . . . . . . . . . . . . . . . . . .432 8-4 Solving Equations with Variables on Each Side . . . . . . .434 8-5 Problem-Solving Investigation: Guess and Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .438 Prerequisite Skills Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .440 • Get Ready for Chapter 8 415 • Get Ready for the Next Lesson 421, 426, 431, 437, 444, 448 8-6 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 8-7 Solving Inequalities by Adding or Subtracting . . . . . . . .445 Reading and Writing Mathematics 8-8 Solving Inequalities by Multiplying or Dividing . . . . . . .449 • Reading in the Content Area 416 • Reading Math 441 • Vocabulary Link 417 • Writing in Math 420, 426, 431, 437, 438, 444, 448, 453 California Standards Practice • Multiple Choice 421, 426, 431, 437, 444, 447, 448, 453 • Worked Out Example 446 H.O.T. Problems Higher Order Thinking • Challenge 420, 426, 430, 437, 444, 448, 453 • Find the Error 426, 444, 453 • Number Sense 439 • Open Ended 420, 430, 437, 448, 453 • Select a Technique 431 • Which One Doesn’t Belong? 420 xviii Michael Newman/PhotoEdit ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . .454 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .459 California Standards Practice . . . . . . . . . . . . . . . . . .460 CH APTER Jonathan Daniel/Getty Image s 9 Algebra: Linear Functions Explore 9-1 Algebra Lab: Functions . . . . . . . . . . . . . . . . . . . .464 9-1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .465 Explore 9-2 Algebra Lab: Graphing Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . 470 9-2 Representing Linear Functions . . . . . . . . . . . . . . . . . . . . 471 9-3 Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 9-4 Direct Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .483 Table of Contents Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .489 Explore 9-5 Algebra Lab: Proportional and Nonproportional Functions . . . . . . . . . . . . . . . . . . . . . . .490 9-5 Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 Extend 9-5 Graphing Calculator Lab: Modeling Linear Behavior . . . . . . . . . . . . . . . . . . . . . . . .496 9-6 Writing Systems of Equations and Inequalities . . . . . . .498 9-7 Problem-Solving Investigation: Use a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .502 9-8 Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .504 Extend 9-8 Graphing Calculator Lab: Scatter Plots . . . . . . . 510 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . 512 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 California Standards Practice . . . . . . . . . . . . . . . . . . 518 Prerequisite Skills • Get Ready for Chapter 9 463 • Get Ready for the Next Lesson 469, 476, 482, 488, 495, 501 Reading and Writing Mathematics • Reading in the Content Area 465 • Reading Math 479 • Writing in Math 468, 475, 482, 487, 495, 501, 508 California Standards Practice • Multiple Choice 469, 474, 476, 482, 488, 495, 501, 509 • Worked Out Example 473 H.O.T. Problems Higher Order Thinking • Challenge 468, 475, 482, 487, 494, 501, 508 • Find the Error 482, 494 • Number Sense 508 • Open Ended 468, 475, 482, 487, 494, 508 • Reasoning 494 • Which One Doesn’t Belong? 475 xix CH APTER 10 Algebra: Nonlinear Functions and Polynomials 10-1 Linear and Nonlinear Functions . . . . . . . . . . . . . . . . . . . 522 10-2 Graphing Quadratic Functions . . . . . . . . . . . . . . . . . . . . . 528 10-3 Problem-Solving Investigation: Make a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 10-4 Graphing Cubic Functions . . . . . . . . . . . . . . . . . . . . . . . . 534 Extend 10-4 Graphing Calculator Lab: Families of Nonlinear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 10-5 Multiplying Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . 539 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543 10-6 Dividing Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .544 10-7 Powers of Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . .549 10-8 Roots of Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . 557 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 • Get Ready for Chapter 10 521 • Get Ready for the Next Lesson 527, 531, 537, 542, 548, 552 Reading and Writing Mathematics • Reading in the Content Area 529 • Reading Math 554 • Writing in Math 526, 531, 532, 537, 542, 548, 552, 556 California Standards Practice • Multiple Choice 527, 531, 537, 542, 546, 548, 552, 556 • Worked Out Example 545 H.O.T. Problems Higher Order Thinking • Challenge 526, 531, 537, 542, 548, 552, 556 • Number Sense 548 • Open Ended 526, 531, 537, 542, 548, 552, 556 • Which One Doesn’t Belong? 526 xx California Standards Practice . . . . . . . . . . . . . . . . . .562 SBI/NASA/Getty Images Prerequisite Skills Michael Newman/PhotoEdit CH APTER 11 Statistics 11-1 Problem-Solving Investigation: Make a Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .568 Table of Contents 11-2 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 Extend 11-2 Graphing Calculator Lab: Histograms . . . . . . . . 575 11-3 Circle Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 Extend 11-3 Spreadsheet Lab: Line, Bar, and Circle Graphs . . . . . . . . . . . . . . . . . . . . . .583 11-4 Measures of Central Tendency and Range . . . . . . . . . . .585 Extend 11-4 Spreadsheet Lab: Mean, Median, and Mode . . . . . . . . . . . . . . . . . . . . . . . . 591 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .592 11-5 Measures of Variation . . . . . . . . . . . . . . . . . . . . . . . . . . .593 11-6 Box-and-Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . .599 Prerequisite Skills • Get Ready for Chapter 11 567 • Get Ready for the Next Lesson 574, 582, 590, 598, 604, 610 Extend 11-6 Reading and Writing Mathematics 11-7 Stem-and-Leaf Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . .606 • Reading in the Content Area 570 • Vocabulary Link 593 • Writing in Math 568, 574, 581, 590, 598, 604, 610, 614 Graphing Calculator Lab: Box-and-Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . .605 11-8 Select an Appropriate Display . . . . . . . . . . . . . . . . . . . . . 611 ASSESSMENT Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . . 616 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 California Standards Practice . . . . . . . . . . . . . . . . . .622 California Standards Practice • Multiple Choice 574, 582, 588, 590, 598, 604, 610, 615 • Worked Out Example 587 H.O.T. Problems Higher Order Thinking • Challenge 574, 590, 598, 610, 614 • Find the Error 589, 603 • Number Sense 581 • Open Ended 573, 581, 589, 598, 603, 614 • Reasoning 581, 590, 604 xxi California Standards Review Tips for Success . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CA1 Multiple-Choice Questions . . . . . . . . . . . . . . . . . . . . . . . CA2 Practice by Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . CA4 CH APTER 12 Looking Ahead to Grade 8: Probability 12-1 Counting Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .626 12-2 Probability of Compound Events . . . . . . . . . . . . . . . . . . 631 12-3 Experimental and Theoretical Probability . . . . . . . . . . . . 637 Extend 12-3 Probability Lab: Fair Games . . . . . . . . . . . . . . . .642 12-4 Problem-Solving Investigation: Act it Out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .644 Mid-Chapter Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . .646 12-5 Using Sampling to Predict . . . . . . . . . . . . . . . . . . . . . . . . 647 ASSESSMENT Prerequisite Skills • Get Ready for Chapter 12 625 • Get Ready for the Next Lesson 630, 636, 641 Reading and Writing Mathematics • Reading in the Content Area 647 • Vocabulary Link 631, 632, 648 • Writing in Math 629, 636, 640, 643, 644, 651 California Standards Practice • Multiple Choice 630, 633, 636, 641, 652 • Worked Out Example 632 H.O.T. Problems Higher Order Thinking • Challenge 629, 636, 640, 651 • Find the Error 635 • Number Sense 629 • Open Ended 629, 635, 640 xxii Terry Eggers/CORBIS Study Guide and Review . . . . . . . . . . . . . . . . . . . . . . .653 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 California Standards Practice . . . . . . . . . . . . . . . . . .658 Student Handbook Built-In Workbooks Prerequisite Skills. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .662 Extra Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 Mixed Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 Reference English-Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . 720 Selected Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 Photo Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 Table of Contents Mission bells, also known as California poppies xxiii Correlation California Content Standards, Grade 7 Correlated to California Mathematics, Grade 7 ©2008 = Key Standards defined by Mathematics Framework for California Public Schools * = Standard assessed on the California High School Exit Exam (CAHSEE) Standard Text of Standard Primary Citations Supporting Citations Number Sense 1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms: 41–49, 51–56, 84–89, 91–112, 114–118, 126–133, 155–159, 252–261, 263–273, 275–294, CA4, CA5, CA6 62–63, 65–73, 91–95, 119–125, 148–153, 155–159, 203, 256–261, 310, 314–315, 319, 631–636 1.1* Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers using scientific notation. 91–95, 130–133, CA5, CA6 256–261 1.2* Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. 41–49, 51–56, 96–112, 62–63, 65–73, 114–118, 126–129, CA4, 119–125, 152–153, 256–261, 631–636 CA6, CA34, CA36, CA37, CA38 1.3* Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. 84–89, 252–261, 263–273, 275–283, CA4, CA6, CA35 91–95, 203, 314–315 1.4 Differentiate between rational and irrational numbers. 155–159, CA6 148–151 1.5 Know that every rational number is either a terminating or repeating decimal and be able to convert terminating decimals into reduced fractions. 84–89, CA6 91–95, 155–159 1.6* Calculate the percentage of increases and decreases of a quantity. 284–289, CA6 310 1.7* Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest. 279–294, CA4, CA5, CA6, CA39 310, 319 2.0 Students use exponents, powers, and roots and use exponents in 35–39, 114–118, 126–129, 144–151, working with fractions: 539–542, 544–548, CA7, CA8, CA9 124, 549–556 2.1* Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base. 126–129, 539–542, 544–548, CA7, CA9 549–552 2.2* Add and subtract fractions by using factoring to find common denominators. 114–118, CA7, CA8, CA9 124 2.3* Multiply, divide, and simplify rational numbers by using exponent rules. 539–542, 544–548, CA7, CA8, CA9 126–129 2.4* Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why. 144–151, CA8, CA9 553–556 xxiv California Content Standards for Mathematics Standard 2.5* Primary Citations Text of Standard Understand the meaning of the absolute value of a number; interpret 35–39, CA9 the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. Supporting Citations 553–556 Algebra and Functions 1.0 Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs: 29–34, 40, 57–61, 102–107, 119–123, 416–421, 427–431, 434–437, 441–453, 470–476, 490, 496–501, 528–531, CA10, CA11, CA12 24–34, 41–49, 51–56, 65–73, 90, 102–112, 154–159, 316–319, 416–426, 432–437, 465–469, 498–501, 522–527, 532–537 1.1* Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g., three less than a number, half as large as area A). 57–61, 119–123, 416–421, 427–431, 434–437, 441–453, 498–501, CA10, CA12, CA35, CA37 29–34, 65–73, 316–319, 438–439, 532–533 1.2* Use the correct order of operations to evaluate algebraic expressions such as 3(2x ⫹ 5)2. 29–34, CA11, CA12, CA39 46–49, 51–56, 416–426, 432–433, 465–469, 534–537 1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. 29–34, 102–107, 416–421, CA10, CA12, CA37 41–45, 51–56, 108–112, 155–159, 498–501, 534–537 1.4 Use algebraic terminology (e.g., variable, equation, term, coefficient, inequality, expression, constant) correctly. 29–34, 57–61, 416–421, CA12 90, 102–107, 154, 422–426, 434–437, 498–501 1.5* Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. 40, 470–476, 490, 496–497, 528–531, CA11, CA12 24–28, 498–501, 522–527, 532–533 2.0 Students interpret and evaluate expressions involving integer powers and simple roots: 126–129, 539–542, 544–556, CA13, CA14, CA15 29–34, 51–56, 96–101, 144–147, 548–556 2.1* Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. 126–129, 539–542, 544–548, CA13, CA14, CA15 29–34, 51–56, 96–101, 144–147, 549–556 2.2* Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. 539–542, 544–556, CA13, CA14, CA15 548, 552, 556 3.0 Students graph and interpret linear and some nonlinear functions: 471–482, 491–495, 528–538, CA16, CA17, CA18 194–197, 230–241, 464–469, 471–476, 483–488, 537, 542, 548, 552, 559 3.1* Graph functions of the form y ⫽ nx2 and y ⫽ nx3 and use in solving problems. 528–531, 534–538, CA16 542, 548 3.2 Plot the values from the volumes of three-dimensional shapes for various values of the edge lengths (e.g., cubes with varying edge lengths or a triangle prism with a fixed height and an equilateral triangle base of varying lengths). 534–537, CA18 552, 559 California Content Standards for Mathematics xxv Primary Citations Supporting Citations Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. 477–482, 491–495, CA16, CA18 230–241, 464–469, 471–476, 483–488 3.4* Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. 471–482, CA17, CA18 194–197, 236–241 4.0 Students solve simple linear equations and inequalities over the rational numbers: 190–193, 198–202, 422–426, 434–437, 445–453, 483–488, CA19, CA20, CA21 204–205, 360–361, 427–431 4.1* Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. 422–426, 434–437, 445–453, CA19, CA20, CA21, CA34 427–431 4.2* Solve multistep problems involving rate, average speed, distance, and time or a direct variation. 190–193, 198–202, 483–488, CA19, CA20, CA21, CA35, CA37, CA38, CA39 204–205, 360–361 Standard Text of Standard 3.3* Measurement and Geometry 1.0 Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems: 96–107, 190–193, 213–219, 224–229, CA22, CA23, CA24 96–107, 192–197, 207, 401 1.1* Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters) 213–223, CA22, CA23, CA24 192, 193, 196 1.2* Construct and read drawings and models made to scale. 224–229, CA24, CA37 207, 401 1.3* Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer. 190–193, 213–218, CA22, CA23, CA24, CA36 96–107, 194–197 2.0 Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale: 220–223, 352–357, 362–367, 373–384, 386–391, 397–404, 674–675, CA25, CA26, CA27 110, 161, 197, 229, 235, 372, 385, 528, 530, 535–536, 542, 549–552 2.1* Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. 352–357, 373–384, 386–391, CA25, CA27, CA35, CA38, CA39 110, 161, 197, 229, 235, 362–367, 372, 380, 385–392, 528, 530, 535–536, 542, 549–552, 674–675 2.2* Estimate and compute the area of more complex or irregular two-and three-dimensional figures by breaking the figures down into more basic geometric objects. 362–367, 373–378, CA26, CA27, CA34, CA35, CA39 372, 385–391 xxvi California Content Standards for Mathematics Standard Primary Citations Text of Standard Supporting Citations 2.3* Compute the length of the perimeter, the surface area of the faces, and 397–404, CA25, CA26 the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor. 380, 385–391 2.4* Relate the changes in measurement with a change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot ⫽ 144 square inches or [1 ft2] ⫽ [144 in2], 1 cubic inch is approximately 16.38 cubic centimeters or [1 in3] ⫽ [16.38 cm3]). 220–223, CA26, CA27 229, 235 3.0 Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures: 161–178, 311–313, 320–325, 327–341, 352–359, 368–372, 380, 385, 388, 392, CA28, CA29, CA30 193, 197, 206–211, 311–313, 324–325, 378, 384, 393 3.1 Identify and construct basic elements of geometric figures (e.g., altitudes, mid-points, diagonals, angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles) by using a compass and straightedge. 311–313, 352–359, CA26, CA27 324–325 3.2* Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections. 172–178, 327–341 161 3.3* 161–178, CA27, CA30 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. 3.4* Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures. 320–325, CA30 206–211, 311–313 3.5 Construct two-dimensional patterns for three-dimensional models, such as cylinders, prisms, and cones. 380, 385, 388, 392 393 3.6 Identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and describe how two or more objects are related in space (e.g., skew lines, the possible ways three planes might intersect). 368–372, CA28, CA30 378, 384 193, 197 Statistics, Data Analysis, and Probablility 1.0 Students collect, organize, and represent data sets that have one or more variables and identify relationships among variables within a data set by hand and through the use of an electronic spreadsheet software program: 504–511, 570–591, 593–615, CA31, CA32, CA33 502–503, 568–569, 599–605, 611–615 1.1* Know various forms of display for data sets, including a stem-andleaf plot or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data. 570–584, 599–615, CA33, CA36 568–569 1.2* 504–511, CA32, CA33 Represent two numerical variables on a scatterplot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables (e.g., between time spent on homework and grade level). 502–503, 611–615 California Content Standards for Mathematics xxvii Standard 1.3 Text of Standard Primary Citations Understand the meaning of, and be able to compute, the minimum, 585–591, 593–598, the lower quartile, the median, the upper quartile, and the maximum CA31, CA32, CA33, CA37 of a data set. Supporting Citations 599–605 Mathematical Reasoning 1.0 Students make decisions about how to approach problems: 1.1* Analyze problems by identifying relationships, distinguishing relevant 24–28, 62–63, 124–125, CA34, CA35 from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. 194–197, 314–315, 324–331, 360–361, 490 1.2* Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed. 203, 314–315, CA35 24–28 1.3 Determine when and how to break a problem into simpler parts. 360–361, CA34, CA35 64, 190–197 2.0 Students use strategies, skills, and concepts in finding solutions: Used throughout the text. For example, 62–63, 124–125, 230–235, CA36, CA37 2.1* Use estimation to verify the reasonableness of calculated results. 62–63, CA36 275–278 2.2 Apply strategies and results from simpler problems to more complex problems. 360–361, CA36, CA37 96–101, 294, 362, 397–398, 490 2.3* Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques. 230–235, CA36 155–159 2.4* Make and test conjectures by using both inductive and deductive reasoning. 124–125, CA37 24–34, 161, 311–315, 324–325, 358–359 2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 152–153, 204–205, 432–433, 465–469, 502–503, 532–533, 568–569, 644–645, CA37 154, 306, CA37 172, 464, 490, 575, 583–584, 591, 605 El Capitan and the Yosemite Valley Used throughout the text. For example, 24–28, 314–315, 360–361, CA34, CA35 642–643 2.6 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.7 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. 148–151 155–159, 162–166, 256–261, 352–357 2.8 Make precise calculations and check the validity of the results from the context of the problem. 438–439, CA37 204–205 3.0 Students determine a solution is complete and move beyond a particular problem by generalizing to other situations: Used throughout the text. For example, 162–166, 272–273, 316–319, CA38, CA39 3.1 Evaluate the reasonableness of the solution in the context of the original situation. 24–28, 272–273, CA38, 62–63, 124–125, CA39 152–153, 204–205, 316–319 3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. 162–166, CA38, CA39 62–63, 124–125, 152–153, 204–205, 219, 272–273, 316–319, 360–361, 441–444, 502–503, 532–533, 568–569, 644–645 3.3* Develop generalizations of the results obtained and the strategies used and apply them to new problem situations. 316–319, CA38, CA39 96–101, 161, 496–497, 538 xxviii California Content Standards for Mathematics Start Smart Be a Better Problem Solver As you gear up to study mathematics, you are probably wondering, “What will I learn this year?” You will focus on these three areas: • Use basic principles of algebra to analyze and represent proportional and nonproportional linear relationships, • Apply operations with rational numbers, and • Use probability and statistics to make predictions. Along the way, you’ll learn more about problem solving, how to use the tools and language of mathematics, and how to THINK mathematically. Start Smart 1 Gabe Palmer/CORBIS Reinforcement of Standard 6MR1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Reinforcement of Standard 6AF3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2, A = ½bh, C = πd—the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). Real-World Problem Solving A lot of planning goes into running a restaurant, and this is the responsibility of the executive chef. Executive chefs have many duties, including planning the menu, setting menu prices, directing food preparation, and managing the budget. With all of these different responsibilities, they must have a plan to make sure that the restaurant runs smoothly. Executive chefs have to be creative problem-solvers and at times have to modify their daily plans. Their final objective is to have a great restaurant for customers like you. In mathematics, there is a plan that will help you solve problems. It’s called the four-step plan. The plan will keep you focused and on track. ART Mrs. Moseley’s art students are designing rectangular collages. Each student is given a piece of trim that is 20 inches long to go around the outside of the collage. Edwina wants to design a collage that will have the largest area possible. What are the dimensions that Edwina should use if all measurements are to the nearest inch? 1 EXPLORE What are you trying to find? Restate the problem in your own words. Use as few words as possible. Edwina needs to find the dimensions of the collage. Do you think you’ll need any additional information such as a formula or measurement conversion? The formula for the perimeter of a rectangle. P = 2( + w) You also need the formula for the area of a rectangle. A = w 2 2 PLAN Start Smart Stewart Cohen/Getty Images Make a table with different combinations of lengths and widths that total 20 inches. Use the lengths and widths to calculate the area of each rectangle. SOLVE Perimeter (inches) Length (inches) Width (inches) Area (inches 2) 20 1 9 9 20 2 8 16 20 3 7 21 20 4 6 24 20 5 5 25 20 6 4 24 Start Smart 3 The dimensions that produce the largest collage are 5 inches by 5 inches. 4 CHECK Is the answer reasonable? Looking at the table, the area of 25 square inches is the largest area in inches. Practice Use the four-step plan to solve each problem. 1. The Boneque family is planning a family Location reunion in the southwest United States. Each family member gets to vote on two locations that they would most like to visit. The table shows the fraction of votes each location received. Which two locations are the most popular? Natural Bridges National Park 2. Using eight coins, how can you make change Rocky Mountain National Park Fraction of Votes Carlsbad Caverns Bryce Canyon _4 5 _11 15 _7 10 _5 6 for 65¢ that will not make change for a quarter? 3. Mrs. Patterson is selecting ceramic tile for her new bathroom. She is choosing between the two tiles shown. Which tile is the better buy per square inch? IN IN IN IN PER TILE PER TILE 4. There are four boxes that weigh a total of 7 10 pounds. One box weighs 1_ pounds, 24 3 pounds. What and another box weighs 2_ 8 are the possible weights of the other two boxes? John Evans swer Is your an ? reasonable e Check to b sure it is. Reinforcement of Standard 6MR1.3 Determine when and how to break a problem into simpler parts. Reinforcement of Standard 6NS2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation. Real-World Problem Solving Teachers are expert problem-solvers. Every day, teachers have to use many problem-solving skills. So, take a tip from the experts! Become an effective problemsolver by using different strategies. There are many different types of problems that you encounter in your life. In mathematics, one type of problem-solving strategy is the solve a simpler problem strategy. DANCE Zoë and her friend Isabella are ordering punch for the spring dance. Fruit punch is sold in 1-gallon containers. A serving size is 8 ounces, and they are expecting about 160 students at the dance. If each student has 2 servings of punch, how many gallons of punch will Zoë and Isabella need to purchase? 1 EXPLORE What are you trying to find? Restate the problem in your own words. Use as few words as possible. You need to know how many gallons of punch to buy. Do you think you’ll need any additional information such as a formula or measurement conversion? You need to know how many ounces are in 1 gallon. 1 gallon = 128 ounces 2 4 PLAN Start Smart (t)Bill Aron/PhotoEdit, (b)John Evans First, solve a simpler problem by finding the total number of ounces of punch needed. Then convert the number of ounces to gallons. PROBLEM -SOLVING STRATEGIE S • Draw a d iagram. • Look for a pattern. • Guess an d check. • Act it ou t. • Solve a s impler problem. • Work ba ckward. SOLVE Multiply the total number of students by number of ounces needed for each person. servings student ounces serving 160 students × 2 _ × 8 _ = 2,560 ounces To find the total number of gallons needed, divide the total number of ounces by the number of ounces in 1 gallon. ounces 2,560 ounces ÷ 128 _ = 20 gallons gallon So, 20 gallons of punch are needed. 4 CHECK Is the answer reasonable? One gallon of punch has 128 ÷ 8 or 16 servings. Since each student has 2 servings, one gallon serves 8 students. So, 20 gallons would serve 160 students. ✔ Practice Use the solve a simpler problem strategy to solve each problem. 1. Two supermarkets are having sales on detergent. The sales are shown in the tables below. Which supermarket has the better buy? 2. How many links are needed to join 30 pieces of chain into one long chain? 3. Five workers can make five chairs in five days. How many chairs can 10 workers working at the same rate make in 20 days? 4. The first three molecules for a certain family of hydrocarbons are shown below. How many hydrogen atoms (H) are in a molecule containing 6 carbon atoms (C)? ( ( # ( ( #( -ETHANE ( ( ( # # ( ( ( #( %THANE ( ( ( ( # # # ( ( ( ( #( 0ROPANE 5. Gabriel is playing Chinese checkers with his brother, Keenan. The brothers are playing until the win-to-loss ratio is at least 3 to 2. So far, Gabriel has won 4 games and Keenan has won 3 games. What is the minimum number of games that will have to be played for either brother to meet the win-to-loss ratio of 3 to 2? Problem-Solving Strategies 5 Start Smart 3 Reinforcement of Standard 6MR2.7 Make precise calculations and check the validity of the results from the context of the problem. Reinforcement of Standard 6AF1.4 Solve problems manually by using the correct order of operations or by using a scientific calculator. Real-World Problem Solving In 2005, Kingda Ka was the world’s tallest and fastest roller coaster. The coaster, located at Six Flags Great Adventure in Jackson, New Jersey, has a height of 465 feet and reaches a maximum speed of 128 miles per hour! Prior to opening the ride, the technicians and engineers encountered many problems that needed to be solved. With hard work and determination, they were able to solve these problems and open the ride. Often the mathematics problems you encounter take more than one step to solve. We call these problems multi-step problems. How do you solve a multi-step problem?—one step at a time, just like engineers and technicians solve their problems. SHOPPING The local department store is having a special on accessories. How much money will you save if you buy 2 box sets instead of buying each item separately? 6 )TEMS 1 EXPLORE What are you trying to find? You need to find how much you’ll save by buying 2 box sets instead of buying the items separately. 2 PLAN What steps do you need to solve the problem? Step 1 Find the total cost of the items. Start Smart Stan Honda/AFP/Getty Images Step 2 Find the difference between the cost of the box set and the total of the items. Step 3 Multiply the difference by 2. "OX 3ET SOLVE Step 1 Step 3 Step 2 $38.48 32.99 _______ $5.49 $9.99 15.50 + 12.99 _______ $38.48 hat scarf gloves $5.49 × 2 ______ $10.98 difference number of box sets total cost cost of box set difference total cost savings So, you will save $10.98 by buying the box set. 4 CHECK Use estimation to check. The three items together cost about $10.00 + $15.50 + $13.00 or about $38.50. The special costs about $33.00. The difference is about $5.50. So, $5.50 + $5.50 is $11.00. ✔ Practice MI 1. WALKING A walking path around a lake is shown at the right. If Nadia walks around the lake two times, how many miles did she walk? MI MI 2. MONEY The Student Council spent $220 MI MI to rent a cotton candy machine and a popcorn machine for the school carnival. They also spent $125 on cotton candy mix, popcorn, bags, and other supplies. If they sold 325 bags of popcorn for $2 each and 385 bags of cotton candy for $3 each, how much money did they make after paying for the machines and the supplies? 3. SKATING The table shows the prices including tax at the local ice-skating rink. Mr. Dewenger bought 2 adult tickets, 2 child tickets, and 1 student ticket. He gave the cashier $20. If he had a coupon for $2 off, how much change should he receive? 3KATING 0RICES #HILD 3TUDENT !DULT 3ENIOR 4. FIELD TRIP Harding Middle School is going on a field trip. Each school bus has 32 seats, and each seat can hold 2 people. If there are 16 homerooms with 18 students each and 32 staff members, how many buses are needed for the field trip? Multi-Step Problems 7 Start Smart 3 Reinforcement of Standard 6AF1.4 Solve problems manually by using the correct order of operations or by using a scientific calculator. This year, you may use an exciting tool to help you visualize and strengthen concepts—a graphing utility or graphing calculator. Graphing Calculator Labs have been included in your textbook so you can use technology to extend concepts. These labs use the TI-83 Plus or TI-84 Plus calculator. A graphing calculator does more than just graph. You can also use it to calculate. is used to enter equations. Press 2nd to access the additional functions listed above each key. The key is used to find the second power of a number or expression. Press ON to turn on your calculator. Press 2nd [OFF] to turn off your calculator. ( ) is used to indicate a negative or opposite value. 8 Start Smart Matt Meadows Press 2nd [TABLE] to display a table of values for equations entered using the key. Press CLEAR once to clear an entry. Press CLEAR twice to clear the screen. Use the operation keys to add, subtract, multiply, or divide. Multiplication is shown as * on the screen and division is displayed as /. The ENTER key acts like an equals button to evaluate an expression. It is also used to select menu items. Start Smart Entering Expressions 1 Evaluate each expression. a. (3 × 6) ÷ (14 ÷ 7) Keystrokes: b. 3 6 µ 14 µ 7 ENTER 6 8 µ 6×8 _ 9+3 Keystrokes: 9 3 ENTER Evaluate each expression. 1. (15 - 5) + (9 × 2) 2. (12 + 20) - (4 × 6) 3. 6 × (9 ÷ 3) - 1 3 × 10 4. _ 9×8 5. _ 6. 2+4 4×6 16 × 7 _ 12 ÷ 3 Evaluating with Integers 2 Evaluate each expression. a. 12 + (-6) + 5 ( ) 6 Keystrokes: 12 5 ENTER b. -4(15) + (-10) Keystrokes: ( ) 4 ( ) 10 15 ENTER Evaluate each expression. 7. -6 + 12 + (-20) 8. 4 + 9 + (-16) 9. 5 + 9 - 3 + (-17) 10. -6 + 8 - 10 + 15 11. -3(-8) - (-20) 12. 12(5) + (-6) Squares and Square Roots 3 Evaluate each expression. a. (-16) 2 Keystrokes: ( ) 16 ENTER b. √ 121 Keystrokes: 2nd ⎡⎣ √ ⎤⎦ 121 ENTER Evaluate each expression. 13. -25 2 14. -(6 2) 15. √ 225 16. - √ 36 The Graphing Calculator 9 Reinforcement of Standard 6MR2.1 Use estimation to verify the reasonableness of calculated results. Reinforcement of Standard 6AF1.4 Solve problems manually by using the correct order of operations or by using a scientific calculator. Solving problems is more than using paper and pencil. Follow the path to choose the best method of computation. 1.99 2.15 4.2785 Start Read the problem carefully. Look for key words and numbers. $19.99 2 is about $40 Need an exact answer? If not, then ESTIMATE. Are the numbers very large or are there lots of calculations? If so, use a Can I do this “in my head?” If so, use CALCULATOR. MENTAL MATH. $10 25 $250 Are the calculations simple? Use PAPER AND PENCIL. Finish PROBLEM SOLVED! 10 Start Smart (tl tr br)John Evans, (bl)Laura Sifferlin $3.99 2.15 $6.14 Choose the best method of computation to solve each problem. Then solve. 1. Evan bought a hat, a scarf, gloves, and a pair of boots. He gave the cashier $100. If the change Evan received was $8.72, what was the total cost including tax of the merchandise? 2. In one day, a coffee shop sold 274 cups of hot chocolate. About how many cups of hot chocolate will they sell in 1 week? For Exercises 3–5, use the information in the table. Boat Rental Fees Company Cost ($/h) A 20.00 B 18.99 C 24.95 3. What is the total cost of renting a boat from Company E for 8 hours? 4. What is the total cost of renting a boat from Company C for 7 hours? 5. How much money would you save if you D 32.39 E 30.00 rented a boat for 12 hours from Company B instead of Company D? 6. Mrs. Coughin is making bread. She needs 4 _ cups of flour. Mrs. 3 4 1 Coughlin already has 1_ cups. How many more cups of flour does she 4 still need? 7. Paige has a bag of apples that weighs 5.5 pounds. If the bag has 12 apples in it, about how much does each apple weigh? 8. The table shows the circumference of various planets. Planets’ Circumferences Planet Circumference (mi) Venus 23,627 Earth 24,901 Mars 13,263 Saturn 235,298 Jupiter 279,118 Source: NASA What is the difference between the circumference of Jupiter and the circumference of Mars? 9. The price of a certain stock has tripled in value since the beginning of the year. If the original value of the stock was $12.64, what is the current value of the stock? 10. An adult male chimpanzee weighs about 55 kilograms. An adult male manatee, by contrast, weighs about 11 times as much. About how much does an adult male manatee weigh? Techniques for Problem Solving 11 StockTrek/Getty Images Start Smart Practice Why do I need my math book? Have you ever been in class and not understood all of what was presented? Or, you understood everything in class, but at home, got stuck on how to solve a couple of problems? Maybe you just wondered when you were ever going to use this stuff? These next few pages are designed to help you understand everything your math book can be used for … besides homework problems! Before you read, have a goal. • What information are you trying to find? • Why is this information important to you? • How will you use the information? Have a plan when you read. • Read the Main IDEA at the beginning of the lesson. • Look over photos, tables, graphs, and opening activities. • Locate words highlighted in yellow and read their definitions. • Find Key Concept and Concept Summary boxes for a preview of what’s important. • Skim the example problems. Keep a positive attitude. • Expect mathematics reading to take time. • It is normal to not understand some concepts the first time. • If you don’t understand something you read, it is likely that others don’t understand it either. 12 Start Smart John Evans Start Smart Doing Your Homework Regardless of how well you paid attention in class, by the time you arrive at home, your notes may no longer make any sense and your homework may seem impossible. It’s during these times that your book can be most useful. • Each lesson has example problems, solved step-by-step, so you can review the day’s lesson material. • has extra examples at ca.gr7math.com to coach you through solving those difficult problems. • Each exercise set has (/-%7/2+ (%,0 boxes that show you which examples may help with your homework problems. • Answers to the odd-numbered problems are in the back of the book. Use them to see if you are solving the problems correctly. If you have difficulty on an even problem, do the odd problem next to it. That should give you a hint about how to proceed with the even problem. For … r Tuto l a n rso Pe h Look Online wit ples m th s • Ma xtra Exa boxe n E p l d e n H a ing o t k r r a o t ss mew swer n • Ho A ected • Sel 743 page Doing Your Homework 13 John Evans Studying for a Test You may think there is no way to study for a math test. However, there are ways to review before a test. Your book can help! • Review all of the new vocabulary words and be sure you understand their definitions. These can be found on the first page of each lesson or highlighted in yellow in the text. • Review the notes you’ve taken on your and write down any questions that you still need to have answered. • Practice all of the concepts presented in the chapter by using the chapter Study Guide and Review. It has additional problems for you to try as well as more examples to help you understand. You can also take the Chapter Practice Test. • Take the Self-Check Quizzes at ca.gr7math.com. Look For … • Self-check Quizzes at ca.gr7math.com • Study Guide and Review at the end of each chapter 14 Start Smart John Evans Start Smart Let’s Get Started Use the Scavenger Hunt below to learn where things are located in each chapter. 1. What is the title of Chapter 1? 2. How can you tell what you’ll learn in Lesson 1-1? 3. In the margin of Lesson 1-2, there is a Vocabulary Link. What can you learn from that feature? 4. What is the key concept presented in Lesson 1-2? 5. Sometimes you may ask “When am I ever going to use this?” Name a situation that uses the concepts from Lesson 1-3. 6. How many examples are presented in Lesson 1-3? 7. What is the title of the feature in Lesson 1-3 that tells you how to read inequality symbols? 8. What is the Web address where you could find extra examples? 9. Suppose you’re doing your homework on page 38 and you get stuck on Exercise 19. Where could you find help? 10. What problem-solving strategy is presented in the Problem-Solving Investigation in Lesson 1-8? 11. List the new vocabulary words that are presented in Lesson 1-9. 12. What is the Web address that would allow you to take a self-check quiz to be sure you understand the lesson? 13. There is a Real-World Career mentioned in Lesson 1-10. What is it? 14. On what pages will you find the Study Guide and Review for Chapter 1? 15. Suppose you can’t figure out how to do Exercise 25 in the Study Guide and Review on page 76. Where could you find help? Scavenger Hunt 15 The following pages contain data about California that you’ll use throughout the book. DESERT TORTOISE, ACORN BREAD • 6 Tbsp. cornmeal • • _1 c cold water • 1 c mashed potatoes • 1 c boiling water • 2 c all-purpose flour • 1 tsp salt • 2 c finely ground leached 2 STATE REPTILE _1 c lukewarm water 4 Weight: 8-15 poun ds Length (carapac e): 9-15 inches Height: 4-6 inch es Incubation Perio d: 90-120 days Number Of Eggs : 4-8 Lifespan: 80-100 years Typical Diet: he rbs, grasses, wildflow ers acorn meal • 1 Tbsp. butter • 1 pkg active dry yeast Source: desertusa .com Source: siouxme.com/acorn SANTA CRUZ BEAC H BOARDWALK Merry-Go-Round 3 $1.95 Limits 56”maximum Jet Copters $1.95 56”maximum Starfish 3 Rugged Buggies $1.95 56”maximum Freefall 3 Ride Tickets Price Bulgy Red Baron Speed Boats 3 $1.95 56”maximum 3 $1.95 56”maximum Speedway 3 $1.95 3 Convoy 3 Sea Dragons under 36” with chaperone ” with $1.95 under 36 chaperone ” with $1.95 under 36 chaperone rdwalk.com Source: beachboa 16 Ride California Data File (tr)Theo Allofs/CORBIS, (b)Gary Crabbe/Alamy Tickets 4 3 4 Limits ” with $2.60 under 36 ne ero chap ” with $1.95 under 36 chaperone ” to 34” $2.60 under 42 with chaperone Price 42” minimum ” to 34” $3.25 under 48 with chaperone ” to 34” $2.60 under 48 with chaperone ” to 34” $2.60 under 48 $3.25 Space Race e Logger ’s Reveng 5 Rock-O-Plane 4 Sea Serpent 4 5 California Data File CALIFORNIA SCIENCE CENTER ia Science Center Located in Los Angeles, the Californ n science center. is the west coast’s largest hands-o HOLLYWOOD Source: californiasciencecenter.org SURFING Source: yahoo.c om 2004 X Games Team Surfing East West Quarter 1 Quarter 2 27.60 22.47 22.79 26.22 Quarter 3 Quarter 4 24.73 21.99 21.91 19.34 97.03 90.02 Total EDUCATION K-12 .com Source: skatelog California Scho ol Enrollment Trends 1990-2011 (th ousands) Type 1990 2001 2011 Higher Education UC, CSU EMPLOYMENT California Employment Outperforms arm payrolls Year-on-year percent change in total non-f Community College s Source: Californ 4,842.2 6,068.9 2,035.7 6,295.3 2.254.2 522.7 1,513.0 2,841.5 567.6 1,666.6 751.4 2,090.1 ia Department of Finance Percent 4.0 3.0 California 2.0 1.0 U.S. 0 -1.0 -2.0 1999 2000 2001 2002 2003 Source: Bureau of Labor Statistics Source: Bureau of Labor Statistics 2004 2005 California Data File 17 Nik Wheeler/CORBIS GRIZZLY BEAR, STATE ANIMAL Yellowish brown to dark brown, often with white-tipped hairs, giving grizzled appearance. Claws of Front Feet: 4 in. (10 cm). ROSE BOWL FACTS _1 Height: 4 4 ft (130 cm) 11 _ Length: 5 12 –7 ft (180–213 cm) –680 kg) Typical Weight: 324–1,499 lb (147 kg) Maximum Weight: 1,700 lb (700 Source: stateanimals.com • Approximate seatin g capacity to date: 90,000 +. • The Rose Bowl has approximately 77 row s of seats. • The stadium measu res 880 feet from north to south rims and 695 feet from east to we st rims. • The perimeter of the rim is 2,430 feet, while the inside perimeter at field level is approximately 1,350 feet. • The turfed area ins ide the bowl measures 79,156 square feet. • The fence around the Rose Bowl is one mile long. • There are over 100 different varieties of ros e bushes situated between the stadium and the fence. • The dimensions of the playing field for footba ll are 53 yd x 100 yd. The dimensions of the pla ying field for soccer are 70 yd x 120 yd. • The stadium itself is approximately 830 fee t abo ve sea level. • It would take approx imately 84,375,000 gal lons of water to fill the Rose Bowl to the rim . • The Rose Bowl Pre ss Box is the highest poi nt of the stadium at 100 feet above ground . Source: pasadena.com BASKETBALL Scoring/ Rebounds ars California Golden Be Free 3-Point Field rows Th FG Goals Points Average 553 20.5 L. Powe A. Ubaka 49.6% 33.3% 71.9% 41.3% 37.2% 83.3% 451 14.5 38.5% 38.6% 78.7% 257 8.6 R. Midgley O. Wilkes 44.9% 67.6% 213 7.6 48.0% 62.5% 225 D. Hardin — 7.3 49.7% ENNIS HOUSE, LOS ANGE LES Source: calbears.com Built in 1923 by Frank Llo house is a monumenta yd Wright, the Ennis l textile-block house. The textile-blocks form pa tterns that are symme trical. Source: greatbuild ings.com 18 California Data File (tl)Michio Hoshino/Minden Pictures, (tr)Ken Levine/Getty Images, (b)Tim Street-Porter/Beateworks/CORBIS California Data File GEOGRAPHY California includes 16 3,707 square miles, or 4.5% of the nation’s total are a. Source: 50states.com WEATHER Forecast for Sacr amento, CA July 10–17 High Temperature Day Monday CES NATURAL RESOUR 98° Tuesday 96° Wednesday Probability of Precipitation 10% 20% Median Home Price 100° Division 0% • The California Thursday 94° of Beaches 10% s a Friday and Parks manage 95° s, 20% total of 188 park Sat urd ay n, tio 102° beaches, recrea 10% d Sunday historic, and relate 103 ° 0% areas. Source: Nationa l Weather Service Division • The California of Fish and life Game has 18 wild as management are nia . under the Califor with 188,780 acres with 70,225 acres ts es for te sta ht • There are eig try. ng and Division of Fores water for swimmi ing t 200,000 acres of ou ab d moor an g hin nc lau at• The State has oximately 500 bo boating and appr . complete, will ies facilit Trail, about half HOUSING king and Riding Hi nges. Ra nia t or as lif Co Ca d e an • Th through the Sierra les mi 00 1,9 d exten Median Prices Source: cr.nps.gov Cloud Cover Sunny Partly cloudy Sunny Sunny Partly cloudy Partly cloudy Sunny $500K $450K $400K $350K $300K $250K $200K $150K $100K $50K 0 1980 1984 1988 1992 1996 2000 2004 eabc.com Source: realestat California Data File 19 Dale Sanders/Masterfile Number and Operations: Rational and Real Numbers Focus Use appropriate operations to solve problems and justify solutions. CHAPTER 1 Use exponents, powers, and roots and use exponents in working with fractions. CHAPTER 2 Algebra: Rational Numbers Know the properties of, and compute with, rational numbers expressed in a variety of forms. Choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems. CHAPTER 3 Real Numbers and the Pythagorean Theorem Know the Pythagorean theorem and understand plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures. 20 Peter Cade/Getty Images Peter Cade/Getty Images Algebra: Integers Express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs. Peter Cade/Getty Images Math and Geography Bon Voyage! All aboard! We’re setting sail on an adventure that will take us to exotic vacation destinations. Along the way, you’ll act as a travel agent for one of three different families, working to meet their vacation needs while still staying within their budget. You will also plan their itinerary and offer choices of activities for them to participate in at their destinations. We’ll be departing shortly, so pack your problem-solving tool kit and hop on board. Log on to ca.gr7math.com to begin. Unit 1 Number and Operations: Rational and Real Numbers 21 1 Algebra: Integers • Standard 7AF1.0 Express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs • Standard 7NS2.0 Use exponents, powers, and roots and use exponents in working with fractions. Key Vocabulary algebraic expression (p. 29) equation (p. 57) integer (p. 35) variable (p. 29) Real-World Link Submarines Integers can be used to describe the depth of a submarine. You can also add and subtract integers to determine a change in depth. Algebra: Integers Make this Foldable to help you organize your notes. Begin with a piece of 11” × 17” paper. 1 Fold the paper in sixths lengthwise. 2 Open and fold a 4” tab along the short side. Then fold the rest in half. 3 Draw lines along the folds and label as shown. 8ORDS "1LANFOR 1ROBLEM4OLVING OF *NTEGERS XOF *NTEGERS 4OLVING &QUATIONS 4OLVINGX &QUATIONS 22 Chapter 1 Algebra: Integers &XAMPLE S GET READY for Chapter 1 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Example 1 Find 14.63 + 2.9. Add. (Prior Grade) 1. 64 + 13 2. 10.32 + 4.7 3. 2.5 + 77 4. 38 + 156 5. SHOPPING Mrs. Wilson spent 14.63 + 2.90 ________ 17.53 Line up the decimal points. Annex a zero. $80.20, $72.10, $68.50, and $60.70 on school clothes for her children. Find the total amount she spent. (Prior Grade) Example 2 Find 82 - 14.61. Subtract. (Prior Grade) 6. 200 - 48 7. 59 - 26 8. 3.3 - 0.7 9. 73.5 - 0.87 82.00 - 14.61 ________ 67.39 Annex two zeroes. Multiply. (Prior Grade) Example 3 Find 8.7 × 6. 10. 3 × 5 × 2 8.7 × 6 ______ 52.2 12. 12.7 × 6 11. 2.8 × 5 13. 4 × 9 × 3 1 decimal place + 0 decimal places _______________ 1 decimal place 14. TRAVEL The Perez family drove for 5.75 hours at 55 miles per hour. How far did they drive? (Prior Grade) Divide. (Prior Grade) 15. 244 ÷ 0.2 16. 72 ÷ 9 17. 96 ÷ 3 18. 100 ÷ 0.5 19. 2 ÷ 5 20. 0.36 ÷ 0.3 21. BAGELS A bag of 8 assorted bagels sells for $6.32. What is the price per bagel? (Prior Grade) Example 4 Find 4.77 ÷ 0.9. 0.9 4.77 09. 47.7 Multiply both 5.3 9 47.7 -45 ____ 27 -27 ____ 0 numbers by the same power of 10. Place the decimal point and divide as with whole numbers. Chapter 1 Get Ready for Chapter 1 23 1-1 A Plan for Problem Solving Main IDEA Suppose you are designing rectangular gardens that are bordered by white tiles. The three smallest gardens you can design are shown below. Solve problems by using the four-step plan. Standard 7MR1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Reinforcement of Standard 6AF2.3 Solve problems involving rates, average speed, distance, and time. Garden 1 Garden 2 Garden 3 1. How many white tiles does it take to border each of these three gardens? 2. Predict how many white tiles it will take to border the next-longest garden. Check your answer by modeling the garden. 3. How many white tiles will it take to border a garden that is 6 tiles long? Explain your reasoning. Some problems, like the one above, can be solved by using one or more problem-solving strategies. No matter which strategy you use, you can always use the four-step plan to solve a problem. 1. Explore NEW Vocabulary • Determine what information is given in the problem and what you need to find. • Do you have all the information you need? conjecture • Is there too much information? 2. Plan • Visualize the problem and select a strategy for solving it. There may be several strategies that you can use. • Estimate what you think the answer should be. • Make an educated guess or a conjecture. 3. Solve • Solve the problem by carrying out your plan. • If your plan doesn’t work, try another. Problem-Solving Strategies • Make a model. • Solve a simpler problem. • Make an organized list. • Make a table. • Find a pattern. • Work backward. • Draw a graph. • Guess and check. 24 4. Check Chapter 1 Algebra: Integers • Examine your answer carefully. • See if your answer fits the facts given in the problem. • Compare your answer to your estimate. • You may also want to check your answer by solving the problem again in a different way. • If the answer is not reasonable, make a new plan and start again. 1 GARDENING Refer to the Mini Lab on page 24. The table shows how the number of blue tiles it takes to represent each garden is related to the number of white tiles needed to border the garden. How many white tiles Blue Tiles will it take to border a garden White Tiles that is 10 blue tiles long? 1 2 3 4 5 6 8 10 12 14 16 18 Explore You know the number of white tiles it takes to border gardens up to 6 tiles long. You need to determine how many white tiles it will take to border a garden 10 tiles long. Plan You might make the conjecture that there is a pattern to the number of white tiles used. One method of solving this problem is to look for a pattern. Solve First, look for the pattern. Then, extend the pattern. Blue Tiles 1 2 3 4 5 6 7 8 9 10 White Tiles 8 10 12 14 16 18 20 22 24 26 +2 +2 +2 +2 +2 +2 +2 +2 +2 It would take 26 white tiles to border a garden that was 10 blue tiles long. Check Reasonableness Always check to be sure your answer is reasonable. If the answer seems unreasonable, solve the problem again. It takes 8 white tiles to border a garden that is 1 blue tile wide. Each additional blue tile needs 2 white tiles to border it, one above and one below. Garden 1 Garden 2 So, to border a garden 10 blue tiles long, it would take 8 white tiles for the first blue tile and 9 × 2 or 18 for the 9 additional tiles. Since 8 + 18 = 26, the answer is correct. a. INTERNET The table shows the number of visitors, rounded to the nearest thousand, to a new Web site on each of the first five days after the owners place an ad in a newspaper. If this pattern continues, about how many visitors should the Web site receive on day 8? Extra Examples at ca.gr7math.com Ed Bock/CORBIS Day Visitors 1 15,000 2 30,000 3 60,000 4 120,000 5 240,000 Lesson 1-1 A Plan for Problem Solving 25 Some problems can be solved by a combination of operations. Use the Four-Step Plan 2 ANIMALS Refer to the Word Problems It is important to read a problem more than once before attempting to solve it. You may discover important details that you overlooked when you read the problem the first time. 8e`dXcJg\\[ graphic. If a cheetah and a giant tortoise travel at their top speed for 1 minute, how much farther does the cheetah travel? #HEETAH 1UARTER (ORSE !NIMAL READING Math $OMESTIC #AT 'IANT 4ORTOISE 4OP 3PEED FEET PER SECOND Explore What do you know? You know the top speeds for a cheetah and a giant tortoise in feet per second. What are you trying to find? You need to find the difference in the distances traveled by a cheetah and a giant tortoise in 1 minute. Plan Begin by determining the distance each animal traveled in 1 minute. Since 1 minute is 60 seconds, multiply each top speed by 60. Then, subtract to find the difference of the distances traveled by the two animals. Estimate Solve 100 × 60 = 6,000 and 0.25 × 60 = 15 6,000 - 15 = 5,985 103 × 60 = 6,180 0.25 × 60 = 15 6,180 - 15 = 6,165 Distance cheetah travels in 1 minute Distance giant tortoise travels in 1 minute Difference in distances traveled A cheetah will travel 6,165 feet farther than a giant tortoise in 1 minute. Check Is your answer reasonable? The answer is close to the estimate, so the answer is reasonable. b. FOOD Almost 90 million jars of a popular brand of peanut butter are sold annually. Use the four-step plan to determine the approximate number of jars sold every second. Personal Tutor at ca.gr7math.com 26 Chapter 1 Algebra: Integers Example 1 (p. 25) 1. CRAFTS A quilt is made using different-sized groups of square patches as shown below. Use the four-step plan to determine how many square patches it will take to create the 10th figure in this pattern. Figure 1 Example 2 (p. 26) Figure 2 Figure 3 Figure 4 ART For Exercises 2 and 3, use the following information. The number of paintings an artist produced during her first four years as a professional is shown in the table at the right. Year Paintings Produced 2. About how many more paintings did she produce 1 59 in the last two years than in the first two years? 2 34 3. Estimate the total number of paintings the artist 3 91 4 20 has produced. (/-%7/2+ (%,0 For Exercises 4–5 6–7 See Examples 1 2 Use the four-step plan to solve each problem. 4. TRAVEL The table shows a portion of the bus Second and Elm Bus Schedule schedule for the bus stop at the corner of Second Street and Elm Street. What is the earliest time that Tyler can catch the bus if he cannot make it to the bus stop before 9:30 A.M.? 6:40 A.M. 6:58 A.M. 7:16 A.M. 7:34 A.M. 7:52 A.M. 8:10 A.M. 5. HOBBIES Owen fills his bird feeder with 4 pounds of sunflower seeds on Sunday morning. On Thursday morning, the bird feeder was empty, so he filled it again. The following Saturday, the seeds were half gone. If this feeding pattern continues, on what day will Owen need to fill his feeder? 6. FIELD TRIP Two 8th-grade teams, the Tigers and the Waves, are going to Washington, D.C. There are 123 students and 4 teachers on the Tigers team and 115 students and 4 teachers on the Waves team. If one bus holds 64 people, how many buses are needed for the trip? 7. HISTORY In 1803, the United States acquired the Louisiana Purchase from France for $15 million. The area of this purchase was 828,000 square miles. If one square mile is equal to 640 acres, about how much did the United States pay for the Louisiana Purchase per acre? Non-U.S. or Disputed Territories Louisiana Purchase United States 1803 Lesson 1-1 A Plan for Problem Solving 27 Use the four-step plan to solve each problem. 8. SCHOOL SUPPLIES A bookstore sells pens for $0.45 and writing tablets for $0.85. How many pens and tablets could you buy for exactly $2.15? 9. JOBS John stocks the vending machines at Rose Hill Elementary every 9 school days and Nassaux Intermediate every 6 school days. In September, he stocked both schools on the 27th. How many school days earlier had he stocked the vending machines at both schools on the same day? %842!02!#4)#% See pages 676, 708. Self-Check Quiz at GEOMETRY For Exercises 10 and 11, draw the next two figures in each pattern. 10. 11. ca.gr7math.com H.O.T. Problems 12. OPEN ENDED Refer to the Mini Lab at the beginning of the lesson. Describe another method you could use to find the number of white tiles it takes to border a garden 12 green tiles long. 13. CHALLENGE Draw the next figure in the pattern at the right. How many white tiles are needed when 21 green tiles are used? Explain. 14. SELECT A TECHNIQUE Handy Crafts will paint a custom design on the back of a cell phone for $3.25. Which of the following techniques should one use to determine the fewest number of phones that will need to be painted in order to earn $58.29 for the painting supplies? Justify your selection(s). Then use the technique(s) to solve the problem. mental math 15. estimation paper/pencil */ -!4( Summarize the four-step problem-solving plan. (*/ 83 *5*/( 16. Mrs. Acosta wants to buy 2 flag pins for each of the 168 band members for the Fourth of July Parade. Pins cost $0.09 each. Which is the best estimate of the cost of the pins? A $8 C $30 B $20 D $50 17. The next figure in the pattern will have what fraction of its area shaded? 3 F _ 8 _ G 1 2 5 H _ 8 3 J _ 4 PREREQUISITE SKILL Add, subtract, multiply, or divide. 18. 15 + 45 28 Chapter 1 Algebra: Integers 19. 1,287 - 978 20. 4 × 3.6 21. 280 ÷ 0.4 1- 2 Variables, Expressions, and Properties Main IDEA Evaluate expressions and identify properties. Standard 7AF1.2 Use the correct order of operations to evaluate algebraic expressions such as 3(2x + 5)2. Standard 7AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g. identity, inverse, distributive, associative, commutative) and justify the process used. Standard 7AF1.4 Use algebraic terminology (e.g. variable, equation, term, coefficient, inequality, expression, constant) correctly. The figures below are formed using toothpicks. If each toothpick is a unit, the perimeter of the first figure is 4 units. 1. Copy and complete the table. What is the relationship between the figure number and the perimeter of the figure? }ÕÀiÊ£ }ÕÀiÊÓ Figure Number 1 2 Perimeter 4 8 }ÕÀiÊÎ 3 4 5 6 2. What would be the perimeter of Figure 10? A variable is a symbol, usually a letter, used to represent a number. You can use the variable n to represent the figure number in the Mini Lab above. figure number 4×n NEW Vocabulary variable algebra algebraic expression evaluate numerical expression order of operations powers property counterexample expression for perimeter of figure The branch of mathematics that involves expressions with variables is called algebra. The expression 4 × n is called an algebraic expression because it contains a variable, a number, and at least one operation. To evaluate or find the value of an algebraic expression, first replace the variable or variables with the known values to produce a numerical expression, one with only numbers and operations. Then find the value of the expression using the order of operations. +%9 #/.#%04 Order of Operations 1. Perform all operations within grouping symbols first; start with the READING in the Content Area For strategies in reading this lesson, visit ca.gr7math.com. innermost grouping symbols. 2. Evaluate all powers before other operations. 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. Lesson 1-2 Variables, Expressions, and Properties 29 Algebra uses special ways of showing multiplication. Since the multiplication symbol × can be confused with the variable x, 4 times n is usually written as 4 • n, 4(n), or 4n. Expressions such as 7 2 and x 3 are called powers and represent repeated multiplication. 72 7 squared or 7 · 7 x3 x cubed or x · x · x Evaluate Algebraic Expressions 1 Evaluate 6(x - y)2 if x = 7 and y = 4. Parentheses Parentheses around a single number do not necessarily mean that multiplication should be performed first. Remember to multiply or divide in order from left to right. 6(x - y)2 = 6(7 - 4)2 = 6(3)2 Replace x with 7 and y with 4. Perform operations in the parentheses first. = 6 · 9 or 54 Evaluate the power. Then multiply. 2 Evaluate g 2 - 2g - 4 if g = 5. g 2 - 2g - 4 = (5)2 - 2(5) - 4 20 ÷ 4(2) = 5(2) or 10 Replace g with 5. = 25 - 2(5) - 4 Evaluate powers before other operations. = 25 - 10 - 4 Multiply 2 and 5. = 15 - 4 or 11 Subtract from left to right. Evaluate each expression if c = 3 and d = 7. b. 4(d - c)2 + 1 a. 6c + 4 - 3d c. d2 + 5d - 6 The fraction bar is another grouping symbol. Evaluate the expressions in the numerator and denominator separately before dividing. Evaluate Algebraic Fractions 3 Evaluate 4 + 6m _ if m = 9 and n = 5. 2n - 8 4 + 6(9) 4 + 6m _ =_ 2n - 8 2(5) - 8 Replace m with 9 and n with 5. 58 =_ Evaluate the numerator. 58 =_ or 29 Evaluate the denominator. Then divide. 2(5) - 8 2 Evaluate each expression if p = 5 and q = 12. 3p - 6 8-p d. _ 4q q + 2(p + 1) e. _ Personal Tutor at ca.gr7math.com 30 Chapter 1 Algebra: Integers 2 f. q _ 4p - 2 A property is a feature of an object or a rule that is always true. The following properties are true for all numbers. Property Algebra Arithmetic Commutative a+b=b+a a·b=b·a 6+1=1+6 7·3=3·7 Associative a + (b + c) = (a + b) + c a · (b · c) = (a · b) · c 2 + (3 + 8) = (2 + 3) + 8 3 · (4 · 5) = (3 · 4) · 5 Distributive a(b + c) = ab + ac a(b - c) = ab - ac 4(6 + 2) = 4 · 6 + 4 · 2 3(7 - 5) = 3 · 7 - 3 · 5 Identity a+0=a a·1=a 9+0=9 5·1=5 Identify Properties 4 Name the property shown by the statement 2 · (5 · n) = (2 · 5) · n. BrainPOP® ca.gr7math.com The order of the numbers and variables did not change but their grouping did. This is the Associative Property of Multiplication. Name the property shown by each statement. g. 42 + x + y = 42 + y + x h. 3x + 0 = 3x You may wonder whether any of the properties applies to subtraction or division. If you can find a counterexample, an example that shows that a conjecture is false, the property does not apply. Find a Counterexample Vocabulary Link Conjecture Everyday Use a guess Math Use an informed guess based on known information. 5 State whether the following conjecture is true or false. If false, provide a counterexample. Division of whole numbers is commutative. Write two division expressions using the Commutative Property. 15 ÷ 3 3 ÷ 15 1 5≠_ 5 State the conjecture. Divide. We found a counterexample. That is, 15 ÷ 3 ≠ 3 ÷ 15. So, division is not commutative. The conjecture is false. i. State whether the following conjecture is true or false. If false, provide a counterexample. The difference of two different whole numbers is always less than either of the two numbers. Extra Examples at ca.gr7math.com Lesson 1-2 Variables, Expressions, and Properties 31 Examples 1–3 (p. 30) Evaluate each expression if a = 2, b = 7, and c = 4. 1. (a + b)2 2. 4(a + b - c)2 3. c2 - 2c + 5 4. b2 - 2a + 10 5. _ 6. _ c2 b-5 bc 2 Example 3 (p. 30) Example 4 (p. 31) Example 5 (p. 31) 7. INSECTS The expression _ + 37 gives the approximate temperature of the c 4 air in degrees Fahrenheit, given the number of chirps c per minute made by a cricket. If Brandon estimates that a cricket has chirped 140 times in the past minute, what is the approximate temperature of the air in degrees Fahrenheit? Name the property shown by each statement. 8. 3(m + n) = 3m + 3n 9. 6(5 · y) = (6 · 5)y 10. State whether the following conjecture is true or false. If false, provide a counterexample. Subtraction of whole numbers is associative. (/-%7/2+ (%,0 For Exercises 11–22 23, 24 25–32 33–36 See Examples 1–3 3 4 5 Evaluate each expression if w = 2, x = 6, y = 4, and z = 5. 11. 2x + y 15. wx2 12. 3z - 2w 16. (wx)2 13. 9 + 7x - y x2 - 3 17. _ 2z + 1 14. 12 + z - x wz2 y+6 18. _ Evaluate each expression if a = 4, b = 3, and c = 6. 19. 3(c - b)2 - a 20. 2(ab - 9)2 ÷ c 21. 3b2 + 2b - 7 22. 2c2 - 4c + 5 23. MEASUREMENT When a temperature in degrees Fahrenheit F is known, 5F - 160 the expression _ can be used to find the temperature in degrees 9 Celsius C. If a thermometer shows that the temperature is 50°F, what is the temperature in degrees Celsius? 24. TRAVEL The cost of renting a car from EZ Rent-A-Car for a day is given by the expression _, where m is the number of miles driven. How much 270 + m 10 would it cost to rent a car for one day and drive 50 miles? Name the property shown by each statement. 32 25. 1(12 · 4) = 12 · 4 26. 14(16 · 32) = (14 · 16)32 27. a + (b + 12) = (b + 12) + a 28. (5 + x) + 0 = 5 + x 29. 15(3 + 6) = 15(3) + 15(6) 30. 16 + (c + 17) = (16 + c) + 17 31. 9(ab) = (9a)b 32. y · 7 = 7y Chapter 1 Algebra: Integers State whether each conjecture is true or false. If false, provide a counterexample. 33. The sum of two even numbers is always even. 34. The sum of two odd numbers is always odd. 35. Division of whole numbers is associative. 36. Subtraction of whole numbers is commutative. PETS For Exercises 37 and 38, use the information below. You can estimate the number of a certain type of pet National Percent of in a community with a population of c people by Households Owning c Pets evaluating the expression _ n · p. The variable n is the number of people per household, and p is the percent Dogs 0.316 of households owning that pet. Cats 0.273 Real-World Link The average dog visits its veterinarian almost twice as many times as the average cat or horse. Source: The American Veterinary Medical Association 37. According to the 2000 U.S. Census, there are approximately 2.62 people per household. Estimate the number of dog-owning households for a community with a population of 50,000. Birds 0.046 Horses 0.015 Source: U.S. Pet Ownership & Demographics Sourcebook 38. Estimate the number of bird-owning households in this community. 39. PHYSICAL SCIENCE The distance in feet an object falls t seconds after it is gt2 released is given by the expression _, where g is the force of gravity. 2 How many feet will a stone fall 3 seconds after it is released from the top of a cliff? Assume a force of gravity of 16 feet per second squared. Write each verbal statement as an algebraic expression. 40. the square of x minus the sum of four times x and 6 41. three times n cubed increased by four times n 42. the product of 3 and r decreased by the quotient of r squared divided by 6 RECREATION For Exercises 43–45, use the following information. A group is planning to go to an amusement park. There are two parks in the area, Fun World and Coaster City. The cost in dollars for n admission tickets to Fun World is 37n. If the group has 15 or more people, the cost at Coaster City is 30n + 75. If the group has fewer than 15 people, the cost at Coaster City is 40n. As few as 10 people or as many as 25 people might go. %842!02!#4)#% 43. Find the cost for each possible group size if they go to Fun World. See pages 676, 708. 44. Find the cost for each possible group size if they go to Coaster City. 45. Write a recommendation that details which park they should go to based Self-Check Quiz at ca.gr7math.com H.O.T. Problems on the number of people they expect to attend. Justify your answer. 46. OPEN ENDED Write an equation that illustrates the Commutative Property of Multiplication. CHALLENGE Decide whether each equation is true or false. If false, copy the equation and insert parentheses to make it true. 47. 8 · 4 - 3 · 2 = 26 48. 8 + 23 ÷ 4 = 4 49. 6 + 7 · 2 + 5 = 55 Lesson 1-2 Variables, Expressions, and Properties Gabe Palmer/CORBIS 33 50. FIND THE ERROR Regina and Camila are evaluating 10 ÷ 2 × 5. Who is correct? Explain. 10 ÷ 2 × 5 = 10 ÷ 10 =1 10 ÷ 2 × 5 = 5 × 5 = 25 Regina Camila */ -!4( Compare the everyday meaning of the term variable (*/ 83 *5*/( 51. with its mathematical definition. 52. The expression 6s 2 can be used to find 53. Which equation is an example of the the surface area of a cube, where s is the length of an edge of the cube. What is the surface area of the cube shown below? Associative Property? F 4·a=a·4 G 5 + (x + y) = (x + y) + 5 H w + (3 + 2) = w + (2 + 3) J d(9 · f) = (d · 9)f 54. If r = 4 and t = 3, then rt - 2r = CM A 4 A 144 cm2 B 6 B 432 cm2 C 19 C 864 cm2 D 40 D 5,184 cm2 Use the four-step plan to solve each problem. 55. DINING Kyung had $17. His lunch cost $5.62, and he gave the cashier a $10 bill. How much change should he receive from the cashier? (Lesson 1-1) 56. BABY-SITTING Kayla earned $30 baby-sitting last weekend. She wants to buy 3 CDs that cost $7.89, $12.25, and $11.95. Does she have enough money to purchase the CDs? Explain your reasoning. (Lesson 1-1) PREREQUISITE SKILL Replace each ● with <, >, or = to make a true sentence. 57. 4 ● 9 34 Chapter 1 Algebra: Integers (l)Cleve Bryant/PhotoEdit, (r)David Young-Wolff/PhotoEdit 58. 7 ● 7 59. 8 ● 5 60. 3 ● 2 1- 3 Integers and Absolute Value Main IDEA Compare and order integers and find absolute value. Standard 7NS2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers. NEW Vocabulary negative number positive number integer coordinate inequality absolute value GEOGRAPHY Badwater, in Death Valley, California, is the lowest point in North America, while Mt. McKinley in Alaska is the highest point. 1. What does an elevation of -86 meters represent? 2. What does a temperature of -35° represent? With sea level as the starting point 0, you can express 86 meters below sea level as negative 86 or -86. A negative number is a number less than zero. A positive number like 125 is a number greater than zero. Numbers like -86 and 125 are called integers. An integer is any number from the set {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...} where ... means continues without end. positive integer negative integer ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 Numbers to the left of zero are less than zero. 0 1 2 3 4 5 6 Numbers to the right of zero are greater than zero. Zero is neither positive nor negative. To graph an integer, locate the point corresponding to the integer on a number line. The number that corresponds to a point is called its coordinate. graph of a point with coordinate 4 graph of a point with coordinate ⫺5 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 READING Math Inequality Symbols < is less than > is greater than 0 1 2 3 4 5 6 Notice that -5 is to the left of 4 on the number line. This means that -5 is less than 4. A sentence that compares two different quantities is called an inequality. Inequalities contain symbols like < and >. -5 is less than 4. -5 < 4 4 > -5 4 is greater than -5. Lesson 1-3 Integers and Absolute Value 35 Compare Integers 1 Replace the ● with < or > to make -2 ● -4 a true sentence. Graph each integer on a number line. Since -2 is to the right of -4, -2 > -4. Replace each ● with < or > to make a true sentence. a. -3 ● 2 b. -5 ● -6 c. -1 ● 1 The distance between a number and 0 on a number line is called its absolute value. On the number line below, notice that -4 and 4 are each 4 units from 0, even though they are on opposite sides of 0. They have the same absolute value, 4. 4 units ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 4 units 0 1 2 3 4 5 6 The symbol for absolute value is two vertical bars on either side of the number. The absolute value of 4 is 4. Absolute Value Since distance cannot be negative, the absolute value of a number is always positive or zero. ⎪4⎥ = 4 ⎪-4⎥ = 4 The absolute value of -4 is 4. Expressions with Absolute Value 2 Evaluate ⎪-7⎥. 7 units ⫺8 ⫺7 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 3 The graph of -7 is 7 units to the left of 0 on the number line. ⎪-7⎥ = 7 3 Evaluate ⎪5⎥ + ⎪-6⎥. ⎪5⎥ + ⎪-6⎥ = 5 + ⎪-6⎥ The absolute value of 5 is 5. =5+6 The absolute value of -6 is 6. = 11 Simplify. 4 Evaluate ⎪5 - 3 ⎥ + ⎪8 - 10⎥. ⎪5 - 3⎥ + ⎪8 - 10⎥ = ⎪2⎥ + ⎪-2⎥ 36 Chapter 1 Algebra: Integers Simplify the absolute value expressions. = 2 + ⎪-2⎥ The absolute value of 2 is 2. =2+2 The absolute value of -2 is 2. =4 Simplify. 5 Evaluate 8 + ⎪n⎥ if n = -12. 8 + ⎪n⎥ = 8 + ⎪-12 ⎥ Replace n with -12. = 8 + 12 or 20 ⎪-12⎥ = 12 Evaluate each expression. d. ⎪14⎥ e. ⎪-8⎥ - ⎪-2⎥ f. ⎪7 - 4⎥ + ⎪12 - 15⎥ g. ⎪a⎥ - 2, if a = -5. You can also use an absolute value expression to find the distance between a number and zero on a number line. 6 SNAKES A tank used to keep a pet snake should be kept at a temperature of 80°F, give or take 5°. Graph the equation ⎪x - 80⎥ = 5 to determine the least and the greatest temperatures. ⎪x - 80⎥ = 5 means that the distance between x and 80 is 5 units. Start at 80 and move 5 units in either direction to find the value of x. 5 units 74 Real-World Link 76 5 units 78 80 The distance from 80 to 75 is 5 units. 82 84 86 The distance from 80 to 85 is 5 units. The solution set is {75, 85}. Snakes are cold-blooded, which means that they cannot regulate their body temperature. Their body temperature will reflect the temperature of their surroundings, and these animals cannot survive temperature extremes. Source: trailquest.net h. The average lifespan of an elephant in the wild is 65 years, give or take 6 years. Graph the equation ⎪y - 65| = 6 on a number line to determine the least and greatest average age of an elephant. Personal Tutor at ca.gr7math.com Example 1 (p. 36) Examples 2-4 (p. 36) Example 5 (p. 37) Example 6 (p. 37) Replace each ● with < or > to make a true sentence. 1. 1 ● -5 2. -1 ● -2 3. -4 ● 3 4. -7 ● -3 7. ⎪6 - 3⎥ - ⎪2 - 4⎥ 8. ⎪-8⎥ - ⎪-2⎥ Evaluate each expression. 5. ⎪5⎥ 6. ⎪-9⎥ Evaluate each expression if x = -10 and y = 6. 9. 3 + ⎪x⎥ 10. ⎪y⎥ + 12 11. ⎪x⎥ - y 12. PROFIT In order to ensure a profit, the average cost of a CD must be $16, give or take $3. Graph the equation ⎪c - 16⎥ = 3 to determine the least and greatest cost of a CD. Extra Examples at ca.gr7math.com © Peter Arnold, Inc./Alamy Lesson 1-3 Integers and Absolute Value 37 (/-%7/2+ (%,0 For Exercises 13–24 25–30 31–34 35–38 See Examples 1 2–4 5 6 Replace each ● with <, >, or = to make a true sentence. 13. 0 ● -1 14. 5 ● -6 15. -9 ● -7 16. -6 ● -1 17. -7 ● -2 18. 0 ● 12 19. -9 ● -10 20. 4 ● -11 21. -3 ● 0 22. -15 ● 14 23. -8 ● -8 24. -13 ● -13 Evaluate each expression. 25. ⎪-14⎥ 26. ⎪25⎥ 27. ⎪0⎥ + ⎪-18⎥ 28. ⎪2⎥ - ⎪-13⎥ 29. ⎪6 - 8⎥ + ⎪9 - 5⎥ 30. ⎪14 - 7⎥ - ⎪5 - 8⎥ Evaluate each expression if a = 5, b = -8, c = -3, and d = 9. 31. ⎪b⎥ + 7 32. a - ⎪c⎥ 33. d + ⎪b⎥ 34. 6⎪b⎥ + d Graph the equation to determine the solutions. 35. ⎪x - 15⎥ = 10 36. ⎪a - 7⎥ = 4 37. SOCCER A professional soccer player is in his prime at age 26 plus or minus 7 years. This range can be modeled by the equation ⎪x - 26⎥ = 7. Graph the equation on a number line to determine the least and the greatest ages. 38. MONEY The Perez family spends an average of $435 per month on groceries give or take $22. This range can be modeled by the equation ⎪y - 435⎥ = 22. Graph the equation on a number line to determine the least and the greatest money spent. CHEMISTRY For Exercises 39–42, use the table at the right. Gas Freezing Point (°F) at Sea Level 39. Which of these gases freezes at the coldest hydrogen -435° krypton -251° oxygen -369° helium -458° argon -309° temperature? 40. Which of these gases freezes at the warmest temperature? 41. The freezing point for xenon at sea level is %842!02!#4)#% See pages 676, 708. about 200 degrees warmer than the freezing point for oxygen. What is the approximate freezing point of xenon? Justify your answer using a number line. 42. How many degrees lower is the freezing point Self-Check Quiz at ca.gr7math.com 38 for oxygen at sea level than the freezing point for argon? Justify your answer using a number line. Chapter 1 Algebra: Integers H.O.T. Problems CHALLENGE Determine whether each statement is always, sometimes, or never true. Explain your reasoning. 43. The absolute value of a positive integer is a negative integer. 44. If a and b are integers and a > b, then ⎪a⎥ > ⎪b⎥. 45. If a and b are integers, a - ⎪b⎥ ≤ a + b. 46. Which One Doesn’t Belong? Identify the phrase that cannot be described by the same integer as the other three. Explain your reasoning. 5° below normal 5 miles above sea level a loss of 5 pounds giving away $5 */ -!4( Explain why the absolute value of a number is never (*/ 83 *5*/( 47. negative. 49. If a = -3 and b = 3, then which of the 48. The table shows the number of laps selected race cars finished behind the winner of a race. following statements is false? F ⎪a⎥ > 2 Car Number Laps Behind Winner G ⎪a⎥ = ⎪b⎥ 3 -1 H ⎪b⎥ < 2 8 -12 J 15 -3 24 0 48 -8 ⎪a⎥ = b 50. Which expression has the greatest value? A ⎪-25⎥ Which list shows the finishing order of the cars from first to fifth? B ⎪-16⎥ A 8, 48, 15, 3, 24 C 24, 3, 15, 48, 8 C ⎪18⎥ B 3, 8, 15, 24, 48 D 48, 24, 15, 8, 3 D ⎪22⎥ ALGEBRA Evaluate each expression if m = 3, n = 2, p = 10, and r = 15. (Lesson 1-2) 51. r - 4n 52. 2m 2 - p + 3 53. 3p + m _ r - 2n 54. CHARITY WALK Krystal knows that she can walk about 1.5 meters per second. If she can maintain that pace, about how long should it take her to complete a 10-kilometer charity walk? (Lesson 1-1) PREREQUISITE SKILL Add or subtract. 55. 9 + 14 56. 100 - 57 57. 47 - 19 58. 18 + 34 + 13 Lesson 1-3 Integers and Absolute Value 39 Extend 1-3 Main IDEA Algebra Lab Graphing Data In this lab, you will investigate the relationship between the height of a chute and the distance an object travels as it leaves the chute. Graph and interpret data. Standard 7AF1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. Standard 7MR2.3 Estimate unknown quantities graphically and solve for them by using logical reasoning and arithmetic and algebraic techniques. Make a meter-long chute for the ball out of cardboard. Reinforce the chute by taping it to one of the metersticks. Use the tape measure to mark off a distance of 3 meters on the floor. Make a 0-meter mark and a 3-meter mark using tape. Place the end of your chute at the edge of the 0-meter mark. Raise the back of the chute to a height of 5 centimeters. Let a tennis ball roll down the chute. When the ball stops, measure how far it is from the 3-meter mark. Copy the table shown and record your results. If the ball stops short of the 3-meter mark, record the distance as a negative number. If the ball passes the 3-meter mark, record the distance as a positive number. Raise the chute by 5 centimeters and repeat the experiment. Continue until the chute is 40 centimeters high. meterstick 5 cm 0m 3m Height h of Chute (cm) 5 10 15 Distance d from 3-meter Mark (cm) ANALYZE THE RESULTS 1. Graph the ordered pairs (h, d) on a coordinate grid. 2. Describe how the points appear on your graph. 3. Describe how raising the chute affects the distance the ball travels. 4. MAKE A PREDICTION Use your graph to predict how far the ball will roll when the chute is raised to the 50-centimeter mark. Then check your prediction. 40 Chapter 1 Algebra: Integers 1- 4 Adding Integers Main IDEA Add integers. Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. Standard 7AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g. identity, inverse, distributive, associative, commutative) and justify the process used. Thank you all for participating in our tournament! You owe us a grand total of $13,200! 1. Write an integer that describes the game show host’s statement. 2. Write an addition sentence that describes this situation. The equation -3,200 + (-7,400) + (-2,600) = -13,200 is an example of adding integers with the same sign. Notice that the sign of the sum is the same as the sign of each addend. NEW Vocabulary Add Integers with the Same Sign opposites additive inverse 1 Find -4 + (-2). Use a number line. REVIEW Vocabulary • Start at zero. addends numbers that are added together sum the result when two or more numbers are added together • Move 4 units left. ⫺2 • From there, move 2 units left. ⫺4 ⫺7 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 So, -4 + (-2) = -6. Add. Use a number line if necessary. a. -3 + (-2) b. 1 + 5 c. -5 + (-4) These examples suggest a rule for adding integers with the same sign. +%9 #/.#%04 Add Integers with the Same Sign Words To add integers with the same sign, add their absolute values. The sum has the same sign as the integers. Examples -7 + (-3) = -10 5+4=9 Lesson 1-4 Adding Integers 41 A number line can also help you add integers with different signs. Add Integers with Different Signs 2 Find 5 + (-2). Use a number line. Adding Integers on a Number Line Always start at zero. Move right to model a positive integer and left to model a negative integer. ⫺2 5 • Start at zero. ⫺1 • Move 5 units right. 0 1 2 3 4 5 6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 0 1 2 • From there, move 2 units left. 5 + (-2) = 3 3 Find -4 + 3. Use a number line. 3 ⫺4 • Start at zero. • Move 4 units left. • From there, move 3 units right. -4 + 3 = -1 Add. Use a number line if necessary. d. 7 + (-5) e. -6 + 4 f. -1 + 8 These examples suggest a rule for adding integers with different signs. +%9 #/.#%04 Words Add Integers with Different Signs To add integers with different signs, subtract their absolute values. The sum has the same sign as the integer with the greater absolute value. Examples 8 + (-3) = 5 -8 + 3 = -5 Add Integers with Different Signs 4 Find -14 + 9. -14 + 9 = -5 To find -14 + 9, subtract ⎪9⎥ from ⎪-14⎥. The sum is negative because ⎪-14⎥ > ⎪9⎥. Add. g. -20 + 4 42 Chapter 1 Algebra: Integers h. 17 + (-6) i. -8 + 27 Interactive Lab ca.gr7math.com Two numbers with the same absolute value but different signs are called opposites. For example, -2 and 2 are opposites. An integer and its opposite are also called additive inverses. +%9 #/.#%04 Words Additive Inverse Property The sum of any number and its additive inverse is zero. Examples Numbers Algebra 7 + (-7) = 0 x + (-x) = 0 The Commutative, Associative, and Identity Properties, along with the Additive Inverse Property, can help you add three or more integers. Add Three or More Integers 5 Find -4 + (-12) + 4. -4 + (-12) + 4 = -4 + 4 + (-12) Commutative Property = 0 + (-12) Additive Inverse Property = -12 Identity Property of Addition Add. j. 33 + 16 + (-33) k. 3 + (-2) + (-10) + 6 Personal Tutor at ca.gr7math.com 6 MONEY The starting balance in a checking account is $75. What is the balance after checks for $12 and $20 are written? Writing a check decreases your account balance, so integers for this situation are -12 and -20. Add these integers to the starting balance to find the new balance. 75 + (-12) + (-20) = 75 + [-12 + (-20)] Associative Property = 75 + (-32) -12 + (-20) = -32 = 43 Simplify. Real-World Link One out of every five Americans does not have a checking account. The balance is now $43. Source: harpers.org l. BANKING A checking account has a starting balance of $130. What is the balance after writing checks for $58 and $62, then making a deposit of $150? Extra Examples at ca.gr7math.com Ross M. Horowitz/Getty Images Lesson 1-4 Adding Integers 43 Add. Examples 1–4 (pp. 41–42) Example 5 1. -4 + (-5) 2. -18 + (-8) 3. -3 + (-12) 4. 10 + (-6) 5. 7 + (-18) 6. -9 + 16 7. 11 + 9 + (-3) 8. 8 + (-6) + 5 9. 3 + (-15) + 1 (p. 43) Example 6 (p. 43) (/-%7/2+ (%,0 For Exercises 11–16 17–22 23–28 29, 30 See Examples 1 2–4 5 6 10. GOLF Suppose a player shot -5, +2, -3, and -2 in four rounds of a tournament. What was the player’s final score? Add. 11. 14 + 8 12. 12 + 17 13. -14 + (-6) 14. -21 + (-13) 15. -5 + (-31) 16. -7 + (-24) 17. 20 + (-5) 18. 45 + (-4) 19. -15 + 8 20. -19 + 2 21. -10 + 34 22. -17 + 28 23. 5 + 18 + (-22) 24. 8 + 13 + (-14) 25. -17 + (-4) + 10 26. -26 + (-8) + 2 27. -12 + 9 + (-15) 28. -34 + 19 + (-16) ANALYZE TABLES For Exercises 29 and 30, use the table below that shows the change in music sales to the nearest percent from 2001 to 2003. 29. What is the percent of music sold in 2003 for each of these musical categories? 30. What was the total percent change in the sale of these types of music? Percent of Music Sold in 2001 Percent Change as of 2003 Rock 24 +1 Rap/Hip Hop 11 +2 Pop 11 -2 Country 11 -1 Style of Music Source: Recording Industry Association of America Write an addition expression to describe each situation. Then find each sum and explain its meaning. 31. FOOTBALL Your team gains 8 yards on its first play. On the next play, your team loses 5 yards. 32. SCUBA DIVING A scuba diver dives 125 feet below the water’s surface. Later, she rises 46 feet. 33. WEATHER The temperature outside is -2°F. The temperature drops by 9°. %842!02!#4)#% See pages 677, 708. Self-Check Quiz at ca.gr7math.com 44 Add. 34. -47 + (-41) + (-33) 35. -51 + (-38) + (-44) 36. -31 + (-26) + (-60) 37. -13 + 6 + (-8) + 13 38. 9 + (-4) + 12 + (-9) 39. -14 + 2 + (-15) + 7 Chapter 1 Algebra: Integers H.O.T. Problems 40. OPEN ENDED Give an example of a positive and a negative integer with a negative sum. Then find their sum. 41. CHALLENGE Determine whether the following statement is always, sometimes, or never true. Give examples to justify your answer. If x and y are integers, then ⎪x + y⎥ = ⎪x⎥ + ⎪y⎥. */ -!4( Find the sum of -8, 25, and -2 mentally by applying (*/ 83 *5*/( 42. the properties of numbers. Justify the process. 43. A stock’s opening value on Monday 44. Marcus started the month with a morning was $52. What was the stock worth at the end of the day on Friday? Day Change Monday -$2 Tuesday +$1 Wednesday +$3 Thursday -$1 Friday -$4 balance of $75 in his checking account. He made a deposit of $12.50 and wrote three checks in the amounts of $25, $58.75, and $32. What is the balance of his checking account? F $3.75 G $0 H -$18.75 -$28.25 J A $41 C $57 B $49 D $63 Replace each ● with <, >, or = to make a true sentence. (Lesson 1-3) 45. -6 ● -11 46. 5 ● -5 47. 5 ● ⎪8⎥ 48. ⎪-7⎥ ● -7 49. WEATHER The time s in seconds between seeing lightning and hearing thunder can be used to estimate a storm’s distance in miles. Use the expression _s to determine how far away a storm is if this time is 5 15 seconds. (Lesson 1-2) of prime-time television viewers in millions for different age groups. Estimate the total number of viewers for all the age groups given. (Lesson 1-1) Prime-Time Viewers (millions) 18 to 24 Age Group 50. STATISTICS The graph shows the number 73.8 25 to 34 81.3 81.1 83.5 85.6 86.7 35 to 44 45 to 54 55 to 64 65 and over 70 PREREQUISITE SKILL Evaluate each expression if x = 3, y = 9, and z = 5. (Lesson 1-2) 51. x + 14 52. z - 2 53. y - z 80 90 54. x + y - z Lesson 1-4 Adding Integers 45 1- 5 Subtracting Integers Main IDEA You can use algebra tiles to model the subtraction of two integers. Follow these steps to model 3 - 5. Remember that subtract means take away or remove. Subtract integers. Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. Place 3 positive tiles on the mat. Add 2 zero pairs to the mat, so you have 5 positive tiles. Remove 5 positive tiles. Since 2 negative tiles remain, 3 - 5 = -2. 1. How does this result compare with the result of 3 + (-5)? 2. Use algebra tiles to find -4 - 2. 3. How does this result compare to -4 + (-2)? 4. Use algebra tiles to find each difference and sum. Compare the results in each group. a. 1 - 5; 1 + (-5) b. -6 - 4; -6 + (-4) When you subtract 5, as shown in the Mini Lab, the result is the same as adding -5. When you subtract 2, the result is the same as adding -2. same integers 3 - 5 = -2 same integers 3 + (-5) = -2 -4 - 2 = -6 opposite integers -4 + (-2) = -6 opposite integers These and other examples suggest a method for subtracting integers. BrainPOP® ca.gr7math.com +%9 #/.#%04 Words Examples 46 Chapter 1 Algebra: Integers Subtract Integers To subtract an integer, add its opposite or additive inverse. Numbers Algebra 4 - 7 = 4 + (-7) or -3 a - b = a + (-b) Subtract a Positive Integer 1 Find 9 - 12. 9 - 12 = 9 + (-12) To subtract 12, add -12. = -3 Add. 2 Find -6 - 8. -6 - 8 = -6 + (-8) To subtract 8, add -8. = -14 Add. Subtract. a. 3 - 8 b. -5 - 4 c. 10 - 7 Subtract a Negative Integer 3 Find 7 - (-15). 7 - (-15) = 7 + 15 or 22 To subtract -15, add 15. 4 CHEMISTRY The melting point of mercury is about -39°C and the melting point of aluminum is about 660°C. Find the difference between these temperatures. 660 - (-39) = 660 + 39 or 699 To subtract -39, add 39. The difference between the temperatures is about 699°C. Subtract. d. 6 - (-7) e. -5 - (-19) f. -14 - (-2) Personal Tutor at ca.gr7math.com Evaluate Algebraic Expressions Common Error In Example 5, a common error is to replace b with 8 instead of its correct value of -8. Prevent this error by inserting a set of parentheses before replacing b with its value. Evaluate each expression if a = 9, b = -8, and c = -2. 5 14 - b 14 - b = 14 - (-8) = 14 + 8 or 22 To subtract -8, add 8. 6 c - a2 ) c - a2 = -2 - 92 = 14 - (-8) = -2 - 81 14 - b = 14 - ( Replace b with -8. Replace c with -2 and a with 9. Simplify 92. = -2 + (-81) or -83 To subtract 81, add -81. Evaluate each expression if x = -5 and y = 7. g. x - (-8) Extra Examples at ca.gr7math.com h. -3 - y i. y2 - x + 3 Lesson 1-5 Subtracting Integers 47 Examples 1– 4 (p. 47) Example 3 (p. 47) Examples 5, 6 (p. 47) (/-%7/2+ (%,0 For Exercises 13–16 17–20 21–24 25–28 29–30 31–38 See Examples 1 2 3 4 1–4 5, 6 Subtract. 1. 8 - 13 2. 5 - 24 3. -4 - 10 4. -6 - 3 5. 7 - (-3) 6. 2 - (-8) 7. -2 - (-6) 8. -18 - (-7) 9. SPACE On Mercury, the temperatures range from 805°F during the day to -275°F at night. Find the change in temperature from day to night. Evaluate each expression if n = 10, m = -4, and p = -12. 10. n - 17 11. m - p 12. p + n - m Subtract. 13. 14 - 8 14. 17 - 12 15. 5 - 9 16. 1 - 8 17. -16 - 4 18. -15 - 12 19. -3 - 14 20. -6 - 13 21. 9 - (-5) 22. 10 - (-2) 23. 5 - (-11) 24. 17 - (-14) 25. -5 - (-4) 26. -18 - (-7) 27. -3 - (-6) 28. -9 - (-20) GEOGRAPHY For Exercises 29 and 30, use the table at the right. Great Lakes 29. How far below the surface Deepest Point (m) Surface Elevation (m) Erie -64 174 elevation is the deepest part of Lake Huron? Huron -229 176 Michigan -281 176 30. Find the difference between the Ontario -244 75 Superior -406 183 deepest part of Lake Erie and the deepest part of Lake Superior. Source: National Ocean Service Evaluate each expression if a = -3, b = 14, and c = -8. 31. b - 20 35. b - a 32. c - 15 33. a - c 34. a - b 36. c - b 37. (b - a)2 + c 38. a - c -b2 ANALYZE TABLES For Exercises 39 and 40, use the table. 39. The wind makes the air outside feel Wind Chill Temperature colder than the actual air temperature. How much colder does a temperature of 0°F with a 30-mile-per-hour wind feel than the same temperature with a 10-mile-per-hour wind? Temperature (°F) Wind (miles per hour) %842!02!#4)#% 40. How much warmer does 20°F feel than See pages 677, 708. -10°F, both with a 30-mile-per-hour wind? Calm 10 20 30 20 9 4 1 10 -4 -9 -12 0 -16 -22 -26 -10 -28 -35 -39 Source: National Weather Service Self-Check Quiz at ca.gr7math.com 48 Simplify. 41. 31 - (-3) - (-18) Chapter 1 Algebra: Integers 42. -20 - [6 + (-28)] 43. (-3 + 8) - (-21 - 10) H.O.T. Problems 44. OPEN ENDED Write an expression involving the subtraction of a negative integer. Then write an equivalent addition expression. 45. FIND THE ERROR Anna and David are finding -5 - (-8). Who is correct? Explain your reasoning. -5 - (-8) = 5 + 8 = 13 -5 - (-8) = -5 + 8 =3 Anna David CHALLENGE For Exercises 46 and 47, determine whether the statement is true or false. If false, give a counterexample. 46. If x and y are positive integers, then x - y is a positive integer. 47. Subtraction of integers is commutative. 48. */ -!4( Write a problem about a real-world situation (*/ 83 *5*/( involving subtraction of integers for which the answer is -4. 49. Use the thermometers to determine how much the temperature increased between 8:00 A.M. and 12:00 P.M. A 14°F C 30°F B 15°F D 31°F !- & 0- & 50. Find the distance between A and B on the number line. A B ⫺5 H 3 units G -3 units J several baseball teams in a recent year. What was the total income of all of these teams? (Hint: A gain is positive income, and a loss is negative income.) (Lesson 1-4) Evaluate each expression. (Lesson 1-3) 53. ⎪20⎥ - ⎪-5⎥ 54. ⎪13 - (-7)⎥ 55. ⎪-12 + (-25)⎥ 2 F -7 units 51. BASEBALL The table shows the income of 52. ⎪-14⎥ + ⎪3⎥ 0 7 units Team Income (thousands) Atlanta Braves -$14,360 Chicago Cubs $4,797 Florida Marlins -$27,741 New York Yankees $40,359 Source: mlb.com PREREQUISITE SKILL Find the mean for each set of data. 56. 1, 2, 3, 6, 8 57. 12, 13, 14, 16, 17, 18 58. 40, 45, 55, 60, 75, 85 Lesson 1-5 Subtracting Integers (l)age fotostock/SuperStock, (r)David Young-Wolff/PhotoEdit 49 CH APTER 1 Mid-Chapter Quiz Lessons 1-1 through 1-5 1. TRAVEL A cruise ship has 148 rooms, with 8. fifty on each of the two upper decks and the rest on the two lower decks. An upper deck room costs $1,000, and a lower deck room costs $900. Use the four-step plan to find the greatest possible room sales on one trip. STANDARDS PRACTICE The table gives several of the highest and lowest elevations, in meters, on Earth’s land surface. Name (Lesson 1-1) 2. STANDARDS PRACTICE A landscaper plants bushes in a row across the back and down two sides of a yard. A bush is planted at each of the four corners and at every 4 meters. Which expression would give the number of bushes that are planted? (Lesson 1-1) Mt. Everest Nepal 8,850 Lake Assal Djibouti –156 Mt. McKinley Alaska 6,194 Death Valley California –86 Dead Sea Israel –400 F -86, -156, -400, 6,194, 8,850 G 8,850, 6,194, -400, -156, -86 H -400, -156, -86, 6,194, 8,850 "ACK M Elevation Choose the group of elevations that is listed in order from least to greatest. (Lesson 1-3) BUSH M Location J M -156, -86, -400, 6,194, 8,850 Add or subtract. (Lessons 1-4, 1-5) A 2 × (36 ÷ 4) + (68 ÷ 4) 9. -7 + 2 + (-1) 10. -3 - (-4) B 2 + 2 × (36 ÷ 4) + (64 ÷ 4) 11. 2 - 6 12. -5 + (-8) C 4 + 2 × (36 ÷ 4) + (68 ÷ 4) 13. -5 + 9 14. -11 + 15 + 11 + (-6) D 2 × (36 ÷ 4) + 2 × (68 ÷ 4) 15. 12 + (-4) - 7 16. -7 + 14 + (-1) + 13 17. -4 + -7 18. (-1) + (-5) + 18 - 3 Evaluate each expression if x = 3, y = 6, and z = 2. (Lesson 1-2) 3. x 2 + y 2 + z 2 xy 4. _ z - 4z 5. MEASUREMENT The expression 2 + 2w gives the perimeter of a rectangle with length and width w. What amount of fencing would Mr. Nakagawa need in order to fence his tomato garden that is 12 feet long and 9 feet wide? (Lesson 1-2) Replace each ● with <, >, or = to make a true sentence. (Lesson 1-3) 6. -3 ● 2 50 7. ⎪-4⎥ ● ⎪4⎥ Chapter 1 Algebra: Integers 19. STANDARDS PRACTICE If ⎪y⎥ = 5, what is the value of y? A -25 or 25 B 0 or 5 C -5 or 5 D -5 or 0 20. ELEVATORS In one hour, an elevator traveled up 5 floors, down 2 floors, up 8 floors, down 6 floors, up 11 floors, and down 14 floors. If the elevator started on the seventh floor, on which floor is it now? (Lessons 1-4, 1-5) 1- 6 Multiplying and Dividing Integers Main IDEA Multiply and divide integers. Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers. Standard 7AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g. identity, inverse, distributive, associative, commutative) and justify the process used. OCEANOGRAPHY A deep-sea submersible descends 120 feet each minute to reach the bottom of Challenger Deep, a trench in the Pacific Ocean. The table shows the submersible’s depth at different times. Time (min) Depth (ft) 1 -120 2 -240 1. Write two different addition sentences that could be used to find the submersible’s depth after 3 minutes. Then find their sums. 2. Write a multiplication sentence that could be used to find this same depth. Explain your reasoning. 3. Write a multiplication sentence that could be used to find the submersible’s depth after 10 minutes. Then find the product. Multiplication is repeated addition. So, 3(-120) means that -120 is used as an addend 3 times. ⫺120 3(-120) = -120 + (-120) + (-120) = -360 REVIEW Vocabulary product the result when two or more numbers are multiplied together ⫺360 ⫺120 ⫺240 ⫺120 ⫺120 0 120 By the Commutative Property of Multiplication, 3(-120) = -120(3). This example suggests the following rule. +%9 #/.#%04 Words Multiply Integers with Different Signs The product of two integers with different signs is negative. Examples 2(-5) = -10 -5(2) = -10 Multiply Integers with Different Signs REVIEW Vocabulary factors numbers that are multiplied together 1 Find 6(-8). 6(-8) = -48 The factors have different signs. The product is negative. 2 Find -9(2). -9(2) = -18 The factors have different signs. The product is negative. Multiply. a. 5(-3) b. -8(6) c. -2(4) Lesson 1-6 Multiplying and Dividing Integers Chris McLaughlin/CORBIS 51 The product of two positive integers is positive. What is the sign of the product of two negative integers? Look at the pattern below. Factor · Factor = Product Negative · Positive = Negative Negative · Negative = Positive +%9 #/.#%04 -3 · 2 = -6 -3 · 1 = -3 -3 · 0 = 0 -3 · (-1) = 3 -3 · (-2) = 6 +3 +3 +3 +3 Multiply Integers with the Same Sign The product of two integers with the same sign is positive. Words Examples 2(5) = 10 -2(-5) = 10 Multiply Integers with the Same Sign 3 Find -4(-3). -4(-3) = 12 The factors have the same sign. The product is positive. Multiply. d. -3(-7) f. (-5) 2 e. 6(4) To multiply more than two integers, use the Commutative and Associative Properties of Multiplication. Multiply More than Two Integers 4 Find -2(13)(-5). METHOD 1 Use the Associative Property. -2(13)(-5) = [-2(13)](-5) = -26(-5) = 130 METHOD 2 Mental Math Look for products that are multiples of ten to make the multiplication simpler. Associative Property -2(13) = -26 -26(-5) = 130 Use the Commutative Property. -2(13)(-5) = -2(-5)(13) = 10(13) = 130 Commutative Property -2(-5) = 10 10(13) = 130 Multiply. g. 4(-2)(-5) h. -1(-3)(-8) Personal Tutor at ca.gr7math.com 52 Chapter 1 Algebra: Integers i. (-2) 3 Examine the following multiplication sentences and their related division sentences. READING Math Division In a division sentence like 12 ÷ 3 = 4, the number you are dividing, 12, is called the dividend. The number you are dividing by, 3, is called the divisor. The result is called the quotient. Multiplication Sentence Related Division Sentences 4(3) = 12 12 ÷ 3 = 4 -4(3) = -12 -12 ÷ 3 = -4 4(-3) = -12 12 ÷ 4 = 3 -12 ÷ -4 = 3 -12 ÷ (-3) = 4 -4(-3) = 12 12 ÷ (-3) = -4 -12 ÷ 4 = -3 12 ÷ (-4) = -3 These examples suggest that the rules for dividing integers are similar to the rules for multiplying integers. +%9 #/.#%04 Words Divide Integers The quotient of two integers with different signs is negative. The quotient of two integers with the same sign is positive. Examples 16 ÷ (-8) = -2 -16 ÷ (-8) = 2 Divide Integers 5 Find -24 ÷ 3. The dividend and the divisor have different signs. -24 ÷ 3 = -8 The quotient is negative. 6 Find -30 . _ The signs are the same. -30 _ =2 The quotient is positive. -15 -15 Divide. j. -28 ÷ (-7) k. _ 36 -2 l. -40 _ 8 You can use all of the rules you have learned for adding, subtracting, multiplying, and dividing integers to evaluate algebraic expressions. Evaluate Algebraic Expressions 7 Evaluate -2a - b if a = -3 and b = -5. -2a - b = -2(-3) - (-5) = 6 - (-5) =6+5 = 11 Replace a with -3 and b with -5. The product of -2 and -3 is positive. To subtract -5, add 5. Add. Evaluate each expression if a = -4, b = 5, and c = -6. m. c + 3a Extra Examples at ca.gr7math.com n. _ -10 a+b o. ab + c 2 Lesson 1-6 Multiplying and Dividing Integers 53 8 GAMES In each round of a game, you can gain or Atepa –10 –30 –20 10 20 lose points. Atepa’s change in score for each of five rounds is shown. Find Atepa’s mean (average) point gain or loss per round. To find the mean of a set of numbers, find the sum of the numbers. Then divide the sum by how many numbers there are in the set. -10 + (-30) + (-20) + 10 + 20 -30 Find the sum of the set of numbers. ___ =_ 5 5 = -6 Divide by the number in the set. Simplify. Atepa lost an average of 6 points per round. p. TEMPERATURE On six consecutive days, the low temperature in Montreal was -6°C, -5°C, 6°C, 3°C, -2°C, and -8°C. What was the average low temperature for the six days? #/.#%04 3UMMARY Multiplying and Dividing Integers • The product or quotient of two integers with the same sign is positive. • The product or quotient of two integers with different signs is negative. Examples 1–4 (pp. 51–52) Multiply. 1. 4(-5) 4. -7(-2) Examples 5, 6 (p. 53) Example 7 (p. 53) 54 3. -3(7) 2. 3(-6) 5. (-3) 2 6. -4(5)(-7) Divide. 7. -16 ÷ 4 8. 21 ÷ (-3) 9. -72 ÷ (-8) 22 10. _ 11 -25 11. _ -5 12. _ -96 12 Evaluate each expression if a = -5, b = 8, and c = -12. 13. 4a + 9 14. _ a b-c Example 8 16. FOOTBALL During a scoring drive, a (p. 54) football team gained or lost yards on each play as shown. What was the average number of yards per play? Chapter 1 Algebra: Integers 15. 3b - a 2 Yards Gained or Lost +6 +5 -2 +12 +8 -4 0 -3 +23 +18 -4 +1 (/-%7/2+ (%,0 For Exercises 17–22, 29, 30 23–26 27, 28 31–40 41–44 45–48 See Examples 1, 2 3 4 5, 6 7 8 Multiply. 17. 7(-8) 18. 8(-9) 19. -5 · 8 20. -12 · 7 21. -4(9) 22. -6(8) 23. -4(-6) 24. -14(-2) 25. (-4) 2 26. (-7) 2 27. -6(-2)(-7) 28. -3(-3)(-4) 29. HIKING For every 1-kilometer increase in altitude, the temperature drops 7°C. Find the temperature change for a 5-kilometer increase in altitude. 30. LIFE SCIENCE Most people lose 100 to 200 hairs per day. If you were to lose 150 hairs per day for 10 days, what would be the change in the number of hairs you have? Divide. 31. 50 ÷ (-5) 32. -60 ÷ 3 33. 45 ÷ 9 34. -34 ÷ (-2) -84 35. _ 28 36. _ -72 37. _ 38. _ 4 -7 64 8 -6 39. FARMING During a seven-day period, the level of a pond receded 28 centimeters. Find the average daily change in the level of the pond. 40. WEATHER The outside temperature is changing at a rate of -8° per hour. At that rate, how long will it take for the temperature change to be -24°? ALGEBRA Evaluate each expression if w = -2, x = 3, y = -4, and z = -5. 41. x + 6y 8y 43. _ z 44. _ 2 w-x 42. 9 - wz x -5 STATISTICS Find the mean of each set of integers. 45. -4, 6, -10, -3, -8, 1 46. 12, -14, -15, 18, -19, -17, -21 47. -2, -7, -6, 5, -10 48. -14, -17, -20, -16, -13 49. AVIATION An atmospheric research aircraft began descending from an altitude of 36,000 feet above its base, at a rate of 125 feet per minute. How long did it take for the aircraft to land at its base? Multiply or divide. 50. (2) 2 · (-6) 2 51. (-4) 3 52. -2(4)(-3)(-10) ALGEBRA Evaluate each expression if a = 12, b = -4, and c = -8. 53. _ a -b 6c %842!02!#4)#% See pages 677, 708. Self-Check Quiz at ca.gr7math.com 54. _ + c -96 b-a 55. -c 2 - 25 57. MOVIES Predict the number of theater admissions in 2010 if the average change per year following 2004 remains the same as the average change per year from 2002 to 2004. Justify your answer. 56. (3b + 2) 2 ÷ (-4) U.S. Theater Admissions Year Number of Admissions (millions) 2002 1,630 2004 1,530 Source: National Association of Theater Owners Lesson 1-6 Multiplying and Dividing Integers 55 H.O.T. Problems 58. OPEN ENDED Name two integers that have a quotient of -7. NUMBER SENSE Find the sign of each of the following if n is a negative number. Explain your reasoning. 59. n 2 60. n 3 61. n 4 62. n 5 CHALLENGE The sum of any two whole numbers is always a whole number. So, the set of whole numbers (0, 1, 2, 3, ...) is said to be closed under addition. This is an example of the Closure Property. State whether each statement is true or false. If false, give a counterexample. 63. The set of whole numbers is closed under subtraction. 64. The set of integers is closed under multiplication. */ -!4( Determine the sign of the product of -2, -3, and -4. (*/ 83 *5*/( 65. Explain your reasoning. 66. A glacier receded at a rate of 350 feet 67. The temperature at 6:00 P.M. was 10°F. per day for two consecutive weeks. How much did the glacier’s position change in all? A -336 ft C -700 ft Between 6:00 P.M. and midnight, the temperature dropped 4° three different times. What was the temperature at midnight? B -348 ft D -4,900 ft F -12° H 0° G -2° J 2° Subtract. (Lesson 1-5) 68. 12 - 18 69. -5 - (-14) 70. -3 - 20 71. 7 - (-15) Add. (Lesson 1-4) 72. -9 + 2 + (-8) 73. -24 + (-11) + 24 74. -21 + 5 + (-14) 75. -7 + (-3) + 6 76. SHOPPING Gabriel went to the store to buy DVDs. Each DVD costs $20. If he buys four DVDs, he can get a fifth DVD free. How much will he save per DVD if he buys four? (Lesson 1-1) PREREQUISITE SKILL Give an example of a word or phrase that could indicate each operation. Example: addition 77. subtraction 56 Chapter 1 Algebra: Integers the sum of 78. multiplication 79. division 1-7 Writing Equations #* 35 ) 5: % " : 1" 3 Main IDEA EldY\if]>l\jkj GXikp:fjk Write algebraic equations from verbal sentences and problem situations. PARTY PLANNING It costs $8 per guest to hold a birthday party at a skating rink. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A.) Standard 7AF1.4 Use algebraic terminology (e.g. variable, equation, term, coefficient, inequality, expression, constant) correctly. 1. What is the relationship between NEW Vocabulary equation define a variable , ,/fi+' . ./fi,- (' ('/fi/' the cost of a party with g guests. () ()/fi0- 3. What does the equation g · 8 = 120 ^ 6 the number of guests and the cost? 2. Write an expression representing represent in this situation? An equation is a mathematical sentence containing two expressions separated by an equals sign (=). An important skill in algebra is modeling situations using equations. 1 2 3 WORDS Describe the situation. Use only the most important words. VARIABLE Define a variable by assigning a variable to represent the unknown quantity. EQUATION Translate your verbal model into an algebraic equation. To translate your verbal model, look for common words or phrases that suggest one of the four operations. Write an Algebraic Equation 1 GAMES Eduardo had a score of –150 points in the first round of a game. His final score after two rounds was 75 points. Write an equation to find his second round score. Words Defining a Variable Any letter can be used as a variable, but it is often helpful to select letters that can be easily connected to the quantity they represent. Example: score s 1st round score 2nd round score plus was final score. Variable Let s represent the 2nd round score. Equation -150 + s = 75 Write an equation to model each situation. a. The winning time of 27 seconds was 2 seconds shorter than Tina’s. b. A drop of 4°F per hour for the last several hours results in a total temperature change of -24°F. Extra Examples at ca.gr7math.com Lesson 1-7 Writing Equations 57 2 FALLS The height of Yosemite Falls is 239 meters less than the height of Angel Falls in Venezuela. Use the information at the left to write an equation that could be used to find the height of Angel Falls. Words Yosemite’s height 239 meters less than is Angel’s height. Variable Let a represent the height of Angel Falls. Equation 740 = a - 239 c. DANCE The change in attendance from last year’s spring dance was -45 students. The attendance this year was 128 students. Write an equation that could be used to find the attendance last year. Real-World Link Yosemite Falls in Yosemite National Park is the fifth highest falls in the world at a height of 740 meters. Source: U.S. National Park Service You can also write an equation with two variables to express the relationship between two unknown quantities. 3 The number of pounds of insects a bat can eat is 2.5 times its own bodyweight. Given b, a bat’s bodyweight in pounds, which equation can be used to find p, the pounds of insects it can eat? A b = 2.5 · p C b = 2.5 + p B p = b + 2.5 D p = 2.5 · b Read the Item Reading Choices Read all answer choices carefully before deciding on the correct answer. Often two choices will look very similar. The phrase 2.5 times its own bodyweight indicates multiplication. So, you can eliminate B and C. Solve the Item Pounds of insects eaten is 2.5 times bodyweight p = 2.5 · b The solution is D. d. A state’s number of electoral votes is 2 more than its number of Representatives. Given r, a state’s number of Representatives, which equation can be used to find e, the state’s number of electoral votes? F e = 2r G e=r÷2 Personal Tutor at ca.gr7math.com 58 CORBIS Chapter 1 Algebra: Integers H e=r+2 J e=2-r Example 1 (p. 57) Define a variable. Then write an equation to model each situation. 1. Kevin’s score of 20 points was four times Corey’s score. 2. The total was $28 after a $4 tip was added to the bill. Example 2 (p. 58) Define a variable. Then write an equation that could be used to solve each problem. 3. SUBMARINES A submarine dived 75 feet below its original depth. If the submarine’s new depth is -600 feet, what was its original depth? 4. TESTING The total time given to take a state test is equally divided among the 3 subjects tested. If the time given for each subject test is 45 minutes, how many minutes long is the entire test? Example 3 (p. 58) 5. STANDARDS PRACTICE Javier is 4 years younger than his sister Rita. Given j, Javier’s age, which equation can be used to find r, Rita’s age? A j=r÷4 (/-%7/2+ (%,0 For Exercises 6–11 12–15 16–19 See Examples 1 2 3 B j=r+4 C j=r-4 D j = 4r Define a variable. Then write an equation to model each situation. 6. After dropping 12°C, the temperature outside was -5°C. 7. Jamal’s score of 82 was 5 points less than the class average. 8. At 30 meters per second, a cheetah’s top speed is three times that of the top speed of the fastest recorded human. 9. A site is excavated to a level of -75 centimeters over several days for an average dirt removal of 15 centimeters each day. 10. A class of 24 students separated into equal-sized teams results in 6 students per team. 11. When the money was divided among the four grade levels, each grade received $235. Define a variable. Then write an equation that could be used to solve each problem. 12. PETS Nikki’s cat is 5 pounds heavier than her sister’s cat. If Nikki’s cat weighs 9 pounds, how much does her sister’s cat weigh? 13. MEASUREMENT A triangle’s base is one-fourth its height. If the base is 15 meters long, what is the height of the triangle? 14. CREDIT For charging the cost of 4 equally priced shirts, Antonio’s father’s credit card statement shows an entry of -$74. What would the statement have shown for a charge of just one shirt? Lesson 1-7 Writing Equations 59 15. GOLF The graphic shows some of the top 20 leaders in a golf tournament after the first round. If the 6th place participant is 5 strokes behind the leader, what was the leader’s score after the first round? 6. Poole -3 7. Shaw -2 8. Kendrick -2 9. Rodriguez 1 Write an equation that could be used to express the relationship between the two quantities. 16. HEALTH Your heart rate r in beats per minute is the number of times your heart beats h in 15 seconds multiplied by 4. Given h, write an equation to find r. 17. CARS Ashley’s car travels 24 miles per gallon of gas. Given d, the distance the car travels, write an equation to find g, the gallons of gas used. 18. FRAMING A mat for a picture frame should be cut so that its width is _ Real-World Link The earliest year a musical group can be inducted into the Rock and Roll Hall of Fame is 25 years after the year its first album debuted. Source: rockhall.com inch less than the frame’s opening. Given p, the width of the frame’s opening, write an equation to find m, the width of the mat. 1 8 19. MEASUREMENT A seam allowance indicates that the total length of fabric 1 needed is _ inch more than that measured. Given t, the total length of 2 fabric needed, write an equation to find m, the length measured. 20. MUSIC Refer to the information at the left. If an artist was inducted in 2005, write an equation that could be used to find the latest year the artist’s first album could have debuted. Write an equation to model the relationship between the quantities in each table. 21. Yards, y Feet, f 1 22. Centimeters, c Meters, m 3 200 2 2 6 300 3 3 9 400 4 4 12 500 5 y f c m %842!02!#4)#% See pages 678, 708. 23. MAPS The scale on a map indicates that 1 inch on the map represents an Self-Check Quiz at actual distance of 20 miles. Create a table of values showing the number of miles represented by 1, 2, 3, 4, and m inches on the map. Given m, a distance on the map, write an equation to find a, the actual distance. ca.gr7math.com H.O.T. Problems CHALLENGE For Exercises 24–26, consider the sequence 2, 4, 6, 8, …. 24. Express the relationship between a number in this sequence and its position using words. For example, 6 is the third number in this sequence. 25. Define two variables and write an equation to express this relationship. 26. Describe how this relationship would change, using words and a new equation, if the sequence were changed to 0, 2, 4, 6, 8, …. 60 Chapter 1 Algebra: Integers Joseph Sohm/CORBIS 27. FIND THE ERROR Zoe and Toshi are translating the verbal sentence 14 is 6 less than a number into an algebraic equation. Who is correct? Explain. 14 = n - 6 14 = 6 - n Zoe Toshi */ -!4( Analyze the meaning of the equations = 2w and (*/ 83 *5*/( 28. w = 2 if represents the length of a rectangle and w its width. Then draw a rectangle that demonstrates each relationship. 29. The length of an actual car is 87 times its corresponding length of a model of the car. Given a, an actual length of the car, which equation can be used to find m, the corresponding model length? A a = 87 + m 30. The sides of each triangle are 1 unit long. Which equation can be used to represent the perimeter of the figure that contains x triangles? }ÕÀiÊ£ }ÕÀiÊÓ }ÕÀiÊÎ B a = 87 - m C a = 87 · m D a = 87 ÷ m F P = 3x H P=x+2 G P = 3x - 2 J P=x-2 Multiply or divide. (Lesson 1-6) 31. -9(10) 32. -5(-14) 34. _ -105 -5 33. 34 ÷ (-17) 35. BUSINESS During January, a small business had an income I of $18,600 and expenses E of $20,400. Use the formula P = I - E to find the business’s profit P for the month of January. (Lesson 1-5) 36. PREREQUISITE SKILL When Jason joined the football team, he had 8 plays memorized. By the end of the 1st week, he had 10 memorized. By the end of the 2nd week, he had 14 memorized. By the end of the 3rd week, he had 20 memorized. If he continues to learn at this pace, how many plays will he have memorized after 8 weeks? (Lesson 1-1) Lesson 1-7 Writing Equations (l)Andrew Olney/Masterfile, (r)Michael Newman/PhotoEdit 61 1- 8 Problem-Solving Investigation MAIN IDEA: Solve problems by working backward. Standard 7MR1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. e-Mail: WORK BACKWARD YOUR MISSION: Work backward to solve the problem. THE PROBLEM: What time will Trent need to start camp activities? EXPLORE PLAN SOLVE ▲ TRENT: The closing day activities at camp must 1 hours to hold be over by 2:45 P.M. I need 1_ 2 the field competitions, 45 minutes for the awards ceremony, and an hour and 15 minutes for the cookout. Then, everyone will need an hour to pack and check out. You know the time that the campers must leave. You know the time it takes for each activity. You need to determine the time the day’s activities should begin. Start with the ending time and work backward. 2:45 p.m. The day is over at 2:45 p.m. Go back 1 hour for checkout. > > > > > > > > > > > > > > > > > > > > > > 1:45 p.m. Go back 1 hour and 15 minutes for the cookout. > > > > > > > > > > > 12:30 p.m. Go back 45 minutes for the awards ceremony. > > > > > > > > > > > > 11:45 a.m. Go back 1_ hours for the field competitions. > > > > > > > > > > > > > 10:15 a.m. 1 2 So, the day’s activities should start no later than 10:15 a.m. CHECK Assume that the day starts at 10:15 a.m. Work forward, adding the time for each activity. 1. Tell why the work backward strategy is the best way to solve this problem. 2. Explain how you can check a solution when you solve by working backward. 3. */ -!4( Write a problem that can be solved by working backward. (*/ 83 *5*/( Then write the steps you would take to find the solution to your problem. 62 Chapter 1 Algebra: Integers Laura Sifferlin 8. ANALYZE TABLES The table gives the average For Exercises 4–6, solve using the work backward strategy. television viewing time, in hours:minutes, for teens and children. 4. FAMILY Mikal’s great-grandmother was 6 years old when her family came to the United States. That was 73 years ago. If the year is now 2006, in what year was Mikal’s great-grandmother born? 5. GRADES Amelia’s test scores are 94, 88, 93, 85, and 91. What is the minimum score she can make on her next test to maintain a test average of at least 90? 6. SHOPPING Janelle has $75 to spend on a Nightly 8–11 P.M. Teens (ages 12–17) 5:38 19:19 Children (ages 2–11) 4:58 21:00 Total per Week Source: Nielsen Media Research How many more minutes each week do children spend watching television at times other than 8–11 P.M. than teens do? 9. FURNITURE Ms. Calzada makes an initial dress. She buys a dress that is on sale for half price and then applies an in-store coupon for $10 off. After paying an additional sales tax of $1.80, she receives $37.20 in change. What was the original price of the dress? down payment of $150 when purchasing a sofa. She pays the remaining cost of the sofa over 12 months, at no additional charge. If her monthly payment is $37.50, what was the original price of the sofa? Use any strategy to solve Exercises 7–9. Some strategies are shown below. G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s . • Work backward For Exercises 10 and 11, select the appropriate operation(s) to solve the problem. Justify your selection(s) and solve the problem. 10. ANALYZE TABLES The table gives information about two different airplanes. 7. ANALYZE GRAPHS Examine the graph below. -R 0ARKERS #AR 4RIP Airplane Top Speed (mph) Flight Length (mi) Operating Cost per Hour B747-400 534 3,960 $8,443 B727-200 430 644 $4,075 Source: The World Almanac $ISTANCE MI Group How much greater is the operating cost of a B747-400 than a B727-200 if each plane flies at its top speed for its maximum length of flight? 11. PEACE PRIZE Mother Teresa !- !- 0- 0- 4IME Mr. Parker’s total trip will cover 355 miles. If he maintains the speed that he set between 11 A.M. and noon, about what time should he reach his destination? of Calcutta, India, received the Nobel Peace Prize in 1979. If she died in 1997 at the age of 87, how old was she when she received the Nobel Prize? Lesson 1-8 Problem-Solving Investigation: Work Backward Chris Bacon/AP/Wide World Photos 63 Simplify the Problem Have you ever tried to solve a long word problem and didn’t know where to start? Always start by reading the problem carefully. Step 1 Look for key words like more or less to understand how the numbers are related. It is estimated that 12.4 million pounds of potato chips were consumed during a recent Super Bowl. This was 3.1 million pounds more than the number of pounds of tortilla chips consumed. How many pounds of tortilla chips were consumed? The potato chips were 3.1 million more than the tortilla chips. The word this refers to the number of pounds of potato chips. Step 2 Now, try to write the important information in only one sentence. The number of pounds of potato chips was 3.1 million pounds more than the number of pounds of tortilla chips. Step 3 Replace any phrases with numbers that you know. 12.4 million was 3.1 million more than the number of pounds of tortilla chips. Before you write an equation, use the three steps described above to simplify the problem. Refer to page 59. For each exercise below, simplify the problem by writing the important information in only one sentence. Replace any phrases with numbers that you know. Do not write an equation. 1. Exercise 3 64 2. Exercise 12 Chapter 1 Algebra: Integers 3. Exercise 13 Standard 7MR1.3 Determine when and how to break a problem into simpler parts. Standard 7AF1.4 Use algebraic terminology (e.g. variable, equation, term, coefficient, inequality, expression, constant) correctly. 1- 9 Solving Addition and Subtraction Equations Main IDEA Solve equations using the Subtraction and Addition Properties of Equality. Reinforcement of Standard 6AF1.1 Write and solve one-step linear equations in one variable. When you solve an equation, you are finding the values of the variable that make the equation true. These values are called the solutions of the equation. You can use algebra tiles and an equation mat to solve x + 4 = 6. 1 1 x 1 1 x 4 1 1 1 1 1 1 1 1 1 x 44 6 1 1 1 1 1 1 64 Remove the same number of tiles from each side of the mat to get the x by itself on the left side. Model the equation. NEW Vocabulary solve solution inverse operations 1 x The number of tiles remaining on the right side of the mat represents the value of x. So, 2 is the solution of the equation x + 4 = 6. Solve each equation using algebra tiles. 1. x + 1 = 4 2. x + 3 = 7 3. x + (-4) = -5 4. Explain how you would find a value of x that makes x + (-3) = -8 true without using algebra tiles. In the Mini Lab, you solved the equation x + 4 = 6 by removing, or subtracting, the same number of positive counters from each side of the mat. This suggests the Subtraction Property of Equality, which can be used to solve addition equations like x + 4 = 6. +%9 #/.#%04 Words Examples Subtraction Property of Equality If you subtract the same number from each side of an equation, the two sides remain equal. Numbers 7=7 Algebra x+4=6 7-3=7-3 x+4-4=6-4 4=4 x=2 Lesson 1-9 Solving Addition and Subtraction Equations 65 You can use this property to solve any addition equation. Remember to check your solution by substituting it back into the original equation. Isolating the Variable When trying to decide which value to subtract from each side of an addition equation, remember that your goal is to get the variable by itself on one side of the equation. This is called isolating the variable. Solve an Addition Equation 1 Solve x + 5 = 3. Check your solution. METHOD 1 Use the vertical method. x+5= 3 x+5= 3 Write the equation. -5 = ___ -5 ________ x = -2 METHOD 2 Subtract 5 from each side. Use the horizontal method. x+5=3 Write the equation. x+5-5=3-5 Subtract 5 from each side. x = -2 The solution is -2. Check x+5=3 Write the original equation. -2 + 5 3 Replace x with -2. Is this sentence true? 3=3 The sentence is true. Solve each equation. Check your solution. a. a + 6 = 2 b. y + 3 = -8 c. 5 = n + 4 Addition and subtraction are called inverse operations because they “undo” each other. For this reason, you can use the Addition Property of Equality to solve subtraction equations like x - 7 = -5. +%9 #/.#%04 Words Examples Addition Property of Equality If you add the same number to each side of an equation, the two sides remain equal. Numbers Algebra 7=7 x-5=6 7+3=7+3 x-5+5=6+5 10 = 10 66 Chapter 1 Algebra: Integers x = 11 Solve an Addition Equation 2 MEASUREMENT Two angles are supplementary if the sum of their measures is 180°. The two angles shown are supplementary. Write and solve an equation to find the measure of angle X. 94⬚ X Words The sum of the measures is 180°. Variable Let x represent the measure of angle X. Equation x + 94 = 180 x + 94 = 180 x + 94 - 94 = 180 - 94 x = 86 Y Write the equation. Subtract 94 from each side. Simplify. The measure of angle X is 86°. d. READING A novel is ranked 7th on a best-seller list. This is a change of -8 from its position last week. Write and solve an equation to determine the novel’s ranking last week. Personal Tutor at ca.gr7math.com Solve a Subtraction Equation 3 Solve -6 = y - 7. METHOD 1 Use the vertical method. -6 = y - 7 Write the equation. -6 = y - 7 Position of the Variable You could also begin solving Example 3 by rewriting the equation so that the variable is in the left side of the equation. -6 = y - 7 y - 7 = -6 +7= +7 ____________ 1=y METHOD 2 Add 7 to each side. -6 + 7 = 1 and -7 + 7 = 0. Use the horizontal method. -6 = y - 7 Write the equation. -6 + 7 = y - 7 + 7 Add 7 to each side. 1=y The solution is 1. -6 + 7 = 1 and -7 + 7 = 0. Check the solution. Solve each equation. e. x - 8 = -3 Extra Examples at ca.gr7math.com f. b - 4 = -10 g. 7 = p - 12 Lesson 1-9 Solving Addition and Subtraction Equations 67 Example 1 (p. 66) Example 2 (p. 67) Example 3 (p. 67) (/-%7/2+ (%,0 For Exercises 8–13 14–19 20–23 See Examples 1 3 2 Solve each equation. Check your solution. 1. a + 4 = 10 2. 2 = z + 7 3. x + 9 = -3 4. RUGS The length of a rectangular rug is 12 inches shorter than its width. If the length is 30 inches, write and solve an equation to find the width. Solve each equation. Check your solution. 5. y - 2 = 5 6. n - 5 = -6 7. -8 = d - 11 Solve each equation. Check your solution. 8. x + 5 = 18 9. n + 3 = 20 10. 9 = p + 11 11. 1 = a + 7 12. y + 12 = -3 13. w + 8 = -6 14. m - 15 = 3 15. b - 9 = -8 16. g - 2 = -13 17. -16 = t - 6 18. -4 = r - 20 19. k - 14 = -7 20. MEASUREMENT Two angles are complementary if the sum of their measures is 90°. The two angles shown are complementary. Write and solve an equation to find the measure of angle A. 37˚ A B 21. BANKING After you withdraw $50 from your savings account, the balance is $124. Write and solve an equation to find your starting balance. 22. TEMPERATURE On one day in Fairfield, Montana, the temperature dropped 84°F from noon to midnight. If the temperature at midnight was -21°F, write and solve an equation to determine the noon temperature that day. 23. TREES Before planting a tree, Manuel digs a hole with a floor 18 inches below ground level. Once planted, the top of the tree is 54 inches above ground. Write and solve an equation to find the height of the tree Manuel planted. ANALYZE TABLES For Exercises 24 and 25, use the table. 24. Lauren Jackson averaged 0.5 point per game more than Tina Thompson. Write and solve an equation to find Thompson’s average points scored per game. 25. Sheryl Swoopes averaged 5.2 fewer points per game than Tina Thompson. Write and solve an equation to find how many points Swoopes averaged per game. 68 Chapter 1 Algebra: Integers 2004 WNBA Regular Season Points Leaders Player AVG Lauren Jackson 20.5 Tina Thompson a Lisa Leslie 17.6 Diana Taurasi 17.0 Source: wnba.com %842!02!#4)#% 26. STOCK MARKET The changes in the price of a certain stock each day from Monday to Thursday of one week were -$2.25, +$0.50, +$1.50, and +$0.75. If the overall change in the stock price for the week was -$0.50, write an equation that can be used to find the change in the price on Friday and explain two methods of solving this equation. Then solve the equation and explain its meaning in the context of the situation. See pages 678, 708. Self-Check Quiz at ca.gr7math.com H.O.T. Problems 27. OPEN ENDED Write one addition equation and one subtraction equation that each have -3 as a solution. 28. Which One Doesn’t Belong? Identify the equation that does not belong with the other three. Explain your reasoning. 4+g=2 a + 5 = -3 m+6=4 1 + x = -1 29. CHALLENGE Solve ⎪x⎥ + 5 = 7. Explain your reasoning. 30. */ -!4( Write a problem about a real-world situation that can (*/ 83 *5*/( be answered by solving the equation x + 60 = 20. Then solve the equation and explain the meaning of its solution in the context of your problem. 31. Dante paid $42 for a jacket, which 32. The record low temperature for included $2.52 in sales tax. Which equation could be used to find the price of the jacket before tax? A x - 2.52 = 42 C x - 42 = 2.52 Virginia is 7°F warmer than the record low for West Virginia. If the record low for Virginia is -30°F, what is West Virginia’s record low? B x + 2.52 = 42 D x + 42 = 2.52 F -37°F H 23°F G -23°F J 37°F 33. TRAVEL James needs to drive an average of 575 miles a day for three days in order to make it to his vacation destination on time. If he drove 630 miles the first day and 480 miles the second day, how many miles does he need to drive on the third day to meet his goal? (Lesson 1-8) ALGEBRA Write an equation to model each situation. (Lesson 1-7) 34. Lindsay, 59 inches tall, is 5 inches shorter than her sister. 35. After cutting the recipe in half, Ricardo needed 3 cups of flour. PREREQUISITE SKILL Multiply. (Lesson 1-6) 36. 3(9) 37. -2(18) 38. -5(-11) 39. 4(-15) Lesson 1-9 Solving Addition and Subtraction Equations 69 Solving Multiplication and Division Equations 1-10 Main IDEA PLANTS Some species of a bamboo can grow 35 inches per day. That is as many inches as the average child grows in the first 10 years of his or her life! Solve equations by using the Division and Multiplication Properties of Equality. Reinforcement of Standard 6AF1.1 Write and solve onestep linear equations in one variable. Bamboo Growth 1. If d represents the number of days the bamboo has been growing, write a multiplication equation you could use to find how long it would take for the bamboo to reach a height of 210 inches. Day Height (in.) 1 35(1) = 35 2 35(2) = 70 3 35(3) = 105 d The equation 35d = 210 models the relationship described above. To undo the multiplication of 35, divide each side of the equation by 35. Solve a Multiplication Equation 1 Solve 35d = 210. 35d = 210 Write the equation. 35d 210 _ =_ 35 35 Divide each side of the equation by 35. 1d = 6 35 ÷ 35 = 1 and 210 ÷ 35 = 6 d=6 Identity Property; 1d = d Solve each equation. Check your solution. Solving Equations When you solve a simple equation like 8x = 72, you can mentally divide each side by 8. a. 8x = 72 b. -4n = 28 In Example 1, you used the Division Property of Equality to solve a multiplication equation. +%9 #/.#%04 Words Examples Division Property of Equality If you divide each side of an equation by the same nonzero number, the two sides remain equal. Numbers 12 = 12 Chapter 1 Algebra: Integers Photowood/CORBIS Algebra 5x = -60 12 12 _ =_ -60 5x _ =_ 3=3 x = -12 4 70 c. -12 = -6k 4 5 5 Extra Examples at ca.gr7math.com READING Math Division Expressions Remember, a _ means a -3 divided by -3. You can use the Multiplication Property of Equality to solve equations. +%9 #/.#%04 Multiplication Property of Equality If you multiply each side of an equation by the same number, the two sides remain equal. Words Examples Algebra Numbers _x = 8 5=5 2 5(-4) = 5(-4) _x (2) = 8(2) -20 = -20 x = 16 2 Solve a Division Equation _ 2 Solve a = -7. -3 a _ = -7 Write the equation. -3 a _ (-3) = -7(-3) -3 a = 21 Multiply each side by -3. -7 · (-3) = 21 Solve each equation. y -4 d. _ = -8 e. _ = -9 m 5 f. 30 = _ b -2 3 REPTILES A Nile crocodile grows to be 4,000 times as heavy as the egg from which it hatched. If an adult crocodile weighs 2,000 pounds, how much does a crocodile egg weigh? Real-World Career How Does a Zoologist Use Math? Zoologists use equations to predict the growth of animal populations. Words Adult weight is 4,000 times egg weight. Variable Let g represent the weight of the crocodile egg. Equation 2,000 = 4,000 · g 2,000 = 4,000g Write the equation. 4,000g 2,000 _ =_ Divide each side by 4,000. 4,000 4,000 0.5 = g 2,000 ÷ 4,000 = 0.5 A crocodile egg weighs 0.5 pound. For more information, go to ca.gr7math.com. g. An adult lizard is about five times longer than a hatchling. If an adult lizard is 11 centimeters long, about how long is a hatchling? Personal Tutor at ca.gr7math.com Lesson 1-10 Solving Multiplication and Division Equations Aaron Haupt 71 Examples 1, 2 (pp. 70, 71) Example 3 (p. 71) (/-%7/2+ (%,0 For Exercises 8–13 14–19 20, 21 See Examples 1 2 3 Solve each equation. Check your solution. 1. 5b = 40 2. -7k = 14 3. -18 = -3n p 4. _ = 9 9 5. _ = -3 6. 22 = _ a 12 m -2 7. LAWN SERVICE Josh charges $15 to mow an average size lawn in his neighborhood. Write and solve a multiplication equation to find how many of these lawns he needs to mow to earn $600. Solve each equation. Check your solution. 8. 4c = 44 9. 9b = 72 10. 34 = -2x 11. 36 = -18y 12. -32 = 8d 13. -35 = 5n 14. _ = 10 15. _ = 6 16. _ = 33 m 7 u 9 q -5 17. 20 = _ h -3 r 19. _ = -3 24 18. -8 = _ c 12 20. ANIMALS An African elephant can eat 500 pounds of vegetation per day. If a zookeeper has 3,000 pounds of vegetation on hand for one elephant, write and solve a multiplication equation to find how many days this supply will last. 21. SCHOOL ACTIVITIES The drama club sold 1,200 tickets for the school musical. If the total ticket sales were $6,000, write and solve a multiplication equation to find the cost per ticket. MEASUREMENT For Exercises 22–26, refer to the table. Write and solve an equation to find each quantity. Customary System Conversions (length) 22. the number of yards in 18 feet 1 foot = 12 inches 23. the number of feet in 288 inches 1 yard = 3 feet 1 yard = 36 inches 24. the number of yards in 540 inches 1 mile = 5,280 feet 25. the number of miles in 26,400 feet 1 mile = 1,760 yards 26. the number of miles in 7,040 yards Solve each equation. 27. 7 = _ z -56 %842!02!#4)#% 28. _ x = -5 10 29. _ a = -21 -126 30. -17 = _ g 136 See pages 679, 708. 31. PHYSICAL SCIENCE The amount of work, measured in foot-pounds, is equal Self-Check Quiz at to the amount of force applied, measured in pounds, times the distance, in feet, the object moved. How far do you have to lift a 45-pound object to produce 180 foot-pounds of work? ca.gr7math.com 72 Chapter 1 Algebra: Integers H.O.T. Problems 32. OPEN ENDED Describe a real-world situation in which you would use a division equation to solve a problem. Then write your equation. 33. NUMBER SENSE Without solving the equation, tell what you know about the x value of x in the equation _ = 300. 25 34. CHALLENGE If an object is traveling at a rate of speed r, then the distance d the object travels after a time t is given by the equation d = rt. Rewrite this equation so that it expresses the value of r in terms of t and d. 35. */ -!4( Explain how to solve -4a = 84. Be sure to state which (*/ 83 *5*/( property you use and why you used it. 36. Grace paid $2.24 for 4 granola bars. All 37. Luis ran 2.5 times the distance 4 granola bars were the same price. How much did each granola bar cost? that Mark ran. If Mark ran 3 miles, which equation can be used to find the distance d in miles that Luis ran? A $0.52 B $0.56 F d = 2.5 + 3 C $1.24 G d + 2.5 = 3 D $1.56 H d = 2.5(3) J 2.5d = 3 38. ARCHITECTURE When the Empire State Building was built, its 185-foot spire was built inside the building and then hoisted to the top of the building upon its completion. Write and solve an equation to find the height of the Empire State Building without its spire. (Lesson 1-9) ALGEBRA Write an equation to model each situation. (Lesson 1-7) 185 ft 1,250 ft x ft 39. Eight feet longer than she jumped is 15 feet. 40. The temperature fell 28°F from 6 A.M. to 17°F at 11 A.M. 41. Three friends shared a $9 parking fee equally. Find each product or quotient. (Lesson 1-6) 42. -23(-12) 43. -25(7) 44. 22 · (-20) 45. 4 · 8 · (-14) 46. -180 ÷ 15 47. 147 ÷ (-21) 48. -162 ÷ 9 49. -208 ÷ (-16) Write an integer for each situation. (Lesson 1-3) 50. a gain of 4 ounces 51. earning $45 52. 2 miles below sea level 53. a decrease of 5 miles per gallon Lesson 1-10 Solving Multiplication and Division Equations 73 CH APTER 1 Study Guide and Review Download Vocabulary Review from ca.gr7math.com Key Vocabulary Be sure the following Key Concepts are noted in your Foldable. 8ORDS &XAMPLE S "1LANFOR 1ROBLEM4OLVING OF *NTEGERS XOF *NTEGERS 4OLVING &QUATIONS 4OLVINGX &QUATIONS absolute value (p. 36) integer (p. 35) additive inverse (p. 43) inverse operations (p. 66) algebra (p. 29) negative number (p. 35) algebraic expression (p. 29) numerical expression (p. 29) coordinate (p. 35) opposites (p. 43) counterexample (p. 31) order of operations (p. 29) define a variable (p. 57) powers (p. 30) Order of Operations (Lesson 1-2) equation (p. 57) solution (p. 65) 1. Do all operations within grouping symbols first. evaluate (p. 29) solve (p. 65) 2. Evaluate all powers before other operations. inequality (p. 35) variable (p. 29) Key Concepts 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. Operations With Integers (Lessons 1-4 to 1-6) • To add integers with the same sign, add their absolute values. The sum has the same sign as the integers. • To add integers with different signs, subtract their absolute values. The sum has the sign of the integer with the greater absolute value. • To subtract an integer, add its opposite or additive inverse. Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. 1. Operations that “undo” each other are called order of operations. 2. The symbol for absolute value is | |. • The product or quotient of two integers with the same sign is positive. 3. A mathematical sentence that contains an • The product or quotient of two integers with different signs is negative. 4. An integer is a number less than zero. equals sign is an inequality. 5. A property is an example that shows that Solving Equations (Lessons 1-9, 1-10) • If you add or subtract the same number to/ from each side of an equation, the two sides remain equal. • If you multiply or divide each side of an equation by the same nonzero number, the two sides remain equal. a conjecture is false. 6. The value of the variable that makes the equation true is called the solution. 7. The number that corresponds to a point is called its coordinate. 8. A power is a symbol, usually a letter, used to represent the number. 9. An expression that contains a variable is an algebraic expression. 74 Chapter 1 Algebra: Integers Vocabulary Review at ca.gr7math.com Lesson-by-Lesson Review 1-1 A Plan for Problem Solving (pp. 24–28) Use the four-step plan to solve each problem. 10. SCIENCE A chemist pours table salt into a beaker. If the beaker plus the salt has a mass of 84.7 grams and the beaker itself has a mass of 63.3 grams, what was the mass of the salt? Example 1 At Smart’s Car Rental, it costs $57 per day plus $0.10 per mile to rent a certain car. How much will it cost to rent the car for 1 day and drive 180 miles? Explore You know the rental cost per day and per mile. You need to find the cost for a 1-day rental for 180 miles. Plan Multiply the numer of miles by the cost per mile. Then add the daily cost. Estimate $0.10 × 200 = $20 and $60 + $20 = $80 Solve $0.10 × 180 = $18 $18 + $57 = $75 11. SPORTS In a basketball game, the Sliders scored five 3-point shots, seven 2-point shots, and fifteen 1-point shots. Find the total points scored. 12. SHOPPING Miguel went to the store to buy jeans. Each pair costs $24. If he buys two pairs, he can get the second pair for half price. How much will he save per pair if he buys two pairs? The cost is $75. Check 1-2 The answer of $75 is close to the estimate of $80, so the answer is reasonable. Variables, Expressions, and Properties (pp. 29–34) Evaluate each expression if a = 6, b = 2, and c = 1. 13. a(b + 4) 15. 3a + 2b + c 14. 3b 2 2 (a + 2) 16. _ bc 17. MEASUREMENT The area A of a triangle can be found using the formula 1 A=_ bh, where b is the base of the 2 Example 2 Evaluate x 2 + yx - z 2 if x = 4, y = 2, and z = 1. x 2 + yx - z 2 2 = 4 + (2)(4) - (1) = 16 + (2)(4) - 1 = 16 + 8 - 1 = 23 Write the expression. 2 x = 4, y = 2, and z = 1 Evaluate powers first. Multiply. Add and subtract. triangle and h is the height. Find the area of the triangle. CM CM Chapter 1 Study Guide and Review 75 CH APTER 1 Study Guide and Review 1-3 Integers and Absolute Value (pp. 35–39) Replace each ● with <, >, or = to make a true sentence. Example 3 Replace the ● in -3 ● -7 with <, >, or = to make a true sentence. 18. -8 ● 7 Graph the integers on a number line. 19. -2 ● -6 20. BASKETBALL On average, the varsity team wins games by a margin of 13 points, give or take 5 points. This range can be modeled by the equation ⎪p - 13⎥ = 5. Graph this equation on a number line to determine the least and the greatest margin of points. Evaluate each expression. 21. ⎪-5⎥ 1-4 1 Since -3 is to the right of -7, -3 > -7. Example 4 Evaluate ⎪-3⎥. Since the graph of -3 is 3 units from 0 on the number line, the absolute value of -3 is 3. Adding Integers (pp. 41–45) Example 5 23. -54 + 21 24. 100 + (-75) 25. -14 + (-20) 26. 38 + (-46) 27. -14 + 37 + (-20) + 2 28. WEATHER At 8:00 A.M., it was -5°F. By noon, it had risen 34°. Write an addition statement to describe this situation. Then find the sum. Find -16 + (-11). -16 + (-11) = -27 Example 6 -7 + 20 = 13 Add ⎪-16⎥ and ⎪-11⎥. Both numbers are negative, so the sum is negative. Find -7 + 20. Subtract ⎪-7⎥ from ⎪20⎥. The sum is positive because ⎪20⎥ > ⎪-7⎥. Subtracting Integers (pp. 46–49) Example 7 Subtract. 29. -2 - (-5) 30. 11 - 15 31. GEOGRAPHY At an elevation of -52 feet, Lake Eyre is the lowest point in Australia. How much lower than Lake Eyre is the Valdes Peninsula in South America, which has an elevation of -131 ft? 76 0 22. ⎪-12⎥ - ⎪4⎥ Add. 1-5 ⫺8 ⫺7 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 Chapter 1 Algebra: Integers Find -27 - (-6). -27 - (-6) = -27 + 6 To subtract -6, add 6. = -21 Add. Mixed Problem Solving For mixed problem-solving practice, see page 708. 1-6 Multiplying and Dividing Integers (pp. 51–56) Example 8 Find 3(-20). 3(-20) = -60 The factors have different Multiply or divide. 32. -4(-25) 33. -7(3) 34. -15(-4)(-1) 35. 180 ÷ (-15) 36. -170 ÷ (-5) 37. -88 ÷ 8 Example 9 38. GAMES José’s score in each of 6 rounds of a game was -2. What was his overall score for these six rounds? 1-7 Find -48 ÷ (-12). -48 ÷ (-12) = 4 The dividend and the divisor have the same sign. The quotient is positive. Writing Equations (pp. 57–61) 39. SPORTS An athlete’s long jump attempt measured 670 centimeters. This was 5 centimeters less than her best jump. Define a variable. Then write an equation that could be used to find the measure of her best jump. 40. ALGEBRA Lauren uses a copier to reduce 1 the length of an image so it is _ of its 4 original size. Given , the length of the image, write an equation to find the length n of the new image. 1-8 signs. The product is negative. Example 10 Tennessee became a state 4 years after Kentucky. If Tennessee became a state in 1796, write an equation that could be used to find the year Kentucky became a state. Tennessee’s year is 4 years after Kentucky year. Variable Let y represent Kentucky’s year. Equation 1796 = y + 4 Words PSI: Work Backward (pp. 62–63) Solve. Use the work backward strategy. 41. TRAVEL Alonzo’s flight to Phoenix departs at 7:15 P.M. It takes 30 minutes to drive to the airport from his home, and it is recommended that he arrive at the airport 2 hours prior to departure. What time should Alonzo leave his house? 42. TICKETS After Candace purchased tickets to the play for herself and her two brothers, ticket sales totaled $147. If tickets were $5.25 each, how many tickets were sold before her purchase? Example 11 Fourteen years ago, Samuel’s parents had their oldest child, Isabel. Six years later, Julia was born. If Samuel was born last year, how many years older than Samuel is Julia? Since Samuel was born last year, he must be one year old. Since Isabel was born fourteen years ago, she must be fourteen years old. Since Julia was born six years after Isabel, she must be eight years old. This means that Julia is seven years older than Samuel. Chapter 1 Study Guide and Review 77 CH APTER 1 Study Guide and Review 1-9 Solving Addition and Subtraction Equations (pp. 65–69) Solve each equation. Check your solution. 43. n + 40 = 90 44. x - 3 = 10 45. c - 30 = -18 46. 9 = a + 31 47. d + 14 = -1 48. 27 = y - 12 49. CANDY There are 75 candies in a bowl after you remove 37. Write and solve a subtraction equation to find how many candies were originally in the bowl. Example 12 Solve 5 + k = 18. 5 + k = 18 Write the equation. 5 - 5 + k = 18 - 5 Subtract 5 from each side. 18 - 5 = 13 k = 13 Example 13 Solve n - 13 = -62. n - 13 = -62 Write the equation. n - 13 + 13 = -62 + 13 Add 13 to each side. -62 + 13 = -49 n = -49 50. WEATHER On August 15, the monthly rainfall for a city was 2 inches below average. On August 31, the monthly total was 1 inch above average. Write and solve an addition equation to determine the amount of rainfall between August 15 and August 31. 1-10 Solving Multiplication and Division Equations (pp. 70–73) Solve each equation. Check your solution. Example 14 60 = 5t Write the equation. 51. 15x = -75 60 5t _ =_ Divide each side by 5. 12 = t Simplify. s 53. _ = 42 7 52. -4x = 52 y 54. _ = -15 -10 55. MONEY Toni borrowed $168 from her father to buy clothes. She plans to pay $28 a month toward this debt. Write and solve an equation to find how many months it will take to repay her father. 56. CARS Mr. Mitchell bought 12 quarts of motor oil for $36. Write and solve an equation to find the cost of each quart of motor oil. 78 Chapter 1 Algebra: Integers 5 5 Example 15 Solve 60 = 5t. _ Solve m = 8. m _ =8 -2 m (-2) = 8(-2) (_ -2 ) m = -16 -2 Write the equation. Multiply each side by -2. Simplify. CH APTER Practice Test 1 1. ANALYZE TABLES The table gives the annual number of hours worked by citizens in four countries in a recent year. Country Annual Hours Worked United States 1,877 Japan 1,840 Canada 1,801 United Kingdom 1,708 On average, how many more hours per week did a person in the United States work that year than a person in the United Kingdom? Evaluate each expression if a = 3, b = 2, and c = -5. 2. (2c + b) ÷ b - 3 2 3. 4a - 5a - 12 4. CELL PHONES The monthly charge in dollars for a specific cell phone company is given x - 500 by the expression 40 + _ where x is 2 the number of minutes of phone usage. Find the charge if a person uses 622 minutes. Replace each ● with <, >, or = to make a true sentence. 5. -8 ● -11 6. ⎪13⎥ ● - 13 STANDARDS PRACTICE Evaluate the following expression: 7. ⎪12 - 7⎥ - ⎪3 - 6⎥ A -8 B -2 Add, subtract, multiply, or divide. 9. -27 + 8 10. -105 ÷ 15 11. _ -70 -5 13. 7(-10)(-4) 12. -4 - (-35) 15. 8(-9) 16. 13 - 61 14. -9 + (-11) STANDARDS PRACTICE What is the absolute value of -7? 17. F -7 1 G -_ 1 H _ 7 7 J 7 18. MEASUREMENT A circle’s radius is half its diameter. Given d, the diameter, write an equation that could be used to find r, the radius. 19. JEANS A store tripled the price it paid for a pair of jeans. After a month, the jeans were marked down by $5. Two weeks later, the price was divided in half. Finally, the price was reduced by $3, down to $14.99. How much did the store pay for the jeans? Solve each equation. Check your solution. 20. x + 15 = - 3 21. -7 = a - 11 22. _ = 16 23. -96 = 8y n -2 24. TRANSPORTATION An airplane flies over a submarine cruising at a depth of -326 feet. The distance between the two is 1,176 feet. Write and solve an equation to find the airplane’s altitude. C 2 25. GAMES After Round 2 in a game, Eneas’ D 8 8. Find the value of ⎪y⎥ - ⎪x⎥ if x = -4 and y = -9. Chapter Test at ca.gr7math.com score was -40. After Round 3, her score was 5. Write and solve an equation to find the number of points scored in Round 3. Chapter 1 Practice Test 79 CH APTER 1 California Standards Practice Chapter 1 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1 Kristy, Megan, and Heather sold a total of 48 magazines this weekend. Megan sold 3 more magazines than Heather, and Kristy sold twice as many magazines as Heather. Which is a reasonable conclusion about the number of magazines sold by the students? A Megan sold the least number of magazines. 5 If ⎪r⎥ = 2, what is the value of r? A -2 or 0 C 0 or 2 B -2 or 2 D -4 or 4 Question 5 In some instances, the quickest and easiest way to answer the question is to simply try each choice to see which one works. 6 B Kristy and Megan sold the same number of magazines. C Heather sold exactly half of the total number of magazines. D Kristy sold the most magazines. Tony received some money from his grandmother for his birthday. He spent $12.75 each for 3 CDs. Then he spent $5.20 for lunch. Later he bought a T-shirt for $8.90. If he had $7.65 left over, which of the following expressions can be used to find how much money Tony received for his birthday? F 3(12.75) + 5.20 + 8.90 + 7.65 2 Two siblings agreed to split the cost of a television and a DVD player evenly. They spent a total of $335.00 on the television and $95.00 on the DVD player. Find the amount that each sibling paid. F $430.00 H $215.00 G $265.00 J $210.00 G 3(12.75) + 5.20 + 8.90 - 7.65 H 3(12.75 + 5.20 + 8.90 + 7.65) J 3(12.75 + 5.20 + 8.90 - 7.65) 7 Abigail evaluated the expression ⎪-27 + 3⎥ ⎪-3 - 5⎥ by performing the following steps. ⎪-27 + 3⎥ - ⎪-3 - 5⎥ = ⎪-24⎥ - ⎪-8⎥ 3 Which of the following numerical expressions results in a positive number? A (-4) + (-7) C (-4) + (7) B (4) + (-7) D (-4) + (7) + (-4) = 24 + 8 = 32 What did Abigail do incorrectly in evaluating the expression? A She evaluated ⎪-24⎥ as 24 when she 4 80 An electrician received d dollars for a job. She had to pay $75 for supplies. On her next job, she received 3m dollars. Which expression represents the amount of money she has now? F d - 75 - 3m H d + 75 - 3m G d + 75 + 3m J d - 75 + 3m Chapter 1 Algebra: Integers should have evaluated ⎪-24⎥ as -24. B She added 24 and 8 when she should have subtracted 8 from 24. C She evaluated ⎪-3 - 5⎥ as ⎪-8⎥ when she should have evaluated ⎪-3 - 5⎥ as ⎪-2⎥. D She added 24 and 8 when she should have subtracted -8 from -24. California Standards Practice at ca.gr7math.com More California Standards Practice For practice by standard, see pages CA1–CA39. 8 Add six to the quotient of a number and three. The answer is 14. Which of the following equations matches these statements? x F 14 = _ + 6 3 x G 6 = 14 + _ 3 x+6 _ H 14 = 3 x + 14 _ J 6= 3 11 Mandy wants to buy a new couch that costs $1,299. For the next 8 months, she plans to save an equal amount of money each month to pay for the couch. About how much will she need to save each month? A $162.50 B $158.50 C $165.75 D $185.00 12 The high temperature on Monday was 9 The table below shows the train travel times from Cleveland (CLE) to Chicago (CHI). Depart CLE 2:30 a.m. 7:45 a.m. 8:20 p.m. 2:00 p.m. -8°F. On Tuesday, the high temperature was 11°F. How much warmer was it on Tuesday than Monday? Arrive CHI 8:45 a.m. 1:45 p.m. 2:25 a.m. 8:20 p.m. F 19°F G 3°F H -3°F J -19°F Pre-AP Which of the following statements about the travel times is true? Record your answers on a sheet of paper. Show your work. A The train leaving at 2:30 A.M. has the least travel time. 13 Below, n, p, r, and t each represent a B The train leaving at 7:45 A.M. has the greatest travel time. different integer. If n = -4 and t ≠ 1, find each of the following values. Explain your reasoning using the properties of integers. C The train leaving at 8:20 P.M. has the least travel time. D The train leaving at 2:00 P.M. has the greatest travel time. n×p=n t×r=r n+t=r 1 10 If x = 5 and y = _ , then y(13 - x) = a. p 4 F 2 H 4 G 3 J 6 b. r c. t NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 10 11 12 13 Go to Lesson... 1-1 1-6 1-4 1-7 1-3 1-1 1-3 1-7 1-1 1-2 1-1 1-5 1-3 MR1.1 NS1.2 NS1.2 AF1.1 NS2.5 MR1.1 NS2.5 AF1.1 MR1.2 AF1.2 MR2.1 NS1.2 NS2.5 For Help with Standard... Chapter 1 California Standards Practice 81 Algebra: Rational Numbers 2 • Standard 7NS1.0 Know the properties of, and compute with, rational numbers expressed in a variety of forms. • Standard 7MG1.0 Choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems. Key Vocabulary exponent (p. 126) rational number (p. 84) reciprocals (p. 102) scientific notation (p. 130) Real-World Link Astronomy Measurements used in astronomy are frequently expressed as powers of 10. For example, the distance from Earth to the Sun can be written as 9.3 × 10 7 miles. Algebra: Rational Numbers Make this Foldable to help you organize your notes. Begin with five 1 sheets of 8 ” × 11” paper. _ 2 1 Place 5 sheets of paper _3 inch apart. 4 3 Staple along the fold. 82 Chapter 2 Algebra: Rational Numbers STScI/NASA/CORBIS 2 Roll up the bottom edges. All tabs should be the same size. 4 Label the tabs with the lesson numbers. Algebra: ers mb Rational Nu 2-1, 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 GET READY for Chapter 2 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. 1. -13 + 4 2. 28 + (-9) Example 1 Find -27 + 13. 3. -8 - 6 4. 23 - (-15) -27 + 13 = -14 Add or subtract. (Lessons 1-4 and 1-5) 5. TEMPERATURE The high temperature for Saturday was 13°F, and the low temperature was -4°F. What was the difference between the high and low temperatures? (Lesson 1-5) ⎪-27⎥ - ⎪13⎥ = ⎪14⎥ The sum is negative because ⎪-27⎥ > ⎪13⎥. Example 2 Find -11 - 8. -11 - 8 = -11 + (-8) -11 + (-8) = -19 To subtract 8, add -8. ⎪-11⎥ + ⎪-8⎥ = 19 Both numbers are negative so the sum is negative. 6. 6(-14) 7. 36 ÷ (-4) Example 3 Find -12(7). 8. -86 ÷ (-2) 9. -3(-9) -12(7) = -84 Multiply or divide. (Lesson 1-6) The factors have different signs. The product is negative. 10. -12x = 144 11. a + 9 = 37 Example 4 Solve -8x = 64. 12. -18 = y - 42 13. 25 = _ -8x = 64 Write the equation. -8x 64 _ =_ Divide each side of the equation by -8. Solve each equation. (Lessons 1-9 and 1-10) n 5 -8 -8 x = -8 Simplify. Find the least common multiple (LCM) of each set of numbers. Example 5 Find the LCM of 9, 12, and 18. (Prior Grade) multiples of 9: multiples of 12: multiples of 18: 14. 12, 16 15. 24, 9 16. 10, 5, 6 17. 3, 7, 9 0, 9, 18, 27, 36, 45, ... 0, 12, 24, 36, 48, ... 0, 18, 36, 54, ... The LCM of 9, 12, and 18 is 36. Chapter 2 Get Ready for Chapter 2 83 2 -1 Rational Numbers Main IDEA Express rational numbers as decimals and decimals as fractions. Standard 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. Standard 7NS1.5 Know that every rational number is either a terminating or repeating decimal and be able to convert terminating decimals into reduced fractions. NEW Vocabulary rational number terminating decimal repeating decimal bar notation WHALE WATCHING The top ten places in the Northern Hemisphere to watch whales are listed below. Viewing Site Sea of Cortez Location Baja California, Mexico Type Seen Blue, Finback, Sei, Sperm, Minke, Pilot, Orca, Humpback, Gray Dana Point California Gray Monterey California Gray San Ignacio Lagoon Baja California, Mexico Gray Churchill River Estuary Manitoba, Canada Beluga Stellwagen Bank National Marine Sanctuary Massachusetts Humpback, Finback, Minke Lahaina Hawaii Humpback Silver Bank Dominican Republic Humpback Mingan Island Quebec, Canada Blue Friday Harbor Washington Orca, Minke 1. What fraction of the sites are in the United States? 2. What fraction of the sites are in Canada? 3. At what fraction of the sites might you see gray whales? 4. What fraction of the humpback viewing sites are in Mexico? Numbers that can be written as fractions are called rational numbers. 8 -7 2 2 Since -7 can be written as _ and 2_ can be written as _ , -7 and 2_ 3 3 3 1 are rational numbers. All integers, fractions, and mixed numbers are rational numbers. +%9 #/.#%04 Words Algebra READING in the Content Area For strategies in reading this lesson, visit ca.gr7math.com. 84 Chapter 2 Algebra: Rational Numbers Peter Guttman/CORBIS Rational numbers are numbers that can be written as fractions. a _ , where a and b are Rational Numbers Model Rational Numbers 0.8 b Integers ⫺6 integers and b ≠ 0. 5.2 1 2 Whole ⫺10 Numbers 2 53 8 13 1 ⫺1.4444... Any fraction, positive or negative, can be expressed as a decimal by dividing the numerator by the denominator. Write a Fraction as a Decimal _ 1 Write 5 as a decimal. 8 _5 means 5 ÷ 8. 8 0.625 8 5.000 - 48 −−−− 20 -16 −−− 40 -40 ____ 0 Divide 5 by 8. Write each fraction or mixed number as a decimal. a. _ b. _ Vocabulary Link Terminating Everyday Use bringing to an end Math Use a decimal that ends c. 4_ -3 5 3 4 13 25 Every rational number can be written as either a terminating or repeating decimal. A decimal like 0.625 is called a terminating decimal because the division ends, or terminates, with a remainder of 0. If the division does not end, a pattern of digits repeats. Repeating decimals have a pattern in their digits that repeats without end. Instead of the three dots at the end of the decimal, bar notation is often used to indicate that a digit or group of digits repeats. − 0.333... = 0.3 −− 60.7151515... = 60.715 −− -0.282828... = -0.28 Write a Repeating Decimal _ 2 Write -1 2 as a decimal. 3 Common Error The bar is placed above the repeating part. To write 8.636363... in bar −− notation, write 8.63, − −−− not 8.6 or 8.636. To write 0.3444... in bar − notation, write 0.34, −− not 0.34. -5 2 can be rewritten as _ . -1_ 3 3 Divide 5 by 3 and add a negative sign. − 2 The mixed number -1_ can be written as -1.6. 3 1.6... 5.0 3 -3 −−− 2.0 -1.8 −−−− 2 Write each fraction as a decimal. d. 7 _ 12 e. -_ 2 9 f. 3_ 1 11 g. -2_ 14 15 Repeating decimals often occur in real-world situations. However, they are usually rounded to a certain place-value position. Extra Examples at ca.gr7math.com Lesson 2-1 Rational Numbers 85 3 BASEBALL Kansas City pitcher Kris Wilson won 6 of the 11 games he started. To the nearest thousandth, find his winning average. To find his winning average, divide the number of wins, 6, by the number of games, 11. Real-World Career How Does a Sports Statistician Use Math? A baseball statistician uses decimal equivalents to determine batting averages and winning averages. A batting average is the number of hits divided by the number of times at bat. For more information, go to ca.gr7math.com. Look at the digit to the right of the thousandths place. Round down since 4 < 5. Kris Wilson’s winning average was 0.545. 0.5 4 5 4... 6.0 0 0 0 11 -5 5 −−−− 50 -4 4 −−−− 60 -5 5 −−−− 50 h. AUTO RACING In a recent season, NASCAR driver Jimmie Johnson won 8 of the 36 total races held. To the nearest thousandth, find the fraction of races he won. Terminating and repeating decimals are also rational numbers because you can write them as fractions. Write Decimals as Fractions 4 Write 0.45 as a fraction. 45 0.45 = _ 0.45 is 45 hundredths. 100 9 =_ 20 Simplify. − 5 ALGEBRA Write 0.5 as a fraction in simplest form. − Assign a variable to the value 0.5. Let N = 0.555... . Then perform operations on N to determine its fractional value. N = 0.555... 10(N) = 10(0.555...) Repeating Decimals If 2 digits repeat, multiply each side by 100. 10N = 5.555... -N = 0.555... −−−−−−−−−−−− 9N = 5 5 N=_ 9 Multiply each side by 10 because 1 digit repeats. Multiplying by 10 moves the decimal point 1 place to the right. Subtract N = 0.555... to eliminate the repeating part. Simplify. Divide each side by 9. − 5 The decimal 0.5 can be written as _ . 9 Write each decimal as a fraction or mixed number in simplest form. − −− i. -0.14 j. 8.75 k. 0.27 l. -1.4 Personal Tutor at ca.gr7math.com 86 Chapter 2 Algebra: Rational Numbers Doug Martin Examples 1, 2 (p. 85) Example 3 Write each fraction or mixed number as a decimal. 1. _ (p. 86) For Exercises 14–21 22–25 26–29 30–33 34–37 See Examples 1 2 3 4 5 29 40 5 6. -7_ 33 262 hits during his 704 at-bats. What was Ichiro Suzuki’s batting average? Round to the nearest thousandth. Write each decimal as a fraction or mixed number in simplest form. 8. 0.6 10. -1.55 9. 0.32 − 11. -0.5 (/-%7/2+ (%,0 3. -1_ 9 16 5 5. 4_ 6 7. BASEBALL In a recent season, Ichiro Suzuki of the Seattle Mariners had (p. 86) Examples 4, 5 2. _ 4 5 5 4. _ 9 −− − 12. -3.8 13. 2.15 Write each fraction or mixed number as a decimal. 14. _ 15. _ 3 4 16. _ 2 5 18. -_ 7 16 22. _ 4 33 19. -_ 5 32 6 23. -_ 11 17. _ 33 40 5 21. 5_ 16 7 80 1 20. 2_ 8 24. -6_ 25. -7_ 13 15 8 45 Students at Carter Junior High FAMILIES For Exercises 26–29, refer to the table at the right. 26. Express the fraction of students with no siblings as a decimal. Number of Siblings Fraction of Students Three _1 15 _1 3 _5 12 _1 Four or More 1 _ None 27. Find the decimal equivalent for the number of One students with three siblings. 28. Write the fraction of students with one sibling Two as a decimal. Round to the nearest thousandth. 6 29. Write the fraction of students with two siblings as a decimal. Round to the nearest thousandth. 60 Write each decimal as a fraction or mixed number in simplest form. 30. –0.4 − 34. 0.2 31. 0.5 −− 35. –0.45 32. 5.55 −− 36. –3.09 33. –7.32 − 37. 2.7 38. ELECTRONICS A computer manufacturer produces circuit chips that are 0.00032 inch thick. Write this measure as a fraction in simplest form. 39. FIND THE DATA Refer to the California Data File on pages 16–19. Choose some data and write a real-world problem in which you would express a fraction as a decimal. Lesson 2-1 Rational Numbers 87 BIOLOGY For Exercises 40–42, write the weight of each animal as a fraction or mixed number. Animal Weight (ounces) 40. queen bee Queen Bee 0.004 41. hummingbird Hummingbird 0.11 Hamster 3.5 42. hamster Source: Animals as Our Companions MEASUREMENT For Exercises 43−46, write the length of each insect as a fraction and as a decimal. 43. 44. in. 1 45. IN CM 46. CM 47. WEATHER Carla recorded the rainfall totals for several months and compared them to the average monthly totals for her town. Her results are shown in the table. Write each decimal as a fraction or mixed number in simplest form. (Hint: -1 means 1 inch less than the average monthly total.) %842!02!#4)#% See pages 679, 709. Self-Check Quiz at ca.gr7math.com 48. FROZEN YOGURT The table shows five popular flavors according to the results of a survey. What is the decimal value of those who liked vanilla, chocolate, or strawberry? Round to the nearest hundredth. Month Above/Below Average (in.) May 1.06 June 0.24 July -2.72 August -3.40 Flavor Fraction Vanilla Chocolate Strawberry H.O.T. Problems 49. OPEN ENDED Give an example of a repeating decimal where two digits repeat. Explain why your number is a rational number. Cookies and Cream Rocky Road _3 10 _1 11 _1 18 _2 55 _1 66 50. Which One Doesn’t Belong? Identify the fraction that does not belong with the other three. Explain your reasoning. 88 4 _ _1 _1 _1 11 2 9 3 Chapter 2 Algebra: Rational Numbers 51. CHALLENGE Explain why any rational number is either a terminating or repeating decimal. 52. */ -!4( Compare 0.1 and 0.−1, 0.13 and 0.−− (*/ 13, and 0.157 and 83 *5*/( −−− 0.157 when written as fractions. Make a conjecture about expressing repeating decimals like these as fractions. 53. Which of the following is equivalent 13 to _ ? 5 A 2.4 C 2.55 B 2.45 D 2.6 54. Felisa made 0.9 of her free throws in 55. Janet wants to buy a pair of jeans that cost $29.99. The sign on the display 1 says that the jeans are _ off. Which 3 expression can be used to estimate the discount? A 0.033 × $30 her last basketball game. Write this decimal as a fraction in simplest form. B 0.33 × $30 4 F _ 8 H _ C 1.3 × $30 J D 33.3 × $30 5 9 G _ 10 9 _3 5 56. The product of two integers is 72. If one integer is –18, what is the other integer? (Lesson 1-10) ALGEBRA Solve each equation. Check your solution. (Lesson 1-9) 57. t + 17 = -5 58. a - 5 = 14 59. 5 = 9 + x 60. m - 5 = -14 61. TIME The time zones of the world are sometimes expressed in relation to Greenwich Mean Time GMT. If Eastern Standard Time is expressed as GMT -5:00 and Pacific Standard Time is expressed as GMT -8:00, what is the difference between Eastern and Pacific Standard Time? (Lesson 1-5) 62. Graph the set of integers { -2, 5, -3, 0, -5, 1} on a number line. Order the integers from least to greatest. (Lesson 1-3) PREREQUISITE SKILL Find the least common multiple for each pair of numbers. (Page 667) 63. 15, 5 64. 6, 9 65. 8, 6 66. 3, 5 Lesson 2-1 Rational Numbers 89 New Vocabulary New vocabulary terms are clues about important concepts and the key to understanding word problems. Your textbook helps you find those clues by highlighting them in yellow, as terminating decimal is highlighted on page 85. Learning new vocabulary is more than just memorizing the definiton. Whenever you see a highlighted word, stop and ask yourself these questions. • How does this fit with what I already know? • How is this alike or different from something I learned earlier? Organize your answers in a word map like the one below. Definition from Text In Your Own Words A terminating decimal is a decimal where the division ends, or terminates, when the remainder is zero. Terminating decimals have a certain number of places past the decimal point and then stop. Word terminating decimal Examples 1.6, 0.75, 0.2875 Nonexamples −− 0.333..., 0.16 Make a word map for each term. The term is defined on the given page. 1. rational number (p. 84) 2. integer (p. 35) 3. greatest common factor (p. 665) 4. least common multiple (p. 667) 90 Chapter 2 Algebra: Rational Numbers Standard 7AF1.4 Use algebraic terminology (e.g. variable, equation, term, coefficient, inequality, expression, constant) correctly. 2-2 Comparing and Ordering Rational Numbers Main IDEA Compare and order rational numbers. Standard 7NS1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10), compare rational numbers in general. RECYCLING The table shows the portion of some common materials and products that are recycled. 1. Do we recycle more or less than half of the paper we produce? Explain. 2. Do we recycle more or less than half of the aluminum cans? Explain. Material 3. Which items have a recycle rate less than one half? Fraction Recycled Paper 4. Which items have a recycle rate greater than one half? Aluminum Cans Glass 5. Using this estimation method, can you order the rates from least to greatest? Scrap Tires _5 11 _5 8 _2 5 _3 4 Source: envirosystemsinc.com REVIEW Vocabulary least common denominator (LCD) the least common multiple (LCM) of the denominators; Example: The LCD of Sometimes you can use estimation to compare rational numbers. Another method is to rename each fraction using the least common denominator and then compare the numerators. _1 and _1 3 is 12. (page 667) 4 Compare Positive Rational Numbers _ _ 1 Replace ● with <, >, or = to make 5 ● 3 a true sentence. 8 4 Rename the fractions using the least common denominator. 5 3 and _ , the least common denominator is 8. For _ 8 4 5·1 5 _5 = _ or _ 8·1 8 8 3·2 6 _3 = _ or _ 4·2 4 8 5 6 _ 3 <_ , 5 <_ . Since _ 8 8 8 4 Replace each ● with <, >, or = to make a true sentence. a. _ ● _ 3 4 7 12 b. _ ● _ 5 6 7 8 c. 1_ ● 1_ 4 9 2 5 Lesson 2-2 Comparing and Ordering Rational Numbers Matt Meadows 91 You can also compare and order rational numbers by expressing them as decimals. Compare Using Decimals _ 2 Replace ● with <, >, or = to make 8 ● 0.8 a true sentence. 9 _8 ● 0.8 9 0.888... ● 0.80 Express _8 as a decimal. In the hundredths place, 8 > 0. 9 8 So, _ > 0.8. 9 Replace each ● with <, >, or = to make a true sentence. d. _ ● 0.3 e. 0.22 ● _ 1 3 f. 2_ ● 2.42 5 12 11 50 Order Rational Numbers 3 HEALTH The average life expectancies of males for several countries are shown in the table. Order the countries from least to greatest male life expectancy. Express each number as a decimal. Australia : 76.9 = 76.90 Real-World Link American males born after 1990 have an average life expectancy of about 74 years. Source: www.cdc.gov 4 France : 74_ = 74.80 5 − 1 = 75.3 Spain : 75_ 3 United Kingdom : 75 = 75.00 Life Expectancy of Males Approximate Age (years) Country Australia 76.9 Spain _4 5 1 75_ United Kingdom 75 United States 74 France 74 3 _1 4 Source: mapquest.com/atlas 1 = 74.25 United States : 74_ 4 From least to greatest life expectancy, the countries are United States, France, United Kingdom, Spain, and Australia. g. ELECTRONICS The overall width in inches of several widescreen − 3 9 2 televisions are 38.3, 38_ , 38_ , 38.4, and 38_ . Order the widths 5 3 16 from least to greatest. h. TOOLS Sophia has five wrenches measuring _ inch, _ inch, 3 8 1 4 5 3 1 _ inch, _ inch, and _ inch. What is the order of the measures 2 16 4 from least to greatest? Personal Tutor at ca.gr7math.com 92 Chapter 2 Algebra: Rational Numbers BananaStock/SuperStock Just as positive and negative integers can be represented on a number line, so can positive and negative rational numbers. You can use a number line to help you compare and order negative rational numbers. Compare Negative Rational Numbers Replace each ● with <, >, or = to make a true sentence. 4 -2.4 ● -2.45 Graph the decimals on a number line. Number Line On a number line, a number to the left is always less than a number to the right. Since -2.4 is to the right of -2.45, -2.4 > -2.45. _ _ 5 -7 ● -6 8 8 Since the denominators are the same, compare the numerators. 6 7 < -_ . -7 < -6, so -_ 8 8 Replace each ● with <, >, or = to make a true sentence. i. -_ ● -_ 9 16 Examples 1–4 (pp. 91–93) 12 16 1. _ ● _ 2. _ ● _ 5 12 1 2 10 18 (p. 93) k. -_ ● -_ 7 10 4 5 Replace each ● with <, >, or = to make a true sentence. 9 25 5. -_ ● -_ Example 5 j. -3.15 ● -3.17 3. _ ● 0.25 3 10 − 6. -_ ● -_ 16 18 4 5 4. 3_ ● 3.625 3 11 7 10 5 8 −− 7. -0.6 ● -0.67 − 8. -2.4 ● -2.42 9. OCEANOGRAPHY The tide heights for several cities are shown in the table. Order the cities from least tide height to greatest. City Baltimore, MD Galveston, TX Gulfport, MS Extra Examples at ca.gr7math.com Tide Height (ft) City − Key West, FL 1.6 _5 12 1 1_ 1 6 Tide Height (ft) − 1.83 Mobile, AL 1.5 Washington, DC 1 _17 20 Lesson 2-2 Comparing and Ordering Rational Numbers 93 (/-%7/2+ (%,0 For Exercises 10, 11 12–15 16, 17 18–23 24–29 See Examples 1 2 3 4 5 Replace each ● with <, >, or = to make a true sentence. 10. _ ● _ 2 3 11. _ ● _ 12. 0.5 ● _ 14. 6_ ● 6.5 15. 2_ ● 2.7 3 5 7 9 13. 0.75 ● _ 5 8 7 12 15 32 11 15 21 30 16. CARPENTRY Rondell has some drill bits marked _, _, _, _, and _. If these 7 3 5 9 16 8 32 16 1 4 are all measurements in inches, how should he arrange them if he wants them from least to greatest? 17. PHOTOGRAPHY Cameras often have multiple shutter speeds. Some common − 1 1 1 , 0.06, _ , 0.125, 0.004, and _ . List these shutter speeds in seconds are _ 60 125 4 speeds in order from the fastest to the slowest. Replace each ● with <, >, or = to make a true sentence. 18. -4.8 ● -4.6 19. -5.25 ● -5.24 20. -22.9 ● -22.09 21. -2.07 ● -2.6 22. -4.3 ● -4.37 23. -2.8 ● -2.86 24. -_ ● -_ 1 11 25. -_ ● -_ 7 10 26. -_ ● -_ 27. -_ ● -_ 28. -1_ ● -1_ 29. -5_ ● -5_ 3 11 3 5 4 10 3 8 7 15 1 6 2 3 1 12 3 5 4 7 Graph the following numbers on a number line. 30. -3_ , -3.68, -3.97, -4_ 31. -2.9, -2.95, -2_, -2_ 32. -5.25, -5_, -4_, -4.6 33. 3.7, 2.9, -4_, 1_ 2 5 1 3 7 8 3 4 1 8 1 5 1 4 1 2 34. STATISTICS If you order a set of numbers from least to greatest, the middle number is the median. Find the median of -18.5°C, -18°C, and 20.2°C. 35. ANALYZE TABLES The table shows the regular season records of five college baseball teams during a recent season. Which team had the best record? (Hint: Divide the number of games won by the number of games played.) Team %842!02!#4)#% See pages 679, 709. Self-Check Quiz at ca.gr7math.com 94 Games Won Games Played University of Alabama 29 55 University of Notre Dame 51 63 University of Southern California 24 56 Florida State University 45 68 Rice University 46 60 36. ATTENDANCE The school play was attended by _ of the 6th grade, _ of the 5 6 4 7th grade, and _ of the 8th grade. Which grade has the greatest part of its 5 class attend the play? Chapter 2 Algebra: Rational Numbers Comstock/SuperStock 3 4 37. NUMBER SENSE Are the fractions _, _ , _ , and _ arranged in order from 5 5 11 12 H.O.T. Problems 5 13 5 14 least to greatest or from greatest to least? Explain. 38. OPEN ENDED Name two fractions that are less than _ and two fractions that 1 2 1 . are greater than _ 2 − 39. CHALLENGE Are there any rational numbers between 0.2 and _? Explain. 40. 2 9 −− */ -!4( Explain why 0.28 is less than 0.28. (*/ 83 *5*/( 41. Which fraction is between -_ and 3 4 2 ? -_ 42. Which point on the number line below is the coordinate of 0.425? 3 P 1 A -_ 2 QR 1 4 0 3 B -_ S 1 2 3 4 1 5 5 C -_ F Point P 7 G Point Q 7 D -_ 8 H Point R J Point S 43. MEASUREMENT The sheet of ice for a hockey rink is created in two layers. 1 First an _ -inch layer of ice is made for the lines to be painted on. 8 6 Then a _ -inch layer of ice is added on top of the painted layer, 8 7 for a total thickness of _ inch. Write the total thickness of the ice 8 as a decimal. (Lesson 2-1) ALGEBRA Solve each equation. Check your solution. (Lesson 1-10) y 7 44. _ = 22 45. 4p = -60 46. 20 = _ 47. 81 = -3d 48. _ = -108 49. -4n = -96 t 15 a 6 50. WEATHER After the temperature had fallen 10°F, the temperature was -8°F. Write and solve a subtraction equation to find the starting temperature. (Lesson 1-9) PREREQUISITE SKILL Multiply. (Lesson 1-6) 51. -4(-7) 52. 8(-12) 53. (-3)17 54. 23(-5) Lesson 2-2 Comparing and Ordering Rational Numbers 95 2-3 Main IDEA Multiply positive and negative fractions. Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers. Standard 7MG1.3 Use measures expressed as rates (e.g. speed, density) and measures expressed as products (e.g. person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer. NEW Vocabulary dimensional analysis Multiplying Positive and Negative Fractions Animation ca.gr7math.com 1 2 1 2 To multiply _ and _, you can use an area model to find _ of _ . 3 5 3 5 2 5 Draw a rectangle with five columns. Shade two fifths of the rectangle blue. 1 3 Divide the rectangle into three rows. Shade one third of the rectangle yellow. 1 2 The green shaded area represents _ of _ . 3 5 1 2 1. What is the product of _ and _ ? 3 5 2. Use an area model to find each product. a. _ · _ b. _ · _ 3 4 1 2 1 3 c. _ · _ 4 5 2 5 2 3 d. _ · _ 2 3 4 5 3. What is the relationship between the numerators of the factors and the numerator of the product? 4. What is the relationship between the denominators of the factors and the denominator of the product? The Mini Lab suggests the rule for multiplying fractions. +%9 #/.#%04 Words Examples Multiply Fractions To multiply fractions, multiply the numerators and multiply the denominators. Numbers _2 · _4 = _8 3 5 15 Algebra ac _a · _c = _ where b and d ≠ 0 b d bd You can use the rules for multiplying integers to determine the sign of the product of any two signed numbers. 96 Chapter 2 Algebra: Rational Numbers Multiply Positive Fractions REVIEW Vocabulary greatest common factor (GCF) the greatest of the common factors of two or more numbers; Example: the GFC of 8 and 12 is 4. (page 665) _ _ 1 Find 4 · 3 . Write in simplest form. 5 9 1 _4 · _3 = _4 · _3 5 9 9 Divide 9 and 3 by their GCF, 3. 5 3 4·1 =_ Multiply the numerators. Multiply the denominators. 4 =_ Simplify. 3·5 15 Multiply. Write in simplest form. a. _ · _ 1 4 b. _ · _ 5 12 2 3 c. _ · _ 3 20 7 10 7 16 Multiply Negative Fractions _ _ 2 Find - 5 · 3 . Write in simplest form. 8 6 Negative Fractions 1 -5 5 __ , and _ are 5 _ -5 _ · 3 =_ ·3 -_ all equivalent fractions. -5 · 1 =_ Multiply the numerators. Multiply the denominators. 5 = -_ The fractions have different signs, so the product is negative. -5, 6 6 -6 8 6 Divide 6 and 3 by their GCF, 3. 8 6 2 2·8 16 Multiply. Write in simplest form. d. _ · -_ 8 9 e. -_ · _ 3 4 3 5 7 9 f. (-_12 )(-_67 ) To multiply mixed numbers, first rename them as improper fractions. Multiply Mixed Numbers _ _ 3 Find 4 1 · 2 2 . Write in simplest form. 2 3 9 _ 1 2 _ _ _ 4 ·2 = · 8 2 3 2 3 3 4 2 3 _1 _9 _2 _8 4 = ,2 = 2 9 _ =_ ·8 1 Estimate 4 × 3 = 12 2 3 3 Divide out common factors. 1 3·4 =_ 1·1 12 =_ or 12 1 Multiply the numerators. Multiply the denominators. Simplify. Compare to the estimate. Multiply. Write in simplest form. g. 1_ · 1_ 1 2 2 3 Extra Examples at ca.gr7math.com h. _ · 1_ 5 7 3 5 i. (-2_16 )(-1_15 ) Lesson 2-3 Multiplying Positive and Negative Fractions 97 4 ROLLER COASTERS A roller coaster at an amusement park is 160 feet _ high. If a new roller coaster is built that is 2 3 times the height of 5 the existing coaster, what is the height of the new roller coaster? 3 The new coaster is 2_ times higher than the current coaster. 3 13 _ · 160 = _ · 160 2_ 5 5 160 13 _3 _ , 160 = _ 2 = 5 5 1 2,080 = _ or 416 5 5 1 The new roller coaster will be 416 feet high. Real-World Link A 757 aircraft has an average cruising speed of 540 miles per hour, a capacity of 242 passengers, and a j. CARPENTRY A piece of lumber is 4_ feet long. If you need a piece of wingspan of 165 feet. Dimensional analysis is the process of including units of measurement when you compute. You can use dimensional analysis to check whether your answers are reasonable. _1 3 Source: Continental Traveler 1 4 2 lumber that is _ this size, how long a piece do you need? 3 Use Dimensional Analysis 5 AIRCRAFT Refer to the information at the left. Suppose a 757 aircraft _ is traveling at its cruising speed. How far will it travel in 1 1 hours? 3 Words Distance equals the rate multiplied by the time. Variable Let d represent the distance. Equation d = 540 miles per hour · 1 hours _1 3 540 miles 1 d=_ · 1_ hours Write the equation. 540 miles _ hours d=_ · 4 ·_ 1 = 3 1 hour Mental Math _1 of 540 is 180. 3 Using the Distributive _1 Property, 1 of 540 3 should equal 540 + 180, or 720. 1 hour 3 1 _1 _4 3 3 180 hours 540 miles _ d =_ · 4 ·_ 1 hour 3 1 1 Divide by common factors and units. d = 720 miles 1 At its cruising speed, a 757 will travel 720 miles in 1_ hours. 3 Check for Reasonableness The problem asks for the distance. When you divide the common units, the answer is expressed in miles. k. AIRCRAFT Refer to the information about the 757 aircraft. What is its wingspan in yards? Personal Tutor at ca.gr7math.com 98 Chapter 2 Algebra: Rational Numbers George Hall/CORBIS Examples 1–3 Multiply. Write in simplest form. (p. 97) 1. _ · _ 2. _ · _ 4. -_ · _ 5. -_ · 3 5 5 7 4 5 3. _ · _ 3 8 6 7 7 6 12 2 6. -_ -_ 3 13 3 7 _ _ 9. -6 · 1 9 4 3 2 _ 8 9 1 2 _ _ 8. 2 · 1 2 5 1 4 8 9 1 1 _ 7. 1 · 5_ 3 2 () ( )( ) Example 4 10. BIOLOGY The giant hummingbird of South (p. 98) America is the largest hummingbird in the Giant Hummingbird 1 world. It is 4_ times larger than the bee 8 hummingbird. If the length of a bee hummingbird is 2 inches, how long is the giant hummingbird? Example 5 5 8 (p. 98) (/-%7/2+ (%,0 For Exercises 12–15 16–19 20–23 24, 25 26–27 See Examples 1 2 3 4 5 x in. 11. FRUIT Terrence bought 2_ pounds of grapes that cost $2 per pound. What was the total cost of the grapes? Use dimensional analysis to check the reasonableness of the answer. Multiply. Write in simplest form. 12. _ · _ 5 4 9 2 1 14. _ · _ 15. _ · _ 9 8 5 10 3 9 2 3 12 15 1 4 1 16. -_ · _ 17. -_ _ 18. -_ -_ 19. -_ -_ 25 32 5 3 7 20 10 3 3 5 1 1 1 1 4 2 20. 3_ · _ 21. 4_ · 3_ 22. -3_ · -_ 23. -_ · -1_ 3 4 3 8 5 4 3 6 1 24. FOOD There are 3_ servings of green beans in a certain can. Each serving 2 1 cup of beans. How many cups of green beans does the can contain? is _ 2 1 12 4 7 13. _ · _ 3 16 ( ) ( )( ) ( ) ( )( ) ( ) 25. MEASUREMENT Minh-Thu has a square photograph of the volleyball team 1 2 that measures 3_ inches on each side. She reduces each dimension to _ its 2 3 size. What is the length of a side of the new photograph? Solve each problem. Use dimensional analysis to check the reasonableness of the answer. 26. BAKING A recipe calls for _ cup of sugar per batch of cookies. If Gabe wants 3 4 to make 6 batches of cookies, how many cups of sugar does he need? 27. POPULATION The population density measures how many people live within a certain area. In a certain city, there are about 150,000 people per square mile. How many people live in an area of 2.25 square miles? Lesson 2-3 Multiplying Positive and Negative Fractions Crawford Greenewalt/VIREO 99 _ _ _ _ ALGEBRA Evaluate each expression if r = - 1 , s = 2 , t = 8 , and v = - 2 . 5 4 28. rs 29. rt 9 30. stv 3 31. rtv Find each product. Write in simplest form. 32. _ · -_ · _ 33. _ · _ · _ 34. 35. 2_ · 1_ · 2_ 36. 3_ · 1_ · 5 37. 10 · 3.78 · _ 38. _ · 0.25 39. -_ · 0.3 ( 38 ) 45 1 3 5 9 2 7 1 2 2 5 1 3 1 5 3 4 2 5 1 2 2 9 (-_25 ) · _16 · (-_25 ) 1 5 − 40. -_ · (-2.375) 7 16 GEOGRAPHY For Exercises 41–43, refer to the table and the information below. Round answers to the nearest whole number. Approximate Fraction of Earth’s Landmass Continent _1 5 _9 100 _3 10 11 _ 200 _7 100 33 _ 200 _3 Africa Antarctica There are about 57 million square miles of land on Earth covering seven continents. Asia 41. What is the approximate land area Australia of Europe? 42. What is the approximate land area Europe of Asia? North America 43. Only about _ of Australia’s land 3 10 South America area is able to support agriculture. What fraction of the Earth’s land is this? _ _ 25 _ _ ALGEBRA Evaluate each expression if a = -1 1 , b = 2 7 , c = -2 1 , and d = 4 1 . 5 9 2 4 Express in simplest form. 44. abd %842!02!#4)#% See pages 680, 709. Self-Check Quiz at ca.gr7math.com 2 46. _a 2d 1 2 45. b 2c 2 47. -3ac(-bd) 48. RESEARCH Use the Internet or other resource to find a recipe for spaghetti 2 of the amount. Then change the recipe sauce. Change the recipe to make _ 1 of the amount. to make 1_ 3 2 H.O.T. Problems 49. FIND THE ERROR Matt and Enrique are multiplying 2_ and 3_. Who is 1 2 correct? Explain your reasoning. 1 4 1 1 2_21 · 3_ = 2 · 3 + _21 · _ 4 4 1 =6+_ 8 = 6_1 8 5 _ 13 1 2_21 · 3_ =_ 2 · 4 4 =_ 8 = 8_1 65 Matt 100 Chapter 2 Algebra: Rational Numbers (l)Royalty-Free/CORBIS, (r)Richard Hutchings/Photo Researchers 8 Enrique 50. OPEN ENDED Select two fractions with a product greater than _ and less 1 2 than 1. Use a number line to justify your answer. 51. CHALLENGE Find the missing fraction. _ · 9 =_ 3 4 52. 14 */ -!4( Explain why the product of _ and _ is less than _. (*/ 83 *5*/( 2 8 2 1 53. What number will make _ · _ = _ · n 3 4 true? 7 8 7 8 7 1 54. Find the area of the triangle. Use the 1 formula A = _ bh. 2 4 A _ 8 3 B _ 4 10 C _ 12 h ⫽ 2 in. 3 b ⫽ 1 1 in. 8 7 D _ 3 2 F _ in 8 3 2 H _ in 4 5 2 G _ in 8 8 _1 in 2 6 J Replace each ● with <, >, or = to make a true sentence. (Lesson 2-2) − −− 1 4 2 4 55. _ ● _ 56. _ ● 0.28 57. -_ ● -0.4 2 7 7 9 58. HISTORY In 1864, Abraham Lincoln won the presidential election with about 0.55 of the popular vote. Write this as a fraction in simplest form. (Lesson 2-1) 59. GOLF After four rounds of golf, Lazaro’s score was 5 under par or -5. Lazaro had improved his overall score during the fourth round by decreasing it by 6 strokes. Write and solve a subtraction equation to find Lazaro’s score after the third round. (Lesson 1-9) Write an equation to model the relationship between the quantities in each table. (Lesson 1-7) 60. 61. Regular Price, p Sale Price, s 300 $8 $6 5 750 $12 $9 7 1,050 $16 $12 s C p s Servings, s Total Calories, C 2 PREREQUISITE SKILL Divide. (Lesson 1-6) 62. 51 ÷ (-17) 63. -81 ÷ (-3) 64. -92 ÷ 4 65. -105 ÷ (-7) Lesson 2-3 Multiplying Positive and Negative Fractions 101 2-4 Dividing Positive and Negative Fractions Main IDEA Divide positive and negative fractions. Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers. Standard 7MG1.3 Use measures expressed as rates (e.g. speed, density) and measures expressed as products (e.g. person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer. ANIMALS The world’s longest snake is the reticulated python. It is approximately one-fourth the length of the blue whale. World’s Largest Animals Largest Animal Blue Whale 110 feet long 1. Find the value of 110 ÷ 4. Largest Reptile Saltwater Crocodile 16 feet long 2. Find the value of 110 × _. Largest Bird Ostrich 9 feet tall 3. Compare the values of Largest Insect Stick Insect 15 inches long 1 4 1 . 110 ÷ 4 and 110 × _ Source: The World Almanac for Kids 4 4. What can you conclude about the relationship between dividing 1 by 4 and multiplying by _ ? 4 Two numbers whose product is 1 are multiplicative inverses, or NEW Vocabulary multiplicative inverses reciprocals 1 reciprocals, of each other. For example, 4 and _ are multiplicative 4 1 inverses because 4 · _ = 1. 4 +%9 #/.#%04 BrainPOP® ca.gr7math.com Words Inverse Property of Multiplication The product of a number and its multiplicative inverse is 1. Examples Numbers Algebra _3 · _4 = 1 4 _a · _b = 1, where a and b ≠ 0 3 b a Find a Multiplicative Inverse _ 1 Write the multiplicative inverse of -5 2 . 3 2 17 -5_ = -_ 3 3 Write -5 _2 as an improper fraction. 3 3 2 17 _ Since -_ - 3 = 1, the multiplicative inverse of -5_ is -_ . 3 ( 17 ) 3 17 Write the multiplicative inverse of each number. a. -2_ 1 3 102 Chapter 2 Algebra: Rational Numbers Paul A. Souders/CORBIS b. -_ 5 8 c. 7 Extra Examples at ca.gr7math.com Multiplicative inverses are used in division. Consider _a ÷ _c , b d which can be written as a fraction. Complex Fractions Recall that a fraction bar represents division. So, a _ c a b _ ÷ _ = _. b d _c d _a _a · _d _c _c · _d Multiply the numerator and d denominator by _ , the c c multiplicative inverse of _ . b c _b = _ d d c d _a · _d b c =_ d _c · _ =1 1 d c d = _a · _ b c +%9 #/.#%04 Divide Fractions To divide by a fraction, multiply by its multiplicative inverse. Words Examples Numbers Algebra _2 ÷ _3 = _2 · _4 5 4 5 _a ÷ _c = _a · _d, where b, c, and d ≠ 0 3 d b b c Divide Fractions and Mixed Numbers Divide. Write in simplest form. _ _ 2 -4 ÷ 6 7 5 6 4 4 _ -_ ÷_ = -_ ·7 7 5 5 6 _6 _7 Multiply by the multiplicative inverse of , which is . 7 6 2 4 _ = -_ ·7 5 6 Divide -4 and 6 by their GCF, 2. 3 14 = -_ Multiply. 15 _ ( _) 3 2 2 1 14 7 _ _ 4 ÷ (-3 ) = _ ÷ (-_ 3 2 3 2) 14 2 =_ · -_ 3 ( 7) 14 2 =_ · -_ 3 ( 7) 3 4 2 ÷ -3 1 _2 _1 _ 14 4 =_ , -3 = - 7 3 3 2 2 _ _ The multiplicative inverse of - 7 is - 2 . 2 7 2 Divide 14 and 7 by their GCF, 7. 1 1 4 = -_ or -1_ 3 Dividing By a Whole Number When dividing by a whole number, rename it as an improper fraction first. Then multiply by its reciprocal. 3 Multiply. Divide. Write in simplest form. d. _ ÷ _ 3 4 e. -_ ÷ _ 1 2 g. 2_ ÷ -2_ 3 4 ( 1 5 ) 7 1 8 4 1 1 _ h. 1 ÷ 2_ 2 3 f. -_ ÷ -_ ( 35 ) 2 3 i. -1_ ÷ 12 1 2 Personal Tutor at ca.gr7math.com Lesson 2-4 Dividing Positive and Negative Fractions 103 Real-World Link The first Flag Day was celebrated in 1877. It was the 100th anniversary of the day the Continental Congress adopted the Stars and Stripes as the official flag. Source: World Book 4 HOLIDAYS Isabel and her friends are making ribbons to give to other campers at their day camp on Flag Day. They have a roll with 20 feet of ribbon. How many Flag Day ribbons as shown at the right can they make? 4 in. 4 1 1 Since 4 inches equals _ or _ foot, divide 20 by _ . 12 20 1 1 20 ÷ _ =_ ÷_ 3 3 1 20 _ 3 _ = · 1 1 60 =_ or 60 1 3 3 20 Write 20 as _. 1 _1 Multiply by the multiplicative inverse of , which is 3. 3 Simplify. Isabel and her friends can make 60 Flag Day ribbons. j. LUMBER Some boards are cut to a thickness of 1_ inches. The shelf 1 2 that holds the boards is 36 inches deep. How many boards can be stacked on the shelf? 5 HOME IMPROVEMENT There were 4 persons working on a _ remodeling project. It took them 6 1 days to finish the job. How 2 long would it take 6 persons to finish a similar project? 1 1 days, the project required 4 × 6_ personIf 4 persons each worked 6_ 2 2 days of work. Divide this number by 6 persons to find the number of days it will take to complete the other project. 1 4 × 6_ person-days ÷ 6 persons 2 1 4 × 6_ person-days 1 2 ×_ = __ 1 26 1 =_ or 4_ days 6 3 6 persons Multiply by the multiplicative 1 inverse of 6, which is _. 6 Simplify. Check for Reasonableness The problem asks for the number of days. When you divide the common units, the answer is expressed in days. Dimensional Analysis You can also use dimensional analysis to check the reasonableness of the answer. 104 Aaron Haupt k. TRAVEL Geoff plans to travel 480 miles. If his car gets an average of 32 miles per gallon of gasoline, approximately how much gasoline will he use? Use dimensional analysis to check the reasonableness of the answer. Chapter 2 Algebra: Rational Numbers Example 1 (p. 102) Example 2 (p. 103) Write the multiplicative inverse of each number. 1. _ 5 7 (p. 103) 4. _ ÷ _ 5. _ ÷ _ 6. _ ÷ -_ 7. -_ ÷ -_ 8. _ ÷ 8 9. _ ÷ 3 3 4 2 3 5 8 ( 109 ) (p. 104) ( 78 ) 9 10 4 5 ( 5 6 1 2 7 16 10. -5_ ÷ -4_ Examples 4, 5 3 4 Divide. Write in simplest form. 3 8 Example 3 3. -2_ 2. -12 2 3 11. -3_ ÷ 6_ ) 5 6 7 12 12. BIOLOGY The 300 million-year-old fossil of a cockroach was recently found in eastern Ohio. The ancient cockroach is shown next to the common German cockroach found today. Common German Cockroach 300-Million-Year-Old Cockroach 1 2 in. 1 3 2 in. How many times longer is the ancient cockroach than the German cockroach? (/-%7/2+ (%,0 For Exercises 13–18 19–26 27–34 35, 36 37, 38 See Examples 1 2 3 4 5 Write the multiplicative inverse of each number. 13. -_ 14. -_ 5 8 2 _ 17. 3 5 7 9 16. 18 15. 15 18. 4_ 1 8 Divide. Write in simplest form. 19. _ ÷ _ 20. _ ÷ _ 21. _ ÷ _ 23. -_ ÷ _ 24. _ ÷ -_ 25. -_ ÷ -_ 26. -_ ÷ -_ 27. _ ÷ 4 28. _ ÷ 3 29. _ ÷ 6 30. _ ÷ 4 3 4 2 5 4 5 3 8 3 4 3 10 ( 23 ) 31. 3_ ÷ 2_ 1 2 2 5 ( 23 ) 5 9 1 10 33. -12_ ÷ 4_ 1 4 1 10 7 12 ( 56 ) 6 7 4 5 32. 7_ ÷ 2_ 1 2 22. _ ÷ _ 5 6 2 3 9 16 2 5 3 4 2 3 2 3 34. 10_ ÷ -_ 1 5 ( 153 ) Lesson 2-4 Dividing Positive and Negative Fractions (l)courtesy Jo McCulty/Ohio State University, (r)Tom Young/CORBIS 105 HUMAN BODY For Exercises 35 and 36, use the information below and at the right. Composition of Human Body Component The table shows the composition of a healthy adult male’s body. Examples of body cell mass are muscle, body organs, and blood. Examples of supporting tissue are blood plasma and bones. Fraction of Body Weight 11 _ 20 _3 10 _3 Body Cell Mass Supporting Tissue Body Fat 35. How many times more of a healthy 20 Source: about.com adult male’s body weight is made up of body cell mass than body fat? 36. How many times more of a healthy adult male’s body weight is made up of body cell mass than supporting tissue? For Exercises 37 and 38, use dimensional analysis to check the reasonableness of each answer. Real-World Link 99% of the mass of the human body is made up of six elements: oxygen, carbon, hydrogen, nitrogen, calcium, and phosphorus. Source: about.com 37. PAINTING It took 3 persons 2_ hours to paint a large room. How long 1 2 would it take 5 persons to paint a similar room? 38. VACATION The Sumner family is planning a vacation. The destination is 350 miles away. If they drive at an average speed of 62 miles per hour, approximately how long will it take to get there? 39. BIOLOGY Use the information below. How many of the smallest grasshoppers need to be laid end-to-end to have the same length as one of the largest grasshoppers? ->iÃÌÊ}À>ÃÃ ««iÀ >À}iÃÌÊ}À>ÃÃ ««iÀ IN IN 40. LIBRARIES Pilar is storing a set of art books on a shelf that has 11_ inches of 1 4 3 inch wide, how many books can be stored on shelf space. If each book is _ 4 the shelf? 41. GEOMETRY The circumference C, or distance around a 44 r, circle, can be approximated using the formula C = _ r 7 %842!02!#4)#% See pages 680, 709. Self-Check Quiz at ca.gr7math.com H.O.T. Problems where r is the radius of the circle. What is the radius of the circle at the right? Round to the nearest tenth. C ⫽ 53.2 m 42. BAKING Emily is baking chocolate cupcakes. Each batch of 20 cupcakes 2 1 cups of cocoa. If Emily has 3_ cups of cocoa, how many full requires _ 3 4 batches of cupcakes will she be able to make and how much cocoa will she have left over? 43. OPEN ENDED Select a fraction between 0 and 1. Identify both its additive and multiplicative inverses. Explain your reasoning. 44. CHALLENGE Give a counterexample to the statement The quotient of two fractions between 0 and 1 is never a whole number. 106 Chapter 2 Algebra: Rational Numbers (tl)Phanie/Photo Researchers, (c)George McCarthy/CORBIS, (cr)Dennis Johnson/Papilio/CORBIS 45. NUMBER SENSE Which is greater: 30 · _ or 30 ÷ _? Explain. 3 4 3 4 CHALLENGE Use mental math to find each value. 46. _ · _ ÷ _ 43 594 48. 641 76 47. _ · _ ÷ _ 641 594 783 241 241 783 72 53 */ -!4( Write a real-world problem that can be solved by (*/ 83 *5*/( dividing fractions or mixed numbers. Solve the problem. 49. A submarine sandwich that is 50. Mr. Jones is doing a science 1 26_ inches long is cut into experiment with his class of 20 2 5 _ 4 -inch mini-subs. How many 12 3 students. Each student needs _ cup 4 of vinegar. If he currently has 15 cups of vinegar, which equation could Mr. Jones use to determine if he has enough vinegar for his entire class? mini-subs are there? F x = 15 ÷ 20 3 G x = 15 ÷ _ 4 IN A 4 C 6 B 5 D 7 H x = 20 – (15) J x = 15(20) Multiply. Write in simplest form. (Lesson 2-3) 51. _ · _ 1 2 52. _ · _ 3 4 7 12 4 7 53. 1_ · 4_ 2 3 54. _ · 3_ 1 5 2 3 1 4 55. SCHOOL In a survey of students at Centerburg Middle School, _ of 13 20 17 of the girls said they ride the bus to school. Of those the boys and _ 25 surveyed, do a greater fraction of boys or girls ride the bus? (Lesson 2-2) 56. ALGEBRA Write an equation using two variables that could be used to determine the population of Asia if it is about three million less than five times the population of Africa. (Lesson 1-7) Write an integer to describe each situation. (Lesson 1-3) 57. 10 candy bars short of his goal 58. 7 bonus points PREREQUISITE SKILL Add or subtract. (Lessons 1-4 and 1-5) 59. -7 + 15 60. -9 + (-4) 61. -3 - 15 62. 12 - (-17) Lesson 2-4 Dividing Positive and Negative Fractions 107 2-5 Adding and Subtracting Like Fractions Main IDEA Add and subtract fractions with like denominators. Standard 7NS1.2 Add, subtract, multiply and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. BAKING A bread recipe calls for the ingredients at the right together with small amounts of sugar, oil, yeast, and salt. Bread 1 1_ cups of whole wheat 3 3 1. What is the sum of (sifted) _1 cup oatmeal 3 _1 cup apricots (diced) 3 _1 cup hazelnuts the whole-number parts of the amounts? 2. How many _ cups 1 3 are there? flour (sifted) 1 2_ cups of white flour 3 3. Can you combine these (chopped) ingredients in a 4-cup mixing bowl? Explain. NEW Vocabulary like fractions Fractions that have the same denominators are called like fractions. +%9 #/.#%04 Words Add and Subtract Like Fractions To add or subtract like fractions, add or subtract the numerators and write the result over the denominator. Examples Algebra Numbers _1 + _3 = _4 5 5 5 _7 - _3 = _4 or _1 8 8 a +b _ac + _bc = _ c , where c ≠ 0 a -b _ac - _bc = _ c , where c ≠ 0 2 8 You can use the rules for adding integers to determine the sign of the sum of any two signed numbers. Add Like Fractions _ ( _) 8 8 1 Find 5 + - 7 . Write in simplest form. Look Back You can review adding integers in Lesson 1-4. 5 + (-7) _5 + -_7 = _ ( 8) 8 Add the numerators. The denominators are the same. 8 -2 1 =_ or -_ 8 Simplify. 4 Add. Write in simplest form. a. _ + _ 5 9 108 7 9 Chapter 2 Algebra: Rational Numbers Julie Houck/Stock Boston b. -_ + _ 5 9 1 9 ( 56 ) c. -_ + -_ 1 6 Extra Examples at ca.gr7math.com Subtract Like Fractions _ _ 2 Find - 8 - 7 . Write in simplest form. 9 9 ( 9) 8 8 7 7 -_ -_ = -_ + -_ 9 9 9 -8 + (-7) 9 -15 2 _ = or -1_ 9 3 =_ Subtract the numerators by adding the opposite of 7. Rename -15 6 2 _ as -1_ or -1_. 9 3 9 Subtract. Write in simplest form. d. -_ - _ e. _ - _ 3 5 4 5 3 8 5 8 f. ( 7) _5 - -_4 7 To add or subtract mixed numbers, add or subtract the whole numbers and the fractions separately. Then simplify. Add Mixed Numbers _ _ 3 Find 5 7 + 8 4 . Write in simplest form. 9 9 (9 9) 7 4 7 4 5_ + 8_ = (5 + 8) + _ +_ 9 9 Alternative Method You can also add the mixed numbers vertically. 7 5_ 9 4 _ +8 9 _____ 11 2 _ 13 or 14_ 9 9 +4 = 13 + 7_ Add the numerators. 9 11 2 = 13_ or 14_ 9 Add the whole numbers and fractions separately. 9 11 2 _ = 1_ 9 9 Add or subtract. Write in simplest form. g. 9_ - 3_ 5 8 h. 8 - 6_ 3 8 ( 29 ) i. -8_ + -6_ 5 9 2 9 Another way to add or subtract mixed numbers is to write the mixed numbers as improper fractions. Subtract Mixed Numbers 3 1 4 HEIGHTS Jasmine is 60_ inches tall. Amber is 58_ inches tall. How 4 4 much taller is Jasmine than Amber? 3 235 1 241 60_ - 58_ =_ -_ 4 4 4 4 Estimate 60 - 59 = 1 Write the mixed numbers as improper fractions. 241 - 235 =_ Subtract the numerators. The denominators are the same. 6 1 =_ or 1_ 1 Jasmine is 1_ inches taller than Amber. 4 4 2 2 3 cups of flour. j. BAKING A recipe for chocolate cookies calls for 2_ 4 1 cups of flour, how much more will she need? If Alexis has 1_ 4 Personal Tutor at ca.gr7math.com Lesson 2-5 Adding and Subtracting Like Fractions 109 Examples 1–3 (pp. 108–109) Add or subtract. Write in simplest form. ( 45 ) (-_79 ) 5 2 6. -_ - (-_) 6 6 1. _ + -_ 2. -_ + _ 4. -_ - _ 5. _ - _ 7. 5_ - 2_ 8. -1_ + -2_ 2 5 3 4 9 10 7 10 4 9 3 8 3. -_ + ( 27 ) 9. 10 - 3_ 4 9 7 8 3 7 2 9 1 4 5 16 Example 4 10. CLOTHING Hat sizes are determined by the distance across a person’s head. (p. 109) 3 inches than How much wider is a person’s head who wears a hat size of 7_ 4 1 inches? someone who wears a hat size of 6_ 4 (/-%7/2+ (%,0 Add or subtract. Write in simplest form. For Exercises 11–14 15–18 19–26 27, 28 11. -_ + _ 12. -_ + -_ 13. -_ + _ 14. _ + -_ 15. -_ - _ 3 5 16. _ - _ 17. _ - _ 18. _ - _ 19. 3_ + 7_ 20. 9_ + 4_ 21. 8_ + -2_ See Examples 1 2 3 4 4 5 5 8 15 16 5 8 9 16 5 9 23. -1_ - 3_ 5 6 ( 27 ) 3 7 4 9 1 9 7 9 3 4 7 12 2 9 ( 59 ) 8 9 ( 109 ) 22. 8_125 + (-5 _1211 ) 1 10 3 4 8 9 7 12 1 12 24. -3_ - 7_ 5 6 5 12 25. 7 - 5_ 26. 9 - 6_ 3 7 2 5 27. HOME IMPROVEMENT Andrew has 42_ feet of molding to use as borders 1 3 2 feet of the molding on the around the windows of his house. If he uses 23_ 3 front windows, how much remains for the back windows? 28. WEATHER One year, Brady’s hometown of Powell received about 42_ 6 10 3 inches of snow fell. What is inches of snow. The following year only 14_ 10 the difference in the amount of snow between the two years? Simplify each expression. 29. ( 5) 3 4 1 -7_ + 3_ - 2_ 5 5 ( ) 30. -8_ - -3 _ + 6_ 1 8 5 8 3 8 MEASUREMENT Find the perimeter of each rectangle. 31. 32. 12 1 in. ? FT 4 25 3 in. 4 ? FT 110 Chapter 2 Algebra: Rational Numbers ALGEBRA Evaluate each expression for the given values. 33. a - b if a = 5_ and b = -2_ 1 3 34. x + y if x = -_ and y = -_ 5 1 12 12 1 1 36. s - t if s = -_ and t = -2_ 2 2 1 3 35. n - m if m = 5_ and n = -2_ 2 3 2 3 37. SPORTS One of the track and field events is the triple jump. In this event, the athlete takes a running start and makes three jumps without stopping. Find the total length of the 3 jumps for the athlete below. FT %842!02!#4)#% FT 38. HOMEWORK Rob recorded the amount of time he Day Time spent on homework last week. Express his total time for the week in terms of hours and minutes. Mon 2_ h Tue 2_ h 5 7 inches long. The plumber cuts 2_ inches 64_ Wed 1_ h off the end of the pipe, then cuts off an Thu 2_ h 3 inches. How long is the remaining additional 1_ Fri 1_ h 39. PLUMBING A plumber has a pipe that is 8 See pages 680, 709. Self-Check Quiz at 8 8 ca.gr7math.com H.O.T. Problems FT 1 6 1 2 3 4 5 12 1 4 pipe after the last cut is made? 40. OPEN ENDED Write a subtraction problem with a difference of _. 2 9 41. FIND THE ERROR Allison and Wesley are adding _ and _. Who is correct? 3 7 1 7 Explain your reasoning. 1+3 3 _1 + _ =_ 7 7 7 4 =_ 7 1+3 3 _1 + _ =_ 7 7 7+7 4 =_ or _2 14 7 Allison Wesley 42. CHALLENGE Explain how you could use mental math to find the following sum. Then find the sum. 5 3 2 2 1 1 3_ + 4_ + 2_ + 2_ + 1_ +_ 3 43. 5 6 6 3 5 */ -!4( Write a real-world situation that can be solved by (*/ 83 *5*/( adding or subtracting mixed numbers. Then solve the problem. Lesson 2-5 Adding and Subtracting Like Fractions (l)CORBIS, (r)Skjold Photographs/The Image Works 111 44. Esteban is 63_ inches tall. Haley 1 8 45. The equal-sized square tiles on a 5 is 59_ inches tall. How much taller 8 bathroom floor are set as shown. EQUAL SPACING is Esteban than Haley? Write in simplest form. IN 1 A 4_ in. 2 1 in. B 4_ IN 4 3 in. C 3_ 4 What is the width of the space between the tiles? 1 in. D 3_ 2 3 F _ in. 3 H _ in. 5 1 G _ in. 5 J 10 _2 in. 5 Divide. Write in simplest form. (Lesson 2-4) 46. _ ÷ _ 3 5 47. _ ÷ 2_ 6 7 7 8 48. -3_ ÷ 2_ 4 5 1 4 1 2 49. Find the product of -_ and -_. (Lesson 2-3) 7 8 6 7 50. NUTRITION There is 2.3 times the recommended daily allowance of vitamin C in a 5.5-ounce serving of kiwifruit. Write an equation to represent the amount of vitamin C recommended for each day. (Lesson 1-7) Fruit Vitamin C (mg in 5.5 oz) Orange 52 Strawberries 63 Kiwifruit 103.5 Source: Food and Drug Administration Evaluate each expression. (Lesson 1-3) 51. ⎪-20⎥ - ⎪17⎥ 52. ⎪31⎥ - ⎪-10⎥ 53. ⎪5 + 9⎥ 54. ⎪8 - 17⎥ 55. FOOD On a typical day, 2 million gallons of ice cream are produced in the United States. About how many gallons of ice cream are produced each year? (Lesson 1-1) PREREQUISITE SKILL Find the least common multiple (LCM) of each set of numbers. (page 667) 56. 14, 21 112 57. 18, 9, 6 Chapter 2 Algebra: Rational Numbers 58. 6, 4, 9 59. 5, 10, 20 CH APTER Mid-Chapter Quiz 2 Lessons 2-1 through 2-5 1. MEASUREMENT One centimeter is about 11. WEATHER The table shows the approximate 0.392 inch. What fraction of an inch is this? (Lesson 2-1) number of sunny days each year for certain 3 cities. Oklahoma City has about _ as many 5 sunny days as Phoenix. About how many sunny days each year are there in Oklahoma City? (Lesson 2-3) 2. Write 1_ as a decimal. (Lesson 2-1) 7 16 − 3. Write 0.4 as a fraction in simplest form. (Lesson 2-1) Sunny Days Per Year City Days Austin, TX 120 5. -_ ● -_ Denver, CO 115 Phoenix, AZ 215 7. -7.833… ● -7.8 Sacramento, CA 195 Santa Fe, NM 175 Replace each ● with <, >, or = to make a true sentence. (Lesson 2-2) 4. _ ● _ 1 3 1 4 −− 4 6. 0.12 ● _ 33 8. 3 10 2 5 Source: National Oceanic and Atmospheric Administration STANDARDS PRACTICE The table gives the durations, in hours, of several human spaceflights. Mission Year Duration (h) Challenger (41–B) 1984 4 191_ Discovery (51–A) 1984 191_ Endeavour (STS–57) 1992 1 190 _ 1999 1 191_ Discovery (STS–103) 15 Divide. Write in simplest form. (Lesson 2-4) 12. _ ÷ -_ 1 2 3 4 2 6 ( 34 ) 13. (-1_13 ) ÷ (-_14 ) STANDARDS PRACTICE A board that is 14. 1 25_ feet long is cut into pieces that are 2 1 each 1_ feet long. Which of the steps below 2 Which of the following correctly orders these durations from least to greatest? would give the number of pieces into which the board is cut? (Lesson 2-4) (Lesson 2-2) 1 1 F Multiply 1_ by 25_ . 3 1 1 4 A 190_ , 191_ , 191_ , 191_ 2 6 4 15 3 1 4 1 B 191_ , 191_ , 191_ , 190_ 4 6 15 2 3 1 1 4 C 190_ , 191_ , 191_ , 191_ 2 6 15 4 3 1 4 1 D 191_ , 191_ , 190_ , 191_ 6 15 2 2 2 1 1 G Divide 25_ by 1_ . 2 2 1 1 to 1_ . H Add 25_ 2 J 2 1 1 Subtract 1_ from 25_ . 2 2 4 Multiply. Write in simplest form. Add or subtract. Write in simplest form. (Lesson 2-3) (Lesson 2-5) 9. (-_13 ) · _78 10. (-2_34 ) · (-_15 ) 15. _ + -_ 1 5 ( 45 ) 16. –3_ - 3_ 4 7 6 7 2-6 Adding and Subtracting Unlike Fractions Ma rt Main IDEA a Add and subtract fractions with unlike denominators. FOOD Marta and Brooke are sharing a Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions and terminating decimals) and take positive rational numbers to wholenumber powers. Standard 7NS2.2 Add and subtract fractions by using factoring to find common denomitors. 3 Brooke eats _ of the pizza. NEW Vocabulary unlike fractions 1 pizza. Marta eats _ of the pizza, and 4 8 1. What are the denominators of the fractions? 2. What is the least common multiple of the denominators? e ok Bro ? 1 3. Find the missing value in _ = _. 4 8 Fractions with unlike denominators are called unlike fractions. To add or subtract unlike fractions, rename the fractions using prime factors to find the least common denominator. Then add or subtract as with like fractions. Add and Subtract Unlike Fractions REVIEW Vocabulary least common denominator (LCD) the least common multiple (LCM) of the denominators (page 667) Add or subtract. Write in simplest form. _ ( _) 1 1 + -2 3 4 _1 + -_2 = _1 · _3 + -_2 · _4 ( 3) 4 ( 3) 4 3 8 =_ + (-_ 12 12 ) 4 The LCD is 3 · 4 or 12. 3 3 + (-8) 12 Rename using the LCD. 5 = _ or -_ _ _ Add the numerators. Then simplify. 12 2 - 8 - (- 7 ) 63 99 8 8 _ 7 7 _ +_ = -_ · 11 + _ ·7 -_ 99 63 63 99 11 63 = 3 · 3 · 7, 99 = 3 · 3 · 11 The LCD is 3 · 3 · 7 · 11 or 693. 7 49 88 = -_ +_ Rename using the LCD. =_ Add the numerators. 693 693 -88 + 49 693 39 13 = -_ or -_ 693 Simplify. 231 Add or subtract. Write in simplest form. a. -_ + -_ 5 6 114 file photo ( 12 ) Chapter 2 Algebra: Rational Numbers b. _ + _ 1 14 3 49 c. -_ + _ 5 16 3 10 Extra Examples at ca.gr7math.com Add and Subtract Mixed Numbers _ _ 3 Find -6 2 + 4 5 . Write in simplest form. Estimation Think: _2 -6 is about -6 and _5 9 4 is about 5. Since 6 -6 + 5 is about -1, the answer is about -1. The answer seems reasonable. 9 6 5 29 56 2 -6_ + 4_ = -_ +_ 9 6 6 9 87 112 = -_ + _ 18 18 Write as improper fractions. - 56 _ 29 3 87 2 _ 112 · = -_ and _ · _ = _ 9 =_ -112 + 87 18 -25 7 = _ or -1_ 18 18 2 6 18 3 18 Add the numerators. Simplify. Add or subtract. Write in simplest form. ( 18 ) e. -3_12 + 8_13 f. 2_ - 6_ d. -_ + -_ 5 12 3 4 1 3 _ _ _ g. -1_ + -3_ 2 5 ( 1 3 ) _ 4 Four telephone books are 2 1 , 1 15 , 1 3 , and 2 3 inches thick. If these 8 16 8 4 books were stacked one on top of another, what is the total height of the books? 3 A 5_ in. 3 C 11_ in. 16 3 in. B 8_ 16 16 3 D 15_ in. 16 Read the Item Use Estimation If a test question would take an excessive amount of time to work, try estimating the answer. Then look for the appropriate answer choice. You need to find the sum of four mixed numbers. Solve the Item It would take some time to change each of the fractions to ones with a common denominator. However, notice that all four of the numbers have a value of about 2. Since 2 × 4 equals 8, the answer will be about 8. Notice that only one of the choices is close to 8. The answer is B. h. Amanda is planning a rectangular vegetable garden using a roll of 3 border fencing that is 45_ feet long. If she makes the width of the 4 1 garden 10_ feet, what must the length be? 3 F 12_ ft 8 1 G 17_ ft 2 3 H 24_ ft J 2 4 1 35_ ft 4 Personal Tutor at ca.gr7math.com Lesson 2-6 Adding and Subtracting Unlike Fractions 115 Examples 1–3 (pp. 114–115) Add or subtract. Write in simplest form. 1. _ + -_ 2. -_ + _ 4. _ - _ 5. _ - _ ( 16 ) 3 4 5 8 3 4 7 8 7 13 7. -3_ + 1_ Example 4 ( 23 ) 14 12 6. _ - (-_) 15 21 7 7 9. -4_ - (-3_) 72 12 4 9 2 9 8. 3_ - 1_ 5 6 2 5 3. -_ + -_ 1 2 5 8 1 3 1 STANDARDS PRACTICE Tamera played a computer game for 1_ hours, 10. 4 1 1 studied for 2_ hours, and did some chores for _ hour. How much (p. 115) 2 4 time did Tamera spend on all of these tasks? 1 h A 2_ 1 B 3_ h 2 (/-%7/2+ (%,0 For Exercises 11–14 15–18 19–26 42, 43 See Examples 1 2 3 4 1 D 4_ h C 4h 4 2 Add or subtract. Write in simplest form. 11. _ + -_ 12. -_ + _ 15. _ - _ ( 127 ) 1 4 1 3 19. 3_ + -8_ ( 1 2 ) 23. -4_ - 5_ 3 4 13. -_ + -_ ( 152 ) 1 2 20. 1_ + (-6_) 6 3 24. -8_ - 4_ 25. -15_ + 11_ 26. -22_ + 15_ 5 6 5 8 1 3 ( 38 ) 7 12 18. -_ - (-_) 15 25 3 1 22. 7_ - (-1_) 8 4 6 7 4 5 7 8 1 5 14. -_ + -_ 16. _ - -_ ( 12 ) 3 2 17. -_ - (-_) 11 9 3 1 21. 8_ - (-6_) 7 2 3 8 5 6 5 8 5 9 2 3 2 5 5 6 27. HIKING The map shows a hiking trail at a campground. If the distance 1 along the trail from the campground to Silver Lake is 4_ miles, how far is 10 it from Glacier Ridge to Silver Lake? 2107 mi 112 mi Nature Center Silver Lake 4 5 mi Youth Camp x Glacier Ridge Cabins 1 2 mi Campground 112 mi ALGEBRA Evaluate each expression for the given values. 28. c - d if c = -_ and d = -12_ 3 4 116 Chapter 2 Algebra: Rational Numbers 7 8 29. r - s if r = -_ and s = 2_ 5 8 5 6 30. HISTORY In the 1824 presidential Candidate election, Andrew Jackson, John Quincy Adams, Henry Clay, and William H. Crawford received electoral votes. Use the information at the right to determine what fraction of the votes William H. Crawford received. Fraction of Vote _3 8 _1 3 _1 Andrew Jackson John Quincy Adams Henry Clay 7 Source: The World Almanac 1 31. PHOTOGRAPHY Two 4-inch by 6-inch 8 2 in. digital photographs are printed on an 1 8_ -inch by 11-inch sheet of photo paper. 2 4 in. After the photos are printed, Aaron cuts them from the sheet. What is the area of the remaining photo paper? 11 in. 4 in. 6 in. MEASUREMENT Find the missing measure for each figure. 32. 33. 3 6 4 ft x in. 1 1 1 4 3 ft 7 9 2 in. 11 8 in. 4 3 ft 1 13 4 in. x ft 3 perimeter 40 4 in. 11 %842!02!#4)#% perimeter 17 12 ft See pages 681, 709. 34. Self-Check Quiz at ca.gr7math.com H.O.T. Problems FIND THE DATA Refer to the California Data File on pages 16–19. Choose some data and write a real-world problem in which you would add or subtract unlike fractions or mixed numbers. 35. OPEN ENDED Write a subtraction problem using unlike fractions with a least common denominator of 12. Find the difference. 36. NUMBER SENSE Without doing the computation, determine whether _4 + _5 is greater than, less than, or equal to 1. Explain. 7 9 37. CHALLENGE Suppose a bucket is placed under two faucets. If one faucet is turned on alone, the bucket will be filled in 5 minutes. If the other faucet is turned on alone, the bucket will be filled in 3 minutes. Write the fraction of the bucket that will be filled in 1 minute if both faucets are turned on. */ -!4( For Exercises 38–41, write an expression for each (*/ 83 *5*/( statement. Then find the answer. 38. _ of _ 3 2 3 4 3 2 40. _ less than _ 3 4 39. _ more than _ 3 2 3 4 3 2 41. _ divided into _ 3 4 Lesson 2-6 Adding and Subtracting Unlike Fractions 117 42. A recipe for snack mix contains 43. Which of the following shows the 1 1 2_ cups of mixed nuts, 3_ cups of 3 2 3 _ granola, and cup raisins. What is 4 next step using the least common 3 2 denominator to simplify _ -_ ? 4 _3 × _5 – _2 × _6 (4 5) (3 6) 3 6 5 2 ×_ – _ ×_ G (_ 6) (3 5) 4 3 3 2 4 ×_ – _ ×_ H (_ 3) (3 4 4) 3 3 4 2 ×_ – _ ×_ J (_ 3) 4 4) (3 the total amount of snack mix? F 2 A 5_ c 3 7 B 5_ c 12 2 C 6_ c 3 7 D 6_ c 3 12 Add or subtract. Write in simplest form. (Lesson 2-5) 44. -_ + _ 7 11 5 11 46. 5_ - 7_ 45. -_ - _ 7 15 4 5 4 15 1 5 47. ALGEBRA Find a ÷ b if a = 3_ and b = -_. (Lesson 2-4) 1 2 7 8 POPULATION For Exercises 48 and 49, use the graphic at the right. (Lesson 1-7) GfglcXk`fe?flicp:_Xe^\ 48. Write and solve a multiplication equation to determine the number of hours it would take for the population of the United States to increase by 1 million. 49. Write and solve a multiplication equation to determine the number of days it would take for the U.S. population to increase by 1 million. Source: U.S. Census Bureau 50. INVESTMENTS Mr. Coffey purchased stock for $50 per share. The next day the value of the stock dropped $12. On the second and third days, the value dropped another $16, then rose $25. What was the value of the stock at the end of the third day? (Lesson 1-4) PREREQUISITE SKILL Solve each equation. Check your solution. (Lessons 1-9 and 1-10) 118 51. d - 13 = -44 52. -18t = 270 53. -34 = y + 22 54. -5 = _ Chapter 2 Algebra: Rational Numbers a 16 2-7 Solving Equations with Rational Numbers Main IDEA Solve equations involving rational numbers. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inquality, or a system of equations or inequalities that represents a verbal descripton (e.g. three less than a number, half as large as an area A. Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers. REVIEW Vocabulary equation a mathematical sentence that contains an equals sign (Lesson 1-7) BIOLOGY An elephant, which can run at a speed of 25 miles per hour, 5 runs _ as fast as a grizzly bear. If s 6 represents the speed of a grizzly 5 bear, you can write the equation 25 = _ s. 6 1. Multiply each side of the equation by 6. Then divide each side by 5. Write the result. 2. Multiply each side of the original equation by the multiplicative 5 . Write the result. inverse of _ 6 3. What is the speed of a grizzly bear? 4. Which method of solving the equation seems most efficient? You have used properties of equality to solve equations with integers. The same properties can also be used to solve equations with rational numbers. Solve by Using Addition or Subtraction 1 Solve p - 7.36 = 2.84. p - 7.36 = 2.84 Write the equation. p -7.36 + 7.36 = 2.84 + 7.36 Add 7.36 to each side. p = 10.2 _ Simplify. _ 2 Solve 1 = t + 3 . 2 4 _1 = t + _3 2 4 4 4 Write the equation. _1 - _3 = t + _3 - _3 Subtract _1 - _3 = t Simplify. _2 - _3 = t Rename . 1 -_ =t Simplify. 2 2 4 4 4 _3 from each side. 4 _1 2 4 4 Solve each equation. Check your solution. a. t - 7.81 = 4.32 Extra Examples at ca.gr7math.com Tom Brakefield/CORBIS b. y + _ = -_ 2 5 1 2 c. 1_ = 2_ + a 5 6 1 3 Lesson 2-7 Solving Equations with Rational Numbers 119 Solve by Using Multiplication or Division _ 3 Solve 4 b = 16. Check your solution. 7 _4 b = 16 7 7 _ 4 b = 7 (16) 4 7 4 _( ) _ b = 28 Write the equation. _7 _4 Multiply each side by , the reciprocal of . 4 7 Simplify. Check the solution. 4 Solve 58.4 = -7.3m. Check your solution. 58.4 = -7.3m Write the equation. 58.4 -7.3m _ =_ Divide each side by -7.3. -7.3 -7.3 -8 = m Simplify. Check the solution. Solve each equation. Check your solution. d. -12 = _ r 4 5 e. -_n = -_ 3 5 2 3 f. 7.2v = -36 5 BASKETBALL Suppose that during her last game, Sue Bird made _ 12 field goals, which were 3 of her field goal attempts. Write and 4 solve an equation to determine her number of field goal attempts that game. Real-World Link During her rookie season for the WNBA, Sue Bird’s field goal average was 0.379, and she made 232 field goal attempts. Source: WNBA.com Words _3 of field goal attempts is 12. Variable Let g represent field goal attempts. Equation _3 g = 12 _3 g = 12 4 4 _ 3 g = 4 (12) 3 4 3 _( ) _ g = 16 4 4 Write the equation. _4 _3 Multiply each side by , the reciprocal of . 3 4 Simplify. Sue Bird had 16 field goal attempts. g. COMMUNICATION Larissa pays $0.25 per minute for long distance calls on her cell phone. Her long distance charge last month was $5. Write and solve an equation that could be used to determine the number of minutes she used to make long distance calls. Personal Tutor at ca.gr7math.com 120 Chapter 2 Algebra: Rational Numbers Elaine Thompson/AP/Wide World Photos Solve each equation. Check your solution. Examples 1, 2 (p. 119) 1. t + 0.25 = -4.12 2. v - 8.34 = -3.77 3. a - _ = -_ 4. c + _ = -1_ 5. -45 = _d 6. -_n = 18 7. -26.5 = -5.3w 8. 2.6x = 22.75 3 4 Examples 3, 4 5 6 (p. 120) Example 5 (p. 120) (/-%7/2+ (%,0 For Exercises 10–15 16–21 22, 23 See Examples 1, 2 3, 4 5 5 8 3 8 9 16 7 10 9. SPACE The planet Jupiter takes 11.9 Earth years to make one revolution around the Sun. Write and solve a multiplication equation to determine the number of revolutions Jupiter makes in 59.5 Earth years. Solve each equation. Check your solution. 10. q + 0.45 = 1.29 11. a - 1.72 = 5.81 12. -_ = m - _ 13. -_ = f + _ 14. g - (-1.5) = 2.35 15. -1.3 = n - (-6.12) 17. -_p = -8 2 9 18. -1.92 = -0.32s 20. _ = -4.5 21. _ = -2.2 5 1 9 3 4 16. -_b = 16 7 t 3.2 19. -8.4 = 1.2t 1 2 2 3 h -5.75 22. MONEY The currency of Egypt is called a pound. One U.S. dollar is equal to 3 Egyptian pounds. Write and solve a multiplication equation to find the 3_ 4 number of U.S. dollars that would equal 21 Egyptian pounds. 23. RECREATION Refer to the graphic. Write and solve an addition equation to determine the number of visitors v that the Golden Gate National Recreation Area needs to equal the number of visitors to the Blue Ridge Parkway. DfjkGfglcXiEXk`feXcGXibj 6ISITORS MILLIONS Solve each equation. Check your solution. 24. 3.5g = -_ 7 8 1 25. -7.5r = -3_ − 1 26. 4_ = -3.3c 6 3 "LUE 2IDGE 0ARKWAY 'OLDEN 'ATE .ATIONAL 2ECREATION !REA 'REAT 3MOKEY -OUNTAINS .ATIONAL 0ARK 27. -4.2 = _ x 7 Lesson 2-7 Solving Equations with Rational Numbers 121 28. FOOTBALL In his rookie season, Ben Roethlisberger completed 196 passes with a season pass-completion rate of 0.664. Write and solve an equation to determine the number of passes Ben Roethlisberger attempted during his rookie season. 29. COMPUTERS Stephan’s CD recorder can write 5.3 megabytes of data per second. If he uses a CD with a 700 megabyte capacity, how long will it take to record the entire CD? MEASUREMENT Find the area of each rectangle. 30. 31. M Real-World Link As of 2006, Ben Roethlisberger held the record as the youngest quarterback to win the Super Bowl. IN 0ERIMETER IN 0ERIMETER M Source: nfl.com 32. TRAVEL Mr. Harris filled the gas tank of his car. Gasoline cost $2.95 per gallon, and Mr. Harris spent a total of $39.53. If his car can travel 32.5 miles per gallon of gasoline, how far can he travel with the gasoline he just purchased? 33. MEASUREMENT Andy has a board that he is going to use to make shelves for 5 a craft fair. The board is 108 inches long. If each shelf is 9_ inches long, 8 write and solve an equation to find how many shelves he can make using this board. MEASUREMENT Find the missing measure in each triangle. %842!02!#4)#% 34. 35. See pages 681, 709. h Self-Check Quiz at 15 in. A 45 in2 ca.gr7math.com H.O.T. Problems 5.5 cm b A 37.73 cm2 36. OPEN ENDED Write an equation with rational numbers that has a solution 1 . of _ 4 37. Which One Doesn’t Belong? Identify the expression that does not have the same value as the other three. Explain your reasoning. _4 _3 x ( ) 3 4 3 _ -_ -2x 2 ( 3) 2 _x -_1 _x ( 21 ) ( ) 1 3 3 38. CHALLENGE During a clearance sale, sweaters were marked at _ the original 1 4 1 price. Patrice had a coupon for _ off the marked price of any sweater. If 3 Patrice paid $24 for a sweater, what was the original price of the sweater? 39. 2 */ -!4( Explain how to solve -_ (*/ x = 14 using properties of 83 *5*/( 3 equality. Use the term multiplicative inverse in your explanation. 122 Chapter 2 Algebra: Rational Numbers Stephen Dunn/Getty Images 40. If the area of the rectangle is 41. The difference of a number x and 2.3 3 22_ square inches, what is the is 1.8. Which equation shows this relationship? 4 width of the rectangle? F x + 2.3 = 1.8 G x – 2.3 = 1.8 x H _ = 1.8 WIDTH 2.3 J x – 1.8 = 2.3 INCHES 42. If a = 6 and b = 4, then 5a – ab = 4 A _ in. 13 A 6 1 B 2_ in. B 24 2 C 30 1 C 3_ in. D 54 4 3 D 3_ in. 4 Add or subtract. Write in simplest form. (Lesson 2-6) 43. _ + _ 1 6 1 7 44. _ - _ 7 8 45. -5_ - 6_ 1 6 1 2 47. GEOMETRY Find the perimeter of the triangle. (Lesson 2-5) 2 5 1 2 2 3 48. VEGETABLES Hudson purchased 3_ pounds 46. 2_ + 5_ 4 5 FT FT of vegetables that cost $3 per pound. What was the total cost of the vegetables? (Lesson 2-3) FT 49. ALGEBRA The sum of two integers is 13. One of the integers is -5. Write and solve an equation to find the other integer. (Lesson 1-9) Add. (Lesson 1-4) 50. -48 + 13 + (-16) 51. 35 + 17 + (-25) 52. -50 + (-62) + 3 53. 27 + (-30) + (-26) 54. PREREQUISITE SKILL Kishi wants to buy a digital music player that costs $250 with tax. So far, she has saved $120. If she saves $15 each week, in how many weeks will she be able to purchase the digital music player? Use the four-step plan. (Lesson 1-1) Lesson 2-7 Solving Equations with Rational Numbers 123 2-8 Problem-Solving Investigation MAIN IDEA: Look for a pattern to solve problems. Standard 7MR2.4 Make and test conjectures by using both inductive and deductive reasoning. Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. e-Mail: LOOK FOR A PATTERN YOUR MISSION: Look for a pattern to solve the problem. THE PROBLEM: How many bounces occurred before Terry’s ball reached a height less than 1 inch? ▲ TERRY: In science class, I dropped a ball from 48 inches above the ground. After the first, second, third, and fourth bounces, the ball reached heights of 24, 12, 6, and 3 inches, respectively. EXPLORE PLAN You know the original height of the ball and the heights after the first 4 bounces. You want to know the number of bounces before the ball reached a height less than 1 inch. Look for a pattern in the heights. Then continue the pattern to find when the ball’s height was less than 1 inch. +1 SOLVE +1 +1 +1 +1 Bounce 1 2 3 4 5 Height (in.) 24 12 6 3 1 6 _1 _3 4 2 _ _ _ _ _ ×1 ×1 ×1 ×1 ×1 2 CHECK 2 2 2 2 After the sixth bounce, the ball reached a height less than 1 inch. Check your pattern to make sure the answer is correct. 1. Describe how to continue the pattern in the second row. Find the fraction of the height after 7 bounces. */ -!4( Write a problem that can be solved by finding a (*/ 83 *5*/( 2. pattern. Describe a pattern. 124 John Evans Chapter 2 Algebra: Rational Numbers For Exercises 3–5, look for a pattern. Then use the pattern to solve the problem. 3. MUSIC The names of musical notes form a pattern. Name the next three notes in the following pattern: whole note, half note, quarter note. 4. GEOMETRY Draw the next two figures in the pattern. 8. TRAVEL Rafael is taking a vacation. His plane is scheduled to leave at 2:20 P.M. He must arrive at the airport at least 2 hours before his flight. It will take him 45 minutes to drive from his house to the airport. When is the latest he should plan to leave his house for the airport? 9. WATER MANAGEMENT A tank is draining at a rate of 8 gallons every 3 minutes. If there are 70 gallons in the tank, when will the tank have just 22 gallons left? 10. THEATER A theater is designed with 12 seats 5. MUSEUMS A science museum offers discount passes for group admission. If this pattern continues, how many people would be admitted if a group buys 31 passes? Passes People Admitted 2 3 5 7 7 10 12 18 in the first row, 17 seats in the second row, 22 seats in the third row, and so on. How many seats are in the ninth row? For Exercises 11–13, select the appropriate operation(s) to solve the problem. Justify your selection(s) and solve the problem. 11. INSECTS The longest insect in the world Use any strategy to solve Exercises 6–10. Some strategies are shown below. G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s rn. • Look for a patte is the stick insect whose length reaches 15 inches. The smallest insect is the fairy fly whose length is only 0.01 inch. How many times longer is the stick insect than the fairy fly? 12. ANALYZE TABLES In computer terminology, a bit is the smallest unit of data. A byte is equal to 8 bits. The table below gives the equivalences for several units of data. 6. GEOMETRY Find the perimeters of the next two figures in the pattern.The length of each side of a triangle is 4 meters. Unit of Data Equivalence 1 byte 8 bits 1 kilobyte (kB) 1,024 bytes 1 megabyte (MB) 1,024 kilobytes 1 gigabyte (GB) 1,024 megabytes How many bits are in 1 MB? 7. MONEY To attend the class trip, each student will have to pay $7.50 for transportation, and $5.00 for food. If there are 360 students in the class, how much money will need to be collected for the trip? 13. PIZZA Lola is planning a party. She plans to order 4 pizzas, which cost $12.75 each. If she has a coupon for $1.50 off each pizza, find the total cost of the pizzas. Lesson 2-8 Problem-Solving Investigation: Look for a Pattern 125 2-9 Powers and Exponents Main IDEA Use powers and exponents in expressions. FAMILY Every person has 2 biological parents. 2 parents Standard 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to wholenumber powers. Standard 7NS2.1 Understand negative wholenumber exponents. Multiply and divide expressions involving exponents with a common base. Standard 7AF2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. 2 · 2 or 4 grandparents 2 · 2 · 2 or 8 great grandparents 1. How many 2s are multiplied to determine the number of great grandparents? great-great grandparents? A product of repeated factors can be expressed as a power, that is, using an exponent and a base. 4 common factors The base is the common factor. 2·2·2·2=2 The exponent tells how many times the base is used as a factor. Write Expressions Using Powers NEW Vocabulary power base exponent 4 Write each expression using exponents. _ _ _ 1 1 · 1 · 1 ·3·3·3·3 2 2 2 _1 · _1 · _1 · 3 · 3 · 3 · 3 = _1 · _1 · _1 · (3 · 3 · 3 · 3) 2 2 (2 2 2) 1 = (_ ·3 2) 2 3 4 Associative Property Definition of exponents 2 a·b·b·a·b a·b·b·a·b=a·a·b·b·b Commutative Property = (a · a) · (b · b · b) Associative Property = a2 · b3 Definition of exponents Write each expression using exponents. a. _ · 7 · _ · _ · 7 · _ 2 3 2 3 2 3 2 3 b. m · m · n · n · n · m Personal Tutor at ca.gr7math.com 126 Chapter 2 Algebra: Rational Numbers c. 3 · a · b · 2 · 3 · a Exponents can also be negative. Consider the pattern in the powers of 10. Negative Exponents Remember that 10 -2 Exponential Form Standard Form 10 3 1,000 2 100 10 1 10 10 0 1 10 Negative powers are the result of repeated division. 10 2 100 ÷ 10 = 10 10 ÷ 10 = 1 1 1 _ or _ 10 10 1 1 1 _ ÷ 10 = _ or _ _1 10 1 _ 10 -1 1 equals _, 1,000 ÷ 10 = 100 10 -2 1 ÷ 10 = 10 1 100 100 not -100 or -20. 10 2 The pattern suggests the following definition for zero exponents and negative exponents. +%9 #/.#%04 Words Zero and Negative Exponents Any nonzero number to the zero power is 1. Any nonzero number to the negative n power is the multipicative inverse of its nth power. Examples Numbers Algebra 0 0 5 =1 _ _1 · _1 or _1 7 -3 = 1 · 7 7 7 73 x = 1, x ≠ 0 x -n = _1 , x ≠ 0 x n Evaluate Powers REVIEW Vocabulary evaluate to find the value of an expression (Lesson 1-2) (_3 ) 4 3 Evaluate 2 . 4 (_23 ) = _23 · _23 · _23 · _23 Write the power as a product. 16 =_ Multiply. 81 4 Evaluate 4 -3. 1 4 -3 = _ 3 Write the power using a positive exponent. 4 1 =_ 4 3= 4 · 4 · 4 or 64 64 5 ALGEBRA Evaluate a 2 · b 4 if a = 3 and b = 5. a2 · b4 = 32 · 54 Replace a with 3 and b with 5. = (3 · 3) · (5 · 5 · 5 · 5) Write the powers as products. = 9 · 625 or 5,625 Multiply. Evaluate each expression. d. 1 (_ 15 ) 3 Extra Examples at ca.gr7math.com e. 5 -4 f. c 3 · d 2 if c = -4 and d = 9 Lesson 2-9 Powers and Exponents 127 Examples 1, 2 (p. 126) Examples 3, 4 (p. 127) Write each expression using exponents. 1. 2 · 2 · 2 · 3 · 3 · 3 2. r · s · r · r · s · s · r · r 3. _ · p · k · _ · p · p · k 1 2 1 2 Evaluate each expression. (_17 ) 3 4. 2 6 5. 6. 6 -3 7. 3 -5 8. EARTH SCIENCE There are approximately 10 21 kilograms of water on Earth. This includes oceans, rivers, lakes, ice caps, and water vapor in the atmosphere. Evaluate 10 21. Example 5 9. ALGEBRA Evaluate x 2 · y 4 if x = 2 and y = 10. (p. 127) (/-%7/2+ (%,0 For Exercises 10–15 16–23 24–27 See Examples 1 2–3 4 Write each expression using exponents. 10. 8 · 8 · a 11. 5 · q · 3 · q · q · 3 1 1 12. m · _ · p · m · _ 4 4 13. d · 2 · 2 · d · k · d · k 14. 2 · 7 · a · 9 · b · a · 7 · b · 9 · b · a 15. x · _ · y · y · _ · 5 · y · 5 · x · _ · y · y 1 6 1 6 1 6 Evaluate each expression. (_13 ) 4 16. 2 3 17. 20. 5 -4 21. 9 -3 18. 3 3 · 4 2 19. 3 2 · 22. 7 –2 23. 4 –3 (_15 ) 2 ALGEBRA Evaluate each expression. 24. g 5 · h, if g = 2 and h = 7 25. x 3 · y 4, if x = 1 and y = 3 26. a 2 · m 6, if a = _ and m = 2 27. k 4 · d, if k = 3 and d = _ 5 6 1 2 ASTRONOMY For Exercises 28–31, refer to the information at the right. How Many Stars Can You See? Unaided Eye in Urban Area 3 · 10 2 stars Unaided Eye in Rural Area 2 · 10 3 stars With Binoculars 3 · 10 4 stars 29. How many stars can be seen with With Small Telescope 2 · 10 6 stars unaided eyes in a rural area? Source: Kids Discover 28. How many stars can be seen with unaided eyes in an urban area? 30. How many stars can be seen with binoculars? 31. How many stars can be seen with a small telescope? %842!02!#4)#% Evaluate each expression. See pages 682, 709. Self-Check Quiz at ca.gr7math.com 128 32. 5 · 2 3 · 7 2 33. 2 2 · 7 · 10 4 34. 2 3 · 7 -2 35. 5 –2 · 2 -7 36. 4 · 2 5 · 5 –3 37. 3 –2 · 5 · 7 –3 3 3 · 10 2 3 · 10 39. _ 3 5 2 38. _ 2 4 Chapter 2 Algebra: Rational Numbers 42 · 35 · 24 4 ·3 ·2 40. (0.2) 3 · (_12 ) 4 H.O.T. Problems 41. NUMBER SENSE Without evaluating the powers, order 6 -3, 6 2, and 6 0 from least to greatest. Explain your reasoning. 42. CHALLENGE Complete the following pattern. , 3 -1 = 3 4 = 81, 3 3 = 27, 3 2 = 9, 3 1 = 3, 3 0 = , 3 -2 = , 3 -3 = 43. OPEN ENDED Write an expression with a negative exponent whose value is 1 between 0 and _ . 2 44. CHALLENGE Select several fractions between 0 and 1. Find the values of each fraction after it is raised to the -1 power. Explain the relationship between the -1 power and the original fraction. 45. */ -!4( Explain the difference between the expressions (*/ 83 *5*/( (-4) 2 and 4 -2. 46. To find the volume of a cube, 47. Which is equivalent to 2 3 · 3 4? multiply its length, its width, and its depth. F 3·3·4·4·4 G 2·2·2·3·3·3·3 H 2·2·2·3·3·3 J 48. IN What is the volume of the cube expressed as a power? A 62 C 64 B 63 D 66 6 · 12 3 (_34 ) = 9 A _ 9 C _ 12 9 B _ 16 64 27 D _ 64 49. BICYCLING The table shows the relationship between the time Melody rides her bike and the distance that she rides. If she continues riding at the same rate, how far will she ride in 1 hour? Use the look for a pattern strategy. (Lesson 2-8) Time (min) Distance (mi) 5 1 15 3 25 5 50. FOOD Suppose hamburgers are cut in squares that are 2_ inches on a 1 2 side. Write and solve a multiplication equation to determine how many hamburgers can fit across a grill that is 30 inches wide. (Lesson 2-7) PREREQUISITE SKILL Write each number. 51. two million 52. three hundred twenty 53. twenty-six hundred Lesson 2-9 Powers and Exponents 129 2-10 Scientific Notation Main IDEA Express numbers in scientific notation. 1. Copy and complete each table below. Expression Standard 7NS1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10), compare rational numbers in general. 8.7 × 10 1 = 8.7 × 10 Product 87 8.7 × 10 2 = 8.7 × 100 Expression 1 8.7 × 10 -1 = 8.7 × _ 10 1 8.7 × 10 -2 = 8.7 × _ Product 0.87 100 3 8.7 × 10 = 8.7 × 8.7 × 10 -3 = 8.7 × 2. If 8.7 is multiplied by a positive power of 10, what relationship exists between the decimal point’s new position and the exponent? 3. When 8.7 is multiplied by a negative power of 10, how does the new position of the decimal point relate to the negative exponent? NEW Vocabulary scientific notation Scientific notation is a compact way of writing numbers whose absolute value is very large or very small. factor greater than or equal to 1, but less than 10 8.7 × 10 -4 power of 10 written in exponential form If the number is negative, a negative sign precedes it. +%9 #/.#%04 Scientific Notation to Standard Form • Multiplying by a positive power of 10 moves the decimal point right. • Multiplying by a negative power of 10 moves the decimal point left. • The number of places the decimal point moves is the absolute value of the exponent. Express Numbers in Standard Form Negative Exponents Negative exponents represent repeated division. So, multiplying by a number with a negative exponent is the same as dividing by that number repeatedly. 130 1 Write 5.34 × 10 4 in standard form. 5.34 × 10 4 = 53,400. The decimal point moves 4 places right. 2 Write -3.27 × 10 -3 in standard form. -3.27 × 10 -3 = -0.00327 The decimal point moves 3 places left. Write each number in standard form. a. 7.42 × 10 5 Chapter 2 Algebra: Rational Numbers b. -6.1 × 10 -2 c. 3.714 × 10 2 Extra Examples at ca.gr7math.com +%9 #/.#%04 Standard Form to Scientific Notation To write a number in scientific notation, follow these steps. 1. Move the decimal point to the right of the first nonzero digit. 2. Count the number of places you moved the decimal point. 3. Find the power of 10. If the absolute value of the original number was between 0 and 1, the exponent is negative. Otherwise, the exponent is positive. Write Numbers in Scientific Notation 3 Write -3,725,000 in scientific notation. -3,725,000 = -3.725 × 1,000,000 The decimal point moves 6 places. = -3.725 × 10 6 Since 3,725,000 > 1, the exponent is positive. 4 Write 0.000316 in scientific notation. 0.000316 = 3.16 × 0.0001 The decimal point moves 4 places. = 3.16 × 10 -4 Since 0 < 0.000316 < 1, the exponent is negative. Write each number in scientific notation. d. -14,140,000 e. 0.00876 f. 0.114 Top U.S. Cities Visited by Overseas Travelers U.S. City Number of Arrivals 5 Boston 7.21 × 10 Las Vegas 1.3 × 10 6 Los Angeles 2.2 × 10 6 Metro DC area 9.01 × 10 5 New York 4.0 × 10 6 Orlando 1.8 × 10 6 San Francisco 1.6 × 10 6 Real-World Link The table lists seven of the top U.S. cities visited by overseas travelers in a recent year. Source: infoplease.com 5 TRAVEL Refer to the table at the right. Order the countries according to the number of visitors from greatest to least. Order the expressions according to their exponents. Then order expressions with the same exponents by comparing their decimal factors. Step 1 Step 2 Canada and Mexico Japan and United Kingdom 1.46 × 10 7 5.1 × 10 6 1.03 × 10 7 1.46 > 1.03 Canada Mexico > International Visitors in the U.S.A. Country Number of Visitors Canada 1.46 × 10 7 Japan 5.1 × 10 6 Mexico 1.03 × 10 7 United Kingdom 4.7 × 10 6 Source: International Trade Association 4.7 × 10 6 5.1 > 4.7 Japan United Kingdom g. TRAVEL Refer to the information at the left. Order the cities according to the number of arrivals from least to greatest. Personal Tutor at ca.gr7math.com Lesson 2-10 Scientific Notation 131 Examples 1, 2 (p. 130) Examples 3, 4 (p. 131) Example 5 (p. 131) Write each number in standard form. 1. 7.32 × 10 4 2. -9.931 × 10 5 3. 4.55 × 10 -1 4. 6.02 × 10 -4 Write each number in scientific notation. 5. 277,000 6. 8,785,000,000 7. -0.00004955 8. 0.524 9. BASEBALL The table at Ballpark the right lists four Major League Ballparks. List the ballparks from least to greatest capacity. Team Capacity H. H. H. Metrodome Minnesota Twins 4.8 × 10 4 Network Associates Coliseum Oakland Athletics 4.7 × 10 4 The Ballpark in Arlington Texas Rangers 4.9 × 10 4 Wrigley Field Chicago Cubs 3.9 × 10 4 Source: www.users.bestweb.net (/-%7/2+ (%,0 For Exercises 10–13 14–17 18–21 22–25 26–29 See Examples 1 2 3 4 5 Write each number in standard form. 10. 2.08 × 10 2 11. 3.16 × 10 3 12. 7.113 × 10 7 13. -4.265 × 10 6 14. 7.8 × 10 -3 15. -1.1 × 10 -4 16. 8.73 × 10 -4 17. 2.52 × 10 -5 Write each number in scientific notation. 18. 6,700 19. 43,000 20. -52,300,000 21. 147,000,000 22. 0.037 23. -0.0072 24. 0.00000707 25. 0.0000901 26. CHEMISTY The table shows the 27. GEOGRAPHY The areas of the mass in grams of one atom of each of several elements. List the elements in order from the least mass to greatest mass per atom. Great Lakes are listed in the table. Order the lakes according to their area from least to greatest. Great Lake Element Mass per Atom Carbon 1.995 × 10 -23 g Erie 9.91 × 10 3 Gold 3.272 × 10 -22 g Huron 2.3 × 10 4 Hydrogen 1.674 × 10 -24 g Michigan 2.23 × 10 4 Oxygen 2.658 × 10 -23 g Ontario 7.32 × 10 3 Silver 1.792 × 10 -22 g Superior 3.17 × 10 4 Source: Chemistry: Concepts and Applications Source: World Book Arrange these numbers in increasing order. 28. 216,000,000, 2.2 × 10 3, 3.1 × 10 7, 310,000 29. -4.56 × 10 -3 , 4.56 × 10 2, -4.56 × 10 2, 4.56 × 10 -2 132 Area (mi 2) Chapter 2 Algebra: Rational Numbers 30. HEALTH The diameter of a red blood cell is about 7.4 × 10 -4 centimeter. Write this number using standard form. 31. MEASUREMENT The smallest unit of time is the yoctosecond, which equals 0.000000000000000000000001 second. Write this number in scientific notation. 32. SPACE The temperature of the Sun varies from 10,900°F on the surface to %842!02!#4)#% See pages 682, 709. 27 billion°F at its core. Write these temperatures in scientific notation. 33. DINOSAURS The giganotosaurus weighed about 1.6 × 10 4 pounds. The microceratops weighed about 1.1 × 10 1. How many times heavier was the giganotosaurus than the microceratops? Write your answer in standard form. Round to the nearest tenth. Self-Check Quiz at ca.gr7math.com 34. NUMBER SENSE Determine whether 1.2 × 10 5 or 1.2 × 10 6 is closer to one H.O.T. Problems million. Explain. 35. CHALLENGE Compute and express each value in scientific notation. a. (130,000)(0.0057) __ b. 0.0004 (90,000)(0.0016) ___ (200,000)(30,000)(0.00012) */ -!4( Determine whether a decimal times a power of 10 (*/ 83 *5*/( 36. is sometimes, always, or never expressed in scientific notation. Explain. 37. Which shows 0.0000035 in scientific 38. The average width of a strand of a spider web is 7.0 × 10 -6 meter. Which expression represents this number in standard form? notation? A 3.5 × 10 6 B 3.5 × 10 5 F 7,000,000 m C 3.5 × 10 -5 G 700,000 m D 3.5 × 10 -6 H 0.00007 m J 0.000007 m 39. ALGEBRA Evaluate a 5 · b 2 if a = 2 and b = 3. (Lesson 2-9) ALGEBRA Solve each equation. Check your solution. (Lesson 2-7) 40. t + 3_ = 2_ 1 3 1 2 41. -_y = 14 2 3 p 1.3 42. _ = 2.4 43. -1_ = n - 4_ 3 4 1 6 44. LANGUAGE There are about one billion people who speak Mandarin. This is 492 million more than those who speak English. How many speak English? (Lesson 1-1) Lesson 2-10 Scientific Notation 133 CH APTER 2 Study Guide and Review Download Vocabulary Review from ca.gr7math.com Key Vocabulary Be sure the following Key Concepts are noted in your Foldable. R Algebra: rs be ational Num 2-1, 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 Key Concepts bar notation (p. 85) power (p. 126) base (p. 126) rational number (p. 84) dimensional analysis reciprocals (p. 102) (p. 98) repeating decimal (p. 85) exponent (p. 126) scientific notation (p. 130) like fraction (p. 108) terminating decimal (p. 85) multiplicative inverses unlike fraction (p. 114) (p. 102) Rational Numbers (Lesson 2-1) • A rational number is any number that can be a expressed in the form _, where a and b are b integers and b ≠ 0. Multiplying and Dividing Fractions (Lessons 2-3 and 2-4) • To multiply fractions, multiply the numerators and multiply the denominators. • To divide by a fraction, multiply by its multiplicative inverse. Adding and Subtracting Fractions (Lessons 2-5 and 2-6) • To add or subtract fractions, rename the fractions using the least common denominator. Then add or subtract and simplify, if necessary. Powers and Scientific Notation (Lessons 2-9 and 2-10) • A number is expressed in scientific notation when it is written as the product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10. Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. 1. Like fractions have the same denominator. 2. The number that is expressed using an exponent is a rational number. 3. Dimensional analysis is the process of including units of measurement in computation. − 4. The number 0.3 is a repeating decimal. 5. Numbers that can be written as fractions are called reciprocals. 6. The number 4.05 × 10 8 is written in bar notation. 7. The number 2.75 is a terminating decimal. 8. The base tells how many times a number is used as a factor. 9. Two numbers whose product is 1 are multiplicative inverses of each other. 10. The number 5 4 is a power. 134 Chapter 2 Algebra: Rational Numbers Vocabulary Review at ca.gr7math.com Lesson-by-Lesson Review 2-1 Rational Numbers (pp. 84–89) Write each fraction or mixed number as a decimal. 11. 1_ 12. -_ 3 13. -2_ 5 14. _ 1 3 5 8 Write each decimal as a fraction or mixed number in simplest form. 15. 0.3 16. -7.14 17. 4.3 18. -5.7 − _ Write 3 as a decimal. 5 _3 means 3 ÷ 5. 5 9 10 Example 1 − 0.6 5 3.0 -___ 30 0 3 The fraction _ can be written as 0.6. 5 Example 2 Write 0.28 as a fraction in simplest form. 19. HISTORY Thirteen of the 50 states in the United States were the original colonies. Write this fraction as a decimal. 20. BIOLOGY The average rate of human 28 0.28 = _ 100 7 =_ 25 0.28 is 28 hundredths. Simplify. 7 The decimal 0.28 can be written as _ . 25 hair growth is about 0.4 inch per month. Write this decimal as a fraction in simplest form. 2-2 Comparing and Ordering Rational Numbers (pp. 91–95) Replace each ● with <, >, or = to make a true sentence. −− 8 8 2 21. _ ● _ 22. -0.24 ● -_ 3 9 55 1 23. -_ ● -_ 2 110 33 5 3 24. _ ● _ 6 4 25. Order -_, 0.75, -_, 0 from least to greatest. 1 2 3 4 26. BOOKS The heights of Olivia’s books 9 5 15 are 4_ inches, 6_ inches, _ inches, 8 2 16 19 _ inches. What would be the and 4 order of the books if Olivia places them on a shelf in order from least to greatest height? Example 3 Replace ● with < , >, or = 2 to make _ ● 0.34 a true sentence. _2 = 0.4 5 5 2 > 0.34. Since 0.4 > 0.34, _ 5 Example 4 Replace ● with < , >, or = 3 7 to make -_ ● -_ a true sentence. 12 4 3 7 For -_ and -_, the least common 4 12 denominator is 12. 3 9 3·3 -_ = -_ or -_ 4 4·3 12 9 7 Since -9 < -7, -_ < -_ . 12 12 3 7 < -_ . So, -_ 4 12 Chapter 2 Study Guide and Review 135 CH APTER 2 Study Guide and Review 2-3 Multiplying Positive and Negative Fractions (pp. 96–101) Multiply. Write in simplest form. 3 2 27. _ · 1_ 2 2 28. -_ · -_ 5 3 29. _ · _ 6 5 1 10 30. _ · _ 2 11 5 3 3 ( 3) _ _ Example 5 form. Find 2 · 5 . Write in simplest 2·5 _2 · _5 = _ Multiply the numerators. Multiply the denominators. 3 3 3·7 7 10 =_ Simplify. 21 31. COOKING Crystal is making 1_ times a 7 1 2 1 recipe. The original recipe calls for 3_ 2 cups of milk. How many cups of milk does she need? 2-4 Dividing Positive and Negative Fractions (pp. 102–107) 32. _ ÷ _ 7 9 1 3 34. -4_ ÷ (-2) 2 5 _ _ Example 6 Find - 5 ÷ 3 . Write in 5 6 simplest form. Divide. Write in simplest form. 33. _ ÷ -_ ( 23 ) 1 2 35. 6_ ÷ (-1_) 6 3 7 12 3 5 5 _ -_ ÷_ = -_ ·5 5 6 25 = -_ 18 36. DESIGN Marcus wishes to space letters 7 = -1_ equally across the top of a page. If each letter is 1.7 inches wide, and the paper 1 is 8_ inches wide, what is the 2 maximum number of letters that he can fit across the top of the page? 2-5 18 Adding and Subtracting Like Fractions (pp. 108–112) Add or subtract. Write in simplest form. 37. _ + _ 5 6 11 11 1 7 39. _ - _ 8 8 38. _ + - _ 1 28 ( 283 ) 5 20 hours. How much longer did Jeremy work on Monday than on Tuesday? Chapter 2 Algebra: Rational Numbers 5 5 5 13 Monday. On Tuesday, he worked 2_ Simplify. Write as a mixed number. _ _ -2 =_ 3 20 Multiply by the multiplicative inverse. Example 7 Find 1 - 3 . Write in 5 5 simplest form. 1-3 _1 - _3 = _ 3 4 40. 12_ - 5_ 5 5 41. JOBS Jeremy worked 5_ hours on 136 3 6 2 = -_ 5 Subtract the numerators. The denominators are the same. Simplify. Mixed Problem Solving For mixed problem-solving practice, see page 709. Lesson-by-Lesson Review 2-6 Adding and Subtracting Unlike Fractions (pp. 114–118) Add or subtract. Write in simplest form. 43. _ - -_ 42. -_ + _ 2 3 3 5 5 12 ( 157 ) 44. -4_ - 6_ 45. 5 - 1_ 46. 7_ + 3_ 47. 5_ - 12_ 3 4 1 2 4 5 2 3 3 5 2 5 _ _ Example 8 Find 3 + 1 . Write in 3 4 simplest form. 9 4 _3 + _1 = _ +_ 4 3 1 2 48. PIZZA A pizza has 3 toppings with no toppings overlapping. Pepperoni 1 tops _ of the pizza and mushrooms 3 Rename the fractions. 12 12 9 + 4 =_ 12 13 =_ 12 1 = 1_ 12 Add the numerators. Simplify. 2 . The remainder is topped with top _ 5 sausage. What fraction is topped with sausage? 2-7 Solving Equations with Rational Numbers (pp. 119–123) Example 9 Solve each equation. 49. d - (-0.8) = 4 50. _ = -2.2 51. _n = _ 52. -7.2 = _ 3 4 7 8 x 4 r 1.6 3 53. AGE Trevor is _ of Maria’s age. Trevor 8 is 15. Write and solve a multiplication equation to find Maria’s age. 2-8 _ _ Solve t + 1 = 5 . 3 5 1 t+_ =_ 6 Write the equation. 3 6 1 5 1 _ _ t + - = - 1 Subtract 13 from each side. 3 3 6 3 1 _ t= Simplify. 2 _ _ _ PSI: Look for a Pattern (pp. 124–125) Solve. Use the look for a pattern strategy. 54. ALGEBRA Find the next two numbers in the sequence 3, 6, 9, 12, … . 55. RUNNING Marcy can run one lap in 65 seconds. Each additional lap takes her 2 seconds longer to run than the previous lap. How many minutes will it take her to run three miles? (1 mile = 4 laps) 56. GEOMETRY What is the total number of rectangles, of any size, in the figure below? Example 10 Raul’s phone plan charges a flat monthly rate of $4.95 and $0.06 per minute. If Raul spent a total of $7.35 last month, how many minutes did he use? Look for a pattern. Minutes Charges Total 0 4.95 + 0(0.06) $4.95 10 4.95 + 10(0.06) $5.55 20 4.95 + 20(0.06) $6.15 30 4.95 + 30(0.06) $6.75 40 4.95 + 40(0.06) $7.35 So, Raul used 40 minutes last month. Chapter 2 Study Guide and Review 137 CH APTER 2 Study Guide and Review 2-9 Powers and Exponents (pp. 126–129) Write each expression using exponents. Example 11 57. 3 · 3 · 3 · 3 · 3 58. 2 · 2 · 5 · 5 · 5 Write 3 · 3 · 3 · 7 · 7 using exponents. 59. x · x · x · x · y 60. 4 · 4 · 9 · 9 3 · 3 · 3 · 7 · 7 = 33 · 72 Example 12 Evaluate each expression. 61. 5 4 63. 5 -3 _1 2 · _2 2 (3) (5) 3 1 64. (_) · (_) 2 4 62. 2 3 65. PHONE TREES To close school for the day, the principal calls six parents, who in turn call six more parents. If each of those parents calls six more parents, how many calls will be made by the parents in this last group? 2-10 Evaluate 7 3. 7 3 = 7 · 7 · 7 or 343 Example 13 Evaluate 3 -6. 1 3 -6 = _ 6 3 Write the power using a positive exponent 1 =_ 729 3 6 = 3 · 3 · 3 · 3 · 3 · 3 or 729 Scientific Notation (pp. 130–133) Write each number in standard form. Example 14 66. 3.2 × 10 -3 67. 6.71 × 10 4 Write 3.21 × 10 -6 in standard form. 68. 1.72 × 10 5 69. 1.5 × 10 -2 3.21 × 10 -6 = 0.00000321 Move the decimal point 6 places to the left. 70. ANIMALS The smallest mammal is the Kitti’s hog-nosed bat weighing about 4.375 × 10 -3 pound. Write this weight in standard form. Example 15 Write 7.25 × 10 5 in standard form. 7.25 × 10 5 = 725000 Write each number in scientific notation. 71. 0.000064 72. 0.000351 73. 87,500,000 74. 7,410,000 75. SPACE The distance from Earth to the Sun is approximately 93 million miles. Write this distance in standard form and in scientific notation. 138 Chapter 2 Algebra: Rational Numbers Move the decimal point 5 places to the right. Example 16 Write 0.004 in scientific notation. 0.004 = 4 × 0.001 The decimal point moves 3 places. = 4 × 10 -3 Since 0 < 0.004 < 1, the exponent is negative. CH APTER Practice Test 2 Write each fraction or mixed number as a decimal. 5 1. 4_ 6 11. BAKING Madison needs 2_ cups of flour, but 2 3 1 measuring cup. How she can only find her _ 3 7 2. -_ 20 many times will she need to fill the measuring cup with flour to get the amount she needs? 3. FROGS The Gold Frog grows to only 0.375 inch. Write this length as a fraction in simplest form. 4. ENERGY The United States produces about 9 _ of the world’s energy and consumes 50 6 about _ of the world’s energy. Does the Solve each equation. Check your solution. 12. x - _ = _ 5 6 STANDARDS PRACTICE A recipe for two 5. 3 dozen cookies calls for 1_ cups of 4 3 · a using exponents. Evaluate each expression. 15. 6 -4 Texas, hosted the first ever summer Global X Games while Whistler, British Columbia, in Canada hosted the winter games. Team USA won the gold medal with a total of 7 2 · 2 2 points. Evaluate the number of points won by Team USA. 1 A 16_ 2 B 14 4 18. Write 8.83 × 10 -7 in standard form. D 7 19. Write 25,000 in scientific notation. Add, subtract, multiply, or divide. Write in simplest form. 6. -5_ · -2_ 7. -6 ÷ _ 8. -_ + _ 9. 1 4 3 8 16. k 3 · g -2 if k = 4 and g = 8 17. EXTREME SPORTS In 2003, San Antonio, flour. In order to make eight dozen cookies, how many cups of flour should be used? 1 C 9_ 2 3 14. Write the expression 4 · 4 · a · a · b · 3 · 4 · 25 United States produce more energy than it uses or vice versa? Explain your reasoning. 13. 16 = _y 1 3 ( 1 3 ) 4 9 1 8 (-1_78 ) - (-3_14 ) 10. ANALYZE TABLES The table shows the time of the back and forth swing of a pendulum and its length. How long is a pendulum with a swing of 5 seconds? Time of Swing Length of Pendulum 1 second 1 unit 2 seconds 4 units 3 seconds 9 units 4 seconds 16 units Chapter Test at ca.gr7math.com 20. STANDARDS PRACTICE The following table gives the approximate diameter, in miles, for several planets. Planet Diameter Mercury 3.032 × 10 3 Saturn 7.4975 × 10 4 Neptune 3.0603 × 10 4 Earth 7.926 × 10 3 Which list below correctly orders these planets from least to greatest diameters? F Mercury, Neptune, Saturn, Earth G Mercury, Earth, Neptune, Saturn H Mercury, Neptune, Earth, Saturn J Neptune, Mercury, Earth, Saturn Chapter 2 Practice Test 139 CH APTER 2 California Standards Practice Cumulative, Chapters 1–2 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1 A carpenter estimates that it will take one person 54 hours to complete a job. He plans to have three people work on the job for two days. How many hours each day will the workers need to work to complete the job? 4 Which number equals (3) -3? 1 F -_ 27 G -9 1 H _ 27 J 9 5 3 _ Which fraction is equivalent to _ + 3? A 8 hours C 12 hours 6 A _ B 9 hours D 18 hours 9 B _ The weight of a paper clip is 9.0 × 10 -4 kilograms. Which of the following represents this weight in standard notation? 6 9 D _ 5 10 50 15 10 2 9 C _ 15 1 A jar of mixed nuts contains 2_ pounds of 2 5 1 pounds of cashews and 1_ peanuts, 1_ 3 F 0.00000009 6 G 0.000009 pounds of walnuts. What is the total weight of the contents of the jar? H 0.00009 1 pounds F 4_ J 0.0009 1 pounds G 4_ After reading the salon prices listed below, Alex chose Special No. 1. She wanted to find her total savings. Her first step was to find the sum of $19 plus 2 times $4. What should Alex do next to find her total savings? 2 H 5_ pounds 6 2 3 3 1 J 6_ pounds 3 Hair Salon Prices Trim $12 Haircut $19 Shampoo $4 Style $4 Highlights $55 Perm $50 Special #1 Haircut, style, and shampoo $25 Special #2 Haircut, style, shampoo, and highlights $75 Question 6 If the test question would take an excessive amount of time to work, try estimating the answer. Then look for the appropriate answer choice. B Divide the sum by 3. The distance from Earth to the Sun is 92,900,000 miles. Which expression represents this number in scientific notation? C Subtract $25 from the sum. A 92.9 × 10 6 C 9.29 × 10 6 D Add $4 to the sum. B 9.29 × 10 7 D 929 × 10 5 A Subtract $75 from the sum. 140 Chapter 2 Algebra: Rational Numbers 7 California Standards Practice at ca.gr7math.com More California Standards Practice For practice by standard, see pages CA1–CA39. 8 The table shows the atomic weights of certain elements. Element 11 Mr. Carr wants to buy a new computer. He will finance the total cost of $1,350 by making 24 equal monthly payments to pay back this amount plus interest. What other information is needed to determine the amount of Mr. Carr’s monthly payment? Atomic Weight (amu) Argon 39.948 Zinc 65.39 Lead 207.2 Oxygen 15.9994 Titanium 47.867 Mercury 200.59 A the brand of the computer B the amount of money Mr. Carr has in his savings account C the interest rate being charged Which element has an atomic weight that is exactly 160.642 less than the atomic weight of Mercury? 9 F argon H oxygen G titanium J zinc D the amount of Mr. Carr’s weekly income 12 Cindy has 55 minutes before she has to leave to go to school. She spends 15 minutes reading the newspaper. Then she spends 4 minutes brushing her teeth and another 15 minutes watching television. Which expression can you use to find the amount of time she has left before she has to leave? A pizzeria sells large pizzas for $11.50, medium pizzas for $8.75, and small pizzas for $6.50. Suppose a scout group orders 3 large pizzas, 2 medium pizzas, and 2 small pizzas. Which equation can be used to find the total cost of the pizzas? F 55 - 15 + 4 - 15 H 55 - 2(15) - 4 G 55 + 2(15) - 4 J 55 + (-2)(15) + 4 A t = (3 + 2 + 2)(11.50 + 8.75 + 6.50) B t = (3)(11.50) + 2(8.75) + 2(6.50) Pre-AP + 6.50 ) ( 11.50 + 8.75 3 C t = (3 + 2 + 2) ____ Record your answers on a sheet of paper. Show your work. D t = (3)(11.50) + 8.75 + 2(6.50) 13 The container for a child’s set of blocks is 9 10 What does y 3 equal when y = -4? F -64 inches by 9 inches by 9 inches. The blocks measure 3 inches by 3 inches by 3 inches. G -12 a. Describe how to determine the number 1 H _ of blocks needed to fill the container. 64 b. Write and simplify an expression to solve 1 J _ 12 the problem. c. How many blocks will it take? NEED EXTRA HELP? If You Missed Question... Go to Lesson... For Help with Standard... 1 2 3 4 5 6 7 8 9 10 11 12 13 2-3 2-10 1-1 2-9 2-6 2-6 2-10 2-7 1-7 2-9 1-1 1-7 2-9 MG1.3 NS1.1 MR1.1 NS2.1 NS2.2 NS2.2 NS1.1 NS1.2 AF1.1 AF2.1 MR1.1 AF1.1 AF2.1 Chapters 1–2 California Standards Practice 141 Real Numbers and the Pythagorean Theorem 3 • Standard 7MG3.0 Know the Pythagorean theorem and understand plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures. Key Vocabulary ordered pair (p. 173) Pythagorean Theorem (p. 162) real number (p. 155) square root (p. 144) Real-World Link Buildings The Transamerica Pyramid in San Francisco, California, is 853 feet high. To determine the approximate distance you can see from the top of the tower, multiply 1.23 by √ 853 . Real Numbers and the Pythagorean Theorem Make this Foldable to help you organize your notes. 1 Begin with two sheets of 8 ” × 11” notebook paper. _ 2 1 Fold one sheet in half from top to bottom. Cut along the fold from edges to margin. 2 Fold the other sheet in half. Cut along the fold between the margins. 3 Insert the first sheet through the second sheet and align the folds. 4 Label each page with a lesson number and title. 142 Chapter 3 Real Numbers and the Pythagorean Theorem Damir Frkovic/Masterfile >«ÌiÀÊÎ\ ,i>Ê ÕLiÀÃ >`ÊÌ i *ÞÌ >}Ài> / iÀi GET READY for Chapter 3 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Graph each point on a coordinate plane. (Prior Grade) 1. A(-1, 3) Example 1 Graph the points P(-3, 4), M(2, -1), R(4, 0), and W(-1, -3). y P 2. B(2, -4) 3. C(-2, -3) R 4. D(-4, 0) x O M The first number in an ordered pair tells you to move left or right from the origin. The second number tells you to move up or down. W Example 2 Evaluate each expression. (Lesson 2-9) 2 2 6. 3 + 3 7. 10 2 + 8 2 8. 7 2 + 5 2 5. 2 + 4 2 2 Find 6 2 + 4 2. 6 2 + 4 2 = 36 + 16 = 52 Evaluate 6 2 and 4 2. Simplify. 9. AGES Find the sum of the squares of Tina’s age and Warren’s age if Tina is 13 years old and Warren is 15 years old. (Lesson 2-9) Solve each equation. Check your solution. (Lesson 1-9) Example 3 10. x + 13 = 45 11. 56 + d = 71 12. 101 = 39 + a 13. 62 = 45 + m 49 + b = 72 -49 -49 _____________ b = 23 Solve 49 + b = 72. Write the equation. Subtract 49 from each side. 14. MARBLES Barry has 18 more marbles than Heidi. If Barry has 92 marbles, how many marbles does Heidi have? (Lesson 1-9) Chapter 3 Get Ready for Chapter 3 143 3-1 Square Roots Main IDEA Find square roots of perfect squares. Standard 7NS2.4 Use the inverse relationship between raising to a power and extracting the root of a perfect square; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why. Interactive Lab ca.gr7math.com Continue the pattern of square tiles until you reach 5 tiles on each side. 1. Copy and complete the following table. Tiles on a Side 1 2 Total Number of Tiles in the Square Arrangement 1 4 3 4 5 2. Suppose a square arrangement has 36 tiles. How many tiles are on a side? 3. What is the relationship between the number of tiles on a side and the number of tiles in the arrangement? NEW Vocabulary perfect square square root radical sign Numbers such as 1, 4, 9, 16, and 25 are called perfect squares because they are squares of integers. Squaring a number and finding a square root are inverse operations. A square root of a number is one of its two equal factors. The symbol √ , called a radical sign, is used to indicate a square root. Every positive number has both a positive and a negative square root. Find Square Roots Find each square root. 1 √ 64 √ 64 indicates the positive square root. Since 8 2 = 64, √ 64 = 8. 25 _ 2 - 36 25 25 indicates the negative square root of _ . -_ 36 36 25 5 2 _ 5 _ Since -_ = 25 , - = -_ . 36 36 6 6 ( ) 3 ± √ 1.21 1.21 indicates both the positive and negative square roots of 1.21. ± √ 1.21 = ±1.1, or 1.1 and -1.1. Since 1.1 2 = 1.21 and (-1.1) 2 = 1.21, ± √ READING in the Content Area For strategies in reading this lesson, visit ca.gr7math.com. 144 Find each square root. a. _ 9 16 b. - √ 49 Chapter 3 Real Numbers and the Pythagorean Theorem c. ± √ 0.81 Extra Examples at ca.gr7math.com By the definition of a square root, if n 2 = a, then n = ± √ a . You can use this relationship to solve equations that involve squares. Use Square Roots to Solve an Equation 4 ALGEBRA Solve t 2 = 169. Check your solution(s). t 2 = 169 Write the equation. t = ± √ 169 Definition of square root t = 13 and -13 Check 13 · 13 = 169 and (-13)(-13) = 169 ✓ The equation has two solutions, 13 and -13. Solve each equation. Check your solution(s). d. 289 = a 2 f. y 2 = _ 4 25 e. m 2 = 0.09 In most real-world situations, a negative square root does not make sense. Only the positive or principal square root is considered. 5 HISTORY The base of the Great Pyramid covers an area of about 562,500 square feet. Determine the length of each side of the base. Real-World Link The Great Pyramid of Khufu is the largest of the ancient pyramids. Source: infoplease.com Words Area is equal to the square of the length of a side. Variable Let s represent the length of a side. Equation s 2 = 562,500 s 2 = 562,500 Write the equation. s = ± √ 562,500 Definition of square root To find √ 562,500 , find two equal factors of 562,500. 562,500 = 2 · 2 · 3 · 3 · 5 · 5 · 5 · 5 · 5 · 5 Find the prime factors. = (2 · 3 · 5 · 5 · 5)(2 · 3 · 5 · 5 · 5) Regroup into two equal factors. So, s = 2 · 3 · 5 · 5 · 5 or 750. Since distance cannot be negative, the length of each side is 750 feet. g. CONCERTS A concert crew needs to set up 900 chairs on the floor level. If the chairs are placed in a square arrangement, how many should be in each row? Personal Tutor at ca.gr7math.com Lesson 3-1 Square Roots CORBIS 145 Examples 1–3 (p. 144) Find each square root. 1. √ 25 2. √ 0.64 3. - √ 1.69 4. -_ 5. ± √ 100 6. ±_ 16 81 Example 4 (p. 145) Example 5 For Exercises 11–14 15–18 19–22 23–30 31, 32 See Examples 1 2 3 4 5 144 ALGEBRA Solve each equation. Check your solution(s). 8. t 2 = _ 1 9 7. p 2 = 36 9. 6.25 = r 2 10. GAMES A checkerboard is a large square that is made up of 32 small red (p. 145) (/-%7/2+ (%,0 49 squares and 32 small black squares. How many small squares are along one side of a checkerboard? Find each square root. 11. √ 16 12. - √ 81 13. - √ 484 14. ± √ 36 15. _ 16. -_ 17. ±_ 18. -_ 19. - √ 2.56 20. ± √ 1.44 21. √ 0.25 22. ± √ 0.0196 121 324 64 16 9 225 25 49 ALGEBRA Solve each equation. Check your solution(s). 23. v 2 = 81 24. b 2 = 100 25. 144 = s 2 26. 225 = y 2 27. w 2 = _ 28. _ = c 2 29. 0.0169 = d 2 30. a 2 = 1.21 36 100 9 64 31. PHOTOGRAPHY A group of 169 students needs to be seated in a square formation for a yearbook photo. How many students should be in each row? 32. MARCHING BAND A marching band wants to form a square in the middle of the field. If there are 81 members in the band, how many should be in each row? ALGEBRA Solve each equation. Check your solution(s). 33. √ x=5 34. √y = 20 35. √z = 10.5 MEASUREMENT The formula for the perimeter of a square is P = 4s, where s is the length of a side. Find the perimeter of each square. %842!02!#4)#% 36. See pages 682, 710. 37. !REA SQUARE INCHES 38. !REA SQUARE FEET Self-Check Quiz at ca.gr7math.com 146 Chapter 3 Real Numbers and the Pythagorean Theorem !REA SQUARE METERS H.O.T. Problems 39. OPEN ENDED Create an equation that can be solved by finding the square root of a perfect square. 40. CHALLENGE Find each value. a. ( √ 36 ) ( ) 25 _ b. 2 2 c. 81 2 ( √ 1.99 ) 2 d. ( √x ) 41. NUMBER SENSE Under what condition is √x 25 ? > √ 42. */ -!4( Analyze the cartoon. Create a cartoon of your own (*/ 83 *5*/( that uses the square root of a perfect square. 43. The area of each square is 4 square 44. Mr Freeman’s farm has a square units. cornfield. Find the area of the cornfield if the sides are measured in whole numbers. F 164,000 ft 2 G 170,150 ft 2 Find the perimeter of the figure. H 170,586 ft 2 A 8 units C 20 units J B 16 units D 24 units 174,724 ft 2 45. SPACE The Alpha Centuari stars are about 2.5 × 10 13 miles from Earth. Write this distance in standard form. (Lesson 2-10) Write each expression using exponents. (Lesson 2-9) 46. 6 · 6 · 6 47. 2 · 3 · 3 · 2 · 2 · 2 48. s · t · t · s · s · t · s 49. What is the absolute value of -18? (Lesson 1-3) PREREQUISITE SKILL Between which two perfect squares does each number lie? (Lesson 2-2) 50. 57 51. 68 52. 33 53. 40 Lesson 3-1 Square Roots Bill Amend/Distributed by Universal Press Syndicate 147 3-2 Estimating Square Roots Main IDEA On dot paper, draw and cut out a square like the one at the right. The 1 area of section A is _(2 · 2) or 2 2 square units. So, the shaded square has an area of 8 square units. Estimate square roots. Standard 7NS2.4 Use the inverse relationship between raising to a power and extracting the root of a perfect square; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why. ! Draw a number line on your dot paper so that 1 unit equals the distance between dots. 1. Place your square on the number line. Between what two consecutive whole numbers is √ 8 , the side length of the square, located? 2. Between what two perfect squares is 8 located? 3. Estimate the length of a side of the square. Verify your estimate by using a calculator to compute the value of √8. In the Mini Lab, you found that √ 8 is not a whole number since 8 is not a perfect square. The number line shows that √ 8 is between 2 and 3. Since 8 is closer to 9 than 4, the best whole number estimate for √8 is 3. Ȗе Ȗе Ȗе Estimate Square Roots 1 Estimate √ 83 to the nearest whole number. √ 81 = 9 • The largest perfect square less than 83 is 81. • The smallest perfect square greater than 83 is 100. √ 100 = 10 READING Math Inequalities 81 < 83 < 100 is read 81 is less than 83 is less than 100 or 83 is between 81 and 100. Plot each square root on a number line. Then estimate √ 83 . 81 < 83 < 100 2 2 9 < 83 < 10 √ 9 2 < √ 83 < √ 10 2 9 < √ 83 < 10 е Ȗ е е Ȗ е е Ȗ ее Write an inequality. 81 = 9 2 and 100 = 10 2 Find the square root of each number. Simplify. So, √ 83 is between 9 and 10. Since √ 83 is closer to √ 81 than √ 100 , the best whole number estimate for √ 83 is 9. 148 Chapter 3 Real Numbers and the Pythagorean Theorem 2 Estimate √ 23.5 to the nearest whole number. Mental Math It is important to memorize common perfect squares. 12 = 1 22 = 4 2 4 = 16 2 5 = 25 6 2 = 36 7 2 = 49 8 2 = 64 9 2 = 81 10 2 = 100 3 =9 2 2 11 = 121 12 2 = 144 • The largest perfect square less than 23.5 is 16. √ 16 = 4 • The smallest perfect square greater than 23.5 is 25. √ 25 = 5 16 < 23.5 < 25 Write an inequality. 4 2 < 23.5 < 5 2 √ 4 2 < √ 23.5 < √ 52 16 = 4 2 and 25 = 5 2 4 < √ 23.5 < 5 Find the square root of each number. Simplify. So, √ 23.5 is between 4 and 5. Since 23.5 is closer to 25 than 16, the best whole number estimate for √ 23.5 is 5. Ȗе ее е Ȗ е е Ȗ е Estimate to the nearest whole number. b. √ 44.8 a. √ 35 c. √ 170 Personal Tutor at ca.gr7math.com 3 ART The Parthenon is an example of a golden rectangle. The length of the longer side divided by the length of the 2 units _ 1 + √ 5 shorter side is equal to . 2 Estimate this value. (1 ⫹ 兹5) units First estimate the value of √ 5. 4 < 5 <9 2 4 and 9 are the closest perfect squares. 2 2 < 5 <3 √ 2 2 < √5 < √ 32 2 < √5 < 3 4 = 2 2 and 9 = 3 2 Find the square root of each number. Simplify. Since 5 is closer to 4 than 9, the best whole number estimate for √5 is 2. Use this value to evaluate the expression. 1 + √ 5 1+2 _ ≈ _ or 1.5 2 2 d. BASEBALL In Little League, the bases are squares with sides of 14 inches. (s 2 + s 2) represents The expression √ IN IN the distance across a square of side length s. Estimate the distance across a base to the nearest inch. Extra Examples at ca.gr7math.com Charles O’Rear/CORBIS Lesson 3-2 Estimating Square Roots 149 Examples 1, 2 (pp. 148–149) Example 3 Estimate to the nearest whole number. 1. √ 28 2. √ 60 3. √ 135 4. √ 13.5 5. √ 38.7 6. √ 79.2 7. SCIENCE The number of swings back and forth of a pendulum of length L, 375 . About how many swings will a 40-inch in inches, each minute is _ (p. 149) √L pendulum make each minute? (/-%7/2+ (%,0 For Exercises 8–15 16, 17 See Examples 1, 2 3 Estimate to the nearest whole number. 8. √ 44 9. √ 23 10. √ 125 11. √ 197 12. √ 15.6 13. √ 23.5 14. √ 85.1 15. √ 38.4 16. GEOMETRY The radius of a circle with area A is approximately _ . A If a pizza has an area of 78 square inches, estimate its radius. 3 √h 4 17. CAVES The formula t = _ represents the time t in seconds that it takes an object to fall from a height of h feet. Suppose a rock falls from a 200-feet high cave ceiling. Estimate how long will it take to reach the ground. Estimate to the nearest whole number. 18. 5_ 1 5 19. 21_ 7 10 20. 17_ 3 4 Order from least to greatest. 21. 7, 9, √ 50 , √ 85 22. √ 91 , 7, 5, √ 38 23. √ 62 , 6, √ 34 , 8 ALGEBRA Estimate the solution of each equation to the nearest integer. 24. y 2 = 55 27. 25. d 2 = 95 FIND THE DATA Refer to the California Data File on pages 16–19. Choose some data and write a real-world problem in which you would estimate a square root. 28. GEOMETRY Egyptian mathematician Heron %842!02!#4)#% See pages 683, 710. Self-Check Quiz at ca.gr7math.com 150 26. p 2 = 6.8 created the formula A = √ s(s - a)(s - b)(s - c) to find the area A of a triangle. In this formula, a, b, and c are the measures of the sides, and s is one-half of the perimeter. Use this formula to estimate the area of the triangle at the right. CM CM CM 29. NUMBER SENSE Without a calculator, determine which is greater, √ 94 or 10. Explain your reasoning. Chapter 3 Real Numbers and the Pythagorean Theorem H.O.T. Problems 30. OPEN ENDED Find two numbers that have square roots between 7 and 8. One number should have a square root closer to 7, and the other number should have a square root closer to 8. Justify your answer. 31. FIND THE ERROR Josephina and Dario are estimating √ 50 . Who is correct? Explain your reasoning. √ 50 ≈ 7 √ 50 ≈ 25 Josephina Dario 32. CHALLENGE If x 3 = y, then x is the cube root of y. Explain how to estimate the cube root of 30. Find the cube root of 30 to the nearest whole number. 33. */ -!4( Explain how to graph √ (*/ 78 on a number line. 83 *5*/( 34. A whole number is squared. The result is between 950 and 1,000. The number is between A 26 and 28. C 30 and 32. B 28 and 30. D 32 and 34. 35. Point N on the number line best represents which square root? N 9 10 11 12 13 F √ 140 H √ 116 G √ 121 J √ 126 36. ALGEBRA Find a number that, when squared, equals 8,100. (Lesson 3-1) 37. GEOGRAPHY The Great Lakes cover about 94,000 square miles. Write this number in scientific notation. (Lesson 2-10) Multiply or divide. (Lesson 1-6) 38. (-5)(-13) 39. (-2)(5)(7) 40. 72 ÷ (-2) 41. -80 ÷ (-16) 42. PREREQUISITE SKILL To attend a field trip to an art museum, each student will have to pay $6.50 for transportation and $10.00 for admission and lunch. Find the total amount of money to be collected for a class of 240 students. (Lesson 1-1) Lesson 3-2 Estimating Square Roots (l)Janie Airey/Getty Images, (r)Michelle D. Bridwell/PhotoEdit 151 3-3 Problem-Solving Investigation MAIN IDEA: Use a Venn diagram to solve problems. STANDARD 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. STANDARD 7NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. e-Mail: USE A VENN DIAGRAM YOUR MISSION: Use a Venn diagram to solve the problem. THE PROBLEM: How many students are not involved in either music or sports? EXPLORE PLAN SOLVE ▲ Amy: Of the 12 students who ate lunch with me today, 9 are involved in music activities and 6 play sports. Of these students, 4 are involved in both music and sports. You know how many students are in involved in each activity and how many are involved in both activities. You want to organize the information. Make a Venn diagram to organize the information. Draw two overlapping circles to represent the two different activities. Since 4 students are involved in both activities, place a 4 in the -«ÀÌÃ ÕÃV section that is a part of both circles. Use subtraction to determine the number for each of the other sections. CHECK only music: 9 - 4 = 5 only sports: 6 - 4 = 2 neither music nor sports: 12 - 5 - 2 - 4 = 1 Check each circle to see if the appropriate number of students is represented. 1. Describe how to determine the number of students who are in either music or sports but not both using the above Venn diagram. */ -!4( Explain what each section of the Venn diagram above (*/ 83 *5*/( 2. represents and the number of students that belong to that category. 152 John Evans Chapter 3 Real Numbers and The Pythagorean Theorem 3. MASCOTS Nick conducted a survey of 85 students about a new school mascot. The results showed that 40 students liked Tigers, and 31 students liked Bears. Of those students, 12 liked both Tigers and Bears. How many students liked neither Tigers nor Bears? 4. MARKETING A survey showed that 70 customers bought white bread, 63 bought wheat bread, and 35 bought rye bread. Of those who bought exactly two types of bread, 12 bought wheat and white, 5 bought white and rye, and 7 bought wheat and rye. Two customers bought all three. How many customers bought only wheat bread? 5. HEALTH Dr. Bagentose is an allergist. Her patients had the following symptoms. How many patients had only watery eyes? Symptom(s) Number of Patients runny nose 22 watery eyes 20 sneezing 28 runny nose and watery eyes 8 runny nose and sneezing 15 watery eyes and sneezing 12 runny nose, watery eyes, and sneezing 5 7. MONEY The soccer team sponsored a car wash to pay for their new uniforms. They charged $3 for a car and $5 for an SUV. During the first two hours they washed 19 vehicles and earned $71. How many of each type of vehicle did they wash? 8. ALGEBRA Emilio created a graph of the data he collected for a science project. If the pattern continues, about how far will the marble roll if the end of the tube is raised to 1 an elevation of 3_ feet? 2 -ARBLE %XPERIMENT $ISTANCE -ARBLE 2OLLED FT Use a Venn diagram to solve Exercises 3–5. %LEVATION OF 4UBE FT 9. SPORTS Student Council surveyed a group of 24 students. The results showed that 14 students liked softball, and 18 liked basketball. Of these, 8 liked both. How many students liked just softball and how many liked just basketball? For Exercises 10 and 11, select the appropriate operation(s) to solve the problem. Justify your selection(s) and solve the problem. Use any strategy to solve Exercises 6–9. Some strategies are shown below. G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s rn. • Look for a patte ram. ag • Use a Venn di 10. JOBS Three after-school jobs are posted on the job board. The first job pays $5.15 per hour for 15 hours of work each week. The second job pays $10.95 per day for two hours of work, 5 days a week. The third job pays $82.50 for 15 hours of work each week. If you want to apply for the best-paying job, which job should you choose? Explain your reasoning. 11. ROLLER COASTERS The Silver Streak roller 6. ALGEBRA What are the next two numbers in the pattern? 864, 432, 216, 108, , coaster can accommodate 1,296 people in one hour. The coaster has 12 vehicles. If each vehicle carries 4 passengers, how many runs are made in one hour? Lesson 3-3 Problem-Solving Investigation: Use a Venn Diagram 153 The Language of Mathematics The language of mathematics is very specific. But many of the words you use in mathematics are also used in everyday language as well as scientific language. Sometimes the everyday or scientific usage can give you clues to the mathematical meaning. Here are some examples. Usage Example Some words are used in English and in mathematics, but have distinct meanings. i} Some words are used in science and in mathematics, but the meanings are different. x + 4 = -2 x = -6 solution Þ«ÌiÕÃi Some words are used only in mathematics. Explain how the mathematical meaning of each word compares to its everyday meaning. 1. factor 2. leg 3. rational 4. root Explain how the mathematical meaning of each word compares to its meaning in science. 5. radical 6. variable Some words are used in English and in mathematics, but the mathematical meaning is more precise. Explain how the mathematical meaning of each word is more precise than the everday meaning. 7. similar 154 8. real Chapter 3 Real Numbers and the Pythagorean Theorem Standard 7AF1.4 Use algebraic terminology (e.g. variable, equation, term, coefficient, inequality, expression, constant) correctly. 3- 4 The Real Number System Main IDEA Identify and classify numbers in the real number system. Standard 7NS1.4 Differentiate between rational and irrational numbers. NEW Vocabulary irrational number real number SPORTS Most sports have rules for the size of the field or court where the sport is played. A diagram of a volleyball court is shown. 2EAR 3PIKERS ,INES FT FT FT FT IN IN FT A NG I ERV !RE 3 q FT 1. The length of the court is 60 feet. Is 60 a rational number? Explain. 2. The distance from the net to the rear spikers line is 7_ feet. Is 7_ 1 2 1 2 a rational number? Explain. 3. The diagonal across the court is √ 4,500 feet. Can this square root be written as a rational number? Explain. REVIEW Vocabulary rational number any number that can be expressed a in the form , where a and b b are integers and b ≠ 0 (Lesson 2-1) _ It is known that √ 4,500 has a decimal value of 67.08203932... . Although this continues on and on, it does not repeat. Since the decimal does not terminate or repeat, √ 4,500 cannot be written as a fraction. Therefore, it is not a rational number. Numbers that are not rational are called irrational numbers. The square root of any number that is not a perfect square number is irrational. +%9 #/.#%04 Words Irrational Numbers An irrational number is a number that cannot be expressed as the quotient _a , where a and b are integers and b ≠ 0. b 2 ≈ 1.414213562… Examples √ - √ 3 ≈ -1.732050807… The set of rational numbers and the set of irrational numbers together make up the set of real numbers. Study the diagram below. 2EAL .UMBERS 2ATIONAL .UMBERS )NTEGERS )RRATIONAL .UMBERS 7HOLE .UMBERS Lesson 3-4 The Real Number System 155 Classify Numbers Classifying Numbers Always simplify numbers before classifying them. Name all sets of numbers to which each real number belongs. 1 0.252525… The decimal ends in a repeating pattern. It is a rational 25 . number because it is equivalent to _ 99 2 √ 36 Since √ 36 = 6, it is a whole number, an integer, and a rational number. 3 - √ 7 - √7 ≈ -2.645751311… Since the decimal does not terminate or repeat, it is an irrational number. b. -2_ 2 5 a. √ 10 c. √ 100 Real numbers follow the properties that are true for whole numbers, integers, and rational numbers. #/.#%04 3UMMARY Real Number Properties Property Arithmetic Algebra Commutative 3.2 + 2.5 = 2.5 + 3.2 5.1 · 2.8 = 2.8 · 5.1 a+b=b+a a·b=b·a Associative (2 + 1) + 5 = 2 + (1 + 5) (3 · 4) · 6 = 3 · (4 · 6) (a + b) + c = a + (b + c) (a · b) · c = a · (b · c) Distributive 2(3 + 5) = 2 · 3 + 2 · 5 a(b + c) = a · b + a · c Identity √ 8 + 0 = √ 8 √ 7 · 1 = √ 7 a+0=a a·1=a Additive Inverse 4 + (-4) = 0 a + (-a) = 0 Multiplicative Inverse _2 · _3 = 1 _a · _b = 1, where a, b ≠ 0 3 2 b a Graph Real Numbers 4 Estimate √ 6 and - √3 to the nearest tenth. Then graph √ 6 and - √ 3 on a number line. √ 6 ≈ 2.449489742… or about 2.4 Use a calculator. - √ 3 ≈ - 1.7320508075… or about -1.7 Use a calculator. – 3 Real Numbers The graph of all real numbers is the entire number line without any “holes.” –3 –1 0 1 2 3 Estimate each square root to the nearest tenth. Then graph the square root on a number line. d. √ 5 156 –2 6 e. - √7 Chapter 3 Real Numbers and the Pythagorean Theorem f. √ 22 Compare Real Numbers Replace each ● with <, >, or = to make a true sentence. Mental Math Remember that a negative number is always less than a positive number. Therefore, you can determine that - √3 is less than 1.7 without computation. _ 5 √ 7 ● 22 3 Write each number as a decimal. √ 7 ≈ 2.645751311… 2 2_ = 2.666666666… 3 2 Since 2.645751311…is less than 2.66666666…, √7 < 2_ . 3 − 6 1.5 ● √ 2.25 2.25 Write √ 2.25 as a decimal. 1.5 1.5 √ 2.25 = 1.5 − 1.5 = 1.555555555… − Since 1.555555555… is greater than 1.5, 1.5 > √ 2.25 . g. √ 11 ● 3_ 1 3 h. √ 17 ● 4.03 7 SKYSCRAPERS On a clear Real-World Career How Does an Architect Use Math? Architects design and draw plans for buildings. They use math to ensure the buildings are structurally sound. 1.6 1 i. √ 6.25 ● 2_ 2 Los Angeles Skyscrapers day, the number of miles 1200 a person can see to the 1000 750 horizon is about 1.23 times 699 800 the square root of his or her distance from the 600 ground, in feet. Suppose 400 Domingo is at the top of 200 Bank of America Tower and Jewel is at the top of 0 Bank of Two Two California Plaza. America California How much farther can Tower Plaza Jewel see than Domingo? Source: National Park Service 1,018 858 Aon Center US Bank Tower Use a calculator to approximate the distance each person can see. For more information, go to ca.gr7math.com. Domingo: 1.23 √ 699 ≈ 32.52 Jewel: 1.23 √ 750 ≈ 33.68 Jewel can see about 33.68 - 32.52 or 1.16 miles farther than Domingo. j. MEASUREMENT How much greater is the perimeter of a square with area 250 square meters than a square with area 125 square meters? Personal Tutor at ca.gr7math.com Extra Examples at ca.gr7math.com Artiga Photo/CORBIS Lesson 3-4 The Real Number System 157 Examples 1–3 (p. 156) Example 4 (p. 156) Name all sets of numbers to which each real number belongs. 1. 0.050505… (p. 157) Example 7 (p. 157) 1 4 Estimate each square root to the nearest tenth. Then graph the square root on a number line. 6. - √ 18 5. √ 2 Examples 5, 6 4. -3_ 3. √ 17 2. - √ 64 Replace each ● with <, >, or = to make a true sentence. 1 8. √ 2.25 ● 1_ 7. √ 15 ● 3.5 2 −− 9. 2.21 ● √ 5.2 10. MEASUREMENT The area of a triangle with all three sides s 2 √3 _ the same length is 6 in. 6 in. , where s is the length of a side. 4 Find the area of the triangle. 6 in. (/-%7/2+ (%,0 Name all sets of numbers to which each real number belongs. For Exercises 11–18 19–22 23–28 29–30 11. 14 12. _ 15. 4.83 16. 7.2 See Examples 1–3 4 5,6 7 2 3 − 13. - √ 16 14. - √ 20 17. - √ 90 18. _ 12 4 Estimate each square root to the nearest tenth. Then graph the square root on a number line. 19. √ 6 21. - √ 22 20. √8 22. - √ 27 Replace each ● with <, >, or = to make a true sentence. 23. √ 10 ● 3.2 24. √ 12 ● 3.5 26. 2_ ● √ 5.76 27. 5_ ● 5.16 2 5 1 6 − 25. 6_ ● √ 40 1 3 − 28. √ 6.2 ● 2.4 29. LAW ENFORCEMENT Police can use the formula s = 5.5 √ 0.75d to estimate the speed of a vehicle, where s is the speed of the vehicle in miles per hour, and d is the length of the skid marks in feet. How fast was the vehicle going for the skid marks at the right? 125 ft 30. FOOTBALL The time t in seconds that a football remains in the air is t= 2y _ , where y is the initial height in meters of the football. Find the 9.8 time to the nearest hundredth of a second that a football remains in the air if the initial height is 2 meters. 158 Chapter 3 Real Numbers and the Pythagorean Theorem %842!02!#4)#% 31. ALGEBRA In the sequence 4, 12, , 108, 324, the missing number can ab where a and b are the numbers on be found by simplifying √ See pages 683, 710. either side of the missing number. Find the missing number. Self-Check Quiz at ca.gr7math.com H.O.T. Problems Order each set of numbers from least to greatest. − −− − 32. √ 5 , √ 3 , 2.25, 2.2 33. 3.01, 3.1, 3.01, √ 9 − , √ 34. -4.1, √ 17 , -4.1, 4.01 35. - √5 6 , -2.5, 2.5 36. OPEN ENDED Give a counterexample for the statement all square roots are irrational numbers. Explain your reasoning. CHALLENGE Tell whether the following statement is always, sometimes, or never true. If a statement is not always true, explain. 37. Integers are rational numbers. 38. Rational numbers are integers. 39. The product of a rational number and an irrational number is an irrational number. 40. */ -!4( Write a real-world problem in which you (*/ 83 *5*/( would need to approximate a square root. Then, solve the problem. 41. Which is an irrational number? 42. Which number represents the point graphed on the number line? A -6 2 B _ 3 C √ 9 D √ 3 F - √ 12 H - √ 15 G - √ 10 J - √8 43. SPORTS Students were surveyed about the sports in which they participate. Thirty-five play baseball, 31 play basketball, and 28 play soccer. Of these, 7 play baseball and basketball, 9 play basketball and soccer, 6 play baseball and soccer, and 5 play all three sports. How many students were surveyed? Use a Venn diagram. (Lesson 3-3) 44. Order 7, √ 53 , √ 32 , and 6 from least to greatest. (Lesson 3-2) ALGEBRA Solve each equation. (Lesson 3-1) 45. t 2 = 25 46. y 2 = _ 1 49 47. 0.64 = a 2 PREREQUISITE SKILL Evaluate each expression. (Lesson 2-9) 48. 3 2 + 5 2 49. 6 2 + 4 2 50. 9 2 + 11 2 51. 4 2 + 7 2 Lesson 3-4 The Real Number System 159 CH APTER 3 Mid-Chapter Quiz Lessons 3-1 through 3-4 Find each square root. (Lesson 3-1) 18. 1. √ 1 2. ± √ 81 3. ± √ 36 4. - √ 121 5. -_ 6. √ 0.09 1 25 STANDARDS PRACTICE Point P on the number line best represents which square root? (Lesson 3-2) P 7 7. MEASUREMENT What is the length of a side of the square? (Lesson 3-1) 8 9 10 11 F √ 85 G √ 81 !REA M H √ 98 J √ 79 19. MARKETING A survey showed 83 customers 8. STANDARDS PRACTICE The area of a square picture frame is 529 square centimeters. How long is each side of the frame? (Lesson 3-1) bought wheat cereal, 83 bought rice cereal, and 20 bought corn cereal. Of those who bought exactly two boxes of cereal, 6 bought corn and wheat, 10 bought rice and corn, and 12 bought rice and wheat. Four customers bought all three. How many customers bought only rice cereal? (Lesson 3-3) A 26 cm B 25 cm C 23 cm D 21 cm 20. FOOD Napoli’s pizza conducted a survey of 75 customers. The results showed that 35 customers liked mushroom pizza, 41 liked pepperoni pizza, and 11 liked both mushroom and pepperoni pizza. How many liked neither mushroom nor pepperoni pizza? Use a Venn diagram. (Lesson 3-3) 9. FOOTBALL A group of 121 football players needs to be in a square formation for practice. How many players should be in each row? (Lesson 3-1) Estimate to the nearest whole number. (Lesson 3-2) 10. √ 90 11. √ 28 12. √ 226 Name all sets of numbers to which each real number belongs. (Lesson 3-4) 13. √ 17 14. √ 21 15. √ 75 21. _ 2 3 22. √ 25 23. - √ 15 24. √3 25. 10 26. - √4 16. ALGEBRA Estimate the solution(s) of x 2 = 50 to the nearest integer. (Lesson 3-2) 17. MEASUREMENT The radius of a circle with A _ . If a pie has an area A is approximately 3 area of 42 square inches, estimate its radius. (Lesson 3-2) 160 Replace each ● with <, >, or = to make a true sentence. (Lesson 3-4) 27. √ 15 ● 4.1 28. 6.5 ● √ 45 29. √ 35 ● 5.75 30. 3.3 ● √ 10 Chapter 3 Real Numbers and the Pythagorean Theorem − Explore 3-5 Main IDEA Find the relationship among the sides of a right triangle. Standard 7MG3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. Standard 7MR2.4 Make and test conjectures by using both inductive and deductive reasoning. Geometry Lab The Pythagorean Theorem Four thousand years ago, the ancient Egyptians used mathematics to lay out their fields with square corners. They took a piece of rope and knotted it into 12 equal spaces. Taking three stakes, they stretched the rope around the stakes to form a right triangle. The sides of the triangle had lengths of 3, 4, and 5 units. 5 4 3 BrainPOP® ca.gr7math.com On centimeter grid paper, draw a triangle as shown at the right. Cut out the triangle. Measure the length of the longest side in centimeters. In this case, it is 5 centimeters. Cut out three squares: one with 3 centimeters on a side, one with 4 centimeters on a side, and one with 5 centimeters on a side. REVIEW Vocabulary right triangle a triangle with one angle that measures 90° Place the edges of the squares against the corresponding sides of the right triangle. Find the area of each square. ANALYZE THE RESULTS 1. What relationship exists among the areas of the three squares? Repeat the activity for each right triangle whose two shorter sides have the following measures. Write an equation to show your findings. Use a ruler to verify your measures. 2. 6 cm, 8 cm 3. 5 cm, 12 cm 4. Write a sentence or two summarizing your findings. 5. MAKE A CONJECTURE Determine the length of the third side of a right triangle if the legs of the triangle are 9 inches and 12 inches long. Explore 3-5 Geometry Lab: The Pythagorean Theorem 161 3-5 The Pythagorean Theorem Main IDEA Use the Pythagorean Theorem. Standard 7MG3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. Standard 7MR3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. NEW Vocabulary legs hypotenuse Pythagorean Theorem converse REVIEW Vocabulary right angle an angle with a measure of 90° SPORTS When viewed from the side, the shape of some wooden skateboarding ramps is a right triangle. The dimensions of four possible ramps of this type are given in the table. Copy this table. Ramp height, H base, B (ft) (ft) Design Draw a side-view model of each ramp on grid paper, letting the width of one grid equal 1 foot. 3 6 5 7 A B C D 4 8 12 24 Cut each ramp out and use your grid paper to find the length of the ramp, which is the longest side of your model. Write these measures in a new column labeled length, L (ft). Finally, add a column labeled H 2 + B 2. Calculate each of these values and place them in your table. 1. What is the relationship between the values in the H 2 + B 2 column and the values in the L column? 2. How could you use a value in the H 2 + B 2 column to find a corresponding value in the L column? A right triangle is a triangle with one right angle. 4HE SIDES THAT FORM THE RIGHT ANGLE ARE CALLED LEGS 4HE HYPOTENUSE IS THE SIDE OPPOSITE THE RIGHT ANGLE )T IS THE LONGEST SIDE OF THE TRIANGLE 4HE SYMBOL INDICATES A RIGHT ANGLE The Pythagorean Theorem describes the relationship between the lengths of the legs and the hypotenuse for any right triangle. +%9 #/.#%04 Words Symbols 162 In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. c2 = a2 + b2 Chapter 3 Real Numbers and the Pythagorean Theorem Pythagorean Theorem Model c a b You can use the Pythagorean Theorem to find the length of a side of a right triangle when you know the other two sides. Find the Length of a Side Write an equation you could use to find the length of the missing side of each right triangle. Then find the missing length. Round to the nearest tenth if necessary. 1 c in. 12 in. 9 in. c2 = a2 + b2 2 2 c = 9 + 12 Pythagorean Theorem 2 Replace a with 9 and b with 12. 2 c = 81 + 144 Evaluate 9 2 and 12 2. c 2 = 225 Add 81 and 144. c = ± √ 225 Definition of square root c = 15 or -15 Simplify. The equation has two solutions, 15 and -15. However, the length of a side must be positive. So, the hypotenuse is 15 inches long. b 2 8m 24 m a2 + b2 = c2 8 2 Pythagorean Theorem + b 2 = 24 2 Replace a with 8 and c with 24. Evaluate 8 2 and 24 2. 64 + b 2 = 576 64 - 64 + b 2 = 576 - 64 Check for Reasonableness The hypotenuse is always the longest side in a right triangle. Since 22.6 is less than 24, the answer is reasonable. Subtract 64 from each side. b 2 = 512 Simplify. b = ± √ 512 Definition of square root b ≈ 22.6 or -22.6 Use a calculator. The length of side b is about 22.6 meters. a. b. c. 3 mi 17 cm c yd 24 yd a cm b mi 8 mi 20 cm 18 yd Personal Tutor at ca.gr7math.com Lesson 3-5 The Pythagorean Theorem 163 If you reverse the parts of the Pythagorean Theorem, you have formed its converse. The converse of the Pythagorean Theorem is also true. +%9 #/.#%04 Converse of Pythagorean Theorem If the sides of a triangle have lengths a, b, and c units such that c 2 = a 2 + b 2, then the triangle is a right triangle. Identify a Right Triangle 3 The measures of three sides of a triangle are 15 inches, 8 inches, and 17 inches. Determine whether the triangle is a right triangle. Draw a Picture When solving a problem, it is often helpful to draw a picture to represent the situation. c2 = a2 + b2 Pythagorean Theorem 17 2 15 2 + 8 2 c = 17, a = 15, b = 8 289 225 + 64 Evaluate 17 2, 15 2, and 8 2. 289 = 289 Simplify. The triangle is a right triangle. Determine whether each triangle with sides of given lengths is a right triangle. Justify your answer. d. 36 mi, 48 mi, 60 mi e. 4 ft, 7 ft, 5 ft Write an equation you could use to find the length of the missing side of each right triangle. Then find the missing length. Round to the nearest tenth if necessary. Example 1 16 m 1. 2. c mm (p. 163) cm 100 mm 12 m 200 mm Example 2 (p. 163) 3. 25 ft 4. 8 yd 7 ft a yd b ft 12 yd Example 1 5. The hypotenuse of a right triangle is 12 inches, and one of its legs is (p. 163) 7 inches. Find the length of the other leg. Round to the nearest tenth if necessary. Example 3 Determine whether each triangle with sides of given lengths is a right triangle. Justify your answer. (p. 164) 6. 5 in., 10 in., 12 in. 164 Chapter 3 Real Numbers and the Pythagorean Theorem 7. 9 m, 40 m, 41 m Extra Examples at ca.gr7math.com (/-%7/2+ (%,0 For Exercises 8, 9 10–13 14–19 See Examples 1 2 3 Write an equation you could use to find the length of the missing side of each right triangle. Then find the missing length. Round to the nearest tenth if necessary. 8. 9. c in. 5 in. c ft 27 ft 10 cm 10. 12 in. a cm 15 cm 36 ft 11. 30 mm 12. 51 yd bm 13. a yd 60 yd 80 mm b mm 8m 18 m Determine whether each triangle with sides of given lengths is a right triangle. Justify your answer. 14. 28 yd, 195 yd, 197 yd 15. 30 cm, 122 cm, 125 cm 16. 24 m, 143 m, 145 m 17. 135 in., 140 in., 175 in. 18. 56 ft, 65 ft, 16 ft 19. 44 cm, 70 cm, 55 cm 20. KITES Paulo is flying a kite as 21. GEOGRAPHY Calculate the length shown below. Find the length of the kite string. of the diagonal of the state of Wyoming. MI c ft 40 ft 79/-).' MI 30 ft Write an equation you could use to find the length of the missing side of each right triangle. Then find the missing length. Round to the nearest tenth if necessary. 22. b, 99 mm; c, 101 mm 23. a, 48 yd; b, 55 yd 24. a, 17 ft; c, 20 ft 25. a, 23 in.; b, 18 in. 26. b, 4.5 m; c, 9.4 m 27. b, 5.1 m; c, 12.3 m 50 %842!02!#4)#% See pages 684, 710. Self-Check Quiz at ca.gr7math.com 28. TRAVEL The Research Triangle in North Carolina is formed by Raleigh, Durham, and Chapel Hill. Is this triangle a right triangle? Explain. Durham 12 mi 98 147 Chapel Hill 29 mi 761 55 24 mi Raleigh 401 40 54 1 NORT H CA ROL I NA 70 Lesson 3-5 The Pythagorean Theorem 165 H.O.T. Problems 29. OPEN ENDED State three measures that could be the side measures of a right triangle. Justify your answer. 30. FIND THE ERROR Catalina and Morgan are finding the length of the third side of the right triangle. Who is correct? Explain your reasoning. IN IN 82 = a2 + 52 a2 = 52 + 82 Catalina Morgan 31. CHALLENGE The whole numbers 3, 4, and 5 are called Pythagorean triples because they satisfy the Pythagorean Theorem. Find three other sets of Pythagorean triples. 32. */ -!4( Explain why you can use any two sides of a right (*/ 83 *5*/( triangle to find the third side. 33. What is the perimeter of the triangle ABC? 34. The base of a ten-foot ladder stands six feet from a house. A 10 in. C FT 24 in. B FT A 26 in. C 60 in. B 34 in. D 68 in. How many feet up the side of the house does the ladder reach? F 4.0 ft H 8.0 ft G 5.8 ft J 11.7 ft Replace each ● with <, >, or = to make each a true sentence. (Lesson 3-4) − − 17 35. √ 12 ● 3.5 36. √ 41 ● 6.4 37. 5.6 ● _ 38. √ 55 ● 7.4 3 39. ALGEBRA Estimate the solution of x 2 = 77 to the nearest integer. (Lesson 3-2) PREREQUISITE SKILL Solve each equation. Check your solution. (Lesson 1-9) 40. 57 = x + 24 41. 82 = 54 + y 42. 71 = 35 + z 43. 64 = a + 27 166 Chapter 3 Real Numbers and the Pythagorean Theorem (l)First Light/ImageState, (r)age fotostock/SuperStock 3-6 Using The Pythagorean Theorem Main IDEA Solve problems using the Pythagorean Theorem. Standard 7MG3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. PARASAILING In parasailing, a towrope is used to attach a parasailor to a boat. 1. What type of triangle is formed towrope (t ) vertical height (h) by the horizontal distance, the vertical height, and the length of the towrope? horizontal distance (d ) 2. Write an equation that can be used to find the length of the towrope. The Pythagorean Theorem can be used to solve a variety of problems. 1 PARASAILING Find the height of the parasailor above the surface of the water. Notice that the vertical and horizontal distances, along with the length of the rope form a right triangle. Use the Pythagorean Theorem. c2 = a2 + b2 2 2 200 = a + 135 40,000 - 18,225 = a 2 + 18,225 - 18,225 Square Roots In most real-world situations, only the positive square root is considered. a 135 ft Pythagorean Theorem 2 40,000 = a 2 + 18,225 21,775 = a 200 ft 2 Replace c with 200 and b with 135. Evaluate 200 2 and 135 2. Subtract 18,225 from each side. Simplify. ± √ 21,775 = a Definition of square root 148 or -148 ≈ a Simplify. The parasailor is about 148 feet above the surface of the water. a. AVIATION Write an equation that can be used to find the distance between the planes. Then solve. Round to the nearest tenth. 7 mi d 10 mi Extra Examples at ca.gr7math.com Profimedia.CZ s.r.o./Alamy Images Lesson 3-6 Using the Pythagorean Theorem 167 2 A circular lawn sprinkler with a range View from Above x x of 25 feet is placed 20 feet from the edge of a lawn. Find the length of the section of the lawn’s edge that is within the range of the sprinkler. 20 feet A 15 ft 25 feet Sprinkler B 20 ft C 25 ft D 30 ft Read the Item From the diagram, you know that the distance of the sprinkler from the lawn’s edge, the sprinkler’s range, and a section of the lawn’s edge all form a right triangle. The section of the lawn’s edge within the range of the sprinkler is twice the section forming the right triangle. Solve the Item Use the Pythagorean Theorem. Pythagorean Triples Look for measures that are multiples of a 3-4-5 right triangle. 25 = 5 · 5 20 = 4 · 5 x = 3 · 5 or 15 a2 + b2 = c2 2 2 20 + x = 25 Pythagorean Theorem 2 a = 20, b = x, and c = 25. 400 + x 2 = 625 Evaluate 20 2 and 25 2. 400 - 400 + x 2 = 625 - 400 Subtract 400 from each side. x 2 = 225 Simplify. x = ± √ 225 Definition of square root x = 15 or -15 Simplify. The length of the section of the lawn’s edge within the sprinkler’s range is x + x or 15 + 15 = 30 feet. Therefore, choice D is correct. b. If the “rise” of the stairs of a building is 5 feet and the “run” is 12 feet, how long is it from point A to point B? B 5 ft A F 13 ft G 12 ft Personal Tutor at ca.gr7math.com 168 Chapter 3 Real Numbers and the Pythagorean Theorem 12 ft H 11 ft J 10 ft Example 1 (p. 167) Write an equation that can be used to answer the question. Then solve. Round to the nearest tenth if necessary. 1. How long is each rafter? r 2. How high does the ladder reach? r 9 ft 12 ft 12 ft 15 ft h 3 ft 3. GEOMETRY An isosceles right triangle is a right triangle in which both legs are equal in length. If one leg of an isosceles triangle is 4 inches long, what is the length of the hypotenuse? IN Example 2 (p. 168) (/-%7/2+ (%,0 For Exercises 5–10 21, 22 See Examples 1 2 4. STANDARDS PRACTICE Abigail designed a stained glass window in the shape of a kite. What is the perimeter of the window? A 108 in. C 162 in. B 114 in. D 168 in. IN IN IN Write an equation that can be used to answer the question. Then solve. Round to the nearest tenth if necessary. 5. How long is the kite string? 6. How far is the 7. How high is the ski helicopter from the car? ramp? 15 ft s 95 yd 14 ft d 150 yd h 40 yd 60 yd 8. How long is 9. How high is the wire the lake? attached to the pole? 10. How high is the wheel chair ramp? ᐉ h 18 mi 10 ft 9.5 ft 24 mi 13 m h 3.5 m Lesson 3-6 Using the Pythagorean Theorem 169 11. VOLLEYBALL Two ropes and two stakes are needed to support each pole holding the volleyball net. Find the length of each rope. 8 ft 3.5 ft 12. GEOGRAPHY Suppose Greenville, Rock Hill, and Columbia form a right triangle. What is the distance from Columbia to Greenville? 85 80 mi Greenville Rock Hill 77 26 385 68 mi South Carolina Columbia 13. ENTERTAINMENT Connor loves to watch movies in the widescreen format Real-World Link Televisions are advertised by their diagonal measure. The most common sizes are 27–32 inches. on his television. He wants to buy a new television with a screen that is at least 25 inches by 13.6 inches. What diagonal size television meets Connor’s requirements? 14. CONSTRUCTION Home builders add corner bracing to give strength to a house frame. How long will the brace need to be for the frame shown? 1 Each board is 1 2 in. wide. 16 in. 8 ft 16 in. A 15. GEOMETRY Find the length of the −− diagonal AB in the rectangular prism at the right. (Hint: First find −− the length of BC.) %842!02!#4)#% 8 cm C 5 cm 12 cm See pages 684, 710. 16. Self-Check Quiz at ca.gr7math.com H.O.T. Problems 16 in. B FIND THE DATA Refer to the California Data File on pages 16−19. Choose some data and write a real-world problem in which you would use the Pythagorean Theorem. 17. OPEN ENDED Write a problem that can be solved by using the Pythagorean Theorem. Then explain how to solve the problem. 18. Which One Doesn’t Belong? Each set of numbers represents the side measures of a triangle. Identify the set that does not belong with the other three. Explain your reasoning. 5–12–13 170 10–24–26 Chapter 3 Real Numbers and the Pythagorean Theorem worldthroughthelens/Alamy Images 5–7–9 8–15–17 19. CHALLENGE Suppose a ladder 100 feet long is a placed against a vertical wall 100 feet high. How far would the top of the ladder move down the wall by pulling out the bottom of the ladder 10 feet? Explain your reasoning. 100 ft 100 ft */ -!4( The length of the (*/ 83 *5*/( 20. 10 ft hypotenuse of an isosceles right triangle is √ 288 units. Explain how to find the length of a leg. 21. Ms. Johnson designed a rectangular 22. A hot air balloon is tethered to the garden. She plans to build a walkway through the garden as shown. ground as shown. M FT M Which measure is closest to the length of the walkway? FT How high above the ground is the balloon? A 8m B 11 m C 17 m D 23 m F 55.0 ft H 123.0 ft G 95.3 ft J 163.5 ft 23. GEOMETRY Determine whether a triangle with sides 20 inches, 48 inches, and 52 inches long is a right triangle. Justify your anwer. (Lesson 3-5) − 24. Order √ 45 , 6.6, 6.75, and 6.7 from least to greatest. (Lesson 3-4) Add or subtract. Write in simplest form. (Lesson 2-6) ( 34 ) 25. -3_ + -5_ 2 3 26. -1_ - 7_ 1 8 3 4 27. _ - 4_ 3 5 1 2 ( 56 ) 28. 4_ + -6_ 7 8 29. ARCHAEOLOGY Stone tools found in Ethiopia are estimated to be 2.5 million years old. That is about 700,000 years older than similar tools found in Tanzania. Write and solve an addition equation to find the age of the tools found in Tanzania. (Lesson 1-9) PREREQUISITE SKILL Graph each point on the same coordinate plane. (Page 688) 30. T(5, 2) 31. A(-1, 3) 32. M(-5, 0) 33. D(-2, -4) Lesson 3-6 Using the Pythagorean Theorem 171 Extend 3-6 Main IDEA Graph irrational numbers. Standard 7MG3.2 Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections. Standard 7MR2.5 Use a variety of methods such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Geometry Lab Graphing Irrational Numbers In Lesson 3-2, you found approximate locations for irrational numbers on a number line. You can also accurately graph irrational numbers. Graph √ 34 on a number line as accurately as possible. Find two numbers with squares that have a sum of 34. 34 = 25 + 9 The hypotenuse of a right triangle with legs that 34 = 5 2 + 3 2 measure 5 and 3 units will measure √ 34 units. Draw a number line on grid paper. Then draw a right triangle whose legs measure 5 and 3 units. 3 units 5 units 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Adjust your compass to the length of the hypotenuse. Place the compass at 0 and draw 34 an arc that 0 1 2 3 4 5 6 7 intersects the number line. The point of intersection corresponds to the number √ 34 . Graph each irrational number. a. √ 10 b. √ 13 c. √ 17 d. √ 8 ANALYZE THE RESULTS 1. Explain how you decide what lengths to make the legs of the right triangle when graphing an irrational number. 2. Explain how the graph of √ 2 can be used to graph √ 3. 3. MAKE A CONJECTURE Do you think you could graph the square root of any whole number? Explain your reasoning. 172 Chapter 3 Real Numbers and the Pythagorean Theorem 3-7 Geometry: Distance on the Coordinate Plane Main IDEA Graph rational numbers on the coordinate plane. Find the distance between two points on the coordinate plane. Standard 7MG3.2 Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections. ARCHAEOLOGY An archaeologist sets up a grid with string to keep careful record of the location of objects she finds at a 3 ( 2) Necklace 1 2 , 2_ ( ) 2 2 1 _ dig. She finds a ring at _ , 1 1 and a 1 necklace at 2, 2_ . y (2 2) 1 (_12 , 1_12 ) Ring x 1. What does each colored line on the 1 0 graph represent? 2 3 2. What type of triangle is formed by the lines? 3. What are the lengths of the two red lines? Recall that you can locate a point by using a coordinate system similar to the grid used by the archaeologist. It is called a coordinate plane. NEW Vocabulary coordinate plane origin y-axis x-axis quadrants ordered pair x-coordinate abscissa y-coordinate ordinate y The point of intersection of the two number lines is the origin, (0, 0). Quadrant II The vertical number line is the y-axis. Quadrant I O x Quadrant III Quadrant IV The number lines separate the coordinate plane into four sections called quadrants. (⫺2, ⫺4) The horizontal number line is the x-axis. Any point on the coordinate plane can be graphed by using an ordered pair of numbers. The first number in the ordered pair is the x-coordinate or abscissa. The second number is the y-coordinate or ordinate. Name an Ordered Pair 1 Name the ordered pair for point P. y • Start at the origin. 2 • Move right to find the x-coordinate 1 1 of point P, which is 3_ . 2 ⫺1 • Move up to find the y-coordinate, which is 2. 1 So, the ordered pair for point P is 3_ ,2 . (2 ) O P 1 2 3 x ⫺1 ⫺2 Lesson 3-7 Geometry: Distance on the Coordinate Plane 173 2 Name the ordered pair for point Q. y • Start at the origin. 2 • Move left to find the x-coordinate 1 1 of point Q, which is -4_ . 2 • Move down to find the y-coordinate, ⫺5 ⫺4 ⫺3 ⫺2 ⫺1 Q 1 which is -1_ . O x ⫺1 2 ⫺2 So, the ordered pair for point Q 1 1 , -1_ . is -4_ ( 2 2 ) Name the ordered pair for each point. y J a. J 2 b. K 1 c. L ⫺2 ⫺1 K O d. M 1 L x ⫺1 ⫺2 M Graphing Ordered Pairs y Graph and label each point. A(0.5, 1.75) 1.5 3 A(0.5, 1.75) 1 • Start at the origin and move 0.5 unit Look Back You can review graphing integers on the coordinate plane on page 668. 0.5 to the right. Then move up 1.75 units. • Draw a dot and label it A(0.5, 1.75). ⫺0.5 O 0.5 1 1.5 x 2 4 x ⫺0.5 ( _) 4 B -2, -3 1 4 y • Start at the origin and move 2 units 4 1 to the left. Then move down 3_ 2 4 or 3.25 units. • Draw a dot and label it B -2, -3_ . ( 1 4 ) ⫺4 O ⫺2 B ⫺2, ⫺3_1 ( 4 ) Graph and label each point. e. R 2_, 3_ ( 14 12 ) f. S(-1.5, 3) g. T -_, -3_ ( 12 3 4 ) You can use the Pythagorean Theorem to find the distance between two points on the coordinate plane. 174 Chapter 3 Real Numbers and the Pythagorean Theorem Extra Examples at ca.gr7math.com Find Distance on the Coordinate Plane 5 Graph the ordered pairs (3, 0) and (7, -5). y Then find the distance c between the two points. x c c2 = a2 + b2 Pythagorean Theorem c2 = 42 + 52 Replace a with 4 and b with 5. c 2 = 41 √c2 = ± √ 41 (3, 0) O 5 4 4 2 + 5 2 = 16 + 25 or 41 (7, ⫺5) Definition of square root c ≈ ±6.4 Use a calculator. The points are about 6.4 units apart. Graph each pair of ordered pairs. Then find the distance between the points. Round to the nearest tenth. h. (2, 0), (5, -4) j. (-3,-4), (2, -1) i. (1, 3), (-2, 4) 6 MAPS On the map, each unit !LBANY represents 45 miles. West Point, (_ ) New York, is located at 1 1 , 2 2 and Annapolis, Maryland, ( _2 _2 ) What is the approximate distance between West Point and Annapolis? Let c represent the distance between West Point and Annapolis. Then a = 3 and b = 3.5. c2 = a2 + b2 2 Real-World Link The United States Military Academy, also known as West Point, graduates more than 900 officers each year. The same is true for the United States Naval Academy, which is located in Annapolis. 2 7EST 0OINT (ARTFORD (ARRISBURG is located at -1 1 , -1 1 . 7ASHINGTON $# !NNAPOLIS $OVER Pythagorean Theorem c = 21.25 √ c 2 = ± √ 21.25 3 2 + 3.5 2 = 9 + 12.25 or 21.25 c ≈ ±4.6 4RENTON Replace a with 3 and b with 3.5. 2 2ICHMOND 2 c = 3 + 3.5 "OSTON Definition of square root The map distance is about 4.6 units. Since each map unit equals 45 miles, the distance between the cities is 4.6 · 45 or about 207 miles. Source: www.usma.edu k. SPORTS On a University of Southern California map, Cromwell 1, 3 _ 1 ) and Dedeaux Field at (1 _ 1, 4 _ 1 ). Graph field is located at (2 _ 2 2 2 2 these points. If each map unit is 0.1 mile, about how far apart are the fields? Personal Tutor at ca.gr7math.com Lesson 3-7 Geometry: Distance on the Coordinate Plane Stan Honda/AFP/Getty Images 175 Examples 1, 2 (pp. 173–174) Examples 3, 4 (p. 174) Example 5 (p. 175) Name the ordered pair for each point. 1. A 2. B 3. C 4. D y A B 1 ⫺1 Graph and label each point. 5. J _, 3_ ( 14 12 ) 6. K -1, -2_ 7. L(4.5, -2.25) 8. M(-2.5, 2.5) ( 3 4 O ) x 1 C D ⫺1 Graph each pair of ordered pairs. Then find the distance between the points. Round to the nearest tenth if necessary. 9. (1, 5), (3, 1) 10. (-1, 0), (2, 7) 11. (-5.5, -2), (2.5, 3) 12. GEOMETRY Square ABCD is graphed on the coordinate plane. What is the length of each side? What is the area? Round to the nearest tenth. Example 6 13. PARKS On a park map, the ranger station is (p. 175) located at (2.5, 3.5) and the nature center is located at (0.5, 4). Each unit in the map is equal to 0.5 mile. Graph the ordered pairs. What is the approximate distance between the ranger station and the nature center? (/-%7/2+ (%,0 For Exercises 14–21 22–27 28–33 34–35 See Examples 1 2, 3 4, 5 6 y B A C D Name the ordered pair for each point. 14. P 15. Q 16. R 17. S 18. T 19. U 20. V 21. W x O y R 1 T Q W ⫺1 O 1 S U ⫺1 V x P Graph and label each point. 22. E _, 2_ ( 34 14 ) 1 4 25. H(-2_, 3_) 4 5 23. F _, 1_ 24. G -3, 4_ 26. J(4.3, -3.1) 27. K(-3.75, -0.5) ( 25 12 ) ( 2 3 ) Graph each pair of ordered pairs. Then find the distance between the points. Round to the nearest tenth if necessary. 176 28. (4, 5), (2, 2) 29. (6, 2), (1, 0) 30. (-3, 4), (1, 3) 31. (-5, 1), (2, 4) 32. (2.5, -1), (-3.5, -5) 33. (4, -2.3), (-1, -6.3) Chapter 3 Real Numbers and the Pythagorean Theorem 34. NAVIGATION A ferry sets sail from an island y located at (4, 12) on the map at the right. Its destination is Ferry Landing B at (6, 2). How far will the ferry travel if each unit on the grid is 0.5 mile? 16 12 8 4 35. GEOGRAPHY On a map of Florida, Clearwater is located at (3, 2.5), and Jacksonville is located at (8.5, 14.5). Each unit on the map equals 16.5 miles. Graph the ordered pairs. What is the approximate distance between the cities? A B O 4 C 8 12 x 16 Find the area of each rectangle. 36. 37. y y A F O B D J O x G H x C 38. TRAVEL Rochester, New York, has a longitude of 77° W and a latitude of 43° N. Pittsburgh, Pennsylvania, is located at 80° W and 40° N. At this longitude/latitude, each degree is about 53 miles. Find the distance between Rochester and Pittsburgh. %842!02!#4)#% See pages 684, 710. Self-Check Quiz at ca.gr7math.com ˚ ˚ 80 W 77 W Rochester, NY ? 39. GEOMETRY If one point is located at (-5, 4) and another point is located at (-8, -2), find the distance between the points. H.O.T. Problems ˚ 43 N Pittsburgh, PA ˚ 40 N 40. CHALLENGE Apply what you have learned about distance on the coordinate plane to determine the coordinates of the endpoints of a line segment that is neither horizontal nor vertical and has a length of 5 units. 41. SELECT A TOOL Kendra needs to find the distance between the points A(-2.4, 3.7) and B(4.6, -1.3). Which of the following tools will be most useful to Kendra? Justify your selection(s). Then use the tool(s) to solve the problem. calculator 42. paper and pencil real objects */ -!4( In your own words, explain how to find the length (*/ 83 *5*/( of a non-vertical and a non-horizontal segment whose endpoints are (x 1, y 1) and (x 2, y 2). Lesson 3-7 Geometry: Distance on the Coordinate Plane 177 43. The map shows the location of the 44. Rectangle ABCD is graphed on the towns of Springfield, Centerville, and Point Pleasant. coordinate plane. y A y 20 D Springfield 24 mi x O 10 10 mi B Centerville Point Pleasant O 10 C 30 x 20 Find the area of rectangle ABCD. What is the shortest distance between Centerville and Point Pleasant? F 30 units 2 A 14 mi C 26 mi H 60 units 2 B 22 mi D 34 mi G 50 units 2 J 100 units 2 45. HIKING Hunter hikes 3 miles south and then turns and hikes 7 miles east. How far is he from his starting point? (Lesson 3-6) GEOMETRY Find the missing side of each right triangle. Round to the nearest tenth if necessary. (Lesson 3-5) 46. a, 15 cm; b, 18 cm 47. b, 14 in.; c, 17 in. 48. a, 36 km; b, 40 km 49. ENERGY Electricity costs 6_¢ per kilowatt-hour. Of that cost, 3_¢ goes 1 2 1 4 toward the cost of the fuel. What fraction of the cost goes toward fuel? (Lesson 2-4) ALGEBRA Write and solve an equation to find each number. (Lesson 1-10) 50. The product of a number and 8 is 56. 51. The quotient of a number and 7 is -14. Math and Geography Bon Voyage! It’s time to complete your project. Use the information and data you have gathered about cruise packages and destination activities to prepare a video or brochure. Be sure to include a diagram and itinerary with your project. Cross-Curricular Project at ca.gr7math.com 178 Chapter 3 Real Numbers and the Pythagorean Theorem CH APTER 3 Study Guide and Review Download Vocabulary Review from ca.gr7math.com Key Vocabulary coordinate plane (p. 173) Be sure the following Key Concepts are noted in your Foldable. >«ÌiÀÊÎ\ ,i>Ê ÕLiÀÃ >`ÊÌ i *ÞÌ >}Ài> / iÀi hypotenuse (p. 162) irrational number (p. 155) legs (p. 162) Key Concepts perfect square (p. 144) Pythagorean Theorem (p. 162) Square Roots and Irrational Numbers radical sign (p. 144) (Lessons 3-1, 3-2, and 3-4) real number (p. 155) • A square root of a number is one of its two equal factors. • An irrational number is a number that cannot be a expressed as , where a and b are integers and b b ≠ 0. _ Pythagorean Theorem (Lessons 3-5 to 3-7) • In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. square root (p. 144) Venn diagram (p. 152) Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. 1. The number 11 is a perfect square. Hypotenuse c a 2. The symbol that is used to indicate a square root is the radical sign. 3. A real number is a number that cannot be expressed as the quotient of two integers. Legs b • If the sides of a triangle have lengths a, b, and c units such that c 2 = a 2 + b 2, then the triangle is a right triangle. 4. If the measures of the sides of a triangle are 6 inches, 8 inches and 10 inches, then the triangle is a right triangle. 5. The opposite of squaring a number is finding a converse. 6. A Venn diagram uses overlapping rectangles to organize information and solve problems. 7. The hypotenuse is the shortest side of a right triangle. 8. The Pythagorean Theorem states that the sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse. Vocabulary Review at ca.gr7math.com Chapter 3 Study Guide and Review 179 CH APTER 3 Study Guide and Review Lesson-by-Lesson Review 3-1 Square Roots (pp. 144–147) Find each square root. 9. √ 81 10. ± √ 225 11. - √ 64 12. √ 6.25 13. FARMING Pecan trees are planted in square patterns to take advantage of land space and for ease in harvesting. For 289 trees, how many rows should be planted and how many trees should be planted in each row? 3-2 14. √ 32 15. √ 42 16. √ 230 17. √ 96 18. √ 150 19. √ 8 20. √ 50.1 21. √ 19.25 22. ALGEBRA Estimate the solution of b 2 = 60 to the nearest integer. Since 6 = 36, √ 36 = 6. Example 2 Find - √ 169 . Since (-13)(-13) = 169, - √ 169 = -13. Example 3 Find ± √ 1.21 . 2 Since (1.1) = 1.21 and (-1.1)2 = 1.21, ± √ 1.21 = ±1.1. 135 to the Example 4 Estimate √ nearest whole number. 121 < 135 < 144 Write an inequality. 11 2 < 135 < 12 2 121 = 11 2 and 144 = 12 2 11 < √ 135 < 12 Take the square root of each number. Since 135 is closer to 144 than to 121, the best whole number estimate is 12. PSI: Use a Venn Diagram (pp. 152–153) 23. APARTMENTS An apartment complex offers 15 apartments with a view of the river, 8 with two bedrooms, and 6 that have both selections. How many have only a view of the river? 24. LANGUAGE At Madison Middle School, 95% of the students speak English fluently, 65% speak Spanish fluently, and 60% speak both English and Spanish fluently. What percent of the students speak only Spanish fluently? 180 Find √ 36 . 2 Estimating Square Roots (pp. 148–151) Estimate to the nearest whole number. 3-3 Example 1 Chapter 3 Real Numbers and the Pythagorean Theorem Example 5 The Venn diagram shows the number of dog and cat owners. }Ã >ÌÃ So, 34 people own only dogs, 26 people own only cats, and 12 own both. Mixed Problem Solving For mixed problem-solving practice, see page 710. 3-4 The Real Number System (pp. 155–159) Example 6 Name all sets of numbers to which - √ 33 belongs. Name all sets of numbers to which each real number belongs. − 25. - √ 19 26. 0.3 27. 7.43 28. -12 29. √ 32 30. 101 33 ≈ -5.744562647 - √ Since the decimal does not terminate or repeat, it is an irrational number. 31. MEASUREMENT The area of a square vegetable garden is 360 square meters. To the nearest hundredth meter, what is the perimeter of the garden? 3-5 The Pythagorean Theorem (pp. 162–166) Example 7 Write an equation you could use to find the length of the hypotenuse of the right triangle. Then find the missing length. Write an equation you could use to find the length of the missing side of each right triangle. Then find the missing length. Round to the nearest tenth if necessary. 32. 33. cm 16 m 3m c in. 18 in. 20 m am 24 in. 34. 5 ft 35. 9.5 m 4m 8 ft c ft bm 5m c2 = a2 + b2 c2 = 32 + 52 c 2 = 9 + 25 c 2 = 34 c = ± √ 34 c ≈ ±5.8 Pythagorean Theorem Replace a with 3 and b with 5. Evaluate 3 2 and 5 2. Simplify. Definition of square root Use a calculator. The hypotenuse is about 5.8 meters long. 36. a, 5 in.; c, 6 in. 37. a, 6 cm; b, 7 cm 38. GEOMETRY Lolita drew a right triangle where the hypotenuse was 17 inches and one of the legs was 8 inches. What was the length of the third side? Chapter 3 Study Guide and Review 181 CH APTER 3 Study Guide and Review 3-6 Using the Pythagorean Theorem (pp. 167–171) Write an equation that can be used to answer the question. Then solve. Round to the nearest tenth if necessary. 39. How tall is the light? Example 8 Write an equation that can be used to find the height of the tree. Then solve. 40. How wide is the window? 53 ft h 25 ft 60 in. h 30 in. 20 ft w 25 ft 41. How long is 42. How far is the Use the Pythagorean Theorem to write the equation 53 2 = h 2 + 25 2. Then solve the equation. the walkway? ᐉ plane from the airport? 53 2 = h 2 + 25 2 2,809 = h 2 + 625 5 ft d 2,809 - 625 = h 2 + 625 - 625 10 km 8 ft 18 km 2,184 = h 2 = h ± √2,184 ±46.7 ≈ h Use a calculator. 43. GEOMETRY A rectangle is 12 meters by 7 meters. What is the length of one of its diagonals? 3-7 Geometry: Distance on the Coordinate Plane (pp. 173–178) Graph each pair of ordered pairs. Then find the distance between the points. Round to the nearest tenth if necessary. Example 9 Graph the ordered pairs (2, 3) and (-1, 1). Then find the distance between the points. 44. (0, -3), (5, 5) 45. (-1, 2), (4, 8) 46. (-2, 1.5), (2, 3.6) 47. (-6, 2), (-4, 5) 48. (3, 4.2), (-2.1, 0) 49. (-1, 3), (2, 4) c2 = a2 + b2 c2 = 32 + 22 c 2 c2 = 9 + 4 (⫺1, 1) 3 c 2 = 13 x O c = √ 13 c ≈ 3.6 The distance is about 3.6 units. 50. GEOMETRY The coordinates of points R and S are (4, 3) and (1, 6). What is the distance between the points? Round to the nearest tenth if necessary. 182 The height of the tree is about 47 feet. Chapter 3 Real Numbers and the Pythagorean Theorem y (2, 3) CH APTER 3 Practice Test Find each square root. 1. √ 225 4. 36 3. ±_ 2. - √ 0.25 49 STANDARDS PRACTICE Which list shows the numbers in order from least to greatest? − 1 A 2.2, 2_ , 2.25, √ 5 5 − 1 , 2.2, √ 5 , 2.25 B 2_ 5 1 − 5 , 2.25, 2_ , 2.2 C √ 5 − 1 D 2.25, √5, 2.2, 2_ Determine whether each triangle with sides of given lengths is a right triangle. Justify your answer. 16. 12 in., 20 in., 24 in. 17. 34 cm, 30 cm, 16 cm 18. 15 ft, 25 ft, 20 ft 19. 7 yd, 14 yd, 35 yd 20. STANDARDS PRACTICE Justin is flying a kite. 5 Estimate to the nearest whole number. 6. √ 118 5. √ 67 YD 7. √ 82 YD Name all sets of numbers to which each real number belongs. −− 8. - √ 64 9. 6.13 10. √ 14 11. FOOD Gino’s Pizzeria conducted a survey of 50 customers. The results showed that 15 people liked cheese pizza and 25 liked pepperoni. Of those customers, 4 people liked both cheese and pepperoni pizza. How many people liked neither cheese nor pepperoni pizza? Use a Venn diagram. Which is closest to the length of the string? F 70 yd G 92 yd 21. MEASUREMENT Find the perimeter of a right triangle with legs of 10 inches and 8 inches. 22. SURVEYING A survey team calculated the distance across a river from point A to point B. How wide is the river at this point? Round to the nearest tenth. Write an equation you could use to find each length of the missing side of each right triangle. Then find the missing length. Round to the nearest tenth if necessary. 12. a 13. 5 cm 8 yd c 10 cm H 108 yd J 146 yd Bridge 21 m 72 m A B Graph each pair of ordered pairs. Then find the distance between points. Round to the nearest tenth if necessary. 23. (-2, -2), (5, 6) 24. 6 yd 14. a, 55 in.; b, 48 in. 15. b, 12 ft; c, 20 ft Chapter Test at ca.gr7math.com (_13 , 1), (-1_13 , 1_23 ) 25. (-0.5, 0.25), (0.25, -0.75) Chapter 3 Practice Test 183 CH APTER 3 California Standards Practice Cumulative, Chapters 1–3 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1 5 Erin jogged along the track around the outer edge of a park. She ran two miles along the one edge and then 3 miles along the other edge. She then cut across the park as shown by the dotted line. How far did she jog to get back to her starting point? The proposed location of a new water tower intersects a section of an existing service road. Find x, the inside length of the section of road that is intersected by the water tower. x road 64 ft 80 ft Water Tower 3 mi 2 mi 2 A 3 miles C 5.2 miles B 3.6 miles D 13 miles Michelle had to choose the number closest to 5. Which irrational number should she choose? B 48 ft D 112 ft G √ 27 Zack, Luke, and Charlie ordered a large pizza for $11.99, breadsticks for $2.99, and chicken wings for $5.99. If the three friends agree to split the cost of the food evenly, about how much will each friend pay? H √ 20 F $20.79 H $7.32 √ 18 G $7.93 J $6.99 J 4 C 96 ft Question 5 Remember that the hypotenuse of a right triangle is always opposite the right angle. 6 F √30 3 A 36 ft The square root of 250 is between Out of 100 students surveyed at Central Middle School, 48 are in the band, 52 play a sport, and 50 are in the drama club. Sixteen students are in both the band and the drama club, 22 students are in the drama club and play a sport, and 18 students are in the band and play a sport. Six students are in the band, play a sport, and are in the drama club. How many students are only in the drama club? F 14 and 15. H 16 and 17. A 20 students C 6 students G 15 and 16. J 17 and 18. B 12 students D 0 students The Moon is about 3.84 × 10 5 kilometers from Earth. Which of the following represents this number in standard notation? A 38,400,000 km C 384,000 km B 3,840,000 km D 38,400 km 184 7 Chapter 3 Real Numbers and the Pythagorean Theorem More California Standards Practice For practice by standard, see pages CA1–CA39. 8 Molly multiplied her age by 3 and subtracted 2 from the product. She then divided the difference by 4, and added 7 to the quotient. The result was 14. Which could be the first step in finding Molly’s age? 12 On Monday, the high temperature in Las Vegas, Nevada, was 101°F, and the high temperature in Columbus, Ohio, was 76°F. How much warmer was it in Las Vegas than Columbus? F Add 14 and 7. F 25°F H -25°F G Subtract 7 from 14. G -15°F J 15°F H Multiply 14 by 4. 5 4 13 Which fraction is between _ and _ ? J Divide 14 by 3. 9 1 A _ 5 7 B _ 8 The diameter of a red blood cell is about 0.00074 centimeter. Which expression represents this number in scientific notation? A 7.4 × 10 4 C 7.4 × 10 -3 B 7.4 × 10 3 D 7.4 × 10 -4 1 2 7 9 D _ 11 F 2(-6) H -2 + (-6) G 2 + (-6) J 2 - (-6) 15 Student admission to the movies is $6.25. What is the total cost of tickets for you and four other students? J 3 6 in a positive number? represents √ 8? G H 5 14 Which of the following expressions results 10 Which point on the number line best F 6 C _ 4 5 F point F H point H G point G J point J A $18.75 C $31.25 B $25.00 D $35.50 Pre-AP Record your answers on a sheet of paper. Show your work. 11 Ms. Leigh wants to organize the desks in 16 Use a grid to graph and answer the the study hall into a square. If she has 64 desks, how many should be in each row? following questions. a. Graph the ordered pairs (3, 4) and (-2, 1). A 7 B 8 C 9 b. Describe how to find the distance between the two points. D 10 c. Find the distance between the points. NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Go to Lesson... 3-6 3-2 2-10 3-2 3-6 1-1 3-3 1-7 2-10 3-2 3-1 1-5 2-2 1-6 1-6 3-7 For Help with Standard... MG3.3 NS2.4 NS1.1 NS2.4 MG3.3 NS1.2 MR2.5 AF1.1 NS1.1 NS2.4 NS2.4 NS1.2 NS1.1 NS1.2 NS1.2 MG3.2 California Standards Practice at ca.gr7math.com Chapters 1–3 California Standards Practice 185 Patterns, Relationships, and Algebraic Thinking Focus Compute with proportions and percents. CHAPTER 4 Choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems. CHAPTER 5 Percent Know the properties of, and compute with, rational numbers expressed in a variety of forms. 186 Rob Gage/Getty Images Proportions and Similarity Solve simple linear equations and inequalities over the rational numbers. Math and Art It’s a Masterpiece! Grab some canvas, paint, and paintbrushes. You’re about to create a masterpiece! On this adventure, you’ll learn about the art of painting the human face. Along the way, you’ll research the methods of a master painter and learn about how artists use the Golden Ratio to achieve balance in their works. Don’t forget to bring your math tool kit and a steady hand. This is an adventure you’ll want to frame! Log on to ca.gr7math.com to begin. Unit 2 Patterns, Relationships, and Algebraic Thinking Rob Gage/Getty Images 187 Proportions and Similarity 4 • Standard 7AF4.0 Solve simple linear equations and inequalities over the rational numbers. • Standard 7MG1.0 Choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems. Key Vocabulary constant of proportionality (p. 200) proportion (p. 198) ratio (p. 190) scale factor (p. 207) Real-World Link Lightning During a severe thunderstorm, lightning flashed an average of 8 times per minute. You can use this rate to determine the number of lightning flashes that occurred during a 15-minute period. Proportions and Similarity Make this Foldable to help you organize your notes. Begin with a plain sheet of 11” by 17” paper. 1 Fold in thirds widthwise. 2 Open and fold the bottom to form a pocket. Glue edges. 3 Label each pocket. Place index cards in each pocket. 1 RO P O R 188 Chapter 4 Proportions and Similarity Jim Zuckerman/CORBIS TION S "LG EB RA ( E OME TRY GET READY for Chapter 4 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Simplify each fraction. (Prior Grade) Example 1 10 1. _ Simplify 54 . 24 36 3. _ 81 88 2. _ 104 49 4. _ 91 5. MONEY Devon spent $18 of the $45 that he saved. Write a fraction in simplest form that represents the portion of his savings he spent. (Prior Grade) Evaluate each expression. (Prior Grade) 6-2 6. _ 5+5 3-1 _ 8. 1+9 7-4 7. _ 8-4 5+7 _ 9. 8-6 _ 81 ÷ 27 54 2 _ =_ Divide the numerator and denominator by their GCF, 27. 3 81 ÷ 27 Example 2 Evaluate 11 + 4 _ . 9-4 11 + 4 Simplify the numerator and 15 _=_ denominator. 9-4 5 =3 Simplify. Solve each equation. (Lessons 1-10) Example 3 10. 5 · 6 = x · 2 11. c · 1.5 = 3 · 7 Solve 4 · 6 = 8 · p. 12. 12 · z = 9 · 4 13. 7 · 2 = 8 · g 4·6=8·p 14. 3 · 11 = 4 · y 15. b · 6 = 7 · 9 8p 24 _ =_ 8 16. NUMBER SENSE The product of a 8 3=p Write the equation. Multiply 4 by 6 and 8 by p. Divide each side by 8. number and four is equal to the product of eight and twelve. Find the number. (Lessons 1-10) Chapter 4 Get Ready for Chapter 4 189 4-1 Ratios and Rates Main IDEA Express ratios as fractions in simplest form and determine unit rates. Standard 7AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation. Standard 7MG1.3 Use measures expressed as rates (e.g. speed, density) and measures expressed as products (e.g. person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer. TRAIL MIX The diagram shows a batch of trail mix that is made using 3 scoops of raisins and 6 scoops of peanuts. peanuts 1. To make the batch of trail mix, how many scoops of raisins should you use for every 1 scoop of peanuts? Explain your reasoning. trail mix A ratio is a comparison of two numbers or quantities by division. If a batch of trail mix contains 3 scoops of raisins and 6 scoops of peanuts, the ratio comparing raisins to peanuts can be written as follows. 3 to 6 NEW Vocabulary raisins 3:6 _3 6 Since a ratio can be written as a fraction, it can be simplified. ratio rate unit rate Write Ratios in Simplest Form Express each ratio in simplest form. READING Math Ratios In Example 1, the ratio 2 out of 7 means that for every 7 cats, 2 are Siamese. 1 8 Siamese cats out of 28 cats 8 cats 2 _ =_ 28 cats 7 Divide the numerator and denominator by the greatest common factor, 4. Divide out common units. 2 The ratio of Siamese cats to cats is _ or 2 out of 7. 7 2 10 ounces of butter to 1 pound of flour When writing ratios that compare quantities with the same kinds of units, convert so that they have the same unit. 10 ounces 10 ounces _ =_ 1 pound 16 ounces 5 ounces =_ 8 ounces Convert 1 pound to 16 ounces. Divide the numerator and the denominator by 2. Divide out common units. 5 The ratio of butter to flour in simplest form is _ or 5:8. 8 a. 16 pepperoni pizzas out of 24 pizzas b. 30 minutes of commercials to 2 hours of programming 190 Chapter 4 Proportions and Similarity A rate is a ratio that compares two quantities with different types of units such as $5 for 2 pounds or 130 miles in 2 hours. When a rate is simplified so it has a denominator of 1, it is called a unit rate. An example of a unit rate is $6.50 per hour, which means $6.50 per 1 hour. Find a Unit Rate 3 TRAVEL Darrell drove 187 miles in 3 hours. What was Darrell’s average rate of speed in miles per hour? Write the rate that expresses the comparison of miles to hours. Then find the average speed by finding the unit rate. ÷3 READING Math Math Symbols The symbol ≈ is read approximately equal to. 187 miles 62 miles _ ≈_ 3 hours Divide the numerator and denominator by 3 to get a denominator of 1. 1 hour ÷3 Darrell drove an average speed of about 62 miles per hour. Express each rate as a unit rate. c. 24 tickets for 8 rides d. 4 inches of rain in 5 hours Personal Tutor at ca.gr7math.com Compare Unit Rates 4 CIVICS In 2000, the population of California was about 33,900,000, and the population of Kentucky was about 4,000,000. There were 53 members of the U.S. House of Representatives from California and 6 from Kentucky. In which state did a member represent more people? Real-World Link In the U.S. House of Representatives, the number of representatives from each state is based on a state’s population in the preceding census. For each state, write a rate that compares the state’s population to its number of representatives. Then find the unit rates. C a l ÷ 53 i f 33,900,000 people 640,000 people __ ≈ __ o 53 representatives r n i a 1 representative ÷ 53 ÷6 Source: www.house.gov +ENTUCKY 4,000,000 people 670,000 people __ ≈ __ 6 representatives 1 representative ÷6 A member represented more people in Kentucky than in California. SHOPPING Decide which is the better buy. Explain your reasoning. e. a 17-ounce box of cereal for $4.89 or a 21-ounce box for $5.69 f. 6 cans of green beans for $1 or 10 cans for $1.95 Extra Examples at ca.gr7math.com Peter Heimsath/Rex USA Lesson 4-1 Ratios and Rates 191 Examples 1, 2 (p. 190) Example 3 (p. 191) Example 4 (p. 191) Express each ratio in simplest form. 1. 12 missed days out of 180 days 2. 12 wins to 18 losses 3. 6 inches of water for 7 feet of snow 4. 3 quarts of soda : 1 gallon of juice Express each rate as a unit rate. 5. $50 for 4 days of work 6. 3 pounds of dog food in 5 days 7. SHOPPING You can buy 4 Granny Smith apples at Ben’s Mart for $0.95. SaveMost sells 6 of the same quality apples for $1.49. Which store has the better buy? Explain your reasoning. (/-%7/2+ (%,0 Express each ratio in simplest form. For Exercises 8–11 12–15 16–21 22–23 8. 14 chosen out of 70 who applied 9. 28 out of 100 doctors disagree 10. 33 stores open to 18 closed 11. 56 boys to 64 girls participated 12. 1 cup vinegar in 8 pints of water 13. 2 yards wide : 10 feet long 14. 20 centimeters out of 1 meter cut 15. 2,500 pounds for 1 ton of steel See Examples 1 2 3 4 16. BASEBALL In 2005, Hank Aaron was still the MLB career all-time hitter, with 3,771 hits in 3,298 games. What was Aaron’s average number of hits per game? 17. CARS Manufacturers must publish a car’s gas mileage or the average number of miles one can expect to drive per gallon of gasoline. The test of a new car resulted in 2,250 miles being driven using 125 gallons of gas. Find the car’s expected gas mileage. Express each rate as a unit rate. 18. 153 points in 18 games 19. 350 miles on 15 gallons 20. 100 meters in 12 seconds 21. 1,473 people entered in 3 hours 22. ELECTRONICS A 20-gigabyte digital music player sells for $249. A similar 30-gigabyte player sells for $349. Which player offers the better price per gigabyte of storage? Explain. Real-World Link Gas mileage can be improved by as much as 3.3% by keeping tires inflated to the proper pressure. Source: www.fueleconomy.gov 192 23. MEASUREMENT Logan ran a 200-meter race in 25.24 seconds, and Scott ran 0.4 kilometer in 52.77 seconds. Who ran faster, Logan or Scott? Explain. 24. MAGAZINES Which costs more per issue, an 18-issue subscription for $40.50 or a 12-issue subscription for $33.60? Explain. Chapter 4 Proportions and Similarity JupiterImages/Comstock %842!02!#4)#% 25. TRAVEL Three people leave at the same time from Rawson to travel to Huntsville. Sarah averaged 45 miles per hour for the first third of the trip, 55 miles per hour for the second third, and 75 miles per hour for the last third. Darnell averaged 55 miles per hour for the first half of the trip and 70 miles per hour for the second half. Megan drove at a steady speed of 60 miles per hour the entire trip. Who arrived at Huntsville first? Explain. See pages 685, 711. Self-Check Quiz at ca.gr7math.com H.O.T. Problems 26. Which One Doesn’t Belong? Identify the phrase that does not represent the same rate as the other two. Explain your reasoning. 36 miles per hour 3,168 miles per minute 52.8 feet per second 27. CHALLENGE Luisa and Rachel have some trading cards. The ratio of Luisa’s cards to Rachel’s cards is 3:1. If Luisa gives Rachel 2 cards, the ratio will be 2:1. How many cards does Luisa have? Explain. 28. */ -!4( Write about a real-world situation that can be (*/ 83 *5*/( represented by the ratio 2:5. 29. Lucy typed 210 words in 5 minutes, 30. Jackson drove 70 miles per hour for and Yvonne typed 336 words in 8 minutes. Based on these rates, which statement is true? 4 hours and then 55 miles per hour for 2 hours to go to a conference. How far did Jackson drive in all? A Lucy’s rate was 3-words-perminute slower than Yvonne’s. F 390 miles B Lucy’s rate was 25.2-words-perminute faster than Yvonne’s. H 320 miles G 360 miles J C Lucy’s rate was about 15.8-wordsper-minute faster than Yvonne’s. 280 miles D Lucy’s rate was equal to Yvonne’s. GEOMETRY Graph each pair of ordered pairs. Then find the distance between the points. Round to the nearest tenth. (Lesson 3-7) 31. (1, 4), (6, -3) 32. (-1, 5), (3, -2) 33. (-5, -2), (-1, 0) 34. (-2, -3), (3, 1) 35. MEASUREMENT A square floor exercise mat measures 40 feet on each side. Find the length of the mat’s diagonal. (Lesson 3-6) PREREQUISITE SKILL Write each expression as a decimal. (Lesson 2-1) 36. _ 19 5 37. _ 3 8 38. _ 12.4 4 39. _ 2.5 5 Lesson 4-1 Ratios and Rates 193 4-2 Proportional and Nonproportional Relationships Main IDEA Identify proportional and nonproportional relationships. PIZZA Ms. Cochran is planning a year-end pizza party for her students. Ace Pizza offers free delivery and charges $8 for each medium pizza. Preparation for Standard 7AF3.4 Plot the values of quantities whose ratios are always the same (e. g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. 1. Copy and complete the table to determine the cost for different numbers of pizzas ordered. Pizzas Ordered 1 Cost ($) 8 2 3 4 2. For each number of pizzas, write the relationship of the cost and number of pizzas as a ratio in simplest form. What do you notice? In the example above, notice that while the number of pizzas ordered and the cost both change or vary, the ratio of these quantities remains the same, a constant $8 per pizza. cost of order 8 16 32 24 __ =_ =_ =_ =_ or $8 per pizza 1 pizzas ordered 2 3 4 This relationship is expressed by saying that the cost of an order is proportional to the number of pizzas ordered. If two quantities are proportional, then they have a constant ratio. For relationships in which this ratio is not constant, the two quantities are said to be nonproportional. NEW Vocabulary proportional nonproportional Identify Proportional Relationships 1 PIZZA Uptown Pizzeria sells medium pizzas for $7 each but charges a $3 delivery fee per order. Is the cost of an order proportional to the number of pizzas ordered? Find the cost for 1, 2, 3, and 4 pizzas and make a table to display numbers and cost. Common Error Even though there may be an adding pattern in both sets of values, a proportional relationship may not exist. In Example 1, as the number of pizzas increases by 1, the cost increases by 7, but the ratio of these values is not the same. 194 Cost ($) 10 17 24 31 Pizzas Ordered 1 2 3 4 For each number of pizzas, write the relationship of the cost and number of pizzas as a ratio in simplest form. cost of order __ pizzas ordered 10 _ or 10 1 17 _ or 8.5 2 24 _ or 8 3 31 _ or 7.75 4 Since the ratios of the two quantities are not the same, the cost of an order is not proportional to the number of pizzas ordered. The relationship is nonproportional. Chapter 4 Proportions and Similarity Extra Examples at ca.gr7math.com 2 BEVERAGES You can use the recipe #OMBINE SUGAR ENVELOPE OF MIX QUARTS OF WATER shown to make a healthier version of a popular beverage. Is the amount of mix used proportional to the amount of sugar used? CUP Find the amount of mix and sugar needed for different numbers of batches and make a table to show these mix and sugar measures. Cups of Sugar _1 1 1 _1 2 Envelopes of Mix 1 2 3 4 Quarts of Water 2 4 6 8 2 2 For each number of cups of sugar, write the relationship of the cups and number of envelopes of mix as a ratio in simplest form. cups of sugar __ envelopes of mix _1 1 1_ 0.5 1.5 1 2 2 _2 = _ or 0.5 _ or 0.5 _ =_ or 0.5 _ or 0.5 2 3 3 1 1 4 Since the ratios between the two quantities are all equal to 0.5, the amount of mix used is proportional to the amount of sugar used. READING in the Content Area For strategies in reading this lesson, visit ca.gr7math.com. a. BEVERAGES In Example 2, is the amount of sugar used proportional to the amount of water used? b. MONEY At the beginning of the school year, Isabel had $120 in the bank. Each week, she deposits another $20. Is her account balance proportional to the number of weeks since she started school? Personal Tutor at ca.gr7math.com Examples 1, 2 1. ELEPHANTS An adult elephant drinks about 225 liters of water each day. (pp. 194–195) Is the number of days that an elephant’s water supply lasts proportional to the number of liters of water the elephant drinks? 2. PACKAGES A package shipping company charges $5.25 to deliver a package. In addition, they charge $0.45 for each pound over one pound. Is the cost to ship a package proportional to the weight of the package? 3. SCHOOL At a certain middle school, every homeroom teacher is assigned 28 students. There are 3 teachers who do not have a homeroom. Is the number of students at this school proportional to the number of teachers? 4. JOBS Andrew earns $18 per hour for mowing lawns. Is the amount of money he earns proportional to the number of hours he spends mowing? Lesson 4-2 Proportional and Nonproportional Relationships 195 (/-%7/2+ (%,0 For Exercises 5–12 See Examples 1, 2 5. RECREATION The Vista Marina rents boats for $25 per hour. In addition to the rental fee, there is a $12 charge for fuel. Is the number of hours you can rent the boat proportional to the total cost? 6. ELEVATORS An elevator ascends or goes up at a rate of 750 feet per minute. Is the height to which the elevator ascends proportional to the number of minutes it takes to get there? 7. PLANTS Kudzu is a vine that grows an average of 7.5 feet every 5 days. Is the number of days of growth proportional to the length of the vine as measured on the last day? 8. TEMPERATURE To convert a temperature in degrees Celsius to degrees 9 Fahrenheit, multiply the Celsius temperature by _ and then add 32°. 5 Is a temperature in degrees Celsius proportional to its equivalent temperature in degrees Fahrenheit? ADVERTISING For Exercises 9 and 10, use the following information. On Saturday, Querida gave away 416 coupons for a free appetizer at a local restaurant. The next day, she gave away about 52 coupons an hour. 9. Is the number of coupons Querida gave away on Sunday proportional to the number of hours she worked that day? 10. Is the total number of coupons Querida gave away on Saturday and Real-World Link Ascending at a speed of 1,000 feet per minute, the five outside elevators of the Westin St. Francis are the fastest glass elevators in San Francisco. Source: sfvisitor.org Sunday proportional to the number of hours she worked on Sunday? SHOPPING For Exercises 11 and 12, use the following information. 1 MegaMart collects a sales tax equal to _ of the retail price of each purchase 16 and sends this money to the state government. 11. Is the amount of tax collected proportional to the cost of an item before tax is added? 12. Is the amount of tax collected proportional to the cost of an item after tax has been added? MEASUREMENT For Exercises 13 and 14, determine whether the measures described for the figure shown are proportional. 13. the length of a side and the perimeter s 14. the length of a side and the area POSTAGE For Exercises 15 and 16, use the table below that shows the price to %842!02!#4)#% mail a first-class letter for various weights. See pages 685, 711. 15. Is the cost to mail a letter proportional to its weight? Explain your reasoning. Self-Check Quiz at ca.gr7math.com 196 16. Find the cost to mail a letter that weighs 5 ounces. Justify your answer. Chapter 4 Proportions and Similarity age fotostock/SuperStock Weight (oz) 1 2 3 4 Cost ($) 0.39 0.63 0.87 1.11 5 H.O.T. Problems 17. OPEN ENDED Give one example of a proportional relationship and one example of a nonproportional relationship. Justify your examples. 18. CHALLENGE This year Andrea celebrated her 10th birthday, and her brother Carlos celebrated his 5th birthday. Andrea noted that she was now twice as old as her brother was. Is the relationship between their ages proportional? Explain your reasoning using a table of values. 19. */ -!4( Luke uses $200 in birthday money to purchase some (*/ 83 *5*/( $20 DVDs. He claims that the amount of money remaining after his purchase is proportional to the number of DVDs he decides to buy, because the DVDs are each sold at the same price. Is his claim valid? If his claim is false, name two quantities in this situation that are proportional. 20. Mr. Martinez is comparing the price of oranges from several different markets. Which market’s pricing guide is based on a constant unit price? A Farmer’s Market Number of Total Oranges Cost ($) 5 3.50 10 6.00 15 8.50 20 11.00 C Central Produce Number of Total Oranges Cost ($) 5 3.00 10 6.00 15 9.00 20 12.00 B The Fruit Place Number of Total Oranges Cost ($) 5 3.50 10 6.50 15 9.50 20 12.50 D Green Grocer Number of Total Oranges Cost ($) 5 3.00 10 5.00 15 7.00 20 9.00 Express each ratio in simplest form. (Lesson 4-1) 21. 40 working hours out of 168 hours 22. 2 inches of shrinkage to 1 yard of material 23. GEOMETRY The vertices of right triangle ABC are located at A(-2, -5), B(-2, 8), and C(1, 4). Find the perimeter of the triangle. (Lesson 3-7) ALGEBRA Write and solve an equation to find each number. (Lesson 1-10) 24. The product of -9 and a number is 45. 25. A number divided by 4 is -16. PREREQUISITE SKILL Solve each equation. Check your solution. (Lesson 1-10) 26. 5 · x = 6 · 10 27. 8 · 3 = 4 · y 28. 2 · d = 3 · 5 29. 2.1 · 7 = 3 · a Lesson 4-2 Proportional and Nonproportional Relationships 197 4-3 Solving Proportions Main IDEA Use proportions to solve problems. NUTRITION Part of the nutrition label from a granola bar is shown at the right. Standard 7AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation. 1. Write a ratio in simplest form that compares the number of calories from fat to the total number of calories. 2. Suppose you plan to eat two such granola bars. Write a ratio comparing the number of calories from fat to the total number of calories. 3. Is the number of calories from fat proportional to the total number of calories for one and two bars? Explain your reasoning. In the example above, the ratios of calories from fat to total calories for one or two granola bars are equal or equivalent ratios because they 2 simplify to the same ratio, _ . One way of expressing a proportional 11 relationship like this is by writing a proportion. 20 calories from fat 40 calories from fat __ = __ 110 total calories 220 total calories +%9 #/.#%04 Words A proportion is an equation stating that two ratios or rates are equivalent. NEW Vocabulary equivalent ratios proportion cross products constant of proportionality Proportion Examples Numbers Algebra _6 = _3 8 _a = _c , b ≠ 0, d ≠ 0 4 b d Consider the following proportion. _a = _c b d 1 _a · bd = _c · bd1 b Cross Products If the cross products of two ratios are equal, then the ratios form a proportion. If the cross products are not equal, the ratios do not form a proportion. 198 Multiply each side by bd and divide out common factors. d 1 1 ad = bc Simplify. The products ad and bc are called the cross products of this proportion. The cross products of any proportion are equal. You can use cross products to solve proportions in which one of the quantities is not known. Chapter 4 Proportions and Similarity _6 = _3 8 4 8 · 3 = 24 6 · 4 = 24 The cross products are equal. Write and Solve a Proportion Interactive Lab ca.gr7math.com 1 TEMPERATURE After 2 hours, the air temperature had risen 7°F. Write and solve a proportion to find the amount of time it will take at this rate for the temperature to rise an additional 13°F. Write a proportion. Let t represent the time in hours. temperature time temperature time 13 _7 = _ t 2 13 _7 = _ t 2 Write the proportion. 7 · t = 2 · 13 Find the cross products. 7t = 26 Multiply. 26 7t _ =_ Divide each side by 7. 7 7 t ≈ 3.7 Simplify. It will take about 3.7 hours to rise an additional 13°F. Solve each proportion. a. _ = _ x 4 b. _ = _ y 9 10 c. _ = _ 5 2 34 7 3 n 2.1 You can use ratios to make predictions in situations involving proportions. 2 BLOOD A microscope slide shows 37 red blood cells and 23 blood cells that are not red blood cells. How many red blood cells would be expected in a sample of the same blood that has 925 blood cells? red blood cells total blood cells 37 37 _ or _ 23 + 37 60 Write and solve a proportion. Let r represent the number of red blood cells in the bigger sample. Real-World Career How Does a Medical Technologist Use Math? A medical technologist uses proportional reasoning to analyze blood samples. red blood cells total blood cells 37 r _ =_ 60 925 37 · 925 = 60 · r red blood cells total blood cells Find the cross products. 34,225 = 60r Multiply. 34,225 60r _ =_ 60 60 Divide each side by 60. 570.4 ≈ r Simplify. You would expect to find about 570 red blood cells. For more information, go to ca.gr7math.com. d. RECYCLING Recycling 2,000 pounds of paper saves about 17 trees. Write and solve a proportion to determine how many trees you would expect to save by recycling 5,000 pounds of paper. Personal Tutor at ca.gr7math.com Lesson 4-3 Solving Proportions Matt Meadows 199 You can also use the constant ratio to write an equation expressing the relationship between two proportional quantities. The constant ratio is also called the constant of proportionality. Write and Use an Equation 3 ALGEBRA Jaycee bought 8 gallons of gasoline for $22.32. Write an equation relating the cost to the number of gallons of gasoline. How much would Jaycee pay for 11 gallons at this same rate? for 20 gallons? Find the constant of proportionality between cost and gallons. cost in dollars 22.32 __ =_ or 2.79 The cost is $2.79 per gallon. 8 gasoline in gallons Checking Your Equation You can check to see if the equation you wrote is accurate by testing the two known quantities. Words The cost is $2.79 times the number of gallons. Variable Let c represent the cost. Let g represent the number of gallons. Equation c = 2.79 · g c = 2.79g 22.32 = 2.79(8) Use this equation to find the cost for 11 and 20 gallons sold at the same rate. 22.32 = 22.32 c = 2.79g Write the equation. c = 2.79g c = 2.79(11) Replace g with the number of gallons. c = 2.79(20) c = 30.69 Multiply. c = 55.80 The cost for 11 gallons is $30.69 and for 20 gallons is $55.80. e. ALGEBRA Olivia typed 2 pages in 15 minutes. Write an equation relating the number of minutes m to the number of pages p typed. If she continues typing at this rate, how many minutes will it take her to type 10 pages? to type 25 pages? Example 1 (p. 199) Solve each proportion. 1. _ = _ p 1.5 6 10 2. _ = _ 3.2 9 n 36 3. _ = _ 41 x 5 2 For Exercises 4 and 5, assume all situations are proportional. 200 Example 2 4. TEETH For every 7 people who say they floss daily, there are 18 people (p. 199) who say they do not. Write and solve a proportion to determine out of 65 people how many you would expect to say they floss daily. Example 3 5. TUTORING Amanda earns $28.50 tutoring for 3 hours. Write an equation (p. 200) relating her earnings m to the number of hours h she tutors. How much would Amanda earn tutoring for 2 hours? for 4.5 hours? Chapter 4 Proportions and Similarity Extra Examples at ca.gr7math.com (/-%7/2+ (%,0 Solve each proportion. For Exercises 6–13 14–19 20–25 6. _ = _ 32 56 7. _ = _ 18 39 _ 8. _ p = 11 5 9. _ = _ 10. _ = _ 11. _ = _ 12. _ = _ 13 . _ = _ See Examples 1 2 3 k 7 6 25 x 13 2.5 6 d 30 h 9 44 3.5 8 2 w 0.4 0.7 48 9 a 3.2 72 n For Exercises 14–21, assume all situations are proportional. 14. COOKING Evarado paid $1.12 for a dozen eggs. Write and solve a proportion to determine the ingredient cost of the 3 eggs Evarado needs for a recipe. 15. TRAVEL A certain vehicle can travel 483 miles on 14 gallons of gasoline. Write and solve a proportion to determine how many gallons of gasoline this vehicle will need to travel 600 miles. 16. ILLNESS For every person who actually has the flu, there are 6 people who have flu-like symptoms resulting from a cold. If a doctor sees 40 patients, write and solve a proportion to determine how many of these you would expect to have a cold. 17. LIFE SCIENCE For every left-handed person, there are about 4 right-handed people. If there are 30 students in a class, write and solve a proportion to predict the number of students who are right-handed. PEOPLE For Exercises 18 and 19, use the following information. The head height to overall height ratio for an adult is given in the diagram at the left. Write and solve a proportion to predict the following measures. 18. the height of an adult who has a head height of 9.6 inches Real-World Link Although people vary in size and shape, in general, people do not vary in proportion. 19. the head height of an adult who is 64 inches tall 20. PHOTOGRAPHY It takes 2 minutes to print out 3 digital photos. Write an equation relating the number of photos n to the number of minutes m. At this rate, how long will it take to print 10 photos? 14 photos? Source: Art Talk 21. MEASUREMENT A 20-pound object on Earth weighs 3_ pounds on the 1 3 Moon. Write an equation relating the weight m of an object on the Moon to the weight a of the object on Earth. How much does an object weigh on the Moon if it weighs 96 pounds on Earth? 128 pounds on Earth? MEASUREMENT For Exercises 22–25, use the table to write an equation relating the two measures. Then find the missing quantity. Round to the nearest hundredth. %842!02!#4)#% See pages 685, 711. 22. 12 in. = 24. 2 L = 26. Self-Check Quiz at ca.gr7math.com cm gal Customary System To Metric System 1 in. ≈ 2.54 cm 1 mi ≈ 1.61 km 23. 20 mi = km 1 gal ≈ 3.78 L 25. 45 kg = lb 1 lb ≈ 0.454 kg FIND THE DATA Refer to the California Data File on pages 16–19. Choose some data and write a real-world problem that could be solved by writing and solving a proportion. Lesson 4-3 Solving Proportions 201 27. MEASUREMENT A 5-pound bag of grass seed covers 2,000 square feet. An opened bag has 3 pounds of seed remaining in it. Will this be enough to seed a 14-yard by 8-yard piece of land? Explain your reasoning. H.O.T. Problems 28. OPEN ENDED List two other amounts of cinnamon and sugar, one larger 1 and one smaller, that are proportional to 1_ tablespoons of cinnamon for 2 every 3 tablespoons of sugar. Justify your answers. CHALLENGE Solve each equation. 29. _ = _ 2 3 18 x+5 30. _ = _ x-4 10 31. _ = _ 4.5 17 - x 7 5 3 8 */ -!4( Explain why it might be easier to write an equation to (*/ 83 *5*/( 32. represent a proportional relationship rather than using a proportion. 33. Michael paid $24 for 3 previouslyviewed DVDs at Play-It-Again Movies. Which equation can he use to find the cost c of purchasing 12 previouslyviewed DVDs from this same store? A c = 12 · 24 C c = 12 · 8 B c = 24 · 4 D c = 72 · 36 34. An amusement park line is moving about 4 feet every 15 minutes. At this rate, approximately how long will it take for a person at the back of the 50foot line to reach the front of the line? F 1 hour G 3 hours H 5 hours J 13 hours 35. The graph shows the results of a survey of 30 Northside students. &AVORITE 4YPE OF -USIC AT .ORTHSIDE -IDDLE 3CHOOL *AZZ 2AP #OUNTRY !LTERNATIVE 2OCK .UMBER OF 3TUDENTS Which proportion can be used to find n, the number preferring country music out of 440 Northside students? 30 n A _ =_ 9 440 440 9 B _=_ 30 n 30 n C _ =_ 9 400 9 n D _=_ 30 440 36. MONEY Cassie deposits $40 in a savings account. The money earns $1.40 per month in simple interest, and she makes no further deposits. Is her account balance proportional to the number of months since her initial deposit? (Lesson 4-2) 37. SHOPPING Which is the better buy: 1 pound 4 ounces of cheese for $4.99 or 2 pounds 6 ounces for $9.75? Explain your reasoning. (Lesson 4-1) 38. PREREQUISITE SKILL Jacquelyn pays $8 for fair admission but then must pay $0.75 for each ride. If she rides five rides, what is the total cost at the fair? (Lesson 1-1) 202 Chapter 4 Proportions and Similarity Extend 4-3 Geometry Lab The Golden Rectangle Main IDEA Find the value of the golden ratio. Standard 7MR1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed. Standard 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. Cut out a rectangle that measures 34 units long by 21 units wide. Using your calculator, find the ratio of the length to the width. Express it as a decimal to the nearest hundredth. Record your data in a table like the one below. length 34 21 width 21 13 ratio decimal Cut this rectangle into two parts, in which one part is the Rectangle Square largest possible square and the other part is a rectangle. Record the rectangle’s length and width. Write the ratio of length to width. Express it as a decimal to the nearest hundredth and record in the table. Repeat the procedure described in Step 2 until the remaining rectangle measures 3 units by 5 units. ANALYZE THE RESULTS 1. Describe the pattern in the ratios you recorded. 2. MAKE A CONJECTURE If the rectangles you cut out are described as golden rectangles, what is the value of the golden ratio? 3. Write a definition of golden rectangle. Use the word ratio in your definition. Then describe the shape of a golden rectangle. 4. Determine whether all golden rectangles are similar. Explain your reasoning. 5. RESEARCH There are many examples of the golden rectangle in architecture. One is shown at the right. Use the Internet or another resource to find three places where the golden rectangle is used in architecture. Extend 4-3 Geometry Lab: The Golden Rectangle Doug Corrance/Taxi/Getty Images 203 4-4 Problem-Solving Investigation MAIN IDEA: Solve problems by drawing a diagram. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Standard 7AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation. e-Mail: DRAW A DIAGRAM YOUR MISSION: Draw a diagram to solve the problem. THE PROBLEM: How long will it take to fill a 120-gallon aquarium? EXPLORE PLAN SOLVE ▲ GABRIELLA: It’s been 3 minutes and this 120-gallon tank is only at the 10-gallon mark. I wonder how much longer it will take. Let’s draw a diagram to help us picture what’s happening. The tank holds 120 gallons of water. After 3 minutes, the tank has 10 gallons of water in it. How many more minutes will it take to fill the tank? Draw a diagram showing the water level after every 3 minutes. The tank will be filled after twelve 3-minute time periods. This is a total of 12 × 3 or 36 minutes. FILL LINE TIME PERIODS WATER LEVEL AFTER MINUTES CHECK The tank is filling at a rate of 10 gallons every 3 minutes, which is about 3 gallons per minute. So, a 120-gallon tank will take about 120 ÷ 3 or 40 minutes to fill. An answer of 36 minutes is reasonable. 1. Describe another method the students could have used to find the number of 3-minute time periods it would take to fill the tank. */ -!4( Write a problem that is more easily solved by drawing (*/ 83 *5*/( 2. a diagram. Then draw a diagram and solve the problem. 204 Chapter 4 Proportions and Similarity J. Strange/KS Studio 9. TILES Three-inch square tiles that are For Exercises 3–5, use the draw a diagram strategy to solve the problem. 3. AQUARIUM Refer to the problem at the beginning of the lesson. Jack fills another 120-gallon tank at the same time Gabriella is filling the first 120-gallon tank. After 3 minutes, his tank has 12 gallons in it. How much longer will it take Gabriella to fill her tank than Jack? 4. LOGGING It takes 20 minutes to cut a log into 5 equal-size pieces. How long will it take to cut a similar log into 3 equal-size pieces? piling oranges in the shape of a square-based pyramid, as shown. If the pyramid is to have five layers, how many oranges will he need? G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s rn. • Look for a patte agram. • Use a Venn di m. • Draw a diagra 15 in. 15 in. 12 in. 10. DESSERTS At a birthday party, 12 people class, 19 like to do chemistry labs, 15 prefer physical science labs, and 7 like to do both. How many students like chemistry labs but not physical science labs? For Exercises 12–14, select the appropriate operation(s) to solve the problem. Justify your selection(s) and solve the problem. 6. MONEY Mi-Ling has only nickels in her pocket. Julian has only quarters in his, and Aisha has only dimes in hers. Hannah approached all three for a donation for the school fund-raiser. What is the least each person could donate so that each one gives the same amount? TECHNOLOGY For Exercises 7 and 8, use the diagram and the information below. Seven closed shapes are used to make the digits 0 to 9 on a digital clock. (The number 1 is made using the line segments on the right side of the figure.) 8. Which line segment is used the least? UP 11. SCHOOL Of the 30 students in a science Use any strategy to solve Exercises 6–11. Some strategies are shown below. segment is used most often? THIS SIDE chose cake for dessert and 8 people chose ice cream. Five people chose both cake and ice cream. How many people had dessert? 5. GEOMETRY A stock clerk is 7. In forming these digits, which line 2 inches high are being packaged into boxes like the one below. If the tiles must be laid flat, how many will fit in one box? 12. MEASUREMENT An amusement park features giant statues of comic strip characters. If you multiply one character’s height by 4 and add 1 foot, you will find the height of its statue. If the statue is 65 feet tall, how tall is the character? 13. SPORTS The width of a tennis court is ten feet more than one-third its length. If the court is 78 feet long, what is its perimeter? 14. FLIGHTS A DC-11 jumbo jet carries 345 passengers with 38 in first-class and the rest in coach. For a day flight, a first-class ticket from Los Angeles to Chicago costs $650, and a coach ticket costs $230. What will be the ticket sales if the flight is full? Lesson 4-4 Problem-Solving Investigation: Draw a Diagram 205 4-5 Similar Polygons Main IDEA Identify similar polygons and find missing measures of similar polygons. Reinforcement of Standard 6NS1.3 Use proportions to solve problems. Use cross multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Follow the steps below to discover how the triangles at the right are related. F Copy both triangles onto tracing paper. D Measure and record the sides of each triangle. J E Cut out both triangles. 1. Compare the angles of the K triangles by matching them up. Identify the angle pairs that have equal measure. L 2. Express the ratios _, _, and _ as decimals to the nearest tenth. DF EF LK JK DE LJ 3. What do you notice about the ratios of these sides of matching triangles? NEW Vocabulary polygon similar corresponding parts congruent scale factor A polygon consists of a sequence of consecutive line segments in a plane, placed end to end to form a simple closed figure. Polygons that have the same shape are called similar polygons. In the figure below, polygon ABCD is similar to polygon WXYZ. This is written as polygon ABCD ∼ polygon WXYZ. B X A W C Y Z D The parts of similar figures that “match” are called corresponding parts. X X W W B A Y Z B A C 206 Y Z C D D Corresponding Angles A W, B X, C Y, D Z Corresponding Sides AB WX, BC XY, CD YZ, DA ZW Chapter 4 Proportions and Similarity READING Math The similar triangles in the Mini Lab suggest the following. Congruence The symbol is read is congruent to. Arcs are used to show congruent angles. +%9 #/.#%04 Similar Polygons If two polygons are similar, then • their corresponding angles are congruent, or have the same measure, and • the measures of their corresponding sides are proportional. Words Model B Y ABC ∼ XYZ A Symbols X C Z ∠A ∠X, ∠B ∠Y, ∠C ∠Z, and BC AC AB _ =_=_ YZ XY XZ Identify Similar Polygons 1 Determine whether rectangle HJKL is H similar to rectangle MNPQ. Explain. 3 First, check to see if corresponding angles are congruent. L Common Error Do not assume that two polygons are similar just because their corresponding angles are congruent. Their corresponding sides must also be proportional. MN JK 3 1 _ =_ or _ 6 NP 10 2 K 7 N 10 6 6 Q Next, check to see if corresponding sides are proportional. HJ 7 _ =_ 3 M Since the two polygons are rectangles, all of their angles are right angles. Therefore, all corresponding angles are congruent. J 7 KL 7 _ =_ 3 LH 1 _ =_ or _ 10 PQ P 10 6 QM 2 7 1 Since _ and _ are not equivalent ratios, rectangle HJKL is not similar 2 10 to rectangle MNPQ. Determine whether these polygons are similar. Explain. a. 8 6 b. 12 6 8 A 6 B J 8 14 D 14 6 3.5 K 1.5 1.5 M L 3.5 C The ratio of the lengths of two corresponding sides of two similar polygons is called the scale factor. You can use the scale factor of similar figures or a proportion to find missing measures. Extra Examples at ca.gr7math.com Lesson 4-5 Similar Polygons 207 Find Missing Measures W A 2 GEOMETRY Given that polygon 24 B WXYZ ∼ polygon ABCD, find the missing measure. 12 m D 10 C READING Math Segment Measure The −− measure of XY is written as XY. It represents a number. Z METHOD 1 X 13 15 Y Write a proportion. −− The missing measure m is the length of XY. Write a proportion that relates corresponding sides of the two polygons. XY YZ _ =_ polygon WXYZ polygon ABCD BC CD 15 m _ =_ XY = m, BC = 12, YZ = 15, and CD = 10. m · 10 = 12 · 15 10m = 180 Find the cross products. m = 18 Divide each side by 10. 12 METHOD 2 polygon WXYZ polygon ABCD 10 Multiply. Use the scale factor to write an equation. Find the scale factor from polygon WXYZ to polygon ABCD by finding the ratio of corresponding sides with known lengths. 15 3 YZ scale factor: _ =_ or _ CD 10 2 A length on Words polygon WXYZ Equation 3 m=_ (12) Write the equation. m = 18 Multiply. _ 3 as a length on polygon WXYZ. _3 times as a corresponding length long as on polygon ABCD. 2 m= 2 _ is −− Let m represent the measure of XY. Variable Scale Factor In Example 2, the scale factor from polygon ABCD to 2 polygon WXYZ is , 3 which means that a length on polygon 2 ABCD is as long The scale factor is the constant of proportionality. _3 · 12 2 Find each missing measure above. c. WZ d. AB Square A ∼ square B with a scale factor of 3:2. Notice the relationship between the scale factor and the ratio of their perimeters. M M -µÕ>ÀiÊ Chapter 4 Proportions and Similarity Perimeter A 12 m B 8m -µÕ>ÀiÊ perimeter of square A perimeter of square B 208 Square 3 12 _ =_ or 3:2 8 2 This and other related examples suggest the following. +%9 #/.#%043 Ratios of Similar Figures If two figures are similar with Words Model a a scale factor of _, then the a b b perimeters of the figures have a ratio of _a . Figure B b L 3 Triangle LMN is similar Similarity Statements In naming similar triangles, the order of the vertices indicates the corresponding parts. Read the similarity statement carefully to be sure that you compare corresponding parts. Figure A P 24 to triangle PQR. If the perimeter of LMN is 64 units, what is the perimeter of PQR? 18 R N M A 108 units C 48 units B 96 units D 36 units Q Read the Item You know that the two triangles are similar, and you know the measures of two corresponding sides and the perimeter of LMN. You need to find the perimeter of PQR. Solve the Item Triangle LMN ∼ triangle PQR with a scale factor of 24 4 4 _ or _ . The ratio of the perimeters of LMN to PQR is also _ . 18 3 3 Write and solve a proportion. Let x represent the perimeter of PQR. perimeter of LMN 64 4 ⎫ _ =_ ⎬ Scale factor relating LMN to PQR perimeter of PQR 3 x ⎭ 64 · 3 = 4 · x Find the cross products. 192 = 4x Multiply. 192 = 4x 4 4 Divide each side by 4. 48 = x Simplify. _ _ The answer is C. e. Rectangle KLMN is similar to rectangle TUVW. If the perimeter of rectangle KLMN is 32 units, what is the perimeter of rectangle TUVW? F 128 units H 64 units G 96 units J L 8 K M N U 16 V 40 units Personal Tutor at ca.gr7math.com T Lesson 4-5 Similar Polygons W 209 Example 1 (p. 207) Determine whether each pair of polygons is similar. Explain. 1. 2. 5 3 18 13 5 6 8 4 7.5 10 6 12 13.5 8 Example 2 3. In the figure at the right, FGH ∼ KLJ. (p. 208) F Write and solve a proportion to find each missing side measure. 6 9 3 L G 6 J y K x H Example 3 4. (p. 209) Y STANDARDS PRACTICE ABC is similar to XYZ. If the perimeter of ABC is 40 units, what is the perimeter of XYZ? A 10 units C 40 units B 20 units D 80 units B A X Determine whether each pair of polygons is similar. Explain. For Exercises 5–8 9–12 18, 19 5. 6. 3 3 3 3 Z C 16 (/-%7/2+ (%,0 See Examples 1 2 3 8 5 5 5 5 7. 8. 18 16 20 5 12 4 15 24 8 6 Each pair of polygons is similar. Write and solve a proportion to find each missing side measure. 9. 10. x 12 8 8 8 x 5 4 4.8 3 10 12 11. 29 x 10 21 210 22.4 12. Chapter 4 Proportions and Similarity 14.5 10.5 14 12.8 12 26 7.5 8 x 13. YEARBOOK The scale factor from the original %842!02!#4)#% See pages 686, 711. proof at the right to the reduced picture for a yearbook will be 8:5. Find the dimensions of the pictures as they will appear in the yearbook. 5 in. 14. MOVIES When projected onto a movie screen, the image from a film is 9 meters wide and 6.75 meters high. If the image from this same film is projected so that it appears 8 meters wide, what is the height of the projected image? Self-Check Quiz at ca.gr7math.com H.O.T. Problems 4 in. A 15. CHALLENGE True or false? If ABC ∼ X x XYZ, then _a = _ . Justify your answer. c z c z */ -!4( Determine whether (*/ 83 *5*/( C Z Y a B each statement is always, sometimes, or never true. Explain your reasoning. 16. Any two rectangles are similar. 17. Any two squares are similar. 18. Triangle FGH is similar to triangle RST. G 19. Quadrilateral ABCD is similar to quadrilateral WXYZ. R 36 in. 18 in. F 34 in. 27 in. S 1 A 13_ inches C 24 inches 2 inches B 22_ 1 D 25_ inches 3 6 in. B W 2 4 in. X Z D ? −− What is the length of TS? 2 A H T x C Y If the area of quadrilateral ABCD is 54 square units, what is the area of quadrilateral WXYZ? F 13.5 inches 2 H 27 inches 2 G 24 inches 2 J 36 inches 2 20. ROCK CLIMBING Grace is working her way up a climbing wall. Every 5 minutes she is able to climb 6 feet, but then loses her footing, slips back 1 foot, and decides to rest for 1 minute. If the rock wall is 30 feet tall, how long will it take her to reach the top? Use the draw a diagram strategy. (Lesson 4-4) 21. BAKING A recipe calls for 4 cups of flour for 64 cookies. How much flour is needed for 96 cookies? (Lesson 4-3) PREREQUISITE SKILL Graph and connect each pair of ordered pairs. (Lesson 3-6) 22. (-2.5, 1.5), (1.5, -3.5) 23. (-2, -1_12 ), (4, 3_12 ) 24. (-2_13 , 1), (2, 3_23 ) Lesson 4-5 Similar Polygons John Evans 211 CH APTER 4 Mid-Chapter Quiz Lessons 4-1 through 4-5 Express each ratio in simplest form. (Lesson 4-1) 13. 1. 32 out of 100 dentists 2. 12 tickets chosen out of 60 tickets 3. 300 points in 20 games Express each rate as a unit rate. (Lesson 4-1) F 12 6. 40 laps in 6 races A 25 B 30 H 24 J 48 1,860,000 miles in 10 seconds. How long will it take light to travel 93,000,000 miles from the Sun to Earth? (Lesson 4-3) 5. $420 for 15 tickets STANDARDS PRACTICE In her last race, Bergen swam 1,500 meters in 30 minutes. On average, how many meters did she swim per minute? (Lesson 4-1) G 16 14. MEASUREMENT Light travels approximately 4. 750 yards in 25 minutes 7. STANDARDS PRACTICE There are 2 cubs for every 3 adults in a certain lion pride. If the pride has 8 cubs, how many adults are there? (Lesson 4-3) 15. TELEVISION A typical 30-minute TV program has about 8 minutes of commercials. At that rate, how many commercial minutes are shown during a 2-hour TV movie? (Lesson 4-3) 16. MOVIES A section of a theater is arranged so that each row has the same number of seats. You are seated in the 5th row from the front and the 3rd row from the back. If your seat is 6th from the left and 2nd from the right, how many seats are in this section of the theater? Use the draw a diagram strategy. (Lesson 4-4) C 40 D 50 8. ICE CREAM In one 8-hour day, Bella’s Ice Cream Shop sold 72 cones of vanilla ice cream. In one hour, they sold 9 cones of vanilla ice cream. Is the total number of cones sold in one hour proportional to the number of cones sold during the day? Determine whether each pair of polygons is similar. Explain. (Lesson 4-5) (Lesson 4-2) 17. 9. DISHES Jack washed 60 plates in 30 minutes. It took him 3 minutes to wash 6 plates. Is the number of plates washed in 3 minutes proportional to the total number of plates he washed in 30 minutes? (Lesson 4-2) 18. Solve each proportion. (Lesson 4-3) 10. _ = _ 33 11 2 r 15 x 11. _ = _ 36 24 5 4.5 12. _ = _ 9 a 212 Chapter 4 Proportions and Similarity 19. MEASUREMENT Dollhouse furniture is similar in shape to full-sized furniture. A dollhouse chair is 6 inches high and 2.5 inches wide. If a full-sized chair is 36 inches tall, how wide is the chair? (Lesson 4-5) 4-6 Measurement: Converting Length, Weight/Mass, Capacity, and Time Main IDEA Convert customary and metric units of length, weight or mass, capacity, and time. Standard 7MG1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g. miles per hour and feet per second, cubic inches to cubic centimeters). Jesse Owens set a record of 9.4 seconds for the 100-yard dash at the Big Ten track meet in Ann Arbor, Michigan, on May 25, 1935. The next year at the 1936 Olympic Games in Berlin, he astounded the world by matching the world record of 10.3 seconds in the 100-meter race. How did the lengths of the races compare? 1. A yard is a unit of length in the customary system. Name another unit of length in the customary system. 2. A meter is a unit of length in the metric system. Name another unit NEW Vocabulary unit ratio of length in the metric system. 3. Explain why the college race was measured in yards and the Olympic race was measured in meters. The relationships among the most commonly used customary and metric units of length, weight or mass, capacity, and time are shown in the table below. +%9 #/.#%04 Measurement Conversions Customary Units Metric Units Length 1 foot (ft) = 12 inches (in.) 1 yard (yd) = 3 feet 1 mile (mi) = 5,280 feet 1 meter (m) = 1,000 millimeters (mm) 1 meter = 100 centimeters (cm) 1 kilometer (km) = 1,000 meters Weight Mass 1 pound (lb) = 16 ounces (oz) 1 ton (T) = 2,000 pounds 1 gram (g) = 1,000 milligrams (mg) 1 kilogram (kg) = 1,000 grams Capacity 1 cup (c) = 8 fluid ounces (fl oz) 1 pint (pt) = 2 cups 1 quart (qt) = 2 pints 1 gallon (gal) = 4 quarts 1 liter (L) = 1,000 milliliters (mL) 1 kiloliter (kL) = 1,000 liters Time 1 minute (min) = 60 seconds (s) 1 hour (h) = 60 minutes 1 day (d) = 24 hours 1 week (wk) = 7 days 1 year (yr) = 365 days Lesson 4-6 Measurement: Converting Length, Weight/Mass, Capacity, and Time Bettmann/CORBIS 213 Each of the relationships in the table can be written as a unit ratio. Like a unit rate, a unit ratio is one in which the denominator is 1 unit. 2,000 lb _ 3 ft _ 1,000 m _ 1T 1 yd 1 km 24 h _ 1d Notice that the numerator and denominator of each fraction above are equivalent, so the value of each ratio is 1. You can multiply by a unit ratio of this type to convert or change from larger units to smaller units. Convert Larger Units to Smaller Units 1 Convert 12 yards to feet. 3 ft Since 1 yard = 3 feet, the unit ratio is _ . You should always write the units to ensure that the correct units are being cancelled. 1 yd 3 ft 12 yd = 12 yd · _ 1 yd _ = 12 yd · 3 ft 1 yd = 12 · 3 ft or 36 ft Multiply by _. 3 ft 1 yd Divide out common units, leaving the desired unit, feet. Multiply. So, 12 yards = 36 feet. Complete each conversion. a. 27 yd = ft 1 b. 3_ qt = pt 2 c. 5 km = m d. 7.5 L = mL To convert from smaller units to larger units, multiply by the reciprocal of the appropriate unit ratio. Convert Smaller Units to Larger Units 2 BANNERS Carleta needs 450 centimeters of material to make a banner for a parade. How many meters of material does she need? 1m 450 cm = 450 cm · _ Since 1 meter = 100 centimeters, multiply 100 cm 1m by _. 1m = 450 cm · _ 100 cm 450 =_ m or 4.5 m 100 100 cm Divide out common units, leaving the desired unit, meter. Multiply. So, Carleta needs 4.5 meters of material. Real-World Link The Rose Bowl, “The Granddaddy of Them All,” has been a sellout attraction every year since 1947. Source: tournamentofroses.com 214 Complete each conversion. e. 56 oz = lb f. 48 in. = ft g. 150 mL = L h. 4,000g = kg Chapter 4 Proportions and Similarity AP Photo/Stefan Paltera Extra Examples at ca.gr7math.com REVIEW Vocabulary dimensional analysis The process of including units of measurement when you compute. (p. 98) You can also use dimensional analysis to convert between measurement systems. The table shows conversion factors for units of length, capacity, and mass or weight. +%9 #/.#%04 Metric/Customary Measurement Conversions Length Capacity and Mass or Weight 1 in. ≈ 2.54 cm 1 fl oz ≈ 29.574 mL 1 ft ≈ 0.305 m 1 pt ≈ 0.473 L 1 yd ≈ 0.914 m 1 qt ≈ 0.946 L 1 mi ≈ 1.609 km 1 gal ≈ 3.785 L 1 cm ≈ 0.394 in. 1 oz ≈ 28.35 g 1 m ≈ 1.094 yd 1 lb ≈ 0.454 kg 1 km ≈ 0.621 mi Convert Between Systems 3 Dimensional Analysis Choose conversion factors that allow you to divide out common units. Convert 9 centimeters to inches. METHOD 1 Use 1 in. ≈ 2.54 cm. 1 in. 9 cm ≈ 9 cm · _ 2.54 cm 1 in. 9 cm ≈ 9 cm · _ 2.54 cm 9 in. ≈_ or 3.54 in. 2.54 Since 1 in. ≈ 2.54 cm, multiply by _. 1 in. 2.54 cm Divide out common units, leaving the desired unit, inch. Multiply. METHOD 2 Use 1 cm ≈ 0.394 in. 0.394 in. 9 cm ≈ 9 cm · _ 1 cm 0.394 in. _ 9 cm ≈ 9 cm · 1 cm Multiply by _. 0.394 in. 1 cm Divide out common units, leaving the desired unit, inch. ≈ 9 · 0.394 in. or 3.54 in. Multiply. So, 9 centimeters is approximately 3.54 inches. Complete each conversion. Round to the nearest hundredth. i. 6 oz = g j. 5 km = mi k. 6 yd = m l. 2 L = qt Personal Tutor at ca.gr7math.com Lesson 4-6 Measurement: Converting Length, Weight/Mass, Capacity, and Time 215 Convert Units Using Multiple Steps 4 ANIMALS A sloth’s top speed is 1.9 kilometers per hour. How fast is this in feet per seconds? To convert kilometers to feet, use conversion factors relating kilometers to miles and miles to feet. To convert hours to seconds, use conversion factors relating hours to minutes and minutes to seconds. 1.9 km _ 5280 ft _ 1 min _ · 1 mi · _ · 1h ·_ 60 min 60 sec 1 mi 1.609 km 1.9 km _ 5280 ft _ 1 min =_ · 1 mi · _ · 1h ·_ 60 min 60 s 1 mi 1h 1.609 km 10,032 ft _ = 5,792.4 s 1.73 ft =_ 1s 1h Divide out common units. Multiply. Divide. The sloth’s top speed is 1.73 feet per second. m. A vehicle can travel 11 kilometers per 1 liter of gasoline. How many miles per gallon is this? Examples 1, 2 (p. 214) Complete. 1. 5 lb = oz 2. 8_ yd = ft 3. 630 min = h 4. 686 cm = m 2 3 5. FISH The average weight of a bass in a certain pond is 40 ounces. About how many pounds does a bass weigh? Examples 3, 4 (p. 215–216) Complete each conversion. Round to the nearest hundredth if necessary. 6. 6 in. ≈ cm 7. 1.6 cm ≈ in. 8. 4 qt ≈ L 9. 50 mL ≈ fl oz 10. 50 mph ≈ ft/s 11. 50 gal/h ≈ L/min 12. 350 cm/s ≈ in./min 13. 15 km/min ≈ mi/h 14. How many inches are in 54 centimeters? 15. Convert 17 miles to kilometers. 16. COOKING For a holiday dinner, Joanna peeled 2 pounds of potatoes in 15 minutes. How many ounces did she peel per minute? 17. MILEAGE A certain vehicle travels an average of 18 miles per gallon of gasoline. How many kilometers can it travel per one liter of gasoline? 216 Chapter 4 Proportions and Similarity (/-%7/2+ (%,0 For Exercises 18–28 29–38 39–46 See Examples 1, 2 3 4 Complete. 18. 22 ft = yd 19. 104 oz = lb 20. 4 lb = oz 21. 6 gal = qt 22. 2_ pt = c 23. 5_ c = fl oz 24. 75 min = h 1 2 3 25. 3_ mi = ft 4 1 2 26. 9,000 lb = T 27. How many pounds are in 76 ounces? 28. Convert 11,400 milligrams to grams. Complete each conversion. Round to the nearest hundredth if necessary. 29. 5 in. ≈ cm 30. 5 gal ≈ L 31. 15 cm ≈ in. 32. 17 m ≈ yd 33. 2 L ≈ qt 34. 10 mL ≈ fl oz 35. 2,000 lb ≈ kg 36. 63.5 kg ≈ lb 37. Convert 1.4 quarts to milliliters. 38. How many pounds are there in 19 kilograms? Complete each conversion. Round to the nearest hundredth if necessary. 39. 20 oz/min ≈ qt/day 40. 70 mi/h ≈ ft/s 41. 16 fl oz/h ≈ mL/min 42. 150 fl oz/day ≈ L/h 43. 52 mi/h ≈ km/min 44. 15 gal/h ≈ L/min 45. In meters per second, how fast is 1,550 feet per minute? 46. A storage bin is being filled at a rate of 2,350 pounds per hour. What is the rate in kilograms per minute? Determine which is greater. 47. 3 gal, 10 L 48. 14 oz, 0.4 kg ROLLER COASTERS For Exercises 50–51, use the table that lists the fastest and tallest roller coasters on three different continents. 50. Order the roller coasters from greatest to least speeds. 51. Order the roller coasters from tallest to shortest. %842!02!#4)#% 52. WATER Which is greater: 64 fluid ounces of water or 2 liters of See pages 686, 711. water? Explain your reasoning. Self-Check Quiz at ca.gr7math.com 49. 4 mi, 6.2 km Fastest Roller Coasters Continent Name Speed Asia Dodonpa 172 kph Europe Stealth 128 kph North America Kingda Ka 128 mph Tallest Roller Coasters Continent Name Height Asia Steel Dragon 2000 97 m Europe Silver Star 73 m North America Kingda Ka 456 ft Source: rcdb.com 53. FOOD Which is greater: a 1.5-pound box of raisins or a 650-gram box of raisins? Explain your reasoning. Lesson 4-6 Measurement: Converting Length, Weight/Mass, Capacity, and Time 217 H.O.T. Problems 54. FIND THE ERROR Pedro and Alex are converting 2 liters. Who is correct? Explain your reasoning. Pedro 2.144 qt Alex 0.946 pt 55. CHALLENGE To make it around the track, a roller coaster must achieve a speed of at least 76 miles per hour. At top speed, the coaster traveled 136 meters in 4.3 seconds. Is the coaster traveling fast enough to make it completely around the track? Explain. */ -!4( Refer to the information at the beginning of the (*/ 83 *5*/( 56. lesson. Explain how you can compare the 100-yard dash and the 100-meter dash. Compare Owens’ records in the two events. 57. How many millimeters are in 5 58. 120 kilometers per hour is the same centimeters? rate as which of the following? A 0.05 F 2 kilometers per second B 0.5 G 2 kilometers per minute C 50 H 12 kilometers per minute D 500 J 720 kilometers per second 59. The triangles at the right are similar. Write and solve a 3 in. proportion to find the missing measure. (Lesson 4-5) 8 in. 4.5 in. Solve each proportion. (Lesson 4-3) y 12 60. _ = _ 5 4 61. _ = _ 120 b 24 60 62. _ = _ n 0.6 5 1.5 m in. 63. TECHNOLOGY A hiker uses her GPS (Global Positioning System) receiver to find how much farther she needs to go to get to her stopping point for the day. She is at the red dot on her GPS receiver screen, and the blue dot shows her destination. How much farther does she need to travel? (Lesson 3-7) PREREQUISITE SKILL Find the area of each rectangle. (p. 674) 64. 65. 11 cm 25 ft 39 ft 7 cm 218 Chapter 4 Proportions and Similarity 2 mi. Extend 4-6 Main IDEA Use a spreadsheet to solve problems involving conversions of measurements within and between systems. Standard 7MG1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g. miles per hour and feet per second, cubic inches to cubic centimeters). Standard 7MR3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. Spreadsheet Lab Converting Measures You can use a spreadsheet to convert measurements. COOKING Your cooking class exchanges recipes with a cooking class in France. The class in France sends the following recipe for a Soufflé au Fromage, or cheese soufflé. Find the amount of Swiss cheese, butter, and flour in ounces. To solve the problem, set up a spreadsheet. Excel sample.xls B A 1 C D E Amount Metric Unit Amount Customary Unit ounce 2 Conversion Relationship 28.35 grams 3 4 5 6 7 8 Ingredient Swiss cheese butter plain flour Amount 70 30 20 Metric Unit Amount Customary Unit grams =D2/B2*B5 ounces grams ounces grams ounces Sheet 1 Sheet 2 Sheet 3 1 ANALYZE THE RESULTS 1. Explain the formula in D5. 2. What formulas should be entered in cells D6 and D7? 3. What would you enter into cells B3, C3, D3, and E3 to convert the amount of milk in the recipe from milliliters to fluid ounces? 4. What would you enter into cells A8, B8, C8, D8, and E8 to convert the amount of milk in the recipe to fluid ounces? 5. OPEN ENDED Find another recipe in which ingredients are given in grams or milliliters. Use a spreadsheet to convert these measures into ounces or fluid ounces. Extend 4-6 Spreadsheet Lab: Converting Measures 219 4-7 Measurement: Converting Square Units and Cubic Units Main IDEA Convert square and cubic units of length, weight or mass, capacity, and time in both customary and metric systems. Standard 7MG1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g. miles per hour and feet per second, cubic inches to cubic centimeters). Standard 7MG2.4 Relate the changes in measurement with a change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = 144 square inches or [1 ft 2] = [144 in 2], 1 cubic inch is approximately 16.38 cubic centimeters or [1 in 3] = [16.38 cm 3]). GAMES A puzzle cube can help you understand how to convert measures of area and volume. 1. Look at one face of a puzzle cube. How many cubes are there along each edge? How many squares are there on one face? How many small cubes are there in all? 2. What is the relationship between the number of cubes along each edge and the number of squares on one face? between the number of cubes along each edge and the total number of small cubes? 3. How is the number of square feet in one square yard related to the number of feet in one yard? Some units of area in the customary system are square inch (in 2), square foot (ft 2), square yard (yd 2), and square mile (mi 2). Some units of area in the metric system are square centimeter (cm 2) and square meter (m 2). Just as you used unit ratios to convert units of length, you can use unit ratios when you convert units of area. Convert Units of Area READING Math Units of Area and Volume Remember that ft 2 is the same as ft × ft and cm 3 is the same as cm × cm × cm. Complete each conversion. 1 2 ft 2 = in 2 12 in. 12 in. 2 ft 2 = 2 × ft × ft × _ ×_ 1 ft Multiply by _. 12 in. 1 ft 1 ft = 288 in 2 2 4,800 cm 2 = m 2 1m 1m 4,800 cm 2 = 4,800 × cm × cm × _ ×_ 100 cm 4,800 m 2 10,000 =_ 100 cm Multiply by _. 1m 100 cm Simplify. = 0.48 m 2 Complete each conversion. 220 a. 1.5 ft 2 = in 2 b. 45 ft 2 = yd 2 c. 24 cm 2 = m 2 d. 3.2 km 2 = m 2 Chapter 4 Proportions and Similarity Todd Yarrington Extra Examples at ca.gr7math.com Some units of volume in the customary system are cubic inch (in 3), cubic foot (ft 3), cubic yard (yd 3), and cubic mile (mi 3). Some units of volume in the metric system are cubic centimeter (cm 3) and cubic meter (m 3). Convert Units of Volume 3 BUILDING How many cubic yards of concrete will a builder need for a rectangular driveway that has a volume of 132 cubic feet? 1 yd 3 ft 1 yd 3 ft 1 yd 3 ft 132 ft 3 = 132 × ft × ft × ft × _ × _ × _ Multiply by _. 132 yd 3 27 1 yd 3 ft =_ Multiply. ≈ 4.89 yd 3 Simplify. The builder needs 4.89 cubic yards of concrete. e. How many cubic meters of concrete are needed for a sidewalk that has a volume of 280,000 cubic centimeters? f. A homeowner needs 150 cubic feet of mulch. Mulch is sold by the Look Back You can review conversion factors in Lesson 4-6. cubic yard. How many cubic yards does he need to buy? Personal Tutor at ca.gr7math.com You can also use conversion factors to convert area and volume between the customary and metric systems. Convert Between Systems 4 Convert 12 square centimeters to square inches. 1 in. 1 in. 12 cm 2 = 12 × cm × cm × _ ×_ 2.54 cm 2 12 in =_ 2.54 cm Multiply by _. 1 in. 2.54 cm Multiply. 6.45 ≈ 1.86 in 2 Simplify. So, 12 square centimeters is approximately 1.86 square inches. 5 Convert 7 cubic inches to cubic centimeters. 2.54 cm 2.54 cm 2.54 cm 7 in 3 = 7 × in. × in. × in. × _ ×_ ×_ 3 114.71 cm =_ 1 ≈ 114.71 cm 3 1 in. 1 in. 1 in. Multiply. Simplify. So, 7 cubic inches is approximately 114.71 cubic centimeters. Complete each conversion. Round to the nearest hundredth. g. 25 mi 2 ≈ km 2. h. 23 in 3 ≈ cm 3 i. 750 ft 2 = m 2. j. 212 km 3 = mi 3 Lesson 4-7 Measurement: Converting Square Units and Cubic Units 221 Examples 1, 2 (p. 220) Complete each conversion. 1. 3 ft 2 = in 2 2. 2 yd 2 = ft 2 3. 15 ft 2 = yd 2 4. 10.8 cm 2 = mm 2 5. 148 mm 2 = cm 2 6. 0.264 km 2 = m 2 7. REMODELING Suppose you have a room that is 270 square feet in area. How many square yards of carpet would cover this room? Examples 3–5 (p. 221) (/-%7/2+ (%,0 For Exercises 16–24 25–32 33–42 See Examples 1, 2 3 4 Complete each conversion. Round to the nearest hundredth. 8. 1.5 ft 3 = in 3 9. 4.3 yd 3 = ft 3 10. 0.006 m 3 = mm 3 11. 2,400 cm 3 = m 3 12. 10 ft 2 ≈ m 2 13. 144 in 2 ≈ cm 2 14. 25 m 3 ≈ yd 3 15. 250 ft 3 ≈ m 3 Complete each conversion. Round to the nearest hundredth if necessary. 16. 1.6 yd 2 = ft 2 17. 10.4 ft 2 = in 2 18. 150 ft 2 = yd 2 19. 504 in 2 = ft 2 20. 1.6 m 2 = cm 2 21. 4,654 cm 2 = m 2 22. 0.058 km 2 = m 2 23. 37,200 m 2 = km 2 24. BIOLOGY The total surface area of the average adult’s skin is about 21.5 square feet. Convert this measurement to square inches. Complete each conversion. Round to the nearest hundredth if necessary. 25. 2 ft 3 = in 3 26. 0.4 ft 3 = in 3 27. 300 yd 3 = ft 3 28. 0.00397 km 3 = m 3 29. 16,000 cm 3 = m 3 30. 22 m 3 = cm 3 31. BALLOONS A standard hot air balloon holds about 2,000 cubic meters of hot air. How many cubic centimeters is this? 32. LANDSCAPING A landscape architect is designing the outside of a new restaurant. She needs 5 cubic yards of stone to cover a certain area. Will 100 cubic feet of stones be enough? If not, how many cubic feet are needed? Complete each conversion. Round to the nearest hundredth. Real-World Link This is a close up of a skin cell. The average person loses about 9 pounds of skin cells a year. Source: kidshealth.org 222 33. 10 ft 3 ≈ m 3 34. 25 m 2 ≈ yd 2 35. 240 in 2 ≈ cm 2 36. 2 mi 3 ≈ km 3 37. 120 cm 2 ≈ in 2 38. 4 yd 3 ≈ m 3 39. 45 in 3 ≈ cm 3 40. 108 ft 2 ≈ m 2 41. 37m 3 ≈ ft 3 42. PAINT One gallon of paint can cover 400 square feet of wall. How many square meters will one gallon of paint cover? Chapter 4 Proportions and Similarity Steve Gschmeissner/Photo Researchers, Inc. 43. MICROWAVES The inside of a microwave oven has a volume of 1.2 cubic %842!02!#4)#% See pages 687, 711. feet and measures 18 inches wide and 10 inches long. Using the formula V = wh, find the depth of the microwave to the nearest tenth of an inch. Self-Check Quiz at 44. MEASUREMENT The density of gold is 19.29 grams per cubic centimeter. To ca.gr7math.com the nearest hundredth, find the mass in grams of a gold bar that is 0.75 inch by 1 inch by 0.75 inch. Use the relationship 1 cubic inch ≈ 16.38 cubic centimeters. H.O.T. Problems 45. Which One Doesn’t Belong? Identify which equivalent measure does not belong with the other three. Explain. 5.2 yd 3 6.8 m 3 15.6 ft 3 242,611.2 in 3 46. CHALLENGE A hectare is a metric unit of area approximately equal to 10,000 square meters or 2.47 acres. The base of the Great Pyramid of Khufu is a 230-meter square. About how many acres does the base cover? 47. 230 m 230 m */ -!4( Describe a real-world situation in which converting (*/ 83 *5*/( units of area or volume is necessary. 48. The area of a roof that needs new 49. Approximately how many cubic shingles is 40 square yards. How many square feet of shingles are needed? feet are there in six cubic meters? Use 1 m 3 ≈ 35.31 ft 3. A 4.44 ft 2 F 5.89 H 41.31 G 29.31 J B 120 ft 2 C 360 ft 2 D 1,600 ft 2 211.86 50. COMPUTERS A notebook computer has a mass of 2.25 kilograms. Approximately how many pounds does the notebook weigh? (Use 1 lb ≈ 0.4536 kg.) (Lesson 4-6) 51. Determine whether the polygons at the right are similar. Explain your reasoning. (Lesson 4-5) 1.5 2 2.4 2 3.2 4.8 Find each product or quotient. Write in simplest form. (Lessons 2-3 and 2-4) 52. _ ÷ _ 5 12 3 20 53. -_ · _ 7 48 9 14 3.2 3 54. 2_ · 1_ 3 4 55. -3_ ÷ -_ 2 3 1 5 ( 23 ) PREREQUISITE SKILL Solve. (Lesson 4-3) 56. _ = _ 3 cm 5 ft x cm 9 ft 57. _ = _ 4 in. 5 mi 5 in. x mi Lesson 4-7 Measurement: Converting Square Units and Cubic Units William Floyd Holdman/Index Stock Imagery 223 4-8 Scale Drawings and Models Main IDEA Standard 7MG1.2 Construct and read drawings and models made to scale. NEW Vocabulary scale drawing scale model scale FLOOR PLANS The blueprint for a bedroom is given below. 1. How many units wide is width the room? 2. The actual width of the closet Solve problems involving scale drawings. room is 18 feet. Write a ratio comparing the drawing width to the actual width. 3. Simplify the ratio you found and compare it to the scale shown at the bottom of the drawing. ⫽2 ft A scale drawing or a scale model is used to represent an object that is too large or too small to be drawn or built at actual size. The scale is the ratio of a length on a drawing or model to its actual length. 1 inch = 4 feet 1 inch represents an actual distance of 4 feet. 1:30 1 unit represents an actual distance of 30 units. Distances on a scale drawing are proportional to distances in real life. Use a Scale Drawing 1 GEOGRAPHY Use the map to find the actual distance between Grenada, Mississippi, and Little Rock, Arkansas. Use a centimeter ruler to measure the map distance. The map distance is about 5.2 centimeters. METHOD 1 Scales Scales and scale factors are always written so that the drawing length comes first in the ratio. 224 -EMPHIS ,ITTLE 2OCK !2+!.3!3 -)33)33)00) 'RENADA +EY CM KM % 7 Write and solve a proportion. Let x represent the actual distance to Little Rock. Scale map actual Chapter 4 Proportions and Similarity Grenada, MS to Little Rock, AR 1 cm 5.2 cm _ =_ 50 km x km 1 · x = 50 · 5.2 x = 260 map actual Find the cross products. Simplify. METHOD 2 Write and solve an equation. 50 km Write the scale as _ , which means 50 kilometers per centimeter. 1 cm The actual distance Words is 50 kilometers per centimeter of map distance. Variables Let a represent the actual distance in kilometers. Let m represent the map distance in centimeters. Equation a = 50 m · a = 50m Write the equation. a = 50(5.2) or 260 Replace m with 5.2 and multiply. The actual distance between the two cities is about 260 kilometers. GEOGRAPHY Use an inch ruler and the map shown to find the actual distance between each pair of cities. Measure to the nearest quarter of an inch. . / 24 ( # ! 2 / , ) . ! #HARLOTTE 'ASTONIA 3/54( #!2/,).! 3PARTANBURG a. Spartanburg and Gastonia +EY IN MI b. Charlotte and Spartanburg Find the Scale 2 MODEL TRAINS A passenger car of a model train is 6 inches long. If the actual car is 80 feet long, what is the scale of the model? Let x represent the actual length of the train in feet corresponding to 1 inch in the model. Use a proportion. Length of Train Real-World Link Some of the smallest model trains are built on the Z scale. Using this scale, models are 1 the size of real model actual trains. 6 in. 1 in. _ =_ 80 ft model actual Find the cross products. x ft 6 · x = 80 · 1 6x 80 _ =_ 6 Multiply. Then divide each side by 6. 6 1 x = 13 _ _ 220 Scale Simplify. 3 1 So, the scale is 1 inch = 13 _ feet. 3 Source: www.nmra.org c. ARCHITECTURE The model Mr. Vicario made of the building he designed is 25.6 centimeters tall. If the actual building is to be 64 meters tall, what is the scale of his model? Extra Examples at ca.gr7math.com Doug Martin Lesson 4-8 Scale Drawings and Models 225 The scale factor for scale drawings and models is the scale written as a unitless ratio in simplest form. Find the Scale Factor 3 Find the scale factor for the model train in Example 2. Scale Factors A scale factor between 0 and 1 means that the model is smaller than the actual object. A scale factor greater than 1 means that the model is larger than the actual object. 1 in. 1 in. _ =_ 1 13 _ ft Convert 13 160 in. _1 feet to inches by multiplying by 12. 3 3 1 1 The scale factor is _ or 1:160. This means that the model train is _ 160 160 the size of the actual train. Find the scale factor for each scale. d. 1 inch = 15 feet e. 10 cm = 2.5 m To construct a scale drawing of an object, find an appropriate scale. Construct a Scale Model 4 SOCIAL STUDIES Each column of the Lincoln Memorial is 44 feet tall. Michaela wants the columns of her model to be no more than 12 inches tall. Choose an appropriate scale and use it to determine how tall she should make the model of Lincoln’s 19-foot statue. Try a scale of 1 inch = 4 feet. x in. 1 in. _ =_ 4 ft 44 ft 1 · 44 = 4 · x model actual Find the cross products. 44 = 4x Multiply. 11 = x Divide each side by 4. Using this scale, the columns would be 11 inches tall. Use this scale to find the height of the statue. y in. 1 in. _ =_ 4 ft 19 ft 1 · 19 = 4 · y 19 = 4y 3 =y 4_ 4 3 inches tall. The statue should be 4 _ 4 f. LIFE SCIENCE Kaliah is making a model of the human ear and wants the stirrup bone to be between 1 and 2 centimeters long. An actual stirrup bone is about 3 millimeters long. Choose an appropriate scale and use it to determine how tall his model of an actual 54-millimeter tall ear should be. Personal Tutor at ca.gr7math.com 226 Chapter 4 Proportions and Similarity Example 1 (p. 224) GEOGRAPHY Use the map and an inch ruler to find the actual distance between each pair of cities. ).$)!.! 1. Evansville and Louisville %VANSVILLE 2. Louisville and Elizabethtown %LIZABETHTOWN +EY IN MI MONUMENTS For Exercises 3 and 4, use the following information. Examples 2 and 3 (pp. 225–226) ,OUISVILLE +%.45#+9 At 555 feet tall, the Washington Monument is the highest all-masonry tower. 3. If a scale model of the monument is 9.25 inches high, what is the scale? 4. What is the scale factor for the model? Example 4 5. DECORATING Before redecorating, Nichelle makes a scale drawing of her (p. 226) bedroom on an 8.5- by 11-inch piece of paper. If the room is 10 feet wide by 12 feet long, choose an appropriate scale for her drawing and find the dimensions of the room on the drawing. (/-%7/2+ (%,0 For Exercises 6–11 12–13 14–15 16–17 See Examples 1 2 3 4 FLOOR PLANS For Exercises 6–11, use the portion of an architectural drawing shown and an inch ruler. Fabulous Homes Master Bath Master Bedroom Kitchen and Dining Area Bedroom 2 Living Room Porch Ranch Style Floor Plan Half Bath Key 1 in. = 12 ft Find the actual length and width of each room. Measure to the nearest eighth of an inch. 6. half bath 7. master bath 8. porch 9. bedroom 2 10. master bedroom 11. living room 12. MOVIES One of the models of a dinosaur used in the filming of a movie was only 15 inches tall. In the movie, the dinosaur appeared to have an actual height of 20 feet. What was the scale of the model? Lesson 4-8 Scale Drawings and Models 227 13. LIFE SCIENCE The paramecium shown at the right is a single-celled organism that is 0.006 millimeter long. Find the scale of the drawing. 4 cm 14. FLOOR PLANS What is the scale Paramecium factor of the floor plan used in Exercises 6–11? Explain its meaning. 15. MOVIES What is the scale factor of the model used in Exercise 12? 16. SPIDERS A tarantula’s body length is 5 centimeters. Choose an appropriate scale for a model of the spider that is to be just over 6 meters long. Then use it to determine how long the tarantula’s 9-centimeter legs should be. Real-World Link Earth has an approximate circumference of 40,000 kilometers, while the Moon has an approximate circumference of 11,000 kilometers. Source: infoplease.com 17. AIRPLANES Dorie is building a model of a DC10 aircraft. The actual aircraft is 182 feet long and has a wingspan of 155 feet. If Dorie wants her model to be no more than 2 feet long, choose an appropriate scale for her model. Then use it to find the length and wingspan of her model. SPACE SCIENCE For Exercises 18 and 19, use the information at the left. 18. Suppose you are making a scale model of Earth and the Moon. You decide to use a basketball to represent Earth. A basketball’s circumference is about 30 inches. What is the scale of your model? 19. Which of the following should you use to represent the Moon in your model so it is proportional to the model of Earth in Exercise 18? (The number in parentheses is the object’s circumference.) Explain. a. a soccer ball (28 in.) %842!02!#4)#% See pages 687, 711. c. a golf ball (5.25 in.) b. a tennis ball (8.25 in.) d. a marble (4 in.) 20. TRAVEL On a map of Illinois, the distance between Champaign and 3 1 Carbondale is 6_ inches. If the scale of the map is _ inch = 15 miles, about 2 4 ca.gr7math.com how long would it take the Kowalski family to drive from Champaign to Carbondale if they drove 60 miles per hour? H.O.T. Problems 21. OPEN ENDED Choose a large or small rectangular item such as a calculator, Self-Check Quiz at table, or room. Find its dimensions and choose an appropriate scale for a scale drawing of the item. Then construct a scale drawing and write a problem that could be solved using your drawing. 22. FIND THE ERROR On a map, 1 inch represents 4 feet. Jacob and Luna are finding the scale factor of the map. Who is correct? Explain. scale factor: 1:4 Jacob 228 Chapter 4 Proportions and Similarity (tl)NASA, (tr)M.I. Walker/Photo Researchers, (bl)RubberBall/SuperStock, (br)PNC/Getty Images scale factor: 1:48 Luna 23. CHALLENGE Describe how you could find the scale on a map that did not have a scale printed on it. 24. */ -!4( One model is built on a 1:75 scale. Another model of (*/ 83 *5*/( the same object is built on a 1:100 scale. Which model is larger? Explain. 25. Jevonte is building a model of a ship 26. The actual width w of a garden is with an actual length of 15 meters. 18 feet. Use the scale drawing of the garden to find the actual length . 22 cm x 3.6 in. 5 in. F 17.2 ft 60 cm G 18 ft What other information is needed to find x, the height of the model’s mast? H 20 ft J A the overall width of the ship 25 ft B the scale factor used C the overall height of the mast D the speed of the ship in the water Complete each conversion. Round to the nearest hundredth if necessary. (Lesson 4-7) 27. 4ft 3 = ? yd 3 28. 160 cm 2 = ? m2 29. 6 m 2 = ? ft 2 30. MEASUREMENT The speed limit on a Canadian highway is 100 kilometers per hour. Approximately how fast can you drive on this highway in miles per hour? (Lesson 4-6) 31. MEASUREMENT Makiah has ten liters of water. She wants to pour the water into gallon jugs. To the nearest hundredth, how many gallons of water does she have? (Lesson 4-6) Estimate each square root to the nearest whole number. (Lesson 3-2) 32. √ 11 33. √ 48 34. - √ 118 PREREQUISITE SKILL Evaluate each expression. (Lesson 1-2) 35. _ 45 - 33 10 - 8 36. _ 85 - 67 2001 - 1995 37. _ 29 - 44 55 - 50 38. _ 18 - 19 25 - 30 Lesson 4-8 Scale Drawings and Models 229 4-9 Rate of Change Main IDEA Find rates of change. Preparation for Standard 7AF3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. NEW Vocabulary rate of change E-MAIL The table shows the number of entries in Alicia’s e-mail contact list at the end of 2004 and 2006. Alicia’s E-mail Contact List Year 2004 2006 Entries 10 38 1. What is the change in the number of entries from 2004 to 2006? 2. Over what number of years did this change take place? 3. Write a rate that compares the change in the number of entries to the change in the number of years. Express your answer as a unit rate and explain its meaning. A rate of change is a rate that describes how one quantity changes in relation to another. Find a Positive Rate of Change 1 E-MAIL Alicia had 62 entries in her e-mail contact list at the end of 2007. Use the information above to find the rate of change in the number of entries in her e-mail contact list between 2004 and 2007. +? The change or difference in the number of years is 2007–2004. Year 2004 2007 Entries 10 62 +? The change or difference in the number of entries is 62–10. Write a rate that compares the change in each quantity. change in entries (62 - 10) entries __ = __ Mental Math You can also find a rate of change, or unit rate, by dividing the numerator by the denominator. change in years (2007 - 2004) years Her contact list changed from 10 to 62 entries from 2004 to 2007. 52 entries =_ 3 years Subtract to find the change in the number of entries and years. 17 entries ≈_ Express this rate as a unit rate. 1 year Since this rate is positive, Alicia’s e-mail contact list increased or grew at an average rate of about 17 entries per year between 2004 and 2007. a. HEIGHTS The table shows Ramon’s height at ages 8 and 11. Find the rate of change in his height between these ages. 230 Chapter 4 Proportions and Similarity Gary Atkinson/Photonica/Getty Images Age (yr) 8 11 Height (in.) 51 58 Broken Line Graph In these 2 line graphs, the lines are broken because there are no data points between the points on the graph. A graph of the data in Example 1 is shown at the right. The data points are connected by a segment. A positive rate of change is shown by a segment slanting upward from left to right. Rates of change can also be negative. Find a Negative Rate of Change 2 MUSIC The graph shows cassette sales from 1994 to 2002. Find the rate of change in sales between 2000 and 2002, and describe how this rate is shown on the graph. Make a table of the data being considered using the coordinates of the points listed on the graph. Year Sales (millions of $) 2000 4.9 2002 2.4 Source: Recording Industry Association of America Use the data to write a rate comparing the change in sales to the change in years. change in sales 2.4 - 4.9 __ =_ change in years Rates of Change On a graph, the rate of change is the ratio of the change in y-values to the change in the x-values between two data points. 2002 - 2000 -2.5 =_ 2 -1.25 =_ 1 Sales changed from $4.9 million to $2.4 million from 2000 to 2002. Subtract to find the change in sales amounts and years. Express as a unit rate. The rate of change was -1.25 million dollars in sales per year. The rate is negative because the cassette sales decreased between 2000 and 2002. This is shown on the graph by a line segment slanting downward from left to right. b. In the graph above, find the rate of change between 1994 and 1996. c. Describe how this rate of change is shown on the graph. On a graph, rates of change can be compared by measuring how fast segments rise or fall when the graph is read from left to right. Extra Examples at ca.gr7math.com Lesson 4-9 Rate of Change 231 Compare Rates of Change 3 MAIL The graph shows the cost Postal Rates in cents of mailing a 1-ounce first-class letter. Compare the rate of change between 1998 and 2000 to the rate of change between 2000 and 2002. During which period was the rate of change greatest? Real-World Link In 1847, it cost 5 cents per _ ounce to 1 2 deliver mail to locations under 300 miles away and 10 cents per _ ounce 1 2 to deliver it to locations over 300 miles away. Source: www.stamps.org 40 Cost (cents) 36 The segment from 2000 to 2002 appears steeper than the segment from 1998 to 2000. So, the rate of change between 2000 and 2002 was greater than the rate of change between 1998 and 2000. 32 28 24 0 ’98 ’00 ’02 ’04 ’06 Year Check Find and compare the rates of change. From 1998 to 2000 From 2000 to 2002 change in cost 33 - 32 __ =_ change in cost 37 - 33 __ =_ 2000 - 1998 change in years 2002 - 2000 change in years 4 =_ or 2¢ per year 1 =_ or 0.5¢ per year 2 2 Since 2 > 0.5, the rate of change between 2000 and 2002 was greater than the rate of change between 1998 and 2000. ✓ d. NATURAL RESOURCES Use the table to make a graph of the data. During which 2-year period was the rate of change in oil production the greatest? Explain your reasoning. Texas Oil Production Year 1996 1998 2000 2002 Barrels (millions) 478.1 440.6 348.9 329.8 Personal Tutor at ca.gr7math.com The table below summarizes the relationship between rates of change and their graphs. Zero Rate of Change If a segment connecting two data points is horizontal, such as the change in the postage rate between 2002 and 2004 in Example 3, there was no change in the quantity over time. #/.#%04 3UMMARY Rate of Change positive negative Real-Life Meaning increase decrease y slants y Graph upward O 232 Doug Martin Rates of Change Chapter 4 Proportions and Similarity x O slants downward x (p. 230) Example 2 (p. 231) Example 3 (p. 232) (/-%7/2+ (%,0 For Exercises 4, 5, 13, 14 7, 8, 10, 11 6, 9, 12, 15 See Examples 1 2 3 2. Find the rate of temperature change between Temperature (°F) Time 6 a.m. 33 8 a.m. 45 12 p.m. 57 3 p.m. 57 4 p.m. 59 8 p.m. 34 4 P.M. and 8 P.M. 3. Make a graph of the data. During which time period was the rate of increase the greatest? Explain. ADVERTISING For Exercises 4–6, use the information in the table at the right that shows Tanisha’s progress in folding flyers for the school play. She started folding at 12:55 P.M. 4. Find the rate of change in flyers folded per minute between 1:00 and 1:20. 5. Find her rate of change between 1:25 and 1:30. Time Flyers Folded 12:55 0 1:00 21 1:20 102 1:25 102 1:30 125 6. Make a graph of the data. During which time period was her folding rate the greatest? Explain. INVESTMENTS For Exercises 7–9, use the following information. The value of a company’s stock over a 5-day period is shown in the table. Day Value ($) 1 2 3 4 5 57.48 53.92 50.25 49.74 44.13 7. Determine the rate of change in value between Day 1 and Day 3. 8. What was the rate of change in value between Day 2 and Day 5? 9. Make a graph of the data. During which 2-day period was the rate of change in the stock value greatest? TELEVISION For Exercises 10–12, use the information below and at the right. The graph shows the number of viewers who watched new episodes of a show. 10. Find the rate of change in viewership between season 1 and season 3. 11. Find the rate of change in viewership between season 2 and season 6. 12. Between which two seasons was the rate of change in viewership greatest? Television Ratings 32 Viewers (millions) Example 1 TEMPERATURE For Exercises 1–3, use the information in the table at the right that shows the outside air temperature at different times during one day. 1. Find the rate of temperature change in degrees per hour from 6 A.M. to 8 A.M. (1, 31.7) 30 28 (2, 26.3) 26 (3, 25.0) (4, 24.7) 24 (5, 22.6) 22 (6, 22.1) 0 1 2 3 4 5 6 Season Lesson 4-9 Rate of Change 233 BIRDS For Exercises 13–15, use the information below and at the right. The graph shows the approximate number of American Bald Eagle pairs from 1963 to 2000. Bald Eagle Population Growth 6,000 (00, 6,471) y Bald Eagle Pairs 5,000 13. Find the rate of change in the number of eagle pairs from 1974 to 1994. 14. Find the rate of change in the number of eagle pairs from 1984 to 2000. (94, 4,400) 4,000 3,000 (84, 1,800) 2,000 1,000 15. During which time period did the eagle population grow at the fastest rate? Explain your reasoning. 0 (63, 400) (74, 800) ’60 ’70 ’80 ’90 x ’00 Year Source: birding.about.com FAST FOOD For Exercises 16 and 17, use the information below. The graph shows the estimated total of U.S. food and drink sales in billions of dollars from 1980 to 2005. <Xkè^Flk $OLLARS BILLIONS xää 16. During which time period was the rate of change in food and drink sales greatest? Explain your reasoning. {ää BILLION Îää BILLION Óää BILLION £ää 17. Find the rate of change during that period. BILLION £nä £ä Óäää Óääx 9EAR Source: National Restaurant Association SALES For Exercises 18 and 19, use the following information. The National Confectioners Association reported $1,418 billion in candy sales during the 1998 winter holidays. For 2003, this figure was $1,440 billion. 18. Find the rate of change in sales from 1998 to 2003. 19. If this rate of change were to continue, what would the total candy sales during the winter holidays be in 2010? Explain your reasoning. Real-World Link On May 20, 1957, an F5 tornado touched down in Jackson, Missouri. It was the deadliest recorded tornado in Missouri’s history. Source: missouri.edu 20. MOLD Nine days ago, the area covered by mold on a piece of bread was 3 square inches. Today the mold covers 9 square inches. Find the rate of change in the mold’s area. TORNADOES For Exercises 21 and 22, refer to the table at the right. 21. Graph the data. During which interval %842!02!#4)#% See pages 687, 711. Self-Check Quiz at was the rate of change in the number of tornadoes the greatest? the least? 22. Is it reasonable to state that between 2000 and 2005, the number of tornadoes in a given year changed very little? Explain. ca.gr7math.com Missouri Tornadoes Year Number of Tornadoes 2000 28 2001 39 2002 29 2003 84 2004 69 2005 32 Source: NOAA’s National Weather Service 234 Chapter 4 Proportions and Similarity Jim Reed/Photo Researchers H.O.T. Problems 23. OPEN ENDED Create a set of gasoline price data that has a rate of change of $0.08 per gallon over a period of 4 days. 24. NUMBER SENSE Does the height of a candle as it burns over time show a positive or negative rate of change? Explain your reasoning. 25. CHALLENGE Liquid is poured at a constant rate into a beaker that is shaped like the one at the right. Draw a graph of the level of liquid in the beaker as a function of time. */ -!4( Explain the difference (*/ 83 *5*/( 26. between the rate of change between a set of data values and the change between data values. 27. The graph shows the altitude of a falcon over time. At this rate, how many hours would she need to work to earn $975? Altitude (ft) y A B E C 28. Sarah earns $52 for 4 hours of work. F 13 h H 75 h G 18.75 h J 243.75 h D Time x 29. Ralph rode his bike an average speed Between which two points on the graph was the bird’s rate of change in altitude negative? of 16 miles per hour for two hours on Saturday and then an average speed of 13 miles per hour for three hours. How many miles did Ralph ride in all? A A and B A 29 miles B B and C B 34 miles C C and D C 71 miles D D and E D 74 miles 30. ARCHITECTURE A certain building is 925 feet tall. On a scale drawing of the building, it is 18.5 inches tall. What is the scale for the drawing? (Lesson 4-8) 31. MEASUREMENT The area of the kitchen floor is 180 square feet. The tiles measure 36 square inches. How many tiles will it take to cover the entire floor? (Lesson 4-7) 32. PREREQUISITE SKILL Michael tutors elementary school students in math 1.5 hours each week. Is the total number of hours that he spends tutoring proportional to the number of weeks he tutors during the year? Explain your reasoning. (Lesson 4-1) Lesson 4-9 Rate of Change 235 4-10 Constant Rate of Change Main IDEA Preparation for Standard 7AF3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. NEW Vocabulary linear relationship constant rate of change EXERCISE Cameron knows that after he has warmed up, he can maintain a constant running speed of 8 feet per second. This is shown in the table and in the graph. Time (s) 0 2 4 6 8 Distance (ft) 0 16 32 48 64 Cameron’s Run 80 Distance (ft) Identify proportional and nonproportional linear relationships by finding a constant rate of change. y 64 48 32 16 x 1. Pick several pairs of points and find the rate of change between them. What is true of these rates? 0 4 8 12 16 Time (s) Relationships that have straight-line graphs, like the one in the example above, are called linear relationships. Notice that as the time in seconds increases by 2, the distance in feet increases by 16. +2 +2 +2 +2 Time (s) 0 2 4 6 8 Rate of Change Distance (ft) 0 16 32 48 64 change in distance 16 __ =_ or 8 ft/s change in time 2 +16 +16 +16 +16 The rate of change between any two points in a linear relationship is the same or constant. A linear relationship has a constant rate of change. Identify Linear Relationships 1 MONEY The balance in an account after Check To check the answer to Example 1, graph the data in the table, with the number of transactions on the x-axis and the balance in the account on the y-axis. If the data points fall on a line, then the relationship is linear. Number of Transactions +3 +3 +3 236 Number of Transactions several transactions is shown. Is the relationship between the balance and 3 number of transactions linear? If so, find 6 the constant rate of change. If not, explain 9 your reasoning. 12 Examine the change in the number of transactions and in the balance of the account. Chapter 4 Proportions and Similarity Balance ($) 170 140 110 80 Balance ($) 3 170 6 140 9 110 12 80 -30 -30 -30 As the number of transactions increases by 3, the balance in the account decreases by $30. Since the rate of change is constant, this is a linear relationship. The -30 or -$10 per transaction. This means constant rate of change is _ 3 that, on average, each transaction involved a $10 withdrawal. Determine whether the relationship between the two quantities described in each table is linear. If so, find the constant rate of change. If not, explain your reasoning. a. b. Cooling Water Wrapping Paper Number of Rolls Total Cost ($) Time (min) Temperature (°F) 5 95 2 8.50 10 90 4 17.00 15 86 6 25.50 20 82 8 34.00 Find a Constant Rate of Change 2 LIBRARIES Find the constant Library Fines $6 Daily Fine rate of change for the daily fine for each overdue book in the graph shown. Interpret its meaning. Choose any two points on the line and find the rate of change between them. y $4 $2 x 0 2 4 6 8 10 Number of Overdue Books (5, 2) 5 books, $2 daily fine (10, 4) 10 books, $4 daily fine Real-World Link With 85 branches, the New York Public Library is the world’s largest public library. It has collections totaling 11.6 million items. change in fine $(4 - 2) __ = __ change in books (10 - 5) books $2 _ = 5 books $0.40 =_ 1 book The daily fine changed from $2 to $4 and the number of overdue books changed from 5 to 10. Subtract to find the change in the daily fine and number of books. Express this rate as a unit rate. The daily fine is $0.40 per overdue book. Source: nupl.org Trash Pickup Project constant rate of change for the time it takes to complete a highway trash pickup project for each number of volunteers in the graph shown. Interpret its meaning. Time (min) 80 c. SERVICE PROJECT Find the y 60 40 x 0 8 16 24 32 Number of Volunteers Lesson 4-10 Constant Rate of Change Zoran Milich/Masterfile 237 Some, but not all, linear relationships are also proportional. Identify Proportional Relationships 3 TEMPERATURE Use the graph to 60 Degrees Fahrenheit Look Back To review identifying proportional relationships, see Lesson 4-2. Celsius to Fahrenheit Conversion determine if there is a proportional linear relationship between a temperature on the Fahrenheit scale and a temperature on the Celsius scale. Explain your reasoning. Since the graph of the data forms a line, the relationship between the two scales is linear. This can also be seen in the table of values created using the points on the graph. y (36, 52) 50 (27, 47) (18, 42) 40 (9, 37) 30 (0, 32) x 0 10 20 30 40 Celsius +9 +9 +9 +9 Constant Rate of Change Degrees Celsius 0 9 18 27 36 Degrees Fahrenheit 32 37 42 47 52 change in °F 5 __ =_ change in °C 9 +5 +5 +5 +5 To determine if the two scales are proportional, express the relationship between the degrees for several columns as a ratio. degrees Fahrenheit 37 52 42 47 __ _ ≈ 4.11 _ ≈ 2.33 _ ≈ 1.74 _ ≈ 1.44 9 27 36 18 degrees Celsius Since the ratios are not all the same, a temperature in degrees Celsius is not proportional to the same temperature in degrees Fahrenheit. Pounds to Kilograms Conversion d. MEASUREMENT Use the graph 40 y (80, 36) 30 Mass (kg) to determine if there is a proportional linear relationship between the weight of an object measured in pounds and the mass of the same object measured in kilograms. Explain your reasoning. (60, 27) 20 (40, 18) 10 (20, 9) x 0 20 40 60 80 Weight (lb) #/.#%04 3UMMARY Words Symbols 238 Chapter 4 Proportions and Similarity Proportional Linear Relationships Two quantities a and b have a proportional linear relationship if they have a constant ratio and a constant rate of change. change in b b _ is constant and _ is constant. a change in a Extra Examples at ca.gr7math.com (p. 236) Determine whether the relationship between the two quantities described in each table is linear. If so, find the constant rate of change. If not, explain your reasoning. 1. Example 2 Paint Needed for Chairs Side Length (cm) Volume (cm 3) Number of Chairs Cans of Paint 2 8 5 6 3 27 10 12 4 64 15 18 5 125 20 24 Find the constant rate of change for each graph and interpret its meaning. 3. 4. Distances on Map Actual Distance (mi) (p. 237) 2. Volume of Cube Fuel Level in Car Tank y 90 Fuel Level (gal) Example 1 60 30 24 y 16 8 x x 0 2 4 1 0 6 Example 3 (p. 238) For Exercises 7–10 11–16 17–22 See Examples 1 2 3 3 4 Determine whether a proportional linear relationship exists between the two quantities shown in each of the indicated graphs. Explain your reasoning. 5. Exercise 3 (/-%7/2+ (%,0 2 Time (min) Map Distance (in.) 6. Exercise 4 Determine whether the relationship between the two quantities described in each table is linear. If so, find the constant rate of change. If not, explain your reasoning. 7. 9. 8. Cost of Electricity to Run Personal Computer Total Number of Customers Helped at Jewelry Store Time (h) Cost (¢) Time (h) Total Helped 5 15 1 12 8 24 2 24 12 36 3 36 24 72 4 60 Distance Traveled by Falling Object Time (s) Distance (m) 1 4.9 2 19.6 3 44.1 4 78.4 10. Italian Dressing Recipe Oil (c) 2 4 6 8 Vinegar (c) _3 1 1_ 1 2_ 3 4 2 4 Lesson 4-10 Constant Rate of Change 239 Find the constant rate of change for each graph and interpret its meaning. 11. 12. Ace Pizza Delivery y y 180 Balance ($) 60 Cost ($) Amount Owed on CD Player 40 20 120 60 x x 0 2 4 0 6 2 14. Aircraft Altitude y Scuba-Diving Pressure Pressure (lb/in2) Altitude (ft) 6,000 6 Number of Payments Number of Pizzas 13. 4 4,000 2,000 y 45 30 15 x x 0 2 4 0 6 22 15. 16. 66 Cost of Party y 300 Total Cost ($) Sale Price ($) Sale Price 90 44 Depth (ft) Time (min) 60 30 y 200 100 x 0 20 40 x 60 5 0 Retail Price ($) 10 15 Number of People Determine whether a proportional relationship exists between the two quantities shown in each of the indicated graphs. Explain your reasoning. 17. Exercise 11 18. Exercise 12 19. Exercise 13 20. Exercise 14 21. Exercise 15 22. Exercise 16 %842!02!#4)#% 23. Who is saving more money each week? Explain your reasoning. See pages 688, 711. 24. Whose savings are proportional Self-Check Quiz at ca.gr7math.com 240 to the number of weeks they have been saving? Explain. Chapter 4 Proportions and Similarity Savings 100 y Pedro 80 Balance ($) SAVINGS For Exercises 23 and 24, use the following information. Both Pedro and Jenna are saving money. Their savings account balances over several weeks are shown. 60 40 Jenna 20 x 0 2 4 6 8 Time (weeks) 10 H.O.T. Problems 25. OPEN ENDED Graph two quantities that have a proportional linear relationship. Justify your answer. 26. CHALLENGE Examine the graphs in Exercises 3, 4, and 11–16, as well as your corresponding answers in Exercises 5 and 17–22. What point do all of the graphs that represent proportional linear relationships have in common? */ -!4( Write a real-world problem in which you would (*/ 83 *5*/( 27. need to find a constant rate of change. Then solve your problem. Is the relationship described in your problem proportional? Explain. 28. Tickets to the school play are $2.50 each. Which table contains values that fit this situation, if c represents the total cost for t tickets? 29. The graph shows the distance Bianca traveled over her 2-hour bike ride. $ISTANCE 4RAVELED $ISTANCE MILES A Cost of Play Tickets t 1 2 3 4 c $2.50 $3.25 $4.00 $4.75 B Cost of Play Tickets t c C 1 $3.50 2 $6.00 3 $8.50 4 4IME HOURS $11.00 Which of the following statements is true? Cost of Play Tickets t 1 2 3 4 c $3.50 $4.00 $4.50 $5.00 D F She traveled at a constant speed of 12 miles per hour for the entire ride. G She traveled at a constant speed of 8 miles per hour for the last hour. Cost of Play Tickets t 1 2 3 4 c $2.50 $5.00 $7.50 $10.00 H She traveled at a constant speed of 4 miles per hour for the last hour. J She traveled at a constant speed of 8 miles per hour for the entire ride. 30. MEASUREMENT Three years ago, an oak tree was 4 feet 5 inches tall. Today it is 6 feet 3 inches tall. How fast did the tree grow in inches per year? (Lesson 4-9) 31. GEOGRAPHY On a map, two cities are 3.25 inches apart. If the scale for the map is 0.5 inch = 40 miles, how many miles apart are the cities? (Lesson 4-8) ALGEBRA Solve each equation. Check your solution(s). (Lesson 3-1) 32. p 2 = 0.49 33. t 2 = _ 1 144 34. 6,400 = r 2 Lesson 4-10 Constant Rate of Change 241 CH APTER Study Guide and Review 4 Download Vocabulary Review from ca.gr7math.com Key Vocabulary congruent (p. 207) Be sure the following Key Concepts are noted in your Foldable. proportional (p. 194) constant of proportionality rate (p. 191) (p. 200) rate of change (p. 230) constant rate of change ratio (p. 190) (p. 236) corresponding parts (p. 206) Key Concepts cross products (p. 198) Proportions (Lessons 4-2 and 4-3) equivalent ratios (p. 198) • If two related quantities are proportional, then they have a constant ratio. linear relationship (p. 236) • A proportion is an equation stating that two ratios or rates are equivalent. polygon (p. 206) proportion (p. 198) • The cross products of a proportion are equal. Similar Polygons (Lesson 4-5) nonproportional (p. 194) scale (p. 224) scale drawing (p. 224) scale factor (p. 207) scale model (p. 224) similar (p. 206) unit rate (p. 191) unit ratio (p. 214) • If two figures are similar with a scale factor of _, b then the perimeters of the figures have a ratio a of _. a b Measurement (Lessons 4-6 and 4-8) • You can multiply by a unit ratio to convert or change from larger units to smaller units. • To convert from smaller units to larger units, multiply by the reciprocal of the appropriate unit ratio. • A scale is determined by the ratio of a given length on a drawing or model to its corresponding actual length. Rates (Lessons 4-1, 4-9, and 4-10) • A rate is a comparison of two quantities with different types of units. • To find the rate of change, divide the difference in the y-coordinates by the difference in the x-coordinates. • Two quantities a and b have a proportional linear relationship if they have a constant ratio and a constant rate of change. Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. 1. Polygons that have the same size are called similar polygons. 2. A unit ratio is one in which the denominator is 1 unit. 3. A ratio of two measurements having similar units is called a rate. 4. In a relationship in which the ratio is not constant, the two quantities are said to be nonproportional. 5. A scale is the ratio of a length on a drawing or model to its corresponding actual length. 6. Comparing two numbers by multiplication is called a proportion. 242 Chapter 4 Proportions and Similarity Vocabulary Review at ca.gr7math.com Lesson-by-Lesson Review 4-1 Ratios and Rates (pp. 190–193) 7. 7 chaperones for 56 students Example 1 Express the ratio 10 milliliters to 8 liters in simplest form. 8. 12 peaches:8 pears 10 milliliters __ Express each ratio in simplest form. 9. 5 inches out of 5 feet 10. SHOPPING An 8-pound bag of cat food sells for $13.89. A 20-pound bag of the same brand sells for $24.79. Which is the better buy? Explain your reasoning. 4-2 1 =_ 8 liters = 8 × 1,000 or 8,000 milliliters. Divide numerator and denominator by 10. 800 1 The ratio in simplest form is _ or 1:800. 800 Proportional and Nonproportional Relationships (pp. 194–197) 11. INTERNET A high-speed Internet company charges $30 a month for Internet services. There is also a $30 installation fee. Is the number of months you can have high-speed Internet proportional to the total cost? 12. WORK On Friday, Jade washed a total of 9 vehicles in 4 hours. The next day she washed a total of 15 vehicles in 6 hours. Is the total number of vehicles she washed over the two days proportional to the time it took her to wash them? 4-3 8 liters 10 milliliters = __ 8,000 milliliters Example 2 Leo earns $28 for every haircut he does. Is the amount of money he earns proportional to the number of haircuts he gives? Haircuts 1 2 3 4 Earnings ($) 28 56 84 112 earnings _ 28 56 _ or 28 _ or 28 2 1 haircuts 84 112 _ or 28 _ or 28 3 4 Since these ratios are all equal to 28, the amount of money he earns is proportional to the number of haircuts he gives. Solving Proportions (pp. 198–202) Solve each proportion. _ 13. _ r = 3 6 8 7 _ _ 15. =n 2 4 y 30 14. _ = _ 0.5 0.25 k 72 _ _ 16. = 5 8 17. ANIMALS A turtle can move 5 inches in 4 minutes. How far will it travel in 10 minutes? Example 3 4 _9 = _ x 18 9 · 18 = x · 4 162 = 4x _ _ _ Solve 9 = 4 . x 18 Write the equation. Find the cross products. Multiply. 162 _ = 4x 4 4 Divide each side by 4. 40.5 = x Simplify. Chapter 4 Study Guide and Review 243 CH APTER 4 Study Guide and Review 4-4 PSI: Draw a Diagram (pp. 204–205) Solve. Use the draw a diagram strategy. 18. CONCERTS Nina, Tyrese, Leslie, and Ethan are going to a rock concert. In how many different orders can they enter the concert? 19. PHYSICAL SCIENCE A tennis ball is dropped from 12 feet above the ground. It hits the ground and bounces up half as high as it fell. This is true for each successive bounce. What height does the ball reach on the fourth bounce? Example 4 A photographer is taking the eighth grade class picture. She places 8 students in the first row. Each additional row has 4 more students in it. If there are a total 80 students, how many rows will there be? Draw a diagram with 8 students in row one and then add 4 more students to each additional row. 88888888 888888888888 8888888888888888 20. MEASUREMENT Jasmine unrolled 88888888888888888888 3 48 feet of carpet. This is _ of the total 888888888888888888888888 4 amount of carpet needed for the library. What is the total amount of carpet needed for the library? 4-5 Similar Polygons (pp. 206–211) Each pair of polygons is similar. Write a proportion to find each missing measure. Then solve. 21. 13 5 1 22. 2 x 3 x P 4.5 H x Q 3 K 9 J 6 perimeter of 49 feet and square F has a perimeter of 64 feet, what is the scale factor of the two squares? 24. PARTY PLANNING For your birthday party, you make a map to your house on a 3-inch wide by 5-inch long index card. How long will your map be if you use a copier to enlarge it so it is 8 inches wide? Chapter 4 Proportions and Similarity Example 5 Rectangle GHJK is similar to rectangle PQRS. Find the value of x. G 23. MEASUREMENT If square D has a 244 There are a total of 5 rows. R S The scale factor from GHJK to PQRS is GK 3 1 _ , which is _ or _ . PR GH 1 _ =_ 3 PQ 4.5 _ = _1 3 x 4.5 · 3 = 1 · x 13.5 = x 9 3 Write a proportion. GH = 4.5 and PQ = x Find the cross products. Multiply. Mixed Problem Solving For mixed problem-solving practice, see page 711. 4-6 Measurement: Converting Length, Weight/Mass, Capacity, and Time (pp. 213–218) Complete each conversion. Round to the nearest hundredth if necessary. 25. 5 in. ≈ cm 26. 25 km ≈ mi 27. Which is greater: a 10-pound weight or a 5-kilogram weight? 28. Which is greater: a 5,000-meter race or Example 6 Which has a greater capacity: a bottle containing 32 fluid ounces of spring water or a bottle containing 1 liter of water? 29.574 mL 1L Use the unit ratios _ and _ . 1,000 mL 1 fl oz 29.574 mL _ 1L 32 fl oz ≈ 32 fl oz · _ · 1 fl oz a 4-mile race? 1,000 mL 29.574 L ≈ 32 · _ or 0.95 L 1,000 The 1-liter bottle contains more water. 4-7 Measurement: Converting Square Units and Cubic Units (pp. 220–223) Complete each conversion. Round to the nearest hundredth if necessary. Example 7 Convert 15 square centimeters to square inches. 29. 120 yd 3 = ft 3 1 in. Use the unit ratio _ . 30. 16 m 2 ≈ ft 2 31. PARKING The area of a parking lot is 375,000 square feet. How many square meters is the parking lot? 32. STORAGE The total capacity of a certain 2.54 cm 1 in. 1 in. 15 cm 2 ≈ 15 cm 2 · _ ·_ 2.54 cm 2.54 cm 15 in 2 ≈_ or 2.33 in 2 2.54 · 2.54 storage unit is about 23 cubic meters. How many cubic feet is the storage unit? 4-8 Scale Drawings and Models (pp. 224–229) The scale on a map is 2 inches = 5 miles. Find the actual distance for each map distance. 33. 12 inches 34. 4 inches 35. HOBBIES Mia’s dollhouse is a replica of her family’s townhouse. The outside dimensions of the dollhouse are 25 inches by 35 inches. If the actual outside dimensions of the townhouse are 25 feet by 35 feet, what is the scale of the dollhouse? Example 8 The scale on a model is 3 centimeters = 45 meters. Find the actual length for a model distance of 5 centimeters. 3 cm 5 cm _ =_ 45 m xm model length actual length 3 · x = 45 · 5 Find the cross products. 3x = 225 Multiply. x = 75 Divide each side by 3. The actual length is 75 meters. Chapter 4 Study Guide and Review 245 CH APTER 4 Study Guide and Review 4-9 Rate of Change (pp. 230–235) MONEY For Exercises 36 and 37, use the following information. The table below shows Victor’s weekly allowance between the ages of 6 and 15. Age (yr) 6 8 10 12 15 $ per week 1.00 2.00 2.00 3.00 5.00 36. Find the rate of change in his allowance between ages 12 and 15. 37. Was the rate of change between ages 8 Example 9 At 5 A.M., it was 54°F. At 11 A.M., it was 78°F. Find the rate of temperature change in degrees per hour. change in temperature (78 - 54)° __ = __ change in hours (11 - 5) hours 24° 4° =_ or _ 6 hours 1 hour Between 5 A.M. and 11 A.M., the temperature increased on average 4 degrees per hour. and 10 positive, negative, or zero? 4-10 Constant Rate of Change (pp. 236–241) 38. RAINFALL The amount of rainfall after several hours is shown. Is the relationship between the amount of rainfall and number of hours linear? If so, find the constant rate of change. If not, explain your reasoning. Number of Hours Rainfall (inches) 1 2 2 4 3 7 4 9 39. PHONE CALL The cost of a long-distance phone call after several minutes is shown. Is the relationship between the cost and number of minutes linear? If so, find the constant rate of change. If not, explain your reasoning. 246 Number of Minutes Cost (¢) 3 7 6 14 9 21 12 28 Chapter 4 Proportions and Similarity Example 10 The distance traveled in a car trip is shown. Is the relationship between the distance traveled and number of hours spent in the car linear? If so, find the constant rate of change. If not, explain your reasoning. Number of Hours Distance (miles) 2 120 4 240 6 360 8 480 +2 +2 +2 ×2 ×2 ×2 As the number of hours increases by two, the distance doubles. Since the rate of change is constant, this is a linear relationship. So, the constant rate of 120 change is _ or 60 miles per hour. This 2 means that for every hour they are in the car they travel 60 miles. CH APTER Practice Test 4 1. Express the ratio 15 inches to 1 foot in 11. COOKING Denise is making recipe for a large simplest form. group in which she needs 12 kilograms of ground beef. How many pounds of ground beef does she need? (1 lb ≈ 0.4536 kg) 2. Express 112 feet in 2.8 seconds as a unit rate. 3. STANDARDS PRACTICE At Flynn’s Apple Orchard, 16 acres of land produced 368 bushels of apples. Which rate represents the number of bushels per acre? A 16:1 B 23:1 C 23:2 D 46:1 12. TRAVEL On a map, 1 inch = 7.5 miles. How many miles does 2.5 inches represent? BUSINESS For Exercises 13 and 14, use the table that shows the number of new customers in The Lucky Diner at different times during one day. 4. MEASUREMENT Nick rides his bike 20 miles every two days. Is the distance Nick rides proportional to the number of days? Solve each proportion. _ 5. _ a = 3 6. _ = _ y 9 12 5 3 20 7. NUTRITION An 8-ounce serving of milk Each pair of polygons is similar. Write a proportion to find each missing measure. Then solve. 9. 10 5 x 2 a 10. 6 3 4.5 Chapter Test at ca.gr7math.com 12 p.m. 30 2 p.m. 6 4 p.m. 15 5 p.m. 32 per hour between 4 P.M. and 5 P.M. 14. Find the rate of change in new customers per hour between 12 P.M. and 2 P.M. Then interpret its meaning. 15. 19 like to cook main dishes, 15 prefer baking desserts, and 7 like to do both. How many students like to cook main dishes, but not bake desserts? Use the draw a diagram strategy. New Customers 13. Find the rate of change in new customers provides 30% of the daily value of calcium. How much milk provides 50% of the daily value of calcium? 8. FOOD Of the 30 students in a life skills class, Time STANDARDS PRACTICE A flag is being made that has an area of six square feet. Approximately how many square meters of fabric is this? (1 ft ≈ 0.3048 m) F 0.56 m 2 G 1.83 m 2 H 19.69 m 2 J 64.58 m 2 16. MEASUREMENT Is the relationship between the weight and number of months linear? If so, find the constant rate of change. If not, explain your reasoning. Number of Months Weight (lb) 4 14 6 18 8 20 10 22 Chapter 4 Practice Test 247 APTER 4 California Standards Practice Cumulative, Chapters 1–4 2 0- 1 0- B 3 0- 14 cm 15 cm 4 0- C Z End zone -3 0 -4 0 50 4 0- 3 0- 2 0- 1 0- What is the length, in yards, of the football field, including the end zones? 300 100 23 w D _=_ 300 100 A 50 10 cm -2 0 2 in. 6 in. F 100 yd G 120 yd H 130 yd Trapezoid ABCD is similar to trapezoid −− WXYZ. Find the length of XY. D -4 0 2 -1 0 23 w C _ =_ 100 w 23 300 B _=_ 100 w -3 0 A jar contains 25% green buttons, 32% yellow buttons, 20% brown buttons, and 23% white buttons. There are 300 buttons in the jar altogether. Which proportion can be used to find w, the total number of white buttons in the jar? 23 300 A _ =_ The scale drawing of a football field was made using a scale of 1 inch = 20 yards. End zone 1 4 -2 0 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. -1 0 CH J 150 yd W Y 5 X Sixty-five miles per hour is the same rate as which of the following? A 1.08 miles per second B 3,900 miles per second F 20 cm C 6.5 miles per minute G 21 cm D 1.08 miles per minute H 24 cm J 27 cm 6 Between which two whole numbers is √ 66 located on a number line? F 6 and 7 G 7 and 8 Question 2 This problem involves similar figures. If two polygons are similar, then you can use a scale factor or a proportion to find the missing measure(s). 3 1 How many seconds are in 1_ hours? 2 H 8 and 9 J 9 and 10 7 Which operation results in the same answer regardless of the numbers involved? A 90 A divide by one B 540 B multiply by one C 3,600 C add zero D 5,400 D multiply by zero 248 Chapter 4 Proportions and Similarity California Standards Practice at ca.gr7math.com More California Standards Practice For practice by standard, see pages CA1-CA39. 8 1 Rebekah is 1_ meters tall. About how tall is 11 A teacher plans to buy 5 pencils for each 2 student in her class. If pencils come in packages of 18 and cost $1.99 per package, what other information is needed to find the cost of the pencils? she in feet and inches? (1 meter ≈ 39 inches) 9 F 3 feet 3 inches H 4 feet 9 inches G 4 feet 0 inches J 4 feet 10.5 inches A the cost of erasers B the number of students in the whole school During a 3-hour period, 2,292 people rode the roller coaster at an amusement park. Which proportion can be used to find x, the number of people who rode the coaster during a 12-hour period if the rate is the same? 3 x A _ =_ 2,292 12 3 12 C _ =_ 2,292 x 3 12 =_ B _ x 12 D _ =_ 2,292 x 3 C the number of students in her class D the name of the store where she is buying the pencils 9 2 12 Which fraction is between _ and _ ? 1 F _ 3 H _ 5 1 G _ 2 2,292 J 10 A park is shaped like a rectangle with the 3 10 4 11 _ 9 Pre-AP dimensions shown below. Which of the following is closest to the length of a diagonal of the park? Record your answers on a sheet of paper. Show your work. 13 The table shows how Time (h) 2 4 6 8 much Susan earns when she works at a fast-food restaurant. a. Graph the data YD from the table and connect the points with a line. Wages ($) 9 18 27 36 b. Find the slope of the line. c. What is Susan’s rate of pay? YD F 165 yd H 340 yd G 290 yd J 405 yd d. If Susan continues to be paid at this rate, how much money will she make for working 10 hours? NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 10 11 12 13 Go to Lesson... 4-3 4-5 4-6 4-8 4-6 3-2 1-2 4-6 4-3 3-6 1-1 2-2 4-10 AF1.3 MG1.1 AF4.2 MG3.3 MR1.1 NS1.1 AF3.4 Chapters 1–4 California Standards Practice 249 For Help with Standard... AF4.2 6NS1.3 MG1.1 MG1.2 MG1.1 NS2.4 5 Percent • Standard 7NS1.0 Know the properties of, and compute with, rational numbers expressed in a variety of forms. Key Vocabulary percent (p. 252) percent equation (p. 279) percent of change (p. 284) percent proportion (p. 263) Real-World Link Agriculture In 2004, California’s income from agriculture was $31.8 billion. Of that income, 30% came from the production of fruits and nuts. You can use percents to determine the income from fruits and nuts produced in California. Percent Make this Foldable to help you organize your notes. Begin with five sheets 1 of 8_" × 11" paper. 2 1 Draw a large circle on one of the sheets of paper. 3 Staple the circles on the left side. Write the chapter title on the front and the four lesson titles on the inside right pages. 250 Chapter 5 Percent Stephen Saks/Getty Images 2 Stack the sheets of paper. Place the one with the circle on top. Cut all five sheets in the shape of a circle. 0ERCENT 4 Turn the circle to the back side so that the staples are still on the left. Write the last four lesson titles on the front and right pages of the journal. ,ESSON GET READY for Chapter 5 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Compute each product mentally. Example 1 (Lesson 2-3) Compute 1 · 820 mentally. 1 1. _ · 303 1 2. 644 · _ 3. 0.1 · 550 4. 64 · 0.5 3 2 _ 4 _1 · 820 is one-fourth of 820, or 205. 4 Write each fraction as a decimal. Example 2 (Lesson 2-1) Write 5 as a decimal. 2 5. _ 7 6. _ 5 3 7. _ 4 _ 8 5 to a decimal, divide 5 by 8. To change _ 8 3 8. _ 8 9. SCHOOL Andrea answered 18 out of 20 questions correctly on a math quiz. Write her score as a decimal. (Lesson 2-1) 0.625 8 5.000 -48 _____ 20 - 16 _____ 40 - 40 _____ 0 8 5 So, _ = 0.625. 8 Solve each equation or proportion. Example 3 (Lessons 2-7 and 4-3) Solve w = 5 . _ _ 10. 0.25d = 130 11. 48r = 12 12. 0.4m = 22 13. 0.02n = 9 6 5 w _ =_ 6 12 14. _ = _ 15. _ = _ 6 · w = 12 · 5 6w = 60 Find cross products. Simplify. 6w 60 _ =_ Divide each side by 6. w = 10 Simplify. x 10 3 5 4 9 14 b 16. RECIPES Ruben’s chocolate chip cookie recipe uses 2 eggs for 2 dozen cookies. How many eggs does Ruben need to make 72 cookies? (Lesson 4-3) 12 6 6 Write the proportion. Chapter 5 Get Ready for Chapter 5 251 5 -1 Ratios and Percents Main IDEA Write ratios as percents and vice versa. Standard 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. POPULATION The table shows the ratio of people under 18 years of age to the total population for various states. 1. Name two states from the table that have ratios in which the second numbers are the same. State Ratio of People Under 18 to Total Population Arkansas 1 out of 4 Hawaii 6 out of 25 Mississippi 27 out of 100 Utah 8 out of 25 Source: Time Almanac 2. How can you determine which of the four states has the greatest NEW Vocabulary percent ratio of people under 18 to total population? Ratios such as 27 out of 100 or 8 out of 25 can be written as percents. +%9 #/.#%04 Words Percent A percent is a ratio that compares a number to 100. Percent Ratio Numbers 27% 27 out of 100 Algebra x% x out of 100 Fraction 27 _ 100 x _ 100 Write Ratios as Percents Write each ratio as a percent. 1 POPULATION In a recent census, 26 out of every 100 people living in Illinois were younger than 18. 26 out of 100 = 26% Definition of percent 2 SPORTS At a recent triathlon, 180 women competed for every Large Percents Notice that some percents, such as 180%, are greater than 100%. Since percent means hundredths or per 100, a percent like 180% means 180 per 100. 100 women who competed ten years earlier. 180 per 100 = 180% Definition of percent Write each ratio as a percent. a. BASEBALL During his baseball career, Babe Ruth had a base hit about 34 out of every 100 times he came to bat. b. TECHNOLOGY In a recent year, 50.5 out of 100 households in the United States had access to the Internet. 252 Chapter 5 Percent To write a fraction as a percent, find an equivalent fraction with a denominator of 100. Write Fractions as Percents Write each ratio or fraction as a percent. 3 CARS About 1 out of 5 sport utility vehicles manufactured in the United States is white. x _1 = _ 100 5 × 20 Real-World Link The first law regulating the speed of cars was passed in the state of New York in 1904. It stated that the maximum speed was 10 miles per hour in populated areas and 20 miles per hour in the country. Source: The World Almanac 20 _1 = _ 5 100 1 5 20 100 × 20 So, 1 out of 5 equals 20%. _ 4 TRAVEL About 1 of travelers use scheduled buses. 200 x 1 _ =_ 200 100 ÷2 0.5 1 _ =_ 200 100 1 200 0.5 100 ÷2 So, 1 out of 200 equals 0.5%. Small Percents In Example 4, notice that 0.5% is less than 1%. Percents can be even smaller, such as 0.001%, which is equal to _. 1 100,000 Write each ratio or fraction as a percent. c. TECHNOLOGY In Finland, almost 3 out of 5 people have cell phones. d. ANIMALS About _ of the mammals in the world are bats. 1 4 Personal Tutor at ca.gr7math.com You can express a percent as a fraction by writing it as a fraction with a denominator of 100. Then write the fraction in simplest form. Write Percents as Fractions 5 ENVIRONMENT The circle graph shows an estimate of the percent of each type of trash in landfills. Write the percent for paper as a fraction in simplest form. 30 Paper: 30% = _ 100 _ = 3 10 Definition of percent Simplify. /À>Ã ÊÊ>`vÃ *>«iÀ Îä¯ *>ÃÌV Ó{¯ "Ì iÀÊ /À>Ã Îx¯ `Ê>` 9>À`Ê7>ÃÌi ££¯ Source: Franklin Associates, Ltd. Write as a fraction in simplest form. e. food and yard waste Extra Examples at ca.gr7math.com Hulton-Deutsch Collection/CORBIS f. other trash g. plastic Lesson 5-1 Ratios and Percents 253 Examples 1, 2 (p. 252) Write each ratio as a percent. 1. 17 out of 100 2. 237 per 100 3. TAXES In Illinois, the sales tax rate is 6.25 cents for every 100 cents spent. Examples 3, 4 (p. 253) Example 5 (p. 253) (/-%7/2+ (%,0 For Exercises 10, 11 12–19 20–27 See Examples 1, 2 3, 4 5 Write each ratio or fraction as a percent. 5. _ 9 20 4. 7 : 10 6. 1 out of 500 Write each percent as a fraction in simplest form. 7. 19% 8. 50% 9. 18% Write each ratio or fraction as a percent. 10. 23 out of 100 11. 110 per 100 12. _ 13. _ 14. 8 out of 25 15. 54 out of 300 16. 2 : 5 17. 9 : 10 17 50 7 20 18. TRAVEL One out of every 50 visitors to the United States is from France. Write this ratio as a percent. 19. PETS Three out of 25 households in the United States have both a dog and a cat. Write this ratio as a percent. Write each percent as a fraction in simplest form. 20. 29% 21. 43% 22. 40% 23. 70% 24. 45% 25. 28% 26. 64% 27. 65% 28. ENERGY Germany uses about 4% of the world’s energy. Write this percent as a fraction in simplest form. 29. MUSIC The influences in the purchase of CDs by buyers ages 12–44 are shown in the graphic. Write each percent as a fraction in simplest form. #$ 0URCHASE )NFLUENCES -OVIE 3OUNDTRACK 30. POPULATION According to a recent census, the population of Houston, 7 Texas, is about _ of the United States 2ADIO 3AW IN 3TORE 1,000 population. Write this fraction as a percent. %842!02!#4)#% See pages 688, 712. 31. TEETH Humans have _ more teeth as Self-Check Quiz at ca.gr7math.com 254 Chapter 5 Percent -USIC 6IDEO #HANNEL &RIENDS2ELATIVES 8 5 adults than when they are children. Write this fraction as a percent. Source: Edison Media Research H.O.T. Problems 32. CHALLENGE Explain how a student can receive an 86% on a test with 50 questions. 33. OPEN ENDED Find a percent that is between _ and _. Justify your answer. 3 4 1 2 34. Which One Doesn’t Belong? Identify the ratio that does not belong with the other three. Explain your reasoning. 2 out of 5 35. 40% 10 _ 2 : 10 25 */ -!4( Refer to the opener for this lesson. Express all of the (*/ 83 *5*/( ratios as percents. Explain how doing so helps determine which state has the greatest portion of its total population under 18. 36. What percent of the circle is shaded? 37. A soccer team played twenty games, of which they won fourteen. What percent of the games did they win? F 30% G 60% H 70% A 10% C 30% B 20% D 40% J 80% ALGEBRA Determine whether the relationship between the two quantities described in each table is linear. If so, find the constant rate of change. If not, explain your reasoning. (Lesson 4-10) 38. 39. Pages Printed Cost of Bagels Page 5 10 15 20 Bagels 6 12 18 24 Time (min) 1 2 3 4 Cost ($) 3 5 8 10 40. CARS After driving 150 miles, Mr. Ruiz has used 5 gallons of gasoline. He uses 3 gallons of gas driving another 100 miles. Find the rate of change in miles per gallon for the given distances. (Lesson 4-9) 41. Write 1.8, 1.07, 1_, and 1_ in order from least to greatest. (Lesson 2-2) 8 9 1 2 PREREQUISITE SKILL Write each fraction as a decimal. (Lesson 2-1) 42. _ 3 5 43. _ 3 4 44. _ 5 8 45. _ 1 3 Lesson 5-1 Ratios and Percents 255 5-2 Comparing Fractions, Decimals, and Percents Main IDEA Write percents as fractions and decimals and vice versa. PETS The table gives the percent of households with various pets. Households with Pets 1. Write each percent as a fraction. Standard 7NS1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10), compare rational numbers in general. Standard 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. Do not simplify the fractions. 2. Write each fraction in Question 1 as a decimal. 3. How could you write a percent as a decimal without writing the percent as a fraction first? dog 39% cat 34% freshwater fish 12% bird 7% small animal 5% Source: American Pet Products Manufacturers Association 39 _ Fractions, percents, and decimals are all different ways to represent the same ratio. #/.#%04 3UMMARY Percent 0.39 39% decimal percent Percents and Decimals Decimal Decimal fraction 100 Remember that percent means per hundred. In Lesson 5-1, you wrote percents as fractions with 100 in the denominator. Similarly, you can write percents as decimals by dividing by 100. Percents and Decimals To divide by 100, move the decimal point two places to the left. Percent of Households Pet Percent To write a percent as a decimal, divide by 100 and remove the percent symbol. To write a decimal as a percent, multiply by 100 and add the percent symbol. 39% = 39% = 0.39 0.39 = 0.39 = 39% Percents as Decimals Write each percent as a decimal. 2 115% 1 35% 35% = 35% = 0.35 Divide by 100. Remove the percent symbol. 115% = 115% = 1.15 Divide by 100. Remove the percent symbol. Write each percent as a decimal. a. 27% 256 Chapter 5 Percent Cydney Conger/CORBIS b. 145% c. 0.2% Extra Examples at ca.gr7math.com Decimals as Percents Write each decimal as a percent. 4 1.66 3 0.2 Decimals Greater Than One Notice that decimals greater than 1 are equivalent to percents greater than 100. 0.2 = 0.20 Multiply by 100. 1.66 = 1.66 Multiply by 100. = 20% Add the percent symbol. = 166% Add the percent symbol. Write each decimal as a percent. d. 0.83 e. 1.764 f. 0.005 You have learned to write a fraction as a percent by finding an equivalent fraction with a denominator of 100. This method works well if the denominator is a factor of 100. If the denominator is not a factor of 100, you can solve a proportion or you can write the fraction as a decimal and then write the decimal as a percent. Fractions as Percents 3 5 Write _ as a percent. 8 Use a proportion. METHOD 1 x _3 = _ 8 Write as decimal. First write as a decimal. Then write as a percent. _3 = 0.375 0.375 8 3.000 8 = 37.5% -2 4 _____ 100 3 · 100 = 8 · x 300 = 8x 300 8x _ =_ 8 METHOD 2 60 -56 ____ 40 -40 ____ 0 8 37.5 = x 3 37.5 So, _ =_ or 37.5%. 8 100 _ 6 Write 2 as a percent. 3 Percents In real-world − situations, 66.6% will usually be rounded to 67% or 66.7%. _2 = 0.66−6 3 − = 66.6% − 2 So, _ = 66.6%. 0.66… 2.0 3 -1 8 _____ 20 -18 ____ 2 3 Write each fraction as a percent. g. _ 7 25 h. _ 3 16 i. _1 9 Personal Tutor at ca.gr7math.com 5-2 Comparing Fractions, Decimals, and Percents 257 Compare Numbers 7 TAXES In a recent survey, 0.6 of the people said they will use their tax refund to pay bills, and 7% said they will just spend it. Do more people pay bills or spend their refund? Since 0.6 = 60% and 60% > 7%, more people plan on using their tax refund to pay bills than for spending. j. GEOGRAPHY About _ of Earth’s land is covered by desert. North 3 20 America is about 16% of Earth’s total land surface. Is the area of Earth’s deserts more or less than the total area of North America? Real-World Link The average tax refund in 2005 was $2,144. Source: irs.gov Order Numbers __ 8 Order 30%, 3 , 7 , and 0.33 from least to greatest. 100 20 3 _ = 3% 100 35 7 _ =_ or 35% 20 0.33 = 33% 100 From least to greatest, the percents are 3%, 30%, 33%, and 35%. 3 7 , 30%, 0.33, and _ . So, from least to greatest, the numbers are _ 20 100 Order each set of numbers from least to greatest. k. 22%, _, _, 0.25 1 3 10 25 Examples 1, 2 (p. 256) Examples 3, 4 (p. 257) Examples 5, 6 (p. 257) 4 _1 , 40%, 0.401, _ 5 25 Write each percent as a decimal. 1. 40% 2. 18% 3. 0.3% Write each decimal as a percent. 4. 0.725 5. 1.23 6. 0.3 Write each fraction as a percent. 7. _ 8. _ 9. _ 13 40 11 25 5 6 Example 7 10. HOMEWORK At Hancock Middle School, 57% of the eighth-grade students (p. 258) spend at least 30 minutes a day on math homework. Of the seventh-grade students, 0.5 study this long. In which grade do a greater percent of students spend at least 30 minutes a day on math homework? Example 8 (p. 258) 258 l. Order each set of numbers from least to greatest. 11. _, 60%, 0.062, _ Chapter 5 Percent David Forbert/SuperStock 17 25 13 20 12. 0.99, _, 9%, _ 9 10 19 20 (/-%7/2+ (%,0 For Exercises 13–22 23–32 33–42 43–46 47–52 See Examples 1, 2 3, 4 5, 6 7 8 Write each percent as a decimal. 13. 90% 14. 80% 15. 172% 16. 245% 17. 0.4% 18. 84.2% 19. 7% 20. 5% 21. ENERGY A recent study indicated that 8.4% of the United States’ energy comes from nuclear power. Write this number as a decimal. 22. WATER Only about 0.5% of the world’s water resources are drinkable by humans, animals, and plants. Write this number as a decimal. Write each decimal as a percent. 23. 0.62 24. 0.94 25. 0.475 26. 0.832 27. 0.007 28. 0.009 29. 2.75 30. 1.38 31. PETS If 0.21 of adults own a cat, what percent of adults own a cat? 32. SURVEYS In a survey, 0.312 of teens said that their favorite sport was soccer or basketball. What percent of the teens chose soccer or basketball as their favorite sport? Write each fraction as a percent. 33. _ 34. _ 17 20 1 37. _ 40 35. _ 36. _ 8 5 4 39. _ 9 12 25 1 38. _ 125 7 4 2 40. _ 3 41. TIME Research shows that _ of Americans set their clocks five minutes 8 25 ahead to keep from being late. What percent of Americans set their clocks five minutes ahead? 42. FOOD About _ of Americans prefer cold pizza over hot pizza. 3 20 What percent of Americans prefer cold pizza? ANIMALS For Exercises 43 and 44, use the information about lions shown below. 43. What percent of a day does a lion spend resting? 44. What percent of a day does a lion spend doing activities? 45. CAMPING About 17% of kids will attend an overnight summer camp, 11 _ will attend a day camp, and 50 0.59 will attend no summer camp at all. Which group of kids is the greatest? A Day in the Life of a Lion Activities 4 hours Resting 20 hours Lesson 5-2 Comparing Fractions, Decimals, and Percents Daryl Benson/Masterfile 259 46. BAND At Jeremy’s high school, about _ of the students are in the band, 3 16 0.31 of high school students play at least one sport, and 13% are in the drama club. Of these three, which type of extra-curricular activity is most popular among students at Jeremy’s high school? Order each set of numbers from least to greatest. 47. _, 0.8, 8%, _ 3 4 7 10 41 4 50. 84%, 0.88, _, _ 50 5 48. 0.2, _, 2%, _ 1 4 3 20 49. _, 7%, 0.09, _ 51. 31%, _, _, 0.305 3 3 10 13 1 20 2 25 3 1 52. 6.5%, _, 0.556, _ 5 6 53. BASEBALL In 2004, Major League Baseball player Jack Wilson’s batting average was 0.308. Melvin Mora hit safely 17 out of every 50 at-bats and Bobby Abreu hit safely 30.1% of the time. Find Mora’s and Abreu’s batting averages and order all three averages from least to greatest. Replace each ● with <, >, or = to make a true sentence. 54. 0.035 ● 3_% 1 2 Real-World Link A batting average is found by dividing the number of hits a batter has by the number of times the batter is at bat, not including times when a batter is walked or hit by a pitch. Source: ehow.com 55. _ ● _% 1 250 3 4 57. ANALYZE TABLES A nutrition label from a popular brand of soda is shown at the right. Would more or 1 less than _ of a person’s daily value 5 of carbohydrates come from this can of soda? Explain your reasoning. 58. TRAVEL The projected number of household trips in 2010 is 50,000,000. About 14,000,000 of these trips will involve air travel. What percent of the trips will involve air travel? 56. _ ● 1_% 7 4 1 4 ÕÌÀÌÊ>VÌÃ 3ERVING 3IZE CAN M, ÕÌÊ*iÀÊ-iÀÛ} >ÀiÃÊ $AILY 6ALUE /Ì>Ê>ÌÊG -`ÕÊMG /Ì>Ê >ÀL Þ`À>ÌiÊG 3UGARS G *ÀÌiÊG 0ERCENT $AILY 6ALUES ARE BASED ON A CALORIE DIET SELECT A FORM For Exercises 59 and 60, use the following information. 1 Lisa ate _ of the cookies, gave 0.25 to her friend Kaitlyn, and gave 37.5% to her 8 %842!02!#4)#% sister. To solve each problem below, select the form of number (fraction, decimal, or percent) that would be easiest to use. Explain your reasoning. Then See pages 688, 712. use that form to solve the problem. Self-Check Quiz at ca.gr7math.com H.O.T. Problems 59. Did Lisa eat more cookies than she gave to Kaitlyn? 60. Who was given more cookies, Kaitlyn or Lisa’s sister? 61. FIND THE ERROR Kristin and Neva are changing 0.7 to a percent. Who is correct? Explain your reasoning. 0. 7 = 7% Kristin 260 Chapter 5 Percent (tl)CORBIS, (cr)Doug Martin, (bl)Bill Aron/PhotoEdit, (br)Masterfile 0. 7 = 70% Neva 62. CHALLENGE Write 1_ as a percent. Justify your answer. 3 5 63. OPEN ENDED Write a percent that is between _ and _. 3 5 64. 2 3 */ -!4( Is 0.04 less than or greater than 40%? Explain your (*/ 83 *5*/( reasoning. 65. Mr. Lee asked his students how many 66. Between which two percents is _? 7 40 hours they watched public television last week. The responses are shown in the table. Which number represents the portion of students who said they watched more than 1 hour? Number of Hours Watched Less than or equal to 1 hour Between 1 and 2 hours More than 2 hours A 0.75 5 C _ 75 B _ D 0.075 100 % of Students 92.5% 5% 2.5% F 15% and 16% G 16% and 17% H 17% and 18% 19% and 20% J 67. What percent of the square is shaded? A 20% IN B 31.25% C 44% IN D 62.5% 100 IN IN Write each ratio as a percent. (Lesson 5-1) 68. 27 out of 100 69. 0.6 out of 100 70. 9 : 20 71. 33 : 50 72. ALGEBRA Determine whether the relationship between the two quantities described in the table is linear. If so, find the constant rate of change. If not, explain your reasoning (Lesson 4-10) Data Transferred Megabits 375 750 1,125 1,500 Time (s) 1 2 3 4 73. FOOD Three-fourths of a pan of lasagna is to be divided equally among 6 people. What part of the lasagna will each person receive? (Lesson 2-4) Order the integers in each set from least to greatest. (Lesson 1-3) 74. {-12, 5, -5, 13, -1} 75. {42, -56, -13, 101, 13} 76. {64, -58, -65, 57, -61} PREREQUISITE SKILL Solve each proportion. (Lesson 4-3) 77. _ = _ 5 6 x 24 78. _ = _ a 12 2 15 79. _ = _ 2 7 5 t 80. _ = _ 3 n 10 8 Lesson 5-2 Comparing Fractions, Decimals, and Percents 261 Comparing Data When you are solving a word problem that involves comparing data, look for words such as more than, times, or percent. They give you a clue about what operation to use. For example, the table shows the final standings for the Western Conference of the United Soccer League’s W-League for the 2005 season. W-League Western Conference Team Games Wins Losses Ties Points Vancouver 14 13 1 0 39 Arizona 14 10 3 1 31 Mile High 14 10 4 0 30 Denver 14 7 6 1 22 Seattle 14 5 8 1 16 Fort Collins 14 2 11 1 7 San Diego 14 0 14 0 0 Source: United Soccer Leagues You can compare the data in several ways. Difference Vancouver had 17 more points than Denver. 39 - 22 = 17 Ratio Mile High won 5 times as many games as Fort Collins. 10 ÷ 2 = 5 Percent Arizona lost about 21% of the games they played. (3 ÷ 14) × 100 ≈ 21 Determine whether each problem asks for a difference, ratio, or percent. Write out the key words in each problem. Then solve each problem. 1. How many more games did Arizona win than lose or tie? 2. What percent of the time did Vancouver win its games? 3. How many times as many games did Mile High win than Seattle? 4. */ -!4( Write three statements comparing the data in the (*/ 83 *5*/( table. One comparison should be a difference, one should be a ratio, and one should be a percent. 262 Chapter 5 Percent Josh Devins Photography Standard 7AF1.4 Use algebraic terminology (e.g. variable, equation, term, coefficient, inequality, expression, constant) correctly. 5-3 Algebra: The Percent Proportion Main IDEA Solve problems using the percent proportion. Standard 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. NEW Vocabulary percent proportion You can use a proportion model to determine the percent represented by 3 out of 5. Draw a 10-by-1 rectangle on grid paper. Label the units on the right from 0 to 100. On the left side, mark equal units from 0 to 5, because 5 represents the whole quantity. Draw a horizontal line from 3 on the left side of the model. The number on the right side is the percent. For Questions 1–3, use the model above. 1. What is 40% of 5? 2. 4 is 80% of what number? 3. Draw a model and find what percent 7 is of 20. In a percent proportion, one ratio compares part of a quantity to the whole quantity, also called the base. The other ratio is the equivalent percent written as a fraction with a denominator of 100. part whole 3 out of 5 is 60%. _3 = 60 ⎬⎫ percent 100 ⎭ 5 _ Find the Percent 1 22 is what percent of 110? Since 22 is being compared to 110, 22 is the part and 110 is the whole. You need to find the percent. Let n represent the percent. part n ⎫ 22 _ =_ Write the percent proportion. ⎬ percent whole 100 110 ⎭ 22 · 100 = 110 · n Find the cross products. 2,200 = 110n Multiply. 2,200 110n _ = ,_ Divide each side by 110. 110 110 20 = n READING in the Content Area For strategies in reading this lesson, visit ca.gr7math.com Simplify. 22 is 20% of 110. a. 17 is what percent of 68? b. 41.4 is what percent of 92? Lesson 5-3 Algebra: The Percent Proportion 263 Find the Part READING Math Percents The whole usually follows the word of. 2 What number is 80% of 500? The percent is 80 and the whole is 500. You need to find the part. Let p represent the part. p part 80 ⎫ _ _ = Write the percent proportion. ⎬ percent whole 100 500 ⎭ p · 100 = 500 · 80 Find the cross products. 100p = 40,000 Multiply. 100p 100 Divide each side by 100. 40,000 100 ,_ = _ p = 400 Simplify. 400 is 80% of 500. c. What number is 35% of 48? d. Find 12.5% of 88. Find the Whole 3 14.4 is 32% of what number? The percent is 32 and the part is 14.4. You need to find the whole. Let w represent the whole. part 32 ⎫ 14.4 _ _ = Write the percent proportion. ⎬ percent whole 1w 100 ⎭ 14.4 · 100 = 32 · w Find the cross products. 1,440 = 32w Multiply. 1,440 32w _ =,_ Divide each side by 32. 32 32 45 = w Simplify. 14.4 is 32% of 45. e. 23.4 is 30% of what number? f. 19 is 62.5% of what number? #/.#%04 3UMMARY 264 Chapter 5 Percent Types of Percent Problems Type Example Proportion Find the Percent 7 is what percent of 10? Find the Part What number is 70% of 10? Find the Whole 7 is 70% of what number? n 7 _ =_ 10 100 w 100 70 p _ =_ 10 100 70 7 _ =_ Extra Examples at ca.gr7math.com Percents Greater than 100 4 6 is what percent of 5? Check for Reasonableness Since the part is greater than the whole, 6 > 5, it makes sense that the percent would be greater than 100. 6 is being compared to 5, so 5 is the whole, and 6 is the part. You need to find the percent. Let n represent the percent. n ⎫ part _6 = _ Write the percent proportion. ⎬ percent whole 100 5 ⎭ 6 · 100 = 5 · n Find the cross products. 600 = 5n Multiply. 600 5n _ =_ Divide each side by 5. 120 = n Simplify. 5 5 6 is 120% of 5. g. 12 is what percent of 6? h. Find 175% of 18. 5 HISTORY The Lewis and Clark Expedition reported that it rained 94 days, which was about 89% of their days in Oregon. How many days did the Lewis and Clark Expedition spend in Oregon? The percent is 89, and the part is 94 days. You need to find the whole number of days. Words 94 days is 89% of what number of days? Variable Let w represent the whole. part whole Proportion 94 89 _ =_ Real-World Link The members of the Lewis and Clark Expedition spent the winter of 1805–1806 in Oregon. Source: Kids Discover 1w 100 94 · 100 = w · 89 ⎫ 89 94 _ = _ ⎬ percent w 100 ⎭ Write the percent proportion. Find the cross products. 9,400 = 89w Multiply. 9,400 89w _ =_ Divide each side by 89. 105.6 ≈ w Simplify. 89 89 The Lewis and Clark Expedition spent 106 days in Oregon. i. SCHOOL Carmila answered 23 questions correctly on her science test and received a grade of 92%. How many questions were on the test? Personal Tutor at ca.gr7math.com Lesson 5-3 Algebra: The Percent Proportion Joseph Sohm/Vision of America/CORBIS 265 Write a percent proportion and solve each problem. Round to the nearest tenth if necessary. Examples 1, 2 (pp. 263–264) Examples 3, 4 (pp. 264–265) Example 5 1. 70 is what percent of 280? 2. What percent of 49 is 7? 3. What number is 60% of 90? 4. Find 72% of 200. 5. 151.5 is 75% of what number? 6. 126 is 30% of what number? 7. 48 is what percent of 30? 8. Find 118% of 19. 9. ANIMALS A tiger can eat food that weighs up to about 15% of its body (p. 265) (/-%7/2+ (%,0 For Exercises 10, 11 12, 13 14, 15 16, 17 18, 19 See Examples 1 2 3 4 5 weight. If a tiger can eat 75 pounds of food, how much does a tiger weigh? Write a percent proportion and solve each problem. Round to the nearest tenth if necessary. 10. 3 is what percent of 15? 11. 120 is what percent of 360? 12. What is 15% of 60? 13. What is 17% of 350? 14. 18 is 45% of what number? 15. 95 is 95% of what number? 16. 15.12 is what percent of 12? 17. Find 250% of 57. 18. BRACES In a recent survey, 34% of kids said they will get dental braces. If nearly 28,800 kids were surveyed, about how many will get braces? 19. PETS There are about 68 million owned dogs in the United States. Of these, 13.6 million were adopted from an animal shelter. About what percent of owned dogs were adopted from an animal shelter? Write a percent proportion to solve each problem. Then solve. Round to the nearest tenth if necessary. 20. What is 2.5% of 95? 21. 4 is what percent of 550? 22. 98 is 22.5% of what number? 23. Find 5.8% of 42. 24. What percent of 110 is 1? 25. 57 is 13.5% of what number? GAMES For Exercises 26–28, use the following information. At the start of a game of chess, each player has the pieces listed at the right. %842!02!#4)#% See pages 689, 712. 26. What percent of each player’s pieces are knights? 27. The king is what percent of each player’s pieces? Self-Check Quiz at ca.gr7math.com 266 28. What piece is 50% of each Chapter 5 Percent Pascal Perret/Getty Images player’s total pieces? Chess Pieces 1 king 1 queen 2 bishops 2 knights 2 rooks 8 pawns H.O.T. Problems 29. CHALLENGE Choose any two numbers, x and y. Find x% of y and y% of x. Will the results always be the same? Explain. */ -!4( Roberto made 56% of his free throws in the first half (*/ 83 *5*/( 30. of the basketball season. If he makes 7 shots out of the next 13 attempts, will it help or hurt his average? Explain your reasoning. 31. A baseball stadium manager expects 32. A pattern of equations is shown below. that 60% of the fans at a game will buy at least $3.00 in concessions. If there are 5,600 fans at a game, which statement does not represent the manager’s expectation? A 3,360 fans each will buy at least $3.00 in concessions. B 2,240 fans each will buy fewer than $3.00 in concessions. 1 C More than _ of the fans each will 2 buy at least $3.00 in concessions. 2 of the fans each will D Less than _ 5 buy fewer than $3.00 in concessions. 1% of 100 = 1 2% of 50 = 1 4% of 25 = 1 8% of 12.5 = 1 16% of 6.25 = 1 Which statement best describes this pattern? F If the percent is doubled and the whole is doubled, the answer is 1. G If the percent is doubled and the whole is halved, the answer is 1. H If the percent is increased by 2 and the whole is halved, the answer is 1. J If the percent remains the same and the whole is halved, the answer is 1. 33. Order the set of numbers _, 16%, and 0.016 from least to greatest. (Lesson 5-2) 1 6 34. LUNCH Forty-eight percent of first period class buys a school lunch. Write this percent as a fraction in simplest form. (Lesson 5-1) MEASUREMENT Complete each conversion. Round to the nearest hundredth if necessary. (Lesson 4-6) 35. 8 lb ≈ _____ ? kg 36. 14 mi ≈ _____ ? km 37. GEOMETRY Find the perimeter 24 in. of the right triangle. (Lesson 3-5) 10 in. PREREQUISITE SKILL Compute each product mentally. (Lesson 2-3) 38. _ · 422 1 2 39. 639 · _ 1 3 40. 0.1 · 722 41. 0.5 · 680 Lesson 5-3 Algebra: The Percent Proportion 267 5-4 Finding Percents Mentally Main IDEA Compute mentally with percents. Standard 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. SCHOOL The table lists the enrollment at Roosevelt Middle School by grade level. 1. 50% of the eighth-grade class are girls. How could you find 50% of 104 mentally? 2. Use mental math to find the number of girls in the eighth-grade class. Roosevelt Middle School Enrollment 3. 25% of the sixth-grade class play basketball. Use mental math to find the number of students in the sixth grade who play basketball. Grade Level Number of Students Sixth 84 Seventh 93 Eighth 104 When you compute with common percents like 50% or 25%, it may be easier to use the fraction form of the percent. This number line shows some fraction-percent equivalents. 0% 0 12.5% 25% 37.5% 50% 62.5% 75% 87.5% 100% 1 8 1 4 +%9 #/.#%04 1 1 25% = _ 20% = _ 5 4 Percents and Fractions Some percents are used more frequently than others. So, it is a good idea to be familiar with these percents and their equivalent fractions. 3 8 1 2 5 8 3 4 7 8 1 Percent-Fraction Equivalents 16 % = _2 _1 12 % = _1 _1 10% = 1 _ 3 6 2 8 50% = _1 40% = _2 33 % = _1 _1 37 % = _1 _3 30% = 3 _ 75% = _3 60% = _3 66 % = _2 _2 62 % = _1 _5 70% = 7 _ 100% = 1 80% = _4 83 % = _1 _5 87 % = _1 _7 90% = 9 _ 2 4 5 5 5 3 3 3 3 3 6 2 2 2 8 8 8 Use Fractions to Compute Mentally 1 Compute 20% of 45 mentally. 1 · 45 or 9 20% of 45 = _ 5 268 Chapter 5 Percent Stephen Simpson/Getty Images 10 _1 Use the fraction form of 20%, which is . 5 10 10 10 _ 2 Compute 33 1 % of 93 mentally. 3 _1 1 1 33_ % of 93 = _ · 93 or 31 3 _1 Use the fraction form of 33 %, which is . 3 3 3 Compute mentally. b. 12_% of 160 1 2 a. 25% of 32 c. 80% of 45 You can also use decimals to find percents mentally. Remember that 10% = 0.1 and 1% = 0.01. Use Decimals to Compute Mentally Compute mentally. Multiplying by Decimals To multiply by 0.1, move the decimal point one place to the left. To multiply by 0.01, move the decimal point two places to the left. 4 1% of 235 3 10% of 98 1% of 235 = 0.01 · 235 or 2.35 10% of 98 = 0.1 · 98 or 9.8 Compute mentally. d. 10% of 65 e. 1% of 450 f. 3% of 22 5 ELECTIONS At Madison Middle School, 60% of the students voted in an election for student council officers. There are 1,500 students. How many students voted in the election? METHOD 1 Use a fraction. 3 60% of 1,500 = _ of 1,500 5 THINK Source: infoplease.com METHOD 2 5 5 is 3 · 300 or 900. 60% of 1,500 is 900. Real-World Link In the 2004 presidential election, about 55% of the nearly 174.8 million registered voters in the U.S. turned out to vote. _1 of 1,500 is 300. So, _3 of 1,500 Use a decimal. 60% of 1,500 = 0.6 of 1,500 60% of 1,500 is 900. THINK 0.1 of 1,500 is 150. So, 0.6 of 1,500 is 6 · 150 or 900. There were 900 students who voted in the election. g. TIPPING Alan and his brother ate lunch at the local café. They left a tip that was 20% of the bill. If the bill was $15.50, how much did Alan and his brother leave for the tip? Personal Tutor at ca.gr7math.com Extra Examples at ca.gr7math.com Patti McConville/ImageState Lesson 5-4 Finding Percents Mentally 269 Examples 1–4 (pp. 268–269) 1. 50% of 120 2. 33_% of 60 1 3 3. 37_% of 72 4. 1% of 52 5. 10% of 350 6. 2% of 630 1 2 Example 5 7. BOOKS An author receives a payment, or commission, equal to 25% of the (p. 269) total sales of her book. Determine the amount of money she will receive if the total sales are $48,000. (/-%7/2+ (%,0 For Exercises 8–15 16–23 24, 25 Compute mentally. See Examples 1–2 3–4 5 Compute mentally. 10. 12_% of 64 1 2 2 14. 66_% of 120 3 11. 16_% of 54 17. 10% of 125 18. 1% of 81 19. 1% of 28.3 21. 7% of 210 22. 10% of 17.1 23. 10% of 10.2 8. 25% of 44 9. 50% of 62 12. 40% of 35 13. 60% of 15 16. 10% of 57 20. 3% of 130 2 3 1 15. 62_% of 160 2 24. PEOPLE The average person has about 100,000 hairs on his or her head. However, people with red hair average only 90% of this number. What is the average number of hairs on the head of a person with red hair? 25. TRAVEL About 10% of travel trips in the United States include a visit to an amusement park. If there were 920 million travel trips in the United States, how many of those included a visit to an amusement park? Replace each ● with <, >, or = to make a true statement. 26. 66_% of 18 ● 60% of 15 2 3 27. 1% of 150 ● 10% of 15 28. MEASUREMENT The Amazon is the second longest river in the world with a length of about 4,000 miles. If the longest river in the world, the Nile, is about 104% of the length of the Amazon, find the length of the Nile River. BASEBALL For Exercises 29 and 30, use the following information. The graphic shows the results of a survey asking women about their interest in Major League Baseball. Suppose 1,000 women were surveyed. 29. How many women said they were interested in Major League Baseball? %842!02!#4)#% .OT )NTERESTED )NTERESTED 30. How many women said they were not interested in Major League Baseball? See pages 689, 712. 31. Self-Check Quiz at ca.gr7math.com 270 7OMENS )NTEREST IN -AJOR ,EAGUE "ASEBALL Chapter 5 Percent FIND THE DATA Refer to the California Data File on pages 16–19. Choose some data and write a real-world problem in which you could mentally compute a percent. Source: ESPN H.O.T. Problems 32. CHALLENGE Find two numbers, a and b, such that 10% of a is the same as 30% of b. Explain your reasoning. 33. OPEN ENDED Suppose you wish to find 33_% of x. List two values of x for 1 3 which you could do the computation mentally. Explain your reasoning. 34. FIND THE ERROR Candace and Pablo are finding 10% of 95. Who is correct? Justify your choice. 10% of 95 = 0.95 10% of 95 = 9.5 Candace 35. Pablo */ -!4( Explain how to find 75% of 40 mentally. (*/ 83 *5*/( 36. Allison, Raul, and Theo drove from 37. Etu bought the items listed below. Austin, Texas, to Los Angeles, California, a distance of 1,224 miles. How much money did he save if each item was 20% off the regular price? 1 Allison drove _ of the total distance, Item Shirt Ties Belt Shoes 3 Raul drove 40%, and Theo drove the remainder. How many miles were driven by the person who drove the greatest distance? Regular Price ($) 19 9 8 29 A 330.5 C 489.6 F $52 H $24 B 408 D 734.4 G $36 J $13 38. FOOTBALL Eleven of the 48 members of the football team are on the field. What percent of the team members are on the field? (Lesson 5-3) Write each fraction as a percent. (Lesson 5-2) 39. _ 9 20 40. _ 7 8 41. _ 3 500 42. _ 2 9 43. MEASUREMENT A snail travels one mile in about 30 hours. At this rate, how far can a snail travel in 1 day? (Lesson 2-4) PREREQUISITE SKILL Draw the next three figures in the pattern. (Lesson 2-8) 44. Lesson 5-4 Finding Percents Mentally (l)Darren Modricker/CORBIS, (r)SW Productions/Getty Images 271 5-5 Problem-Solving Investigation MAIN IDEA: Determine a reasonable answer. STANDARD 7MR3.1 Evaluate the reasonableness of the solution in the context of the original situation. STANDARD 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. e-Mail: REASONABLE ANSWERS YOUR MISSION: Determine a reasonable answer to solve the problem. THE PROBLEM: Will Carla have to pay more or less than $25? EXPLORE PLAN SOLVE CHECK ▲ Carla: Because I work at the Jean Shack, I can buy a $50 jacket there for 60% of its price. You know the cost of the jacket. Carla can buy the jacket for 60% of the price. You want to know if the jacket will cost more or less than $25. Use mental math to determine a reasonable answer. THINK 25 1 _ = _ or 50% 50 2 Since Carla will pay 60% of the cost, she will have to pay more than $25. Find 60% of $50. _3 of 50 5 3 1 Since _ of 50 is 10, _ of 50 is 3 × 10 or 30. 60% of 50 = 5 5 Carla will pay $30, which is more than $25. 1. Explain why determining a reasonable answer was an appropriate strategy for solving the above problem. */ -!4( Explain why mental math skills are important when (*/ 83 *5*/( 2. finding the reasonable answers. 272 Laura Sifferlin Chapter 5 Percent For Exercises 3–5, determine a reasonable answer. 8. GEOMETRY What percent of the large rectangle is green? 3. SCHOOL There are 750 students at Monroe Middle School. If 64% of the students have purchased yearbooks, would the number of yearbooks purchased be about 200, 480, or 700? Explain. IN IN IN IN IN IN 9. CARS Seth is saving to buy a car. He wants 4. MONEY Spencer took $40 to the mall. He spent $12.78 at the music store. He wants to buy two items at the bookstore for $7.25 and $15.78. Does he have enough money with him to make these two purchases? Explain. to have a down payment of 10% for a car that costs $13,000. So far he has saved $850. If he saves $75 each week for the down payment, how soon can he buy the car? 10. PETS In a recent survey, 44% of students at 5. BABY-SITTING Cameron is paid $8.50 an hour to watch his nephew. If he is saving to buy a new skateboard that costs $325, should he baby-sit for about 20, 30, or 40 hours? Explain. Use any strategy to solve Exercises 6–11. Some strategies are shown below. G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s . • Work backward rn. tte • Look for a pa Davison High School own a cat. If there are 1,532 students in the school, is 600, 675, or 715 a reasonable estimate for the number of students who own a cat? Explain. 11. POPULATION About 12.25% of the people in the U.S. live in California. If the U.S. population is about 297,000,000, estimate the population of California. For Exercises 12 and 13, select the appropriate operation(s) to solve the problem. Justify your selection(s) and solve the problem. 6. BAKING Desiree spilled 1_ cups of sugar, 1 2 which she discarded. She then used half of the remaining sugar to make cookies. If she 1 had 4_ cups left, how much sugar did she 2 have initially? 7. NUMBER THEORY Study the pattern. 12. MEASUREMENT Juanita is designing isosceles triangles for a mosaic. The sides of the larger 1 triangle are 1_ times larger than the sides of 2 the triangle shown. Find the perimeter of the larger triangle. 5 cm 1×1 =1 11 × 11 = 121 111 × 111 = 12,321 1111 × 1111 = 1,234,321 5 cm 6.5 cm 13. MEASUREMENT The entrance of a new Without doing the multiplication, find 1111111 × 1111111. convention center will need 2.3 × 10 5 square feet of ceramic tile. The tiles measure 2 feet by 2 feet and are sold in boxes of 24. How many boxes of tiles are needed to tile the entrance? Lesson 5-5 Problem-Solving Investigation: Reasonable Answers 273 CH APTER Mid-Chapter Quiz 5 Lessons 5-1 through 5-5 Write each ratio or fraction as a percent. (Lesson 5-1) 1. 3 out of 16 2. 8 : 10 3. _ 4. _ 13 25 5. 12. 63 is what percent of 84? 7 20 13. Find 41% of 700. STANDARDS PRACTICE Fifteen percent of the dogs at a show were Labrador retrievers. Which is not true? (Lesson 5-1) 3 A _ of the dogs were Labrador retrievers. 20 B 15 out of every 100 dogs were Labrador retrievers. C 85% of the dogs were not Labrador retrievers. D 1 out of every 15 dogs was a Labrador retriever. 7. _ 8. _ 3 200 4 15 14. 294 is 35% of what number? 15. What number is 134% of 62? 16. STANDARDS PRACTICE A study showed that 37.5% of residents in a certain neighborhood use public transportation. If there are 168 residents in the neighborhood, which statement is not supported by this study? (Lesson 5-3) F More than half the residents do not use public transportation. G Less than 62.5% of the residents do not use public transportation. Write each decimal or fraction as a percent. (Lesson 5-2) 6. 0.325 Write a percent proportion to solve each problem. Then solve. Round to the nearest tenth if necessary. (Lesson 5-3) 9. 1.72 H 63 residents use public transportation. J 10. SCHOOL Santos answered 37 out of 40 questions correctly on an English exam. On the same exam, Chantal scored 87.5% 9 and David correctly answered _ of the 10 questions. Which student correctly answered the most questions? (Lesson 5-2) 2 Less than _ of the residents use public 5 transportation. Compute mentally. (Lesson 5-4) 17. 25% of 64 18. 1% of 58.5 19. 66_% of 45 20. 3% of 600 2 3 21. HOMEWORK Sean has 192 pages of reading 11. TIME Use the graph below. Does Leah spend more of her day sleeping or at school? Explain your reasoning. (Lesson 5-2) i> ½ÃÊ >Þ -ii«} ÎÎ¯ "Ì iÀ Î Óx 274 -V Î £ä 7À £Ç¯ Chapter 5 Percent to do in the next three days. He wants to 1 complete 33_ % of the reading tonight. 3 Compute mentally how many pages Sean should read tonight. Explain your reasoning. (Lesson 5-4) 22. FOOD In one month, the Schaffer family -ÌÕ`Þ} Ó Óx spent $121.59, $168.54, $98.67, and $141.78 on groceries. If their grocery budget is $500 per month, did they stay within their budget? Explain. (Lesson 5-5) 5-6 Percent and Estimation Main IDEA Estimate by using equivalent fractions and percents. Standard 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. BrainPOP® ca.gr7math.com GEOGRAPHY The total area of Earth is 196,800,000 square miles. The graphic shows the percent of the area of Earth that is land and the percent that is water. OF %ARTH IS LAND OF %ARTH IS WATER 1. Round the total area of Source: World Book Earth to the nearest hundred million square miles. NEW Vocabulary 2. Round the percent of Earth that is land to the nearest ten percent. compatible numbers 3. Use mental math to estimate the area of the land on Earth. When an exact answer is not needed, you can estimate a percent of a number by using compatible numbers. Compatible numbers are two numbers that are easy to divide mentally. Estimate Percents of Numbers 1 Estimate 19% of 30. READING Math 1 19% ≈ 20% or _ . ≈ is approximately equal to _1 of 30 is 6. So, 19% of 30 is about 6. 5 and 30 are compatible numbers. 5 5 2 25% of 41 1 25% is _ , and 41 is about 40. 4 _1 and 40 are compatible numbers. 4 _1 of 40 is 10. So, 25% of 41 is about 10. 4 3 Estimate 65% of 76. 2 2 65% ≈ 66_ % or _ , and 76 is about 75. 3 3 3 and 75 are compatible numbers. _1 of 75 is 25, and _2 of 75 is 2 · 25 or 50. So, 65% of 76 is about 50. 3 3 Estimate. Justify your answer. a. 24% of 44 Extra Examples at ca.gr7math.com b. 40% of 49 c. 13% of 65 Lesson 5-6 Percent and Estimation 275 Sometimes estimation provides the best answer to a real-world problem. 4 LEFT–HANDEDNESS About 11% of the population is left-handed. If there are about 36 million people in California, about how many Californians are left-handed? 1 11% of 36 million ≈ 10% or _ of 36 million 10 = 3.6 million 11% is about 10%. 1 _ × 36 = 3.6 10 So, about 3.6 million Californians are left-handed. d. MONEY A circulating $5 bill in the United States lasts about 22% as long as a $100 bill. If a $100 bill lasts nine years, estimate how long a $5 bill lasts. Personal Tutor at ca.gr7math.com You can use similar techniques to estimate a percent. Estimate Percents Estimate each percent. 5 8 out of 25 8 8 1 _ ≈_ or _ 25 is close to 24. 3 24 25 _1 = 33_1 % 3 3 1 %. So, 8 out of 25 is about 33_ 3 6 14 out of 25 Estimation When estimating, estimate so that you change the ratio the least. 15 3 14 _ ≈_ or _ 14 is close to 15. 25 5 25 _3 = 60% 5 So, 14 out of 25 is about 60%. 7 89 out of 121 90 3 89 _ ≈_ or _ 120 121 3 _ = 75% 4 4 89 is close to 90, and 121 is close to 120. So, 89 out of 121 is about 75%. Estimate each percent. Justify your answer. e. 7 out of 57 276 Chapter 5 Percent f. 9 out of 25 g. 7 out of 79 Examples 1–3 (p. 275) 2. 66_% of 20 2 3 1. 49% of 160 3. 73% of 65 Example 4 4. SCHOOL Math is the favorite subject of about 28% of students, according to (p. 276) a recent study. If there are 30 students in your class, estimate the number of students who would pick math as their favorite subject. Examples 5–7 (p. 276) (/-%7/2+ (%,0 For Exercises 8–15 16–23 24–25 Estimate. See Examples 1–3 5–7 4 Estimate each percent. 5. 6 out of 35 6. 8 out of 79 7. 14 out of 19 Estimate. 8. 29% of 50 9. 67% of 93 10. 20% of 76 11. 25% of 63 12. 21% of 71 13. 92% of 41 14. 48% of 159 15. 73% of 81 Estimate each percent. 16. 7 out of 29 17. 6 out of 59 18. 2 out of 15 19. 5 out of 36 20. 8 out of 23 21. 7 out of 11 22. 4 out of 21 23. 9 out of 55 24. MEASUREMENT The length of an object, in inches, is about 39% of its length in centimeters. Estimate the length, in inches, of an object 50 centimeters long. 25. SPORTS A place kicker made 73% of his field goal attempts last season. If he had 46 attempts, estimate the number of field goals that he made. 26. ANIMALS In 2003, 1,072 species of animals were endangered or threatened. Of these species, 342 were mammals. Estimate the percent of endangered or threatened animals that were mammals. 27. ANALYZE TABLES Estimate the percent of the population of each state that lives in each city. Then determine which city has the greatest percent of its state’s population. 2003 Population %842!02!#4)#% See pages 690, 712. City City Population Entire State Population New York, NY 8,085,742 19,190,115 Los Angeles, CA 3,819,951 35,484,453 Chicago, IL 2,869,121 12,653,544 Source: World Almanac Self-Check Quiz at ca.gr7math.com Estimate. 28. 26.5% of 123 29. 124% of 41 30. 249% of 119 Lesson 5-6 Percent and Estimation Purestock/SuperStock 277 H.O.T. Problems 31. NUMBER SENSE Use mental math to determine which is greater: 24% of 240 or 51% of 120. Explain your reasoning. CHALLENGE Determine whether each statement about estimating percents of numbers is sometimes, always, or never true. Explain. 32. If both the percent and the number are rounded up, the estimate will be greater than the actual answer. 33. If the percent is rounded up and the number is rounded down, the estimate will be greater than the actual answer. 34. */ -!4( Explain how you could use fractions and compatible (*/ 83 *5*/( numbers to estimate 26% of $98.98. 35. Rick took his father to dinner for 36. There are 150 students who participate his birthday. When the bill came, Rick’s father reminded him that it is customary to tip the server 15% of the bill. If the bill was $19.60 and Rick estimated the tip to be $3, which of the following shows his method of estimation? in athletics at Southland High School. If there are 325 total students, about what part of the student body participates in athletics? F 40% G 45% H 50% A 15% of $19.60 ≈ 15% of $15. J B 15% of $19.60 ≈ 10% of $20. 55% C 15% of $19.60 ≈ 20% of $20. D 15% of $19.60 ≈ 15% of $20. 37. LIFE EXPECTANCY The average life expectancy in the United States is about 77 years of age. In 1901, the average life expectancy was about 63% of this number. Would 30, 48, or 60 years of age be a reasonable life expectancy for the year 1901? Explain. (Lesson 5-5) 38. BUSES Of the 840 students at Moyer Middle School, 75% ride the bus. Use mental math to find the number of students who ride the bus. (Lesson 5-4) 39. SEATING A teacher would like to make a square seating chart. If there are 25 students in the class, how many students should be in each row? (Lesson 3-1) PREREQUISITE SKILL Solve each equation. (Lesson 2-7) 40. 0.2a = 7 278 Chapter 5 Percent 41. 20s = 8 42. 0.35t = 140 43. 30n = 3 5-7 Algebra: The Percent Equation Main IDEA Solve problems using a percent equation. Standard 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. Standard 7NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest. NEW Vocabulary GEOGRAPHY The approximate area of New York is 55,000 square miles. Of this area, 13% is water. Total Area (sq mi) Percent of Area Occupied by Water 55,000 13% Source: infoplease.com 1. Use a percent proportion to find the area of water in New York. 2. Express the percent for New York as a decimal. Multiply the total area of New York by this decimal. 3. How are the answers for Exercises 1 and 2 related? A percent equation is an equivalent form of a percent proportion in which the percent is written as a decimal. percent equation part _ = percent whole REVIEW Vocabulary equation a mathematical sentence that contains two expressions separated by an equals sign (Lesson 1-2) The percent is written as a decimal. part _ · whole = percent · whole whole Multiply each side by the whole. part = percent · whole This form is called the percent equation. Find the Part 1 Find 6% of 525. Estimate 1% of 500 is 5. So, 6% of 500 is 6 · 5 or 30. The percent is 6. The whole is 525. You need to find the part. Let p represent the part. part = percent · whole p = 0.06 · 525 Write the percent equation. Note that the percent has been written as a decimal. p = 31.5 Multiply. Check for Reasonableness 31.5 ≈ 30 ✓ Write a percent equation to solve each problem. Then solve. Round to the nearest tenth if necessary. a. What number is 35% of 88? b. Find 15% of 275. Lesson 5-7 Algebra: The Percent Equation David Muench/CORBIS 279 In some instances the percent or the whole are unknown. Solve the percent equation for the missing value. Find the Percent 2 420 is what percent of 600? Decimals and Percents When finding the percent, be sure to place the decimal point correctly when writing your answer. Estimate 420 400 2 _ ≈ _ or 66 _% 600 600 3 The part is 420. The whole is 600. You need to find the percent. Let n represent the percent. part = percent · whole 420 = · 600 n Write the percent equation. 420 _ _ = n · 600 600 600 Divide each side by 600. 0.7 = n Simplify. Since 0.7 = 70%, 420 is 70% of 600. Note that the answer, a decimal, must be converted to a percent. 2 Check for Reasonableness 70% ≈ 66_% ✓ 3 Write a percent equation to solve each problem. Then solve. Round to the nearest tenth if necessary. c. 62 is what percent of 186? d. What percent of 750 is 6? Find the Whole 3 65 is 52% of what number? Estimate 65 is 50% of 130. The part is 65. The percent is 52. You need to find the whole. Let w represent the whole. part = percent · whole 65 = 0.52 · w Write the percent equation. Note that the percent has been written as a decimal. 65 0.52w _ =_ 0.52 0.52 Divide each side by 0.52. 125 = w Simplify. So, 65 is 52% of 125. Check for Reasonableness 125 ≈ 130 ✓ Write a percent equation to solve each problem. Then solve. Round to the nearest tenth if necessary. e. 210 is 75% of what number? 280 Chapter 5 Percent f. 54 is 18% of what number? Extra Examples at ca.gr7math.com #/.#%04 3UMMARY The Percent Equation Type Example Equation Find the Part What number is 25% of 60? p = 0.25(60) Find the Percent 15 is what percent of 60? 15 = n(60) Find the Whole 15 is 25% of what number? 15 = 0.25w 4 SALES TAX A television costs $350. If a 7% sales tax is added, what is the total cost of the television? METHOD 1 Find the amount of tax first. The whole is $350. The percent is 7%. You need to find the amount of the tax, or the part. Let t represent the amount of tax. part = percent · whole Real-World Link State sales tax rates range from 0% in Alaska, Delaware, Montana, New Hampshire, and Oregon, to 7.25% in California. Source: www.taxadmin.org t = 0.07 · 350 Write the percent equation, writing 7% as a decimal. t = 24.5 Multiply. The tax is $24.50. The total cost of the television is $350.00 + $24.50 or $374.50. METHOD 2 Find the total percent first. Find 100% + 7% or 107% of $350 to find the total cost, including tax. Let t represent the total cost. part = percent · whole t = 1.07 · 350 Write the percent equation, writing 107% as a decimal. t = 374.5 Multiply. The total cost of the television is $374.50. g. PROFIT Mrs. Dunn bought a house for $275,000. Three years later, she sold it for a 35% profit. What was the sale price of the house? h. PAYCHECKS Paige earned $250 before taxes working at a movie theater. If 23% of her pay is withheld for taxes, how much is her take-home pay? Personal Tutor at ca.gr7math.com Lesson 5-7 Algebra: The Percent Equation Juan Silva/Getty Images 281 Examples 1–3 (pp. 279–280) Example 4 Solve each problem using a percent equation. 1. Find 85% of 920. 2. What number is 4% of 30? 3. 25 is what percent of 625? 4. What percent of 800 is 2? 5. 680 is 34% of what number? 6. 25% of what number is 10? 7. PROFIT A dealership sets car prices so that there is a 40% profit. If the (p. 281) (/-%7/2+ (%,0 For Exercises 8, 9, 14, 15 10, 11, 16, 17 12, 13, 18, 19 20–23 dealership paid $5,300 for a car, for how much should they sell the car? Solve each problem using a percent equation. See Examples 1 8. Find 60% of 30. 9. What is 40% of 90? 10. What percent of 90 is 36? 11. 45 is what percent of 150? 12. 75 is 50% of what number? 13. 15% of what number is 30? 2 14. What number is 13% of 52? 15. Find 24% of 84. 16. 6 is what percent of 3,000? 17. What percent of 5,000 is 6? 18. 3% of what number is 9? 19. 50 is 10% of what number? 3 4 20. CLOTHING A sweater costs $45. If a 6.5% sales tax is added, what is the total cost of the sweater? 21. FUEL MILEAGE A car can travel 32 miles per gallon of gasoline. When the tires are under-inflated, the car gets 15% fewer miles per gallon. What is the fuel mileage of the car with under-inflated tires? 22. REAL ESTATE A commission is a fee paid to a salesperson based on a percent of sales. Suppose a real estate agent earns a 3% commission. How much commission would be earned on the sale of a $150,000 house? 23. BASKETBALL In a recent National Basketball Association season, LeBron James made about 47.18% of his field-goal attempts. If he made 386 field goals, about how many attempts did he make? 24. MUSEUMS Which museum uses the greater percent of its area for exhibits? %842!02!#4)#% See pages 690, 712. Self-Check Quiz at ca.gr7math.com 282 Guggenheim Museum in New York Total area: 79,600 square feet Exhibition space: 49,600 square feet Chapter 5 Percent (l)R. Kord/Robertstock.com, (r)Steve Vidler/SuperStock Guggenheim Museum in Bilbao, Spain Total area: 257,000 square feet Exhibition space: 110,000 square feet Solve each problem using a percent equation. 25. Find 6_% of 150. 1 4 H.O.T. Problems 26. 360 is what percent of 270? 27. CHALLENGE Determine whether a% of b is sometimes, always, or never equal to b% of a. Explain your reasoning. 28. CHALLENGE Mrs. McGary budgeted a certain amount of money for new shoes. Before she could buy them, there was a 20% increase in price. She waited for a month, and the store discounted the shoes 20%. She bought the shoes, thinking that they would cost less than the original price. Was she correct? Explain your reasoning. */ -!4( Explain, using an example, how a 5% discount plus (*/ 83 *5*/( 29. 5% sales tax on an item does not result in the original price of the item. 30. Mr. Dempsey receives a 7% 31. Shirley purchased an antique dresser commission for every appliance he sells. If he sells a refrigerator for $1,299, what is his commission? for $350. She restored the dresser and sold it for a 50% profit. For how much did Shirley sell the dresser? A $9.09 C $92.93 F $175 H $525 B $90.93 D $909.30 G $367.50 J $700 32. FOOTBALL A quarterback completed 19 out of 30 attempts to pass the football. Estimate his percent of completion. (Lesson 5-5) Compute mentally. (Lesson 5-4) 33. 20% of $200 34. 62.5% of 96 35. 75% of 84 36. 6% of 150 GEOMETRY Find the distance between each pair of points. Round to the nearest tenth, if necessary. (Lesson 3-7) 37. S(2, 3), T(0, 6) 38. E(-1, 1), F(3, -2) 39. W(4, -6), V(-3, -5) 40. WEATHER Ruben read that the low temperature for the day was expected to be -5°F and the high temperature was expected to be 8°F. What was the difference in the expected high and low temperatures? (Lesson 1-5) ALGEBRA Evaluate each expression if f = -9, g = -6, and h = 8. (Lesson 1-2) 41. -5fg 43. -10fh 42. 2gh PREREQUISITE SKILL Evaluate each expression. (Lesson 1-3) 44. ⎪17 - 24⎥ 45. ⎪340 - 253⎥ 46. ⎪531 - 487⎥ 47. ⎪352 - 581⎥ Lesson 5-7 Algebra: The Percent Equation 283 5-8 Percent of Change Main IDEA Find and use the percent of increase or decrease. Standard 7NS1.6 Calculate the percentage of increases and decreases of a quantity. Standard 7NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest. MONEY MATTERS Over the years, some prices increase. Study the change in gasoline prices from 1930 to 1960. Price of a Gallon of Gasoline Year Price (¢) 1930 10 1940 15 1950 20 1960 25 Source: Senior Living 1. How much did the price increase from 1930 to 1940? NEW Vocabulary percent of change percent of increase percent of decrease markup selling price discount 2. Write the ratio __. Then write the ratio as a percent. amount of increase price in 1930 3. How much did the price increase from 1940 to 1950? Write the amount of increase ratio __ . Then write the ratio as a percent. price in 1940 4. How much did the price increase from 1950 to 1960? Write the amount of increase ratio __ . Then write the ratio as a percent. price in 1950 5. MAKE A CONJECTURE Why are the amounts of increase the same but the percents different? The percent that an amount changes from its original amount is called the percent of change. +%9 #/.#%04 Percent of Change Words A percent of change is a ratio that compares the change in quantity to the original amount. Symbols percent of change = amount of change __ original amount To find the percent of change, do the following: Step 1 Subtract to find the amount of change. Step 2 Write the ratio amount of change __ as a decimal. original amount Step 3 Write the decimal as a percent. 284 Chapter 5 Percent Underwood & Underwood/CORBIS When the new amount is greater than the original, the percent of change is a percent of increase. When the new amount is less than the original, the percent of change is called a percent of decrease. Find Percent of Change 1 CLUBS The Science Club had 25 members. Now it has 30 members. Percent of Change When finding percent of change, always use the original amount as the whole. Find the percent of change. State whether the change is an increase or decrease. Step 1 The amount of change is 30 - 25 or 5. Step 2 amount of change percent of __ change = original amount 5 =_ The amount of change is 5. The original amount is 25. = 0.2 Divide. 25 Step 3 Definition of percent of change The decimal 0.2 written as a percent is 20%. So, the percent of change is 20%. Since the new number of members is greater than the original, it is a percent of increase. 2 COMIC BOOKS Consuela had 20 comic books. She gave some to her friend. Now she has 13 comic books. Find the percent of change. State whether the percent of change is an increase or a decrease. Step 1 The amount of change is 20 - 13 or 7. Step 2 amount of change percent of = __ change original amount 7 =_ The amount of change is 7. The original amount is 20. = 0.35 Divide. 20 Real-World Link In 1940, the average comic book sold for $0.10, but today it is worth more than $700. That’s a 6,999% increase in value! Step 3 Definition of percent of change The decimal 0.35 written as a percent is 35%. So, the percent of change is 35%. The new amount is less than the original. It is a percent of decrease. Source: antiqueweb.com Find each percent of change. Round to the nearest tenth if necessary. State whether the percent of change is an increase or a decrease. a. original: 6 hours b. original: 80 water bottles new: 10 hours new: 55 water bottles c. original: 15 meters d. original: 1.25 hours new: 6 meters new: 3.5 hours Personal Tutor at ca.gr7math.com Lesson 5-8 Percent of Change Ted Streshinsky/CORBIS 285 A store sells an item for more than it paid for that item. The extra money is used to cover the expenses and to make a profit. The increase in the price is called the markup. The percent of markup is a percent of increase. The amount the customer pays is called the selling price. Find the Selling Price 3 MARKETING Shonny is selling some embroidered jackets on a Web site. She wants to price the jackets 25% over her cost, which is $35. Find the selling price for a jacket. METHOD 1 Check for Reasonableness To estimate the selling price, think 25% of 35 is about _ 1 4 of 36 or 9. The selling price should be about $35 + $9, or $44. Find the amount of the markup first. The whole is $35. The percent is 25. You need to find the amount of the markup, or the part. Let m represent the amount of the markup. part = percent · whole m = 0.25 · 35 Write the percent equation. m = 8.75 Multiply. Add the markup $8.75 to Shonny’s cost $35 to find the selling price. $35 + $8.75 = $43.75 METHOD 2 Find the total percent first. The customer will pay 100% of Shonny’s cost plus an extra 25% of the cost. Find 100% + 25% or 125% of Shonny’s cost. Let p represent the price. part = percent · whole p = 1.25 · 35 Write the percent equation. p = 43.75 Multiply. The selling price of the jacket is $43.75. Find the selling price for each item given the percent of markup. e. digital camera: $120, 55% markup f. sunglasses: $7, 30% markup g. SHIPPING Cheng-Yu ordered a book that cost $24 from an online store. Her total with the shipping charge was $27. What was the percent of markup charged for shipping? The amount by which a regular price is reduced is called the discount. The percent of change is a percent of decrease. 286 Chapter 5 Percent Extra Examples at ca.gr7math.com Find the Sale Price 4 SHOPPING The Sport Chalet is having a sale. A snowboard has an original price of $95. It is on sale for 35% off the original price. Find the sale price of the snowboard. METHOD 1 Find the amount of the discount first. The percent is 35, and the whole is 95. We need to find the amount of the discount, or the part. Let d represent the amount of discount. part = percent · whole d = 0.35 · 95 Write the percent equation. d = 33.25 Multiply. Subtract the amount of the discount from the original price to find the sale price. $95 - $33.25 = $61.75. Real-World Link Snowboarding is one of the fastest growing sports with over 7.2 million participants. Source: about.com METHOD 2 Find the percent paid first. If the amount of the discount is 35%, the percent paid is 100% - 35% or 65%. Find 65% of $95. Let s represent the sale price. part = percent · whole s = 0.65 · 95 Write the percent equation. s = 61.75 Multiply. The sale price of the snowboard is $61.75. Find the sale price of each item to the nearest cent. h. CD: $14.50, 10% off Examples 1–2 (p. 285) Example 3 (p. 286) Example 4 (p. 287) i. sweater: $39.95, 25% off Find each percent of change. Round to the nearest tenth if necessary. State whether the percent of change is an increase or a decrease. 1. original: $40 2. original: 25 CDs 3. original: 325 miles new: $32 new: 32 CDs new: 400 miles Find the selling price for each item given the percent of markup. 4. roller blades: $60, 35% markup 5. coat: $87, 33% markup 6. BICYCLES Find the sale price of a bicycle that is regularly $140 and is on sale for 40% off the original price. Lesson 5-8 Percent of Change blickwinkel/Alamy Images 287 (/-%7/2+ (%,0 For Exercises 7–14 15–18 19–22 See Examples 1, 2 3 4 Find each percent of change. Round to the nearest tenth if necessary. State whether the percent of change is an increase or a decrease. 7. original: 6 tickets 8. original: 27 guests 9. original: $80 new: 9 tickets new: 39 guests new: $64 10. original: $560 11. original: 68°F 12. original: 150 e-mails new: $420 new: 51°F new: 98 e-mails 13. TELEVISION On Tuesday night, 17.8 million households watched a popular television show. On Wednesday night, 16.6 million households watched the same show. Find the percent of decrease in the number of households watching the show from Tuesday to Wednesday. 14. STOCK Patrice invested $300 into a particular stock. The amount doubled within a few weeks. Find the percent of increase. Find the selling price for each item given the cost to the store and the markup. 15. computer: $700, 30% markup 16. CD player: $120, 20% markup 17. jeans: $25, 45% markup 18. baseball cap: $12, 48% markup Find the sale price of each item to the nearest cent. 19. video game: $75, 25% off 20. trampoline: $399, 15% off 21. skateboard: $119.95, 30% off 22. earrings: $19.50, 35% off 23. INTERNET An Internet service provider offers connection speed that is 35% faster than dial-up. If it takes Brad 8 seconds to connect to the Internet using dial-up, how long would it take using this provider? 24. ANIMALS At birth, a giraffe was 62 inches tall and grew at the highly unusual rate of 0.5 inch per hour. By what percent did the height of the giraffe increase in the first 24 hours? 25. FIND THE DATA Refer to the California Data File on pages 16–19. Choose some data and write a real-world problem in which you would need to find the percent of change. 26. BUSINESS The table gives the price of milk for various years. During which ten-year period did the price of milk have the greatest percent of increase? %842!02!#4)#% See pages 690, 712. 27. ALGEBRA Students receive a 20% discount Self-Check Quiz at ca.gr7math.com 288 Chapter 5 Percent off the price of an adult ticket at the theater. If a student ticket is $6.80, find the price of an adult ticket. (Hint: Let p represent the part and p + 6.80 represent the whole.) Price of a Gallon of Milk Year Price ($) 1970 1.23 1980 1.60 1990 2.15 2000 2.78 Source: Senior Living H.O.T. Problems 28. CHALLENGE Blake bought a computer listed for $x at a 15% discount. He also paid a 5% sales tax. After 6 months, he decided to sell the computer for $y, which was 55% of what he paid originally. Express y in terms of x. 29. FIND THE ERROR Jared and Sidney are solving the following problem: The price of a school play ticket rose from $5.75 to $6.25. What is the percent of increase for the price of a ticket? Who is correct? Explain your reasoning. _ percent 0.50 of change = 5.75 = 0.087 or 8.7% _ percent 0.50 of change = 6.25 = 0.08 or 8% Jared Sidney */ -!4( Write and solve a real-world problem involving a 25% (*/ 83 *5*/( 30. increase or decrease in some quantity. 31. A television originally cost $1,250. 32. Grace and her two brothers shared the Samuel bought it at 30% off. How much was deducted from the original amount? C $425 cost of a new video game system equally. The original price of the system was $179. They received a 15% discount off the original price and paid 7.5% sales tax on the discounted price. Find the approximate amount that each paid for the video game system. D $375 F $51 H $60 G $55 J A $875 B $675 $66 33. TAXES An average of 40% of the cost of gasoline goes to state and federal taxes. If gasoline sells for $2.15 per gallon, how much goes to taxes? (Lesson 5-7) Estimate. (Lesson 5-6) 34. 21% of 60 35. 25% of 83 36. 12% of 31 37. 34% of 95 Express each rate as a unit rate. (Lesson 4-1) 38. $36 in 3 hours 39. 1.5 inches of rain in 5 months PREREQUISITE SKILL Solve each equation. (Lesson 2-7) 40. 45 = 300 · a · 3 41. 24 = 200 · 0.04 · y 42. 21 = 60 · m · 5 Lesson 5-8 Percent of Change First Light 289 5-9 Simple Interest Main IDEA Standard 7NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest. NEW Vocabulary interest principal COLLEGE SAVINGS Hector received $1,000 from his grandparents. He plans to save it for college expenses. The graph shows rates for various investments for one year. Rates for Investments Rate as a Percent Solve problems involving simple interest. 1. If Hector puts his money in a savings account, he will receive 2.5% of $1,000 in interest for one year. Find the interest Hector will receive. 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 4.45% 3.25% 2.5% 0 Savings Money Market Certificate of Deposit (CD) Type of Investment 2. Compare the interest Hector can receive in one year from a money market and from a certificate of deposit. Interest is the amount of money paid or earned for the use of money. For a savings account, you earn interest from the bank. For a credit card, you pay interest to the bank. To solve problems involving simple interest, use the following formula. Interest is the amount of money paid or earned. The annual interest rate is expressed as a decimal. I = prt The principal is the amount of money invested or borrowed. The time is written in years. Find Simple Interest 1 Find the simple interest for $500 invested at 6.25% for 3 years. Reading Math I = prt is read interest equals principal times rate times time. I = prt Write the simple interest formula. I = 500 · 0.0625 · 3 Replace p with 500, r with 0.0625, and t with 3. I = 93.75 The simple interest is $93.75. Find the simple interest to the nearest cent. a. $400 at 3.67% for 2 years 290 Chapter 5 Percent b. $770 at 16% for 6 months Find the Total Amount 2 What is the total amount of money in dollars in an account where $95 is invested at a simple interest rate of 7.5% for 8 months? A $152.50 B $152 C $142.50 D $99.75 Read the Item You need to find the total amount in an account. Converting Units When using the formula I = prt, remember that time is in years. Eight months is _ or _ year. 8 12 2 3 Solve the Item I = prt Simple interest formula 2 I = 95 · 0.075 · _ p = 95, r = 0.075, t = I = 4.75 Simplify. 3 _2 3 The amount in the account is 95 + 4.75 or $99.75. The answer is D. c. What is the total amount of money owed on a credit card with a balance of $1,500 at a simple interest rate of 22% after 1 month? F $1,502.75 G $1,527.50 H $1,533 J $1,830 Personal Tutor at ca.gr7math.com Find the Interest Rate 3 CAR SALES Tonya borrowed $3,600 to buy a used car. She will be paying $131.50 each month for the next 36 months. Find the simple interest rate for her loan. Use the formula I = prt. First find the total that Tonya will pay. $131.50 · 36 = $4,734 She will pay $4,734 - $3,600 or $1,134 in interest. So, I = 1,134. Real-World Career How Does a Car Salesperson Use Math? A car salesperson must calculate the price of a car including any discounts, dealer preparation cost, and state taxes. They may also help customers by determining the amount of their car payments. The principal is $3,600. So, p = 3,600. The loan will be for 36 months or 3 years. So, t = 3. I = prt Write the simple interest formula. 1,134 = 3,600 · r · 3 Replace I with 1,134, p with 3,600, and t with 3. 1,134 = 10,800r Simplify. 1,134 10,800r _ =_ Divide each side by 10,800. 10,800 10,800 0.105 = r The simple interest rate is 0.105 or 10.5%. For more information, go to ca.gr7math.com d. SAVINGS BOND Louie purchased a $200 savings bond. After 5 years, it is worth $232.50. Find the simple interest rate for his bond. Extra Examples at ca.gr7math.com Aaron Haupt Lesson 5-9 Simple Interest 291 Example 1 (p. 290) Example 2 (p. 291) Find the simple interest to the nearest cent. 1. $300 at 7.5% for 5 years Find the total amount in each account to the nearest cent. 3. $660 at 5.25% for 2 years 5. For Exercises 7–10 11–14 15, 16 See Examples 1 2 3 B $110 C $111 D $112 6. LOANS Jose’s brother paid off his $5,000 student loan in 1_ years. If he paid 1 2 (p. 291) (/-%7/2+ (%,0 4. $385 at 12.6% for 9 months STANDARDS PRACTICE Nina invested $100 in a savings account for 4 years. Find the total amount in her account if it earns a simple interest of 2.75%. A $109 Example 3 2. $230 at 12% for 8 months a total of $5,225, what was the simple interest rate for the loan? Find the simple interest to the nearest cent. 7. $250 at 6% for 3 years 8. $725 at 4.5% for 4 years 9. $834 at 7.25% for 2 months 10. $3,070 at 8.65% for 24 months Find the total amount in each account to the nearest cent. 11. $2,250 at 5% for 3 years 12. $5,060 at 7.2% for 5 years 13. $575 at 4.25% for 6 months 14. $950 at 7.85% for 10 months 15. INVESTMENTS Over the summer, Booker earned $1,200, which he invested in stocks that increased in value to $1,335 in only 9 months. Find the simple interest rate for the investment. 16. HOUSING The prices of a ranch in Levittown, New York, are given at the right. Determine the simple interest rate for the investment of a ranch in Levittown from 1947 to 2007. Year Price ($) 1947 9,500 2007 280,000 17. CARS Felicia took out a 5-year loan for $15,000 to buy a car. If the simple interest rate was 11%, how much total will she pay including interest? Find the simple interest to the nearest cent. 18. $1,000 at 7_% for 30 months 1 2 19 $5,200 at 13_% for 1_ years 1 5 1 2 20. CREDIT CARDS The balance on a credit card was $500. Mr. Cook paid the minimum monthly payment of $25. The remaining balance was charged a simple interest rate of 18%. If no additional purchases were made, what was the balance the next month? %842!02!#4)#% See pages 691, 712. 21. HOUSING The Turners need to borrow $100,000 to purchase a home. The Self-Check Quiz at ca.gr7math.com 292 Chapter 5 Percent credit union is offering a 30-year mortgage loan at 5.38% interest while the community bank has a 25-year mortgage loan at 6.12% interest. Assuming simple interest, which loan will result in less total interest? H.O.T. Problems 22. CHALLENGE What will be the monthly payments on a loan of $25,000 at 9% interest so that it will be paid off in 15 years? How much will the total interest be? 23. OPEN ENDED Give a principal and interest rate where the amount of simple interest earned in two years would be $50. Justify your answer. */ -!4( Explain what each variable in the simple interest (*/ 83 *5*/( 24. formula represents. 25. Mr. and Mrs. Owens placed $1,500 in a 26. Dave borrowed $4,000 at 9% simple college savings account with a simple interest rate of 4% when Lauren was born. How much will be in the account in 18 years when Lauren is ready to go to college? Assume no more deposits or withdrawals were made. interest for one year. He made no payments during that year. How much interest is owed at the end of the year? F $90 G $180 A $1,080 H $270 B $2,580 J C $10,800 $360 D $12,300 27. SALES What is the sale price of a $200 cell phone on sale at 10% off the regular price? (Lesson 5-8) Solve each problem using a percent equation. (Lesson 5-7) 28. What percent of 70 is 17.5? 29. 18 is 30% of what number? 30. HEALTH Shashawn’s heart beats 18 times in 15 seconds. Write and solve a proportion to determine how many times her heart beats in 1 minute or 60 seconds. (Lesson 4-3) 31. Express 0.000084 in scientific notation. (Lesson 2-10) Math and Art It’s a Masterpiece It’s time to complete your project. Use the information and data you have gathered about your artist and the Golden Ratio to prepare a Web page or poster. Be sure to include your reports and calculations with your project. Cross-Curricular Project at ca.gr7math.com Lesson 5-9 Simple Interest 293 Extend 5-9 Main IDEA Find compound interest. Spreadsheet Lab Compound Interest Simple interest, which you studied in Lesson 5-9, is paid only on the initial principal of a savings account or a loan. Compound interest is paid on the initial principal and on interest earned in the past. You can use a spreadsheet to investigate the growth of compound interest. Standard 7NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest. Standard 7MR2.2 Apply strategies and results from simpler problems to more complex problems. SAVINGS Find the value of a $2,000 savings account after four years if the account pays 8% interest compounded semiannually. 8% interest compounded semiannually means that the interest is paid twice a year. The interest rate is 8% ÷ 2 or 4% for each 6 months. #OMPOUND )NTERESTXLS NEW Vocabulary compound interest The interest rate is entered as a decimal. The spreadsheet evaluates the formula A4 × B1. The interest is added to the principal every 6 months. The spreadsheet evaluates the formula A4 + B4. ! " # $ 2ATE 0RINCIPAL )NTEREST .EW 0RINCIPAL 4IME 92 3HEET 3HEET 3HEET The value of the savings account after four years is $2,737.14. EXERCISES 1. Use a spreadsheet to find the value of a savings account if $2,000 is invested for four years at 8% interest compounded quarterly. 2. Suppose you leave $1,000 in each of three bank accounts paying 6% interest per year. One account pays simple interest, one pays interest compounded semiannually, and one pays interest compounded quarterly. Use a spreadsheet to find the amount of money in each account after three years. 3. MAKE A CONJECTURE How does the amount of interest change if the compounding occurs more frequently? Explain your reasoning. 294 Chapter 5 Percent CH APTER 5 Study Guide and Review Download Vocabulary Review from ca.gr7math.com Key Vocabulary compatible numbers Be sure the following Key Concepts are noted in your Foldable. percent of change (p. 284) (p. 275) percent of decrease (p. 285) compound interest (p. 294) percent equation (p. 279) 0ERCENT Key Concepts Percent (Lessons 5-1 and 5-2) discount (p. 286) percent of increase (p. 285) interest (p. 290) percent proportion (p. 263) markup (p. 286) principal (p. 290) percent (p. 252) selling price (p. 286) • A percent is a ratio that compares a number to 100. • To write a percent as a decimal, divide by 100 and remove the percent symbol. • To write a decimal as a percent, multiply by 100 and add the percent symbol. Percent Proportion (Lesson 5-3) • A percent proportion is part _ = percent, where whole 3 1 1 50% = _ 75% = _ 100% = 1 25% = _ 4 4 2 3 1 4 2 20% = _ 40% = _ 60% = _ 80% = _ 5 5 5 5 5 1 1 1 1 2 2 2 _ _ _ _ _ _ _ 16 % = 33 % = 66 % = 83 % = _ 3 3 3 3 3 3 6 6 3 5 1 1 1 1 1 7 _ _ _ _ _ _ _ _ 37 % = 62 % = 87 % = 12 % = 2 8 2 8 2 8 2 8 3 9 1 7 _ _ _ _ 30% = 70% = 90% = 10% = 10 1. A (proportion, percent) is a ratio that 2. (Percents, Compatible numbers) are Percent–Fraction Equivalents 10 Choose the correct term or numbers to complete each sentence. compares a number to 100. the percent is written as a fraction. 10 Vocabulary Check 10 numbers that are easy to divide mentally. 3. A (markup, discount) is an increase in price. 4. 25% of 16 is (4, 40). 5. The (interest, principal) is the amount borrowed. 6. In the proportion _ = _, the (part, whole) is 6. 6 5 120 100 7. A (markup, discount) is a decrease in Percent Equation (Lesson 5-7) • A percent equation is part = percent · whole, where the percent is written as a decimal. price. 8. The interest formula is (I = prt, p = Irt). 9. The number 0.015 written as a percent is Percent of Change (Lesson 5-8) • A percent of change is a ratio that compares the change in quantity to the original amount. Vocabulary Review at ca.gr7math.com (0.15%, 1.5%). 10. The (interest, principal) is the money paid for the use of money. Chapter 5 Study Guide and Review 295 CH APTER 5 Study Guide and Review Lesson-by-Lesson Review 5-1 Ratios and Percents (pp. 252–255) Write each ratio or fraction as a percent. 11. _ 4 5 12. 16.5 out of 100 13. WEATHER There is a 1 in 5 chance of rain tomorrow. Write this as a percent. Write each percent as a fraction in simplest form. 14. 90% 5-2 25 1 _1 = _ So, _ = 25%. 100 4 4 × 25 Example 2 Write 35% as a fraction in simplest form. 35 7 35% = _ or _ 20 Comparing Fractions, Decimals, and Percents (pp. 256–261) Write each percent as a decimal. Example 3 Write 24% as a decimal. 16. 4.3% 24% = 24% = 0.24 Divide by 100 and remove Example 4 Write 0.04 as a percent. 0.04 = 0.04 = 4% Multiply by 100 and add the Example 5 Write 9 as a percent. 9 _ = 0.36 Write as a decimal. 19. 0.7 17. 147% 18. 0.7% 20. 0.015 21. 2.55 Write each fraction as a percent. 22. _ 3 40 23. _ 24 25 24. _ 1 6 7 25. CELL PHONES Adam used _ of his total 8 monthly minutes while Andrea used 88%. Which friend used the greater part of his or her minutes? the percent symbol. percent symbol. _ 25 25 Change the decimal to percent. = 36% Algebra: The Percent Proportion (pp. 263–267) Write a percent proportion and solve each problem. Round to the nearest tenth if necessary. 26. 15 is 30% of what number? 27. Find 45% of 18. 28. 75 is what percent of 250? 29. SCHOOL A band charges $3,000 and requires a 20% deposit to play at a school. How much money does the school need for the deposit? 296 4 × 25 100 15. 120% Write each decimal as a percent. 5-3 _ Write 1 as a percent. Example 1 Chapter 5 Percent Example 6 18 is what percent of 27? The whole is 27, and the part is 18. Let n represent the percent. 18 n _ = _ 27 100 18 · 100 = 27 · n 1,800 = 27n 1,800 27n _ = _ 27 27 Percent proportion Find the cross products. Multiply. Divide each side by 27. 66.7 ≈ n Simplify. So, 18 is 66.7% of 27. Mixed Problem Solving For mixed problem-solving practice, see page 712. 5-4 Finding Percents Mentally (pp. 268–271) Compute mentally. 30. 90% of 100 31. 10% of 18.3 2 32. 66_% of 24 33. 6% of 200 3 Example 7 mentally. Compute 50% of 42 1 50% of 42 = _ of 42 or 21 2 50% = _1 2 34. ANIMALS Compute mentally the number of hours a day a Koala bear 1 sleeps if it spends 83_ % of a day asleep. 3 5-5 PSI: Reasonable Answers (pp. 272–273) Determine a reasonable answer. 35. ECOLOGY In a survey of 1,413 consumers, 6% said they would be willing to pay more for recycled products in order to protect the environment. Is 8.4, 84, or 841 a reasonable estimate for the number of consumers willing to pay more? Explain. 36. PIZZA Twelve friends share three large pizzas. If they split the cost evenly among themselves, and each pizza cost $11.95, will each person pay about $2, $3, or $4? Explain. 5-6 Example 8 Philip’s flight departed at 9:10 A.M. and arrived at 3:15 P.M., Eastern Standard Time. While in flight, Philip checked his watch and estimated that he had completed about 63% of the trip. Is 11 A.M., 12 P.M., or 1 P.M. a reasonable estimate for the time that Philip checked his watch? The total duration of the trip is 365 minutes, or 6 hours and 5 minutes. One half, or 50%, of the trip would be 1 3 hours and 2_ minutes after departure, 2 or about 12:12 P.M. Since 63% is greater than 50%, 1 P.M. is the only reasonable answer. Percent and Estimation (pp. 275–278) Example 9 Estimate. 37. 67% of 60 38. 41% of 39 Estimate each percent. 39. 33 out of 98 40. 19 out of 52 Estimate 8% of 104. 104 is about 100. 8% of 100 is 8. So, 8% of 104 is about 8. 41. MEASUREMENT The average temperature of Earth is about 8% of Venus’ average temperature of 850°F. Estimate Earth’s average temperature. Chapter 5 Study Guide and Review 297 CH APTER 5 Study Guide and Review 5-7 Alegebra: The Percent Equation (pp. 279–283) Solve each problem using the percent equation. 42. What is 66% of 7,000? 43. Find 15% of 82. 44. 25 is what percent of 125? 45. MOVIES India produces an average of 1,000 movies each year. The United States averages 63.3% of this amount. On average, how many movies are produced in the United States yearly? 5-8 Example 10 70 is 25% of what number? The part is 70, and the percent is 25. You need to find the whole. Let n represent the whole. 70 = 0.25n Write the percent equation. 70 0.25n _ =_ Divide each side by 0.25. 280 = n Simplify. 0.25 0.25 So, 70 is 25% of 280. Percent of Change (pp. 284–289) Find each percent of change. Round to the nearest tenth if necessary. State whether the percent of change is an increase or a decrease. Example 11 Find the percent of change if the original amount is 900 and the new amount is 725. Round to the nearest tenth. 46. original: 10 47. original: 8 The amount of change is 900 - 725 or 175. new: 15 new: 10 48. original: 37.5 49. original: 18 new: 30 new: 12 amount of change original amount 175 =_ 900 percent of change = __ 50. HOBBIES Mariah collects comic books. ≈ 0.194 or 19.4% Last year she had 50 comic books. If she now has 74 comic books, what is the percent of increase? 5-9 Simple Interest (pp. 290–293) Find the simple interest to the nearest cent. Example 12 Find the simple interest for $250 invested at 5.5% for 2 years. 51. $100 at 8.5% for 2 years I = prt I = 250 · 0.055 · 2 I = 27.50 52. $350 at 5% for 3 years 53. $260 at 17.5% for 18 months 54. RETIREMENT At age 20, Mark invested $500 into a retirement account with a simple interest rate of 6.5%. He makes no more deposits or withdrawals. Find the account value at age 65. 298 Chapter 5 Percent Simple interest formula Write 5.5% as 0.055. Simplify. The simple interest is $27.50. CH APTER Practice Test 5 Write each ratio or fraction as a decimal and as a percent. 1. 74 per 100 16. TAXES Sandra estimated that about 35% of her $420 paycheck was deducted for taxes and insurance. Did about $100, $150, or $200 get deducted from her pay? 4 3. _ 22 2. 3:50 4. FIELD TRIPS Seventeen students brought their permission slips to go to the zoo. If there are 18 students in the class, what percent of the class brought their permission slip? Round to the nearest tenth. Write a percent proportion and solve each problem. Round to the nearest tenth. 17. What is 2% of 3,600? 18. 62 is 90% of what number? Express each percent as a decimal. Solve using the percent equation. 5. 135% 19. Find 45% of 600. 6. 14.6% 7. 0.97% 8. Order the set of numbers 38%, _, and 0.038 3 8 from least to greatest. 20. 75 is what percent of 30? 21. MEDICINE About 37% of the people in the United States have type O + blood. If there are 250 million people in the United States, how many have type O + blood? Compute mentally. 9. 30% of 60 10. 1% of 99 11. 33_% of 90 12. 62_% of 48 1 3 13. 1 2 STANDARDS PRACTICE The figure below shows 8 shaded isosceles triangles formed by the diagonals of three adjacent squares. Find each percent of change and state whether it is an increase or decrease. Round to the nearest tenth if necessary. 22. original: $15 23. original: 40 cars new: $12 new: 55 cars 24. BUSINESS A sporting goods store prices items at a 30% markup rate. If the store purchases a tennis racket for $165, find the selling price of the racket. If the total area of the figure is 12 square feet, which statement is true? A The shaded area is more than 75% of the area of the figure. 2 B The unshaded area is _ of the area of the 3 figure. C The shaded area is 6 square feet. D The unshaded area is 4 square feet. Estimate. 14. 23% of 16 15. 9% of 81 Chapter Test at ca.gr7math.com 25. STANDARDS PRACTICE Kevin invested $125 into a savings account that earns 5.2% simple interest annually. If he does not deposit or withdraw any money for 18 months, which statement is not supported by this information? F The interest earned will be $117. G The interest earned will be $9.75. H The total amount will be $134.75. J The interest earned in this time will be greater than 5.2% of the principal. Chapter 5 Practice Test 299 CH APTER 5 California Standards Practice Cumulative, Chapters 1–5 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1 5 If a pair of inline skates is on sale for 35% off the regular price of $120, what is the sale price of the skates? A 4(12.99 + 25.99 + 2.70 + 4.35) A $48 B 12.99 + 25.99 + 2.70 + 4.35 B $78 C 4(12.99) + 25.99 + 2.70 + 4.35 C $94 D 4(12.99) + 4(25.99) + 4(2.70) + 4(4.35) D $140 6 2 Jeanne’s grandfather gave her money for her birthday. She bought 4 CDs at $12.99 each and a sweater for $25.99. Then she spent $2.70 on an ice cream cone. She had $4.35 left over. Which expression can be used to find how much money Jeanne received from her grandfather? Alan is buying a television that is regularly Find the height, in feet, of the skateboarding ramp shown below. 1 off the priced at $149.99. It is on sale for _ 5 original price. Which expression can he use to estimate the discount on the television? FT F 0.02 × $150 G 0.05 × $150 H 0.2 × $150 FT J 0.5 × $150 3 Andrew purchased a coat for $67.20 that regularly sells for $84.00. What was the percent discount that Andrew received? A 16.8% C 25% B 20% D 80% 7 8 Question 3 To find the percent of discount, you can use the proportion amount of discount percent discount ___ = ____. 100% 4 regular price Find -17 - (5). F 10 ft H 25 ft G 22 ft J 34 ft Rosa can read about 21 pages in about 20 minutes. If she continues to read at this rate, about how many pages can she read in 4 hours? A 220 C 240 B 230 D 250 Eliza purchased a dress off the clearance rack. The original cost for the dress was $35. The dress had been marked down 50%, but the sign on the rack said to take an additional 20% off the discounted price. What was the final sale price Eliza paid for the dress? F -22 H 22 F $3.50 H $14.00 G -12 J 85 G $10.50 J $17.50 300 Chapter 5 Percent California Standards Practice at ca.gr7math.com More California Standards Practice For practice by standard, see pages CA1–CA39. 9 Adrian swam 75 meters in 45 seconds, and Carlos swam 125 meters in 75 seconds. Based on these rates, which statement is true? 12 The widths of a race track are shown below. What is the percent of increase in the track width from the straightaway to the turn? Part of Track straightaway turn A Adrian’s average speed was 2 meters per second faster than Carlos’ average speed. B Carlos’ average speed was equal to Adrian’s average speed. C Carlos’ average speed was 2 meters per second faster than Adrian’s average speed. Width (feet) 50 60 F 8.3% H 16.7% G 10% J 20% 13 If m = 7 and n = 4, then 3(2m – 3n) = A 2 D Adrian’s average speed was 3 meters per second faster than Carlos’ average speed. B 6 C 12 10 In 2003, a new planet was discovered D 30 beyond Pluto. This new planet is 10 10 miles from the sun. Which of the following represents this number in standard notation? Pre-AP F 10,000,000,000 mi G 10,000,000 mi Record your answers on a sheet of paper. Show your work. H 10,000 mi 14 The Dow Jones Average is used to measure changes in stock values on the New York Stock Exchange. Three major drops in the Dow Jones Average for one day are listed in the table. J 100 mi 11 Martin and his sister agreed to split the cost of a new board game. They received a 25% discount on the board game and paid 5.5% sales tax on the discounted price. If the original price of the board game was $30, how much did Martin and his sister each put toward the cost of the board game? Date 10-29-1929 10-19-1987 9-17-2001 Opening 261.07 2246.74 9605.51 Closing 230.07 1738.74 8920.70 a. Which day had the greatest decrease in A $20.57 amount? B $11.87 b. Did this decrease represent the biggest C $10.29 percent of decrease of the three drops? Explain your reasoning. D $9.77 NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Go to Lesson... 5-8 5-6 5-3 1-5 1-2 3-5 4-1 5-8 4-1 2-10 5-8 5-8 1-2 5-8 For Help with Standard... NS1.7 NS1.3 NS1.7 NS1.2 AF1.1 MG3.3 AF4.2 NS1.7 AF4.2 NS1.1 NS1.7 NS1.6 AF1.2 NS1.6 Chapters 1–5 California Standards Practice 301 Geometry and Measurement Focus Demonstrate an understanding of geometry, spatial reasoning, and measurement. CHAPTER 6 Geometry and Spatial Reasoning Understand plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures. CHAPTER 7 Measurement: Area and Volume Compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. Know how perimeter, area, and volume are affected by changes of scale. 302 Jon Hicks/CORBIS Math and Architecture Under Construction You’ve been selected to head the architectural and construction teams on a house of your own design. You’ll create the uniquely-shaped floor plan, research different floor coverings, and finally research different loans to cover the cost of purchasing these floor coverings. So grab a hammer and some nails, and don’t forget your geometry and measurement tool kits. You’re about to construct a cool adventure! Log on to ca.gr7math.com to begin. Unit 3 Geometry and Measurement Jon Hicks/CORBIS 303 Geometry and Spatial Reasoning 6 • Standard 7MG3.0 Deepen understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures. Key Vocabulary congruent polygons (p. 320) reflection (p. 332) transformation (p. 332) translation (p. 337) Real-World Link Architecture The Marin Civic Center, located in San Rafael, California, uses geometric shapes and properties such as symmetry to create balance. Geometry and Spatial Reasoning Make this Foldable to help you organize your notes. Begin with 1 7 sheets of plain 8 ” × 11” paper. 2 _ 1 Fold a sheet of paper in half lengthwise. Cut a 1” tab along the left edge through one thickness. 3 Repeat Steps 1–2 for the remaining sheets of paper. Staple together to form a booklet. 304 Chapter 6 Geometry and Spatial Reasoning © David Frazier/Photo Edit 2 Glue the 1” tab down. Write the title of the lesson on the front tab. ,INE AND !NGLE 2ELATIONSHIPS ,INE AND !NGLE 2ELATIONSHIPS GET READY for Chapter 6 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Solve each equation. (Lesson 1-9) Example 1 1. 49 + b + 45 = 180 Solve 82 + g + 41 = 180. 2. t + 98 + 55 = 180 82 + g + 41 = 180 123 + g = 180 - 123 - 123 ________ g = 57 3. 15 + 67 + k = 180 4. LAWNS Lawrence made $60 on Write the equation. Add 82 and 41. Subtract 123 from each side. Monday and $48 on Tuesday mowing lawns. How much did he make on Wednesday if his threeday total was $180? (Lesson 1-9) Evaluate each expression. (Lesson 1-2) Example 2 5. (3 - 2)180 6. (7 - 2)180 Evaluate (8 - 2)180. 7. (9 - 2)180 8. (11 - 2)180 (8 - 2)180 = (6)180 = 1,080 9. NUMBER SENSE Find the product Subtract 2 from 8. Multiply. of the difference of 5 and 2 and 180. (Lesson 1-2) Find the value of x in each triangle. Example 3 (Lesson 1-9) Find the value of x in ABC. 10. 11. x⬚ x⬚ 26⬚ 72⬚ 55⬚ 12. 38⬚ x⬚ 13. 81⬚ 40⬚ 47⬚ x⬚ A C The sum of the measures of the angles of a triangle is 180°. 60 + 40 + x = 180 100 + x = 180 -100 = -100 ________ x= 80 x⬚ 60⬚ 40⬚ B m∠A = 60, m∠B = 40 Add. Subtract 100 from each side. Chapter 6 Get Ready for Chapter 6 305 6-1 Line and Angle Relationships Main IDEA Draw a horizontal line on notebook paper and a line that intersects the horizontal line at one point. Identify special pairs of angles and relationships of angles formed by two parallel lines cut by a transversal. Standard 7MR2.6 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 4 1 2 3 Label the angles formed as shown. 1. Suppose that the measures of angles 2 and 4 are each 60°. Using angle relationships you have previously learned or a protractor, find and record the measure of each numbered angle. Explain your reasoning. 2. Congruent angles are angles that have the same measure. Describe NEW Vocabulary congruent angles point line collinear plane ray angle vertical angles complementary angles supplementary angles the pairs of angles that appear to be congruent. 3. What do you notice about the measures of angles that are side by side? In the Mini Lab, you drew lines and points. You have also used these words in everyday language. In mathematics, they have very specific definitions. •A point is simply a location. •A line is made up of points, has no thickness or width, and has infinite length. Points on the same line are said to be collinear. •A plane is a flat surface made up of points. A plane has no depth and extends infinitely in all directions. +%9 #/.#%04 Points, Lines, and Planes Point Model P Line A Plane B n X Z Y T READING in the Content Area Symbols point P For strategies in reading this lesson, visit ca.gr7math.com. 306 Chapter 6 Geometry and Spatial Reasoning ‹___› line n, line‹___ AB or AB, › line BA or BA plane T, plane XYZ, plane XZY, plane YXZ, plane YZX, plane ZXY, plane ZYX Name Lines and Planes Use the figure to name each of the following. 1 a line containing point B E There are three points on the line. Any two of the points can be used to name Lines and Planes There is exactly one line through any two points. There is exactly one plane through any three noncollinear points. the line. ‹___› BC ‹___› CB ‹___› BD ‹___› ‹___› DB CD D C B N ‹___› DC The line can also be named as line . 2 a plane containing point C The plane can be named as plane . You can also use the letters of any three noncollinear points to name the plane. plane BCE plane BDE plane CDE a. Use the figure to name a plane containing points B and D. Other geometry terms are related to points, lines, and planes. For example, a ray is a part of a line having one endpoint and extending indefinitely in one direction. An angle is made up of two noncollinear rays sharing a common endpoint. endpoint The rays are called sides of the angle. side side Pairs of angles can be classified by their relationship to each other. #/.#%04 3UMMARY Angles For the examples in this chapter, assume angles that appear straight actually are straight. Special Pairs of Angles Vertical angles are opposite angles formed by intersecting lines. ∠1 and ∠2 are vertical angles. ∠3 and ∠4 are vertical angles. Vertical angles are congruent. READING Math Naming Angles Angles are named using the endpoint as the middle letter and a point from each side, such as ∠ABD. If there is no confusion, they can be named using only the endpoint, such as ∠F. The sum of the measures of complementary angles is 90°. ∠ABD and ∠DBC are complementary angles. The sum of the measures of supplementary angles is 180°. Extra Examples at ca.gr7math.com D B 50˚ 40˚ C F 125˚ ∠F and ∠G are supplementary angles. Angle Measure The measure of ∠ABD is 50°. In symbols, m∠ABD = 50°. A G 55˚ Lesson 6-1 Line and Angle Relationships 307 3 CARPENTRY You are building a End View bench for a picnic table. Classify the relationship between ∠1 and ∠2. If m∠1 = 32°, find m∠2. Justify your method. 1 2 ∠1 and ∠2 are supplementary. So, the sum of their measures is 180°. m∠1 + m∠2 = 180 32 + x = 180 Real-World Career How Does a Carpenter Use Math? Carpenters use angle relationships when cutting lumber to build anything from furniture to houses. Write an equation. m∠1 = 32 and m∠2 = x - 32 = -32 Subtract 32 from each side. ____________ x = 148 Simplify. So, m∠2 = 148°. b. PARKING Engineers angled the parking spaces along a downtown street so that cars could park and back out easily. All of the lines marking the parking spaces are parallel. If ∠1 ∠2 and m∠1 = 55°, find m∠3. Explain your reasoning. For more information, go to ca.gr7math.com. Find a Missing Angle Measure C 4 Find the value of x in the figure. Angles GBD and FBE are vertical angles. So, the angles are congruent. Congruent angles have equal measures. READING Math Congruent and Equals The symbol is used to show that two angles are congruent. 95⬚ B G m∠GBD = m∠FBE Write an equation. x⬚ D 150⬚ E F m∠GBC + m∠CBD = m∠FBE 95 + x = 150 - 95 = -95 ______________ x = 55 ∠GBD ∠FBE The = symbol is used to show that two measures are equal. m∠GBC = 95 and m∠CBD = x Subtract 95 from each side. Simplify. m∠GBD = m∠FBE Find the value of x in each figure. c. d. x˚ 38˚ Personal Tutor at ca.gr7math.com 308 Aaron Haupt Chapter 6 Geometry and Spatial Reasoning e. x˚ 150˚ 75⬚ 110⬚ x⬚ Examples 1, 2 (p. 307) For Exercises 1 and 2, refer to the figure at the right. R E 1. Name a line containing point E. C B 2. Name a plane containing points B and D. Examples 3, 4 q D Find the value of x in each figure. (p. 308) 3. 153⬚ x⬚ 4. 5. 94⬚ 6. 76⬚ x⬚ 86⬚ x⬚ x⬚ 7. CONSTRUCTION Jack cuts a piece of tile at a 148⬚ Xª ª 135° angle. What is the measure of the other angle formed by the cut? (/-%7/2+ (%,0 For Exercises 8-11, refer to the figure at the right. For Exercises 8–11 12–19 20–23 8. Name a line that contains point S. See Examples 1 2, 3 4 n P m Q W S R 9. Name the plane containing lines n and m. F 10. Name a point not contained in lines m or n. 11. What is another name for line m? Find the value of x in each figure. 12. 13. 129⬚ x⬚ x⬚ 14. 15. x⬚ 77⬚ 131⬚ 88⬚ 16. 17. x⬚ 144⬚ 18. x⬚ 68⬚ 19. 125⬚ 88⬚ x⬚ 64⬚ x⬚ For Exercises 20–23, refer to the figure at the right. x⬚ 167⬚ A C 20. Classify the relationship between ∠CBD and B ∠ABF. D 21. Classify the relationship between ∠ABF and ∠ABC. F E 22. If m∠ABC = 145°, find m∠CBD. Real-World Link The Leaning Tower of Pisa is located in the town of Pisa, Italy. Source: NOVA Online 23. If m∠ABF = 35°, find m∠CBD. 24. ARCHITECTURE Refer to the image at the left. If m∠1 = 84.5° and ∠1 ∠3, classify the relationship between ∠2 and ∠3. Then find m∠2. Lesson 6-1 Line and Angle Relationships age fotostock/Superstock 309 %842!02!#4)#% Find the value of x in each figure. See pages 691, 713. 25. 26. 135 Self-Check Quiz at ca.gr7math.com H.O.T. Problems (x 18) 27. 42 90 (x 27) (4x 22) 28. OPEN ENDED Draw and label three collinear points on a plane. Then draw three noncollinear points on the same plane. 29. REASONING How many planes are determined by three collinear points? Justify your response. 30. */ -!4( Describe a real-world example of a plane containing (*/ 83 *5*/( points, lines, and angles. 31. In the figure below, the two angles are 32. Find the value of x. congruent. Find the value of x. 4x 118 (x 23) 120˚ A 30 C 116 F 39° H 141° B 40 D 124 G 62° J 157° 33. BANKING A savings account starts with $560. If the simple interest rate is 3%, find the total amount after 18 months. (Lesson 5-9) Find each percent of change. Round to the nearest tenth if necessary. State whether the percent of change is an increase or a decrease. (Lesson 5-8) 34. original: 20 35. original: 45 36. original: 620 new: 27 new: 18 new: 31 37. ARCHAEOLOGY Two artifacts were found at a dig. On a coordinate plane, one artifact was found at (1, 5), and the other artifact was found at (3, 1). How far apart were the two artifacts? Round to the nearest tenth if necessary. (Lesson 3-7) PREREQUISITE SKILL Evaluate each expression using the given value. Round to the nearest tenth if necessary. (Lesson 1-2) (n + 1)25 38. 110n - 250; n = 4 39. (n - 1)40; n = 10 40. _ ;n=8 n 310 Chapter 6 Geometry and Spatial Reasoning Extend 6-1 Geometry Lab Constructions You can use a compass and a straightedge to construct basic elements of geometric figures. For example, a line segment is a straight path between two endpoints. Line segments that have the same length are called congruent segments. Main IDEA Construct basic elements of geometric figures using a compass and straightedge. Standard 7MG3.1 Identify and construct basic elements of geometric figures (e.g., altitudes, midpoints, diagonals, angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles) by using a compass and straightedge. Standard 7MR2.4 Make and test conjectures using inductive and deductive reasoning. Congruent Segments 1 −− Draw JK. Then use a straightedge J to draw a line segment longer −− −− L than JK. Label it LM. Place the compass at J and adjust the compass setting so you can place the pencil tip on K. The compass setting equals −− the length of JK. Using this setting, place the compass tip at L. Draw an −− arc to intersect LM. Label the intersection P. NEW Vocabulary line segment perpendicular lines perpendicular bisector midpoint angle bisector altitude K M J K L P M −− −− LP is congruent to JK. a. Draw a line segment. Construct a line segment congruent to the one drawn. Use a ruler to verify the segments are congruent. READING Math Segments The symbol −− for line segment JK is JK. Perpendicular lines are lines that form right angles. A perpendicular bisector is a perpendicular line that divides a line segment into two congruent segments at the midpoint. Perpendicular Bisectors 2 −− Draw AB. Then place the compass at point A. Using a setting greater than one −− half the length of AB, draw −− an arc above and below AB. A B (continued on the next page) Extend 6-1 Geometry Lab: Constructions 311 Using this setting, place the compass at point B. Draw another set of arcs above and −− below AB as shown. Segment Bisectors There can be an infinite number of bisectors of a line segment, but only one perpendicular bisector. A B Label the intersection of these arcs X and Y as shown. −− Draw XY. Label the intersection −− of AB and this new line M. X A −− −− XY is the perpendicular bisector of AB. M B Y b. Draw a line segment. Construct the perpendicular bisector of the segment. An angle bisector is a ray that divides an angle into two congruent angles. Angle Bisectors 3 J Draw ∠JKL. X Place the compass at point K and draw an arc that intersects both sides of the angle. Label the intersections X and Y. K With the compass at point X, draw an arc in the interior of ∠JKL. Using this setting, place the compass at point Y. Draw another arc. L Y J X K L Y Label the intersection of these . arcs H. Then draw KH J X KH is the angle bisector of ∠JKL. H K L Y 312 Chapter 6 Geometry and Spatial Reasoning c. Draw an obtuse angle. Then bisect the angle. An altitude of a triangle is a segment from one vertex to the line containing the opposite side and perpendicular to that side. Construct an Altitude READING Math Segments The height of a triangle is the length of the altitude. 4 Place the compass at vertex B and draw two arcs intersecting AC . Label the points where the arcs intersect the side X and Y. B A Adjust the compass to an 1 opening greater than _XY. 2 Place the compass on point X −− and draw an arc above AC. Using the same setting, place the compass on point Y and draw A −− X another arc above AC. Label the point of intersection of the arcs H. . Use a straightedge to draw BH Label the point where BH −− intersects AC as D. −− BD is an altitude of ΔABC. C B H Y C B H A −− Y X X D Y C −− d. Construct the altitudes to AB and BC. (Hint: You may need to extend the lines containing the sides of your triangle.) What do you notice about the altitudes of the triangle? ANALYZE THE RESULTS 1. Draw two different line segments. Then construct a right triangle in which the given segments are the legs. 2. Draw a large acute triangle. Construct the perpendicular bisectors for each side. What do you notice about the perpendicular bisectors? 3. Repeat Exercise 2, constructing the angle bisectors for each angle. What do you notice? 4. MAKE A CONJECTURE Predict a relationship involving the altitudes, perpendicular bisectors, and angle bisectors for any kind of triangle. Extend 6-1 Geometry Lab: Constructions 313 6-2 Problem-Solving Investigation MAIN IDEA: Solve problems by using the logical reasoning strategy. Standard 7MR1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed. Standard 7NS1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. e-Mail: USE LOGICAL REASONING YOUR MISSION: Use logical reasoning to solve the problem. THE PROBLEM: How can Zach find another property that is true for rectangles, but not parallelograms? EXPLORE PLAN SOLVE Try investigating the diagonals of rectangles and parallelograms to see whether there is any pattern. Draw several different rectangles and parallelograms, measure the diagonals, and look for a pattern. A B A B D C D C AC BD CHECK ▲ Zach: I know that a rectangle is a parallelogram with four right angles. Both parallelograms and rectangles have opposite sides that are congruent and parallel. AC BD A D B C AC q BD It appears that the diagonals of a rectangle are congruent, but the diagonals of a parallelogram are not. You can try several more examples to see whether your conjecture appears to be true. But at this point, it is just a conjecture, not an actual proof. 1. Inductive reasoning is the process of making a conjecture after observing several examples. Determine where Zach used inductive reasoning. Explain. 2. */ -!4( Write about a situation in which you use inductive reasoning. (*/ 83 *5*/( 314 Chapter 6 Geometry and Spatial Reasoning For Exercises 3–5, solve each problem using logical reasoning. 3. GEOMETRY Draw several parallelograms and measure their angles. What seems to be true about opposite angles of parallelograms? 4. MEASUREMENT You need to measure 2 pints of juice for a punch recipe. You have a large container of pineapple juice, an empty 5-pint container, and an empty 4-pint container. Explain how you can use only these containers to measure 2 pints of juice. 5 pt 4 pt 5. NUMBER SENSE Write Fraction Decimal 8. LAUNDRY You need two clothespins to hang one towel on a clothesline. One clothespin can be used on a corner of one towel and a corner of the towel next to it. What is the least number of clothespins you need to hang 8 towels? and Bianca were the first five finishers of a race. From the given clues, give the order in which they finished. • Nuna passed Mackenzie just before the finish line. • Bianca finished 5 seconds ahead of Nuna. • Brianna crossed the finish line after Mackenzie. 11 3 _ 9 fractions _ , 6 , and _ . 11 11 Marcus counted their money to see how much they had left. Alex said, “If I had $4 more, I would have as much as you.” Marcus replied, “If I had $4 more, I would have twice as much as you.” How much does each boy have? 9. SPORTS Nuna, Brianna, Mackenzie, Evelina, _1 11 _4 11 _8 each fraction in the table as a decimal. Then use logical reasoning to write the decimal equivalents for the 7. MONEY After a trip to the mall, Alex and • Evelina was fifth at the finish line. 11 Use any strategy to solve Exercises 6–9. Some strategies are shown below. For Exercises 10 and 11, select an appropriate operation(s) to solve the problem. Justify your selection(s) and solve the problem. G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s rn. • Look for a patte m. ra • Draw a diag asoning. • Use logical re 10. MEASUREMENT The circumference of Earth 6. GEOMETRY Some pentagons are arranged according to the pattern below. If the sides of the pentagons shown are each 1 unit long, find the perimeter of the pattern formed by 8 pentagons. around the equator is 24,901.55 miles. The circumference through the North and South Poles is 24,859.82 miles. How much greater is the circumference of Earth around the equator than through the poles? 11. BIRDS The arctic tern has the longest migration of any bird. Each year, it flies over 21,750 miles. If the average lifespan of an arctic tern is 20 years, on average, how many miles will it have flown in the course of its life? Lesson 6-2 Problem-Solving Investigation: Use Logical Reasoning 315 6-3 Polygons and Angles Main IDEA Find the sum of the angle measures of a polygon and the measure of an interior angle of a regular polygon. Standard 7MR3.3 Develop generalizations of the results obtained and the strategies used and apply them to new problem situations. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A.) Copy and complete the table. The sum of the angle measures of a triangle is 180°. Number of Sides 1. Predict the number of triangles and the sum of the angle measures in a polygon with 8 sides. Sketch of Figure Number of Triangles Sum of Angle Measures 3 1 1(180°) = 180° 4 2 2(180°) = 360° 5 6 2. Write an algebraic expression that could represent the number of triangles in an n-sided polygon. Then write an expression to represent the sum of the angle measures in an n-sided polygon. In the Mini Lab, you used the sum of the angle measures of a triangle to find the sum of the interior angle measures of various polygons. An interior angle is an angle that lies inside a polygon. NEW Vocabulary interior angle equilateral equiangular regular polygon +%9 #/.#%04 Interior Angle Sum of a Polygon Words The sum of the measures of the interior angles of a polygon is (n - 2)180, where n represents the number of sides. Symbols S = (n - 2)180. Find the Sum of Interior Angle Measures 1 ALGEBRA Find the sum of the measures of the interior angles of a Naming Polygons Polygons are named by the number of sides. 5 sides: pentagon 6 sides: hexagon 7 sides: heptagon 8 sides: octagon 9 sides: nonagon 10 sides: decagon decagon. S = (n - 2)180 Write an equation. S = (10 - 2)180 A decagon has 10 sides. Replace n with 10. S = (8)180 or 1,440 Simplify. The sum of the measures of the interior angles of a decagon is 1,440°. Find the sum of the angle measures of each polygon. a. hexagon 316 Chapter 6 Geometry and Spatial Reasoning b. octagon c. 15-gon A polygon that is equilateral (all sides congruent) and equiangular (all angles congruent) is called a regular polygon. Since all the angles of a regular polygon are congruent, their measures are equal. equilateral triangle regular pentagon square regular hexagon 2 ARCHITECTURE The Ennis-Brown House in Los Angeles, California, shown at the right was designed by architect Frank Lloyd Wright. The exterior of the house consists of repeating regular quadrilaterals. Find the measure of an interior angle of a regular quadrilateral. Step 1 Real-World Link Frank Lloyd Wright designed 1,141 buildings during his 70-year career. He used obtuse angles and circles, creating unusually shaped structures such as the spiral Guggenheim Museum in New York City. Find the sum of the measures of the angles. S = (n - 2)180 Write an equation. S = (4 - 2)180 Replace n with 4. S = (2)180 or 360 Simplify. The sum of the measures of the interior angles is 360. Step 2 Divide 360 by 4, the number of interior angles, to find the measure of one interior angle. So, the measure of one interior angle of a regular quadrilateral is 360° ÷ 4 or 90°. Source: architecture. about.com Find the measure of one interior angle in each regular polygon. Round to the nearest tenth if necessary. d. octagon e. heptagon f. 20-gon Personal Tutor at ca.gr7math.com Example 1 (p. 316) Example 2 (p. 317) Find the sum of the angle measures of each polygon. 1. quadrilateral 2. nonagon 3. 12-gon 4. QUILTING The quilt pattern shown is made of repeating equilateral triangles. What is the measure of one interior angle of a triangle? Extra Examples at ca.gr7math.com (l)Roger Wood/CORBIS, (r)Bill Aron/PhotoEdit Lesson 6-3 Polygons and Angles 317 (/-%7/2+ (%,0 For Exercises 5–10 11–16 See Examples 1 2 Find the sum of the measures of the interior angles of each polygon. 5. pentagon 6. heptagon 7. 11-gon 8. 14-gon 9. 19-gon 10. 24-gon 11. ART The sculpture below 12. NATURE Each chamber of a consists of repeating regular pentagons and hexagons. Find the measure of one interior angle of a pentagon. bee honeycomb is a regular hexagon. What is the measure of an interior angle in the honeycomb? Find the measure of one interior angle in each regular polygon. Round to the nearest tenth if necessary. 13. nonagon 14. decagon 15. 13-gon 16. 16-gon ART For Exercises 17 and 18, use the following information. A tessellation is a repetitive pattern of polygons that fit together without overlapping and without gaps between them. For each tessellation, find the measure of each angle at the circled vertex. Then find the sum of the angles. 17. 18. 19. ARCHITECTURE The surface of the dome of Spaceship Earth in Orlando, Florida, consists of repeating equilateral triangles as shown. Find the measure of each angle in each outlined triangle. Then make a conjecture about the interior angle measures in equilateral triangles of different sizes. %842!02!#4)#% See pages 692, 713. Self-Check Quiz at ca.gr7math.com H.O.T. Problems 20. CHALLENGE How many sides does a regular polygon have if the measure of an interior angle is 157.5°? Justify your answer. 21. */ -!4( Explain the relationship between the number of sides (*/ 83 *5*/( of a regular polygon and the measure of each interior angle. 318 Chapter 6 Geometry and Spatial Reasoning (tl)George W. Hart, (tr)E.S. Ross/Visuals Unlimited, (b)Allan Friedlander/SuperStock 22. The following statements are true 23. Which statement is not true about the about GHK. pattern of repeating regular octagons and rectangles? • m∠G = m∠H + m∠K. • ∠H and m∠K are complementary. • The measure of each angle is evenly divisible by 15. Which choice does not fit all three statements for angles G, H, and K? A m∠G = 90° C m∠H = 45° m∠H = 50° m∠K = 45° m∠K = 40° B m∠G = 90° D F The sum of the angles in each rectangle is 360°. m∠G = 90° G The sum of the angles in each octagon is 1,080°. H The sum of the angles at the circled vertex is 270°. m∠G = 90° m∠H = 75° m∠H = 60° m∠K = 15° m∠K = 30° J The measure of each interior angle of an octagon is 135°. Classify each pair of angles as complementary, supplementary, or vertical. (Lesson 6-1) 24. ∠3 and ∠6 25. ∠2 and ∠3 26. ∠5 and ∠6 27. ∠1 and ∠4 1 2 6 5 3 4 28. Marisa put $580 in a savings account at a simple interest rate of 5.5%. How much interest will the account earn in 3 years? (Lesson 5-9) SCHOOL For Exercises 29 and 30, use the following information. A recent survey asked parents to grade themselves based on their involvement in their children’s education. The results are shown at the right. (Lesson 5-2) GXi\ekJlim\p !3UPERIOR "!BOVE !VERAGE 29. Write the percent of parents who gave #!VERAGE themselves an “A” as a decimal and as a fraction in simplest form. 30. Did more or less than _ of the parents 2 5 give themselves a “B”? $"ELOW !VERAGE &&AILING PREREQUISITE SKILL Decide whether the figures are congruent. Write yes or no and explain your reasoning. (Lesson 4-4) 31. 5 in. 32. 5 in. 130˚ 130˚ 33. 8 mm 4 mm Lesson 6-3 Polygons and Angles 319 6-4 Congruent Polygons Main IDEA Identify congruent polygons. Standard 7MG3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures. NEW Vocabulary congruent polygon QUILTING A template, or pattern, for a quilt block contains the minimum number of shapes needed to create the pattern. 1. How many different kinds of triangles are shown in the Winter Stars quilt at the right? Explain your reasoning and draw each triangle. 2. Copy the quilt and label all matching triangles with the same number, starting with 1. Polygons that have the same size and shape are called congruent polygons. +%9 #/.#%04 Words Congruent Polygons If two polygons are congruent, their corresponding sides are congruent and their corresponding angles are congruent. Model B G A Symbols C F H Congruent angles: ∠A ∠F, ∠B ∠G, ∠C ∠H −− −− −− −− −− −− Congruent sides: BC GH, AC FH, AB FG In a congruence statement, the letters identifying each polygon are written so that corresponding vertices appear in the same order. For example, for the diagram below, write CBD PQR. C P B Q CBD PQR D R Vertex C corresponds to vertex P. Vertex B corresponds to vertex Q. Vertex D corresponds to vertex R. Two polygons are congruent if all pairs of corresponding angles are congruent and all pairs of corresponding sides are congruent. 320 Chapter 6 Geometry and Spatial Reasoning Identify Congruent Polygons 1 Determine whether the triangles Y shown are congruent. If so, name the corresponding parts and write a congruence statement. L 4 cm M 9 cm 6 cm 6 cm 9 cm The arcs indicate that ∠X ∠M, X 4 cm Z N ∠Y ∠N, and ∠Z ∠L. −− −−− −− −− −− −−− The side measures indicate that XY MN, YZ NL, and XZ ML. Congruence Statements Other possible congruence statements for Example 1 are YZX NLM, ZXY LMN, YXZ NML, XZY MLN, and ZYX LNM. Since all pairs of corresponding angles and sides are congruent, the two triangles are congruent. One congruence statement is XYZ MNL. Determine whether the polygons shown are congruent. If so, name the corresponding parts and write a congruence statement. 40˚ Q a. b. E B 50˚ 8 ft 40˚ V 4 ft 50˚ P T F C H 7 ft G X R W 4 ft D Personal Tutor at ca.gr7math.com Find Missing Measures In the figure, AFH QRN. 2 Find m∠Q. 13 in. A According to the congruence statement, ∠A and ∠Q are corresponding angles. So, ∠A ∠Q. Since m∠A = 40°, m∠Q = 40°. READING Math Recall that symbols like RN refer to the measure of the segment with those endpoints. 40˚ Q F 65˚ 9 in. H R N 3 Find RN. −− −−− −− −−− FH corresponds to RN. So, FH RN. Since FH = 9 inches, RN = 9 inches. In the figure, quadrilateral ABCD is congruent to quadrilateral WXYZ. Find each measure. 4m c. m∠X d. YX e. m∠Y Extra Examples at ca.gr7math.com B 3m A C Y X 145˚ 70˚ D Z W Lesson 6-4 Congruent Polygons 321 Example 1 (p. 321) Determine whether the polygons shown are congruent. If so, name the corresponding parts and write a congruence statement. 1. C G 85˚ A 45˚ F (p. 321) (/-%7/2+ (%,0 For Exercises 7–8 9–12 See Examples 1 2, 3 15 in. P 3. m∠X 4. YW 7 yd 5. XY 6. m∠W M 10 yd X Q 61˚ 73˚ W R Y Determine whether the polygons shown are congruent. If so, name the corresponding parts and write a congruence statement. 7. J B A 9m H 5 cm 3 cm 6 cm 5 cm 8. P K M 3 cm Q Z 9. AD 10. DC 11. m∠G 12. m∠H C D A B 76˚ G E Source: greatbuildings.com 322 D 81˚ F H 13 in. G C 14. ARCHITECTURE The Bank of China Tower shown at the left was designed by architect I.M. Pei and consists of congruent glass triangles. If WXY VWZ, and m∠V = 60°, and m∠VWZ = 50°, find m∠Y. Chapter 6 Geometry and Spatial Reasoning F 11 in. created by David Smith, is located at the Hirshhorn Museum and Sculpture Garden in Washington, D.C. If quadrilaterals JMKL and PSNO are both squares, write one statement you would need to know in order to show that the quadrilaterals are congruent. Explain your reasoning. (l)Peter Bowater/Photo Researchers, (r)Gjon Mili//TIME Life Pictures/Getty Images E 18 in. 13. ART The structure shown at the right, Cubi XII, Real-World Link The Bank of China Tower uses triangular bracing as protection against high winds caused by typhoons. 6m 6 cm Y V 15 in. N H In the figure, quadrilateral ABCD is congruent to quadrilateral HEFG. Find each measure. W L 12 in. 9 in. In the figure, PQR YWX. Find each measure. H X K 12 in. 85˚ E Examples 2, 3 J 45˚ 50˚ 50˚ 2. M K J L S P N O Z %842!02!#4)#% 15. INSECTS The wings of a monarch butterfly are shaped as congruent quadrilaterals. Write a congruence statement. Then find m∠A if m∠Z = 45°, m∠Y = 145°, and m∠X = 90°. See pages 692, 713. Self-Check Quiz at W A Y X D ca.gr7math.com H.O.T. Problems B C 16. CHALLENGE State whether the following statement is sometimes, always or never true. Explain your reasoning. If the perimeters of two triangles are equal, then the triangles are congruent. */ -!4( Explain how you could determine whether two (*/ 83 *5*/( 17. similar polygons were also congruent. 19. In the scaffolding below, ABC 18. Which statement must be true if −− −− DCB, AC is 2.5 meters long, BC is −− 1 meter long, and AB is 2.7 meters −− long. What is the length of BD? PQR TUV? −− −−− A PQ UV −−− −− B QR TV C ∠P ∠T D ∠R ∠U F 1 meter H 2.5 meters G 2 meters J 2.7 meters ALGEBRA Find the measure of one interior angle in each regular polygon. Round to the nearest tenth if necessary. (Lesson 6-3) 22. heptagon 23. nonagon 24. CITY SERVICES The street maintenance vehicles for the city of N Centerburg cannot safely make turns less than 70°. Should the proposed site of the new maintenance garage at the northeast corner of Park and Main be approved? Explain. (Lesson 6-1) Park n 21. pentagon Mai 20. triangle 108° First PREREQUISITE SKILL Which figure cannot be folded so one half matches the other half? 25. A 26. B C D A B C D Lesson 6-4 Congruent Polygons (t)Pete Turner/Getty Images, (b)CORBIS, (b)Doug Martin 323 Extend 6-4 Main IDEA Investigate which three pairs of corresponding parts can be used to show that two triangles are congruent. Standard 7MG3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures. Standard 7MR2.4 Make and test conjectures using inductive and deductive reasoning. Geometry Lab Investigating Congruent Triangles In this lab, you will investigate whether it is possible to show that two triangles are congruent without showing that all six pairs of corresponding parts are congruent. Animation ca.gr7math.com 1 Draw a triangle on a piece of patty paper. Copy the sides onto another piece of patty paper and cut them out. Arrange and tape the pieces together so that they form a triangle. ANALYZE THE RESULTS 1. Is the triangle you formed congruent to the original triangle? Explain. 2. Try to form another triangle with the given sides. Is it congruent to the original triangle? 3. MAKE A CONJECTURE Based on this activity, can three pairs of congruent sides be used to show that two triangles are congruent? 2 Draw a triangle on a piece of patty paper. Copy each angle of the triangle onto separate pieces of patty paper. Extend each ray of each angle to the edge of the patty paper. Arrange and tape the pieces together so that they form a triangle. 324 Chapter 6 Geometry and Spatial Reasoning ANALYZE THE RESULTS 4. Is the triangle you formed congruent to the original triangle? Explain. 5. Try to form another triangle with the given angles. Is it congruent to the original triangle? 6. MAKE A CONJECTURE Based on this activity, can three pairs of congruent angles be used to show that two triangles are congruent? 3 Draw a triangle on a piece of patty paper. Copy two sides of the triangle and the angle between them onto separate pieces of patty paper and cut them out. Arrange and tape the pieces together so that the two sides are joined to form the rays of the angle. Then tape these joined pieces onto a piece of construction paper and connect the two rays to form a triangle. ANALYZE THE RESULTS 7. Is the triangle you formed congruent to the original triangle? Explain. 8. Try to form another triangle with the given sides and angle. Is it congruent to the original triangle? 9. MAKE A CONJECTURE Based on this activity, can two pairs of congruent sides and the pair of congruent angles between them be used to show that two triangles are congruent? 10. EXTENSION Use patty paper to investigate and make a conjecture about whether each of these additional cases can be used to show that two triangles are congruent. Case 4 two pairs of congruent sides and a pair of congruent angles not between them Case 5 two pairs of congruent angles and the pair of congruent sides between them Case 6 two pairs of congruent angles and a pair of congruent sides not between them Extend 6-4 Geometry Lab: Investigating Congruent Triangles 325 CH APTER 6 Mid-Chapter Quiz Lessons 6-1 through 6-4 Find the value of x in each figure. (Lesson 6-1) 1. 14. 2. 77⬚ 35⬚ x⬚ x⬚ 3. 4. 147⬚ ⬚ x A The sum of the angle measures in each square is 180°. 154⬚ ⬚ 78⬚ x B The sum of the angle measures at each vertex is 1,080°. For Exercises 5–8, use the figure. Classify each pair of angles as complementary, supplementary, or vertical. (Lesson 6-1) C The measure of the angle at each vertex is 90°. D The measure of each interior angle of an octagon is 135°. 5. ∠1 and ∠2 6. ∠2 and ∠5 2 7. ∠4 and ∠5 1 8. ∠1 and ∠5 3 5 STANDARDS PRACTICE Mrs. Lytle’s kitchen tile is made up of a pattern of repeating regular octagons and squares. Which statement is true concerning the pattern? (Lesson 6-3) 4 15. FLAGS The blue portions of the flag below are triangular. Determine whether the indicated triangles are congruent. If so, name the corresponding parts and write a congruence statement. (Lesson 6-4) 9. LINES Refer to the figure below. Classify the relationship between ∠J and ∠K. Then find the measure of ∠J. A B C 115 J F K G 10. NUMBERS Consider the following pattern. 12 = 1 In the figure, quadrilateral QRST is congruent to quadrilateral JKLM. Find each measure. 11 2 = 121 (Lesson 6-4) 111 2 = 12,321 Use logical reasoning to find the next equation. Explain your reasoning. (Lesson 6-2) 16. QT 17. QR 18. m∠M 19. m∠K ALGEBRA Find the sum of the measures of the interior angles of each polygon. (Lesson 6-3) 11. pentagon 12. 20-gon 13. 15-gon 326 CORBIS H Chapter 6 Geometry and Spatial Reasoning 20. Q T 58⬚ 125⬚ 42⬚ L 3 yd K 7 yd 6 yd 135⬚ R S M 7 yd J STANDARDS PRACTICE Which statement is not true if ABC DEF? (Lesson 6-4) −− −− F BC EF H ∠F ∠B −− −− J ∠A ∠D G AB DE 6-5 Symmetry Main IDEA Identify line symmetry and rotational symmetry. Standard 7MG3.2 Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections. NEW Vocabulary line symmetry line of symmetry rotational symmetry angle of rotation ARCHITECTURE The Pentagon is the headquarters of the United States Department of Defense and is located near Washington, D.C. Trace the outline of the Pentagon onto both a piece of tracing paper and a transparency. 1. Draw a line through the center and one vertex of the Pentagon. Then fold your paper across this line. What do you notice about the two halves? 2. Are there other lines you can draw that will produce the same result? If so, how many? 3. Place the transparency over the outline on your tracing paper. Use your pencil point at the center of the Pentagon to hold the transparency in place. How many times can you rotate the transparency from its original position so that the two figures match? 4. Find the first angle of rotation by dividing 360° by the total number of times the figures matched. 5. List the other angles of rotation by adding the measure of the first angle of rotation to the previous angle measure. Stop when you reach 360°. A figure has line symmetry if it can be folded over a line so that one half of the figure matches the other half. This fold line is called the line of symmetry. vertical line of symmetry horizontal line of symmetry no line of symmetry Some figures, such as the Pentagon in the Mini Lab above, have more than one line of symmetry. The figure at the right has multiple lines of symmetry: one vertical, one horizontal, and two diagonal. Lesson 6-5 Symmetry spaceimaging.com/Getty Images 327 Identify Line Symmetry Determine whether each figure has line symmetry. If it does, trace the figure and draw all lines of symmetry. If not, write none. 1 2 This figure has one vertical line of symmetry. a. The figure has five lines of symmetry. b. c. Personal Tutor at ca.gr7math.com A figure has rotational symmetry if it can be rotated or turned less than 360° about its center so that the figure looks exactly as it does in its original position. The degree measure of the angle through which the figure is rotated is called the angle of rotation. Some figures have just one angle of rotation, while others, like the Pentagon, have several. Identify Rotational Symmetry 3 LOGOS Determine whether the figure has rotational symmetry. Write yes or no. If yes, name its angle(s) of rotation. Yes, this figure has rotational symmetry. It will match itself after being rotated 180°. Real-World Link Many companies and nonprofit groups, such as the American Red Cross, use a logo so people can easily identify their products or services. They often design their logo to have line or rotational symmetry. 328 d. Chapter 6 Geometry and Spatial Reasoning (l)Michael Newman/PhotoEdit, (bc)Photick/SuperStock e. f. Extra Examples at ca.gr7math.com Use a Rotation 4 FOLK ART Copy and complete the barn sign shown so that the completed figure has rotational symmetry with 90°, 180°, and 270° as its angles of rotation. Use the procedure described above and the points indicated to rotate the figure 90°, 180°, and 270° counterclockwise. A 90° rotation clockwise produces the same rotation as a 270° rotation counterclockwise. 90° counterclockwise Real-World Link The Pennsylvania Dutch, or Pennsylvania Germans, created signs that were painted on the sides of barns or houses. Many feature designs that have rotational symmetry. 180° counterclockwise 90° clockwise Source: folkart.com g. SYMBOLS Copy and complete the symbol for recycling shown so that the completed figure has rotational symmetry with 120° and 240° as its angles of rotation. SPORTS For Exercises 1 and 2, complete parts a and b for each figure. Examples 1, 2 a. Determine whether the logo has line symmetry. If it does, trace the figure (p. 328) Example 3 and draw all lines of symmetry. If not, write none. b. Determine whether the logo has rotational symmetry. Write yes or no. (p. 328) If yes, name its angle(s) of rotation. 1. Example 4 (p. 329) 2. 3. ARCHITECTURE Copy and complete the window for the partial window shown so that the completed window has rotational symmetry with 45°, 90°, 135°, 180°, 225°, 270°, and 315° as its angles of rotation. Lesson 6-5 Symmetry Courtesy Ramona Maston/FolkArt.com 329 (/-%7/2+ (%,0 For Exercises 4a–7a, 8 4b–7b, 9 10, 11 See Examples 1, 2 3 4 JAPANESE FAMILY CRESTS For Exercises 4–7, complete parts a and b for each family crest. a. Determine whether the crest has line symmetry. If it does, trace the crest and draw all lines of symmetry. If not, write none. b. Determine whether the crest has rotational symmetry. Write yes or no. If yes, name its angle(s) of rotation. 4. 5. 6. 7. ROAD SIGNS For Exercises 8 and 9, use the diagrams below. a. b. c. d. 8. Determine whether each sign has line symmetry. If it does, trace the sign and draw all lines of symmetry. If not, write none. 9. Which of the signs above could be rotated and still look the same? If any, name the angle(s) of rotation. 10. HUBCAPS A partial hubcap is shown. Copy and complete the figure so that the completed hubcap has rotational symmetry of 90°, 180°, and 270°. 11. PIZZA A piece of pizza is shown. Copy and complete the figure so that the entire pizza has rotational symmetry of 60°, 120°, 180°, 240°, and 300°. How many slices are needed to complete the pizza? Real-World Link The stained glass window of the Notre Dame Cathedral in Paris, France, has a diameter of 10 meters. The Cathedral itself is 35 meters high, 48 meters wide, and 130 meters long. Source: parisdigest.com 330 12. ARCHITECTURE Determine whether the Taj Mahal in Agra, India, has line symmetry. If it does, state the number of lines of symmetry and describe each one. If not, write none. 13. ART Describe the kind(s) of symmetry shown in the stained glass window at the left. Chapter 6 Geometry and Spatial Reasoning (l to r, t to b)Doug Martin, (2 3 4)Doug Martin, (5)Vanni Archive/CORBIS, (6)Samuel R. Maglione/Photo Researchers Taj Mahal %842!02!#4)#% 14. ALPHABET Which capital letters of the alphabet produce the same letter after being rotated 180°? See pages 692, 713. Self-Check Quiz at 15. TRIANGLES Which types of triangles—scalene, isosceles, equilateral—have line symmetry? Which have rotational symmetry? ca.gr7math.com H.O.T. Problems CHALLENGE For Exercises 16 and 17, determine whether each statement is true or false. If false, give a counterexample. 16. If a figure has one horizontal and one vertical line of symmetry, then it also has rotational symmetry. 17. If a figure has rotational symmetry, it also has line symmetry. 18. */ -!4( Explain the difference between line symmetry and (*/ 83 *5*/( rotational symmetry. 19. The figures below have a repeating pattern. Which shows a 180° rotation of the 17th figure in the pattern? A B C D 20. ALGEBRA Find the value of x in the two congruent triangles. (Lesson 6-4) 3x˚ 39˚ ALGEBRA Find the sum of the measures of the interior angles of each polygon. (Lesson 6-3) 21. hexagon 22. octagon 23. 14-gon 24. 20-gon 25. SHOPPING A pair of boots costs $130 and is on sale for 15% off this price. Find the amount of the discount. (Lesson 5-8) PREREQUISITE SKILL Find the coordinates of the vertices of polygon HJKL after polygon HJKL is dilated using the given scale factor. Then graph polygon HJKL and its dilation. (Lesson 4-5) 26. H(-6, 2), J(4, 4), K(7, -2), L(-2, -4); scale factor _ 1 2 Lesson 6-5 Symmetry 331 6-6 Reflections A Main IDEA Graph reflections on a coordinate plane. Standard 7MG3.2 Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections. PHOTOGRAPHY The undisturbed surface of a pond acts like a mirror and can provide the subject for beautiful photographs. C B 1. Compare the shape and size of the bird to its image in the water. C' B' 2. Compare the perpendicular distance from the water line to each of the points shown. What do you observe? A' 3. The points A, B, and C appear counterclockwise on the bird. How are these points oriented on the bird’s image? NEW Vocabulary reflection line of reflection transformation image The mirror image produced by flipping a figure over a line is called a reflection. This line is called the line of reflection. A reflection is one type of transformation or mapping of a geometric figure. In mathematics, an image is the position of a figure after a transformation. Draw a Reflection 1 Copy JKL at the right on graph paper. J K Then draw the image of the figure after a reflection over the given line. Step 1 Step 2 READING Math Notation Read J as J prime. It is the image of point J. Step 3 Count the number of units between each vertex and the line of reflection. For each vertex, plot a point the same distance away from the line on the other side. L J' 1 1 J K' K 4 Connect the new vertices to form the image of JKL, JKL. 4 2 2 L' L a. Copy the figure on a piece of graph paper. Then draw the image of the figure after a reflection over the given line. Z Y Personal Tutor at ca.gr7math.com 332 Chapter 6 Geometry and Spatial Reasoning Darrell Gulin/CORBIS X Reflect a Figure Over an Axis REVIEW Vocabulary vertex the point where two sides of a figure intersect; Example: point P in PQR is formed −− by the −− intersection of PQ and PR. 2 Graph PQR with vertices P(-3, 4), Q(4, 2), and R(-1, 1). Then graph the image of PQR after a reflection over the x-axis, and write the coordinates of its vertices. y P Q R R' The coordinates of the vertices of the image are P(-3, -4), Q(4, -2), and R(-1, -1). Examine the relationship between the coordinates of each figure. x O same opposites Q' P' P(-3, 4) P(-3, -4) Q(4, 2) Q(4, -2) R(-1, 1) R(-1, -1) Notice that the y-coordinate of a point reflected over the x-axis is the opposite of the y-coordinate of the original point. 3 Graph quadrilateral ABCD with vertices A(-4, 1), B(-2, 3), C(0, -3), and D(-3, -2). Then graph the image of ABCD after a reflection over the y-axis, and write the coordinates of its vertices. y B B' A' A O Points on Line of Reflection Notice that if a point lies on the line of reflection, the image of that point has the same coordinates as those of the point on the original figure. The coordinates of the vertices of the image are A(4, 1), B(2, 3), C(0, -3), and D(3, -2). Examine the relationship between the coordinates of each figure. x D opposites same D' C C' A(-4, 1) A(4, 1) B(-2, 3) B(2, 3) C(0, -3) C(0, -3) D(-3, -2) D(3, -2) Notice that the x-coordinate of a point reflected over the y-axis is the opposite of the x-coordinate of the original point. Graph FGH with vertices F(1, -1), G(5, -3), and H(2, -4). Then graph the image of FGH after a reflection over the given axis, and write the coordinates of its vertices. b. x-axis c. y-axis If a figure touches the line of reflection as it does in Example 3, then the figure and its image together form a new figure that has line symmetry. The line of reflection is then also a line of symmetry. Extra Examples at ca.gr7math.com Lesson 6-6 Reflections 333 Use a Reflection Interactive Lab ca.gr7math.com 4 MASKS Copy and complete the mask shown so that the completed figure has a vertical line of symmetry. You can reflect the half of the mask shown over the indicated vertical line. Find the distance from each vertex on the figure to the line of reflection. Then plot a point that same distance away on the opposite side of the line. Connect vertices as appropriate. Real-World Link Many cultures use masks in rituals as well as theatrical performances. Many masks were created to look like human or animal faces, which display vertical line symmetry. Source: The History Channel d. ART Copy and complete the portion of the animal shown so that the completed picture has horizontal line symmetry. What is the animal? Examples 1–3 (pp. 332–333) Graph the figure with the given vertices. Then graph the image of the figure after a reflection over the x-axis and y-axis and write the coordinates of the image’s vertices. 1. ABC with vertices A(3, 5), B(4, 1), and C(1, 2) 2. WXY with vertices W(-1, -2), X(0, -4), and Y(-3, -5) 334 Example 4 3. ART Copy and complete the pattern (p. 334) shown so that the completed figure has vertical line symmetry. Chapter 6 Geometry and Spatial Reasoning Vanessa Vick/Photo Researchers (/-%7/2+ (%,0 For Exercises 4, 5 6–9 10, 11 See Examples 1 2, 3 4 Copy each figure onto graph paper. Then draw the image of the figure after a reflection over the given line. 4. 5. C G F H B D J Graph the figure with the given vertices. Then graph the image of the figure after a reflection over the given axis, and write the coordinates of the image’s vertices. 6. triangle ABC with vertices A(-1, -1), B(-2, -4), and C(-4, -1); x-axis 7. triangle FGH with vertices F(3, 3), G(4, -3), and H(2, 1); y-axis 8. square JKLM with vertices J(-2, 0), K(-1, -2), L(-3, -3), and M(-4, -1); y-axis 9. quadrilateral PQRS with vertices P(1, 3), Q(3, 5), R(5, 2), and S(3, 1); x-axis 10. PATTERNS The drawing shows the pattern for the left half of the front of the shirt. Copy the pattern onto grid paper. Then draw the outline of the pattern after it has been flipped over a vertical line. Label it “Right Front”. ,EFT &RONT 11. ART The top half of a Ukranian decorative egg is shown. Copy the figure onto a piece of paper. Then draw the egg design after it has been reflected over a horizontal line. 12. ARCHITECTURE Describe in what ways the Real-World Link The Fogong Monastery in Yingxian, China, is an example of a pagoda, a popular style of traditional Chinese architecture. The monastery has five stories and each story is octagonal. Source: chinaknowledge. org symmetry of the Fogong Monastery, shown at the left, is similar to that of the Eiffel Tower in Paris, France, shown at the right. 13. Triangle XYZ has vertices X(-1, 3), Y(2, 5), and Z(3, -2). Find the coordinates of the image after a reflection over the x-axis and then the y-axis. 14. FIND THE DATA Refer to the California Data File on pages 16–19. Choose an image that illustrates a reflection. Lesson 6-6 Reflections (l)Liu Liqun/CORBIS, (r)Daryl Benson/Masterfile 335 Copy each figure onto graph paper. Then draw the image of the figure after a reflection over the given line. %842!02!#4)#% 15. Q R S T See pages 693, 713. V 16. A U B Self-Check Quiz at ca.gr7math.com H.O.T. Problems F C D 17. OPEN ENDED Draw a right triangle ABC in the first quadrant of a coordinate plane. Then draw the image after a reflection over the x-axis. 18. CHALLENGE Suppose point P with coordinates (-4, 5) is reflected so that the coordinates of its image are (-4, -5). Without graphing, which axis was this point reflected over? Explain your reasoning. 19. */ -!4( Find the coordinates of the point (x, y) after it has (*/ 83 *5*/( been reflected over the x-axis. Then find the coordinates of the point (x, y) after it has been reflected over the y-axis. Explain your reasoning. 20. Which of the following is the reflection of ABC with vertices A(1, -1), B(4, -1), and C(2, -4) over the x-axis? A B y y C y y D x O O x O x O x Determine whether each regular polygon has rotational symmetry. Write yes or no. If yes, name its angle(s) of rotation. (Lesson 6-5) 21. 22. 23. 24. ALGEBRA Find the value of x if the triangles at the right are B 12 ft C D congruent. (Lesson 6-4) 16 ft x ft 20 ft A PREREQUISITE SKILL Add. (Lesson 1-4) 25. -4 + (-1) 336 26. -5 + 3 Chapter 6 Geometry and Spatial Reasoning 27. -1 + 4 E 6-7 Translations Main IDEA Graph translations on a coordinate plane. Standard 7MG3.2 Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections. CHESS In chess, there are rules governing how many spaces and in what direction each game piece can be moved during a player’s turn. The diagram at the right shows one legal move of a knight. 1. Describe the motion involved in moving the knight. 2. Compare the shape, size, and orientation of the knight in its original position to that of the knight in its new position. NEW Vocabulary translation A translation (sometimes called a slide) is the movement of a figure from one position to another without turning it. Draw a Translation 1 Copy trapezoid WXYZ at the right on X graph paper. Then draw the image of the figure after a translation 4 units left and 2 units down. W Z Step 1 Move each vertex of the trapezoid 4 units left and 2 units down. Step 2 Connect the new vertices to form the image. X X W Y X' W' Y W' X' W Y Y' Z Y' Z Z' Z' a. Copy square EFGH at the right on graph paper. Then draw the image of the figure after a translation 5 units right and 3 units up. F E G H Lesson 6-7 Translations 337 Translation in the Coordinate Plane 2 Graph JKL with vertices J(-3, 4), K(1, 3), and L(-4, 1). Then graph the image of JKL after a translation 2 units right and 5 units down. Write the coordinates of its vertices. Translations In the coordinate plane, a translation can be described using an ordered pair. A translation up or to the right is positive. A translation down or to the left is negative. (2, -5) means a translation 2 units right and 5 units down. J y y J K K L L J' x O x O K' L' The coordinates of the vertices of the image are J '(-1, -1), K '(3, -2), and L '(-2, -4). Notice that these vertices can also be found by adding 2 to the x-coordinates and -5 to the y-coordinates, or (2, -5). Original Add (2, -5) Image J(-3, 4) (-3 + 2, 4 + (-5)) J '(-1, -1) K(1, 3) (1 + 2, 3 + (-5)) K '(3, -2) L(-4, 1) (-4 + 2, 1 + (-5)) L '(-2, -4) Graph ABC with vertices A(4, -3), B(0, 2), and C(5, 1). Then graph its image after each translation, and write the coordinates of its vertices. b. 2 units down c. 4 units left and 3 units up 3 If triangle PQR is translated 2 units Q left and 3 units down, what are the coordinates of point R'? A (2, 2) C (4, 2) B (4, -1) D (2, -1) R P O Read the Item You are asked to determine the coordinates of point R' after the original figure has been translated 2 units left and 3 units down. Solve the Item You can answer this question without translating the entire triangle. 338 Chapter 6 Geometry and Spatial Reasoning Eliminate the Possibilities Since point R was translated 2 units left, the x-coordinate of point R' is 4 - 2 or 2. This eliminates choices B and C. The coordinates of point R are (4, 2). Original Figure The x-coordinate of R is 4, so the x-coordinate of R' is 4 - 2 or 2. Translating 2 units left is the same as subtracting 2 from the x-coordinate. The y-coordinate of R is 2, so the y-coordinate of R' is 2 - 3 or -1. Translating 3 units down is the same as subtracting 3 from the y-coordinate. The coordinates of R' are (2, -1). The answer is D. d. If ABC with vertices A(-3, -4), B(-1, -3), and C(-3, 1) is translated 3 units to the right and 4 units up, what are the coordinates of B'? F (2, 1) H (-4, 1) G (-4, -7) J C B (2, -7) A Personal Tutor at ca.gr7math.com Example 1 (p. 337) For Exercises 1 and 2, copy the figure at the right. A 1. Draw the image of ABC after a translation B 4 units left and 1 unit up. 2. Draw the image of ABC after a translation 2 units right and 3 units down. C Example 2 (p. 338) Graph XYZ with vertices X(-4, -4), Y(-3, -1), and Z(2, -2). Then graph the image of XYZ after each translation, and write the coordinates of its vertices. 3. 3 units right and 4 units up Example 3 (p. 338) 5. 4. 2 units left and 3 units down STANDARDS PRACTICE Triangle PQR is translated 5 units left and 3 units down. If the coordinates of P' are (-3, 8), find the coordinates of P. A (-8, 11) C (-8, 5) B (-6, 3) D (2, 5) Extra Examples at ca.gr7math.com P' Q' R' Lesson 6-7 Translations 339 (/-%7/2+ (%,0 For Exercises 6–9 10–11 19, 20 See Examples 1 2 3 Copy each figure onto graph paper. Then draw the image of the figure after the indicated translation. 6. 5 units right and 3 units up 7. 3 units right and 4 units down G Q P H F R E 8. 2 units left and 5 units down 9. 1 unit left and 2 units up X N Y M O W Z Graph the figure with the given vertices. Then graph the image of the figure after the indicated translation, and write the coordinates of its vertices. 10. ABC with vertices A(1, 2), B(3, 1), and C(3, 4) translated 2 units left and 1 unit up 11. rectangle JKLM with vertices J(-3, 2), K(3, 5), L(4, 3), and M(-2, 0) translated by 1 unit right and 4 units down 12. ARCHITECTURE The arches in the first three stories of the Coliseum in Rome, Italy, are translations of one another. Describe the minimum number of translations of the indicated arch needed to create the section shown in the photo at the right. FT FT 13. MUSIC The sound wave of a tuning fork is shown below. Look for a pattern in the sound wave. Copy the sound wave and indicate where this pattern repeats or is translated. Find how many translations of the original pattern are shown in the diagram. %842!02!#4)#% See pages 693, 713. Self-Check Quiz at ca.gr7math.com 340 14. WALLPAPER The wallpaper design at the 4 in. right is a traditional Japanese design. Describe the minimum number of translations of the original pattern, A, needed to create the section shown. A Chapter 6 Geometry and Spatial Reasoning Silvia Otte/Getty Images 3 in. 15. GEOMETRY Triangle RST has vertices R(4, 2), S(-8, 0), and T(6, 7). When translated, R ' has coordinates (-2, 4). Find the coordinates of S ' and T '. Then describe the translation of triangle RST. H.O.T. Problems 16. REASONING A figure is translated by (-5, 7). Then the result is translated by (5, -7). Without graphing, what is the final position of the figure? Explain your reasoning. 17. CHALLENGE What are the coordinates of the point (x, y) after being translated m units left and n units up? */ -!4( Write a real-world problem in which you would need (*/ 83 *5*/( 18. to translate a figure. Then solve your problem. 19. If PQR is translated 4 units 20. Find the coordinates of C ' of trapezoid right and 3 units up, what are the coordinates of R ' ? ABCD after a translation 3 units right and 7 units down. y y P Q D x O A C B x O R A (-1, -6) C (-1, 0) F (1, 3) H (1, -3) B (7, 0) D (7, -6) G (5, 7) J (-9, 1) 21. Graph polygon ABCDE with vertices A(-5, -3), B(-2, 1), C(-3, 4), D(0, 2), and E(0, -3). Then graph the image of the figure after a reflection over the y-axis, and write the coordinates of its vertices. (Lesson 6-6) LIFE SCIENCE A diatom is a microscopic algae. For Exercises 22 and 23, use the diagram of the diatom at the right. (Lesson 6-5) 22. Does the diatom have line symmetry? If so, trace the figure and draw any lines of symmetry. If not, write none. 23. Does the diatom have rotational symmetry? Write yes or no. If yes, name its angle(s) of rotation. Order each set of numbers from least to greatest. (Lesson 5-2) 24. 16%, _, 1.6, _ 1 6 1 16 25. _, 0.65, 38%, _ 3 8 5 8 26. 0.44, _, _, 88% 4 4 5 9 Lesson 6-7 Translations 341 CH APTER 6 Study Guide and Review Download Vocabulary Review from ca.gr7math.com Key Vocabulary Be sure the following Key Concepts are noted in your Foldable. ,INE AND !NGLE 2ELATIONSHIPS angle (p. 307) line symmetry (p. 327) collinear (p. 306) plane (p. 306) complementary angles point (p. 306) (p. 307) Key Concepts Angle Relationships (Lesson 6-1) ray (p. 307) congruent angles (p. 306) reflection (p. 332) image (p. 332) rotational symmetry • The sum of the measures of complementary angles is 90°. inductive reasoning • The sum of the measures of supplementary angles is 180°. interior angles (p. 316) line (p. 306) transformation (p. 332) • Vertical angles are opposite angles formed by intersecting lines and are congruent. line of reflection (p. 332) translation (p. 337) line of symmetry (p. 327) vertical angles (p. 307) (p. 328) (p. 314) supplementary angles (p. 307) Polygons (Lessons 6-3 and 6-4) • The sum of the measures of the interior angles of a polygon is (n – 2)(180). • In congruent polygons, corresponding sides and angles are congruent. Symmetry (Lesson 6-5) • A figure with line symmetry can be folded over a line so that the two halves match. Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. • A figure with rotational symmetry can be rotated about its center so that it looks exactly as it does in its original position. 1. m∠1 is read as the measure of ∠1. Transformations (Lessons 6-6 and 6-7) 3. If ABC DEF, then ∠C ∠E. • In a reflection, the image is congruent to the original figure, but the orientation of the image is different from that of the original figure. • In a translation, the image is congruent to the original figure, and the orientation of the image is the same as that of the original figure. 2. A translation of (3, -2) means a translation 3 units left and 2 units down. 4. Inductive reasoning is the process of making a rule after observing several examples and using that rule to make a decision. 5. A polygon whose angles are all congruent is said to be equilateral. 6. A rectangle will match itself after being rotated 90°, 180°, and 270°. 7. P, the image of point P, is read as P prime. 8. When a transformation occurs, the resulting figure is called a line of reflection. 342 Chapter 6 Geometry and Spatial Reasoning Vocabulary Review at ca.gr7math.com Lesson-by-Lesson Review 6-1 Line and Angle Relationships (pp. 306–310) Example 1 Find the value of x in the figure. Find the value of x in each figure. 9. 10. 125˚ 43˚ x˚ 11. Since the angle labeled x° and the angle labeled 108° are vertical angles, they are congruent. Therefore, x = 108. x˚ 12. 122⬚ x⬚ x⬚ 87⬚ 139⬚ Refer to the figure below. Classify each pair of angles as complementary, supplementary, or vertical. 15. ∠1 and ∠5 2 2 3 16. ∠4 and ∠5 4 5 17. ARCHITECTURE On the skyscraper at the right, m∠1 = 86° and ∠1 ∠3. Classify the relationship between ∠2 and ∠3. Then find m∠2. 6-2 3 1 4 1 x⬚ Example 2 Classify ∠2 and ∠3 as complementary, supplementary, or vertical. 13. ∠1 and ∠4 14. ∠2 and ∠3 108⬚ Since ∠2 and ∠3 form a straight line, they are supplementary angles. PSI: Use Logical Reasoning (pp. 314–315) Solve each problem using logical reasoning. 18. GEOMETRY Draw several squares and connect the opposite vertices. Then measure the four angles that are formed by the intersecting diagonals on each square. What seems to be true about the diagonals of a square? Example 3 Use logical reasoning to find the next number. 3, 5, 8, 12, 17, … 3 5 8 12 17 ⁺2 ⁺3 ⁺4 ⁺5 Since the numbers increase by 2, 3, 4, and 5, the next number will increase by 6. The next number is 23. Chapter 6 Study Guide and Review 343 CH APTER 6 Study Guide and Review 6-3 Polygons and Angles (pp. 316–319) Find the sum of the measures of the interior angles of each polygon. Example 4 Find the measure of one interior angle of a regular hexagon. 19. decagon 20. 32-gon Find the sum of the measures of the angles. Find the measure of one interior angle in each regular polygon. Round to the nearest tenth if necessary. S = (n - 2)180 S = (6 - 2)180 S = (4)180 S = 720 21. heptagon 22. pentagon Subtract. Multiply. Divide 720° by 6, the number of interior angles. So, the measure of one interior angle of a regular hexagon is 720° ÷ 6 or 120°. angle of a rug shaped like a regular octagon. Congruent Polygons (pp. 320–323 ) Determine whether the polygons shown are congruent. If so, name the corresponding parts and write a congruence statement. 24. N D E 8 cm 14 in. A Y 6 cm 45˚ B 65˚ 4 ft B C 5 ft P Q C P G H Example 5 In the figure below, ABC RPQ. Find PQ. A 25. V 14 in. S Q R 26. ART BGY MGK in the art design below. If m∠Y = 55°, find m∠K. B Y G K M 344 Replace n with 6. The sum of the measures of the interior angles is 720°. 23. RUGS Find the measure of an interior 6-4 Write an equation. Chapter 6 Geometry and Spatial Reasoning R −− −− PQ corresponds to BC. Since BC = 5 feet, PQ = 5 feet. Mixed Problem Solving Plate decorated with flame pattern. 16th century, Islamic School, Turkey. Ceramic./Louvre, Paris, France, Giraudon/Bridgeman Art Library For mixed problem-solving practice, see page 713. 6-5 Symmetry (pp. 327–331) BOATING Determine whether each signal flag has line symmetry. If it does, trace the figure and draw all lines of symmetry. If not, write none. Example 6 Determine whether the logo below has line symmetry. If it does, trace the figure and draw all lines of symmetry. If not, write none. 27. The logo has line symmetry. 28. 29. 30. Which of the figures above has rotational symmetry? Name the angle(s) of rotation. 31. ART Determine whether the plate design below has rotational symmetry. If it does, name the angle(s) of rotation. Example 7 Determine whether the logo above has rotational symmetry. If it does, name its angle(s) of rotation. The logo has rotational symmetry. Its angles of rotation are 90°, 180°, and 270°. 6-6 Reflections (pp. 332–336) Graph parallelogram QRST with vertices Q(2, 5), R(4, 5), S(3, 1), and T(1, 1). Then graph its image after a reflection over the given axis, and write the coordinates of its vertices. 32. x-axis 33. y-axis 34. ANIMALS Copy and complete the starfish shown so that the completed figure has a vertical line of symmetry. Example 8 Graph FGH with vertices F(1, -1), G(3, 1), and H(2, -3) and its image after a reflection over the y-axis. The x-coordinate of a point reflected over the y-axis is the opposite of the x-coordinate of the original point. So, the coordinates of the vertices of the image are F(-1, -1), G(-3, 1), and H(-2, -3). y G G' O x F' F H' H Chapter 6 Study Guide and Review Plate decorated with flame pattern. 16th century, Islamic School, Turkey. Ceramic./Louvre, Paris, France, Giraudon/Bridgeman Art Library 345 CH APTER 6 Study Guide and Review 6-7 Translations (pp. 337–341) Copy the figure at the right onto graph paper. Then draw the image of the figure after the indicated translation. H 35. 4 units left and 2 units up F G 36. 3 units right and 1 unit down Graph ABC with vertices A(2, 2), B(3, 5), and C(5, 3). Then graph its image after the indicated translation, and write the coordinates of its vertices. 37. 1 unit right and 4 units down 38. 2 units left and 3 units up 39. HIKING From her car, Marjorie hiked 2 miles north and 3 miles west before she decided to stop and rest. If her starting point can be represented by the point P(1, 4), what are the coordinates of her resting point? Assume that each unit in the coordinate plane is equal to one mile. 346 Chapter 6 Geometry and Spatial Reasoning Example 9 Graph XYZ with vertices X(-3, -1), Y(-1, 0), and Z(-2, -3) and its image after a translation 4 units right and 1 unit up. The coordinates of the vertices of the image can be found by adding 4 to the x-coordinates and 1 to the y-coordinates. The coordinates of the image are X (1, 0), Y (3, 1), and Z (2, -2). y Y' Y X' x O X Z' Z CH APTER Practice Test 6 1. ALGEBRA Find the 77⬚ x⬚ value of x. 119⬚ MUSIC Determine whether each figure has line symmetry. If it does, trace the figure and draw all lines of symmetry. If not, write none. 9. 10. 11. 2. ALGEBRA Angles P and Q are supplementary. Find m∠P if m∠Q = 139°. ALGEBRA Find the sum of the measures of the interior angles of each regular polygon. Then, find the measure of one interior angle. 12. MUSIC Which figure above has rotational 3. octagon 13. 4. 15-gon 5. STANDARDS PRACTICE Which of the following statements is not true concerning the quadrilaterals in the stained glass window? F A E symmetry? Name its angle(s) of rotation. STANDARDS PRACTICE A portion of an archway is shown. Which of the following shows the completed archway with vertical line symmetry? F H G J B D C A The sum of the angle measures in quadrilateral ADEF is 360°. B Quadrilateral ABCD is a regular polygon. C The quadrilaterals are congruent. D The sum of the angle measures in quadrilateral ABCD is 360°. In the figure below, MNP ZYX. Find each measure. 6.3 m 28⬚ 5.7 m 35⬚ M 14. reflection over the x-axis 15. translation 2 units left and 5 units up 16. CHESS Describe the minimum number of translations needed to create the pattern of the chess board shown from the original square B if each square has a side length of 1 inch. N X Z P Graph JKL with vertices J(2, 3), K(-1, 4), and L(-3, -5). Then graph its image and write the coordinates of its vertices after each transformation. Y 6. ZY 7. ∠X 8. ∠Z B Chapter Test at ca.gr7math.com Aaron Haupt Chapter 6 Practice Test 347 CH APTER 6 California Standards Practice Cumulative, Chapters 1–6 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1 4 The graph of rectangle LMNP is shown below. x - If LMN is translated 5 units up and 7 units to the right, what are the coordinates of point L’ ? , M 2 3 Ó Î { x 0 Ó { x 1 2 3 4x ⫺2 ⫺3 ⫺4 ⫺5 ⫺6 N ⫺7 A (-10, 2) C (2, 5) B (4, 2) D (4, -3) A refrigerator costs $560. If the refrigerator is on sale for 30% off the regular price, how much is the discount? F $392 H $175 G $260 J $168 What is the area, in square units, of rectangle LMNP? 5 6 F 24 H 12 G 18 J 9 Find -18 - (-7). A -25 C -9 B -11 D 25 A circle with a radius of 4 units has its center at (1, -2) on a coordinate grid. If the circle is translated 5 units up and 4 units left, what will be the coordinates of the new center? A microscope slide shows 35 red blood cells out of 60 blood cells. How many red blood cells would be expected in a sample of the same blood that has 840 blood cells? A 2.5 C 510 B 490 D 1,440 Question 3 The ratio of the number of red blood cells to the total blood cells on the first slide is the same as the ratio on the second slide. Use a proportion. 348 £ Î 4 3 2 1 L . Î Ó £ x { Î Ó ££ y ⫺7⫺6⫺5⫺4⫺3⫺2⫺1O { Chapter 6 Geometry and Spatial Reasoning 7 6 5 4 3 2 1 ⫺7⫺6⫺5⫺4⫺3⫺2⫺1O y 1 2 3 4 5 6 7x ⫺2 ⫺3 ⫺4 ⫺5 ⫺6 ⫺7 F (-5, 2) H (-3, 3) G (-4, 2) J (5, 3) California Standards Practice at ca.gr7math.com More California Standards Practice For practice by standard, see pages CA1–CA39. 7 Which figure is congruent to the figure below? 9 6 3 Dannie can make 3 bracelets in 55 minutes. At this rate, how many hours will it take her to make 18 bracelets? A 3.3 C 9.17 B 5.5 D 330 10 Stu saved $19.75 when he purchased shoes. If the sale price was 25% off the regular price, what was the original price? 6 A 3 F $79 H $25 G $35 J $20 3 B Pre-AP 3 Record your answers on a sheet of paper. Show your work. 3 C 11 Use the figure in the coordinate grid. 1.5 7 6 5 4 3 2 1 4 D 4 ⫺7⫺6⫺5⫺4⫺3⫺2⫺1O 8 y 1 2 3 4 5 6 7x ⫺2 ⫺3 ⫺4 ⫺5 ⫺6 ⫺7 Jesse purchased a new digital camera for $499 and a printer for $299 including tax. If he plans to pay the total amount in 6 equal monthly payments, what is a reasonable estimate of the amount he will pay each month? a. Graph the figure after a reflection over the x-axis. F $66.50 b. Graph the figure after a reflection over the y-axis. G $133.00 c. Graph the figure if it is reflected over the H $155.00 line y = 2 and then over the line y = -2. What transformation is this the same as? J $165.00 NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 10 11 Go to Lesson... 6-7 5-8 4-3 3-7 1-5 6-7 6-4 1-1 4-1 5-8 6-6 For Help with Standard... MG3.2 NS1.7 AF4.2 MG3.2 NS1.2 MG3.2 MG3.4 MR2.1 AF4.2 NS1.7 MG3.2 Chapters 1–6 California Standards Practice 349 Measurement: Area and Volume 7 • Standard 7MG2.0 Compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. Know how perimeter, area, and volume are affected by changes of scale. Key Vocabulary cone (p. 381) cylinder (p. 374) pyramid (p. 369) prism (p. 369) Real-World Link Caverns Stalactites and stalagmites are cone-shaped formations found in caverns. If you know the diameter of the base and the height of the formation, you can determine the volume of rocks and minerals in the formation. Measurement: Area and Volume Make this Foldable to help you organize your notes. Begin with a piece of 11” × 17” paper. 1 Fold in half widthwise. 2 Open and fold the bottom to form a pocket. Glue edges. 3 Label each pocket. Place several index cards in each pocket. "REA 350 Chapter 7 Measurement: Area and Volume Todd Gipstein/CORBIS 7OLUME GET READY for Chapter 7 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Multiply. (Lessons 2-3 and 2-9) Example 1 1 1. _ · 8 · 12 Multiply 1 · 5 · 6 2. _ 1 2. _ · 4 · 9 2 3 3 3. RUNNING Julian runs 4 miles a day for 6 days each week. If he 1 decides to run _ of this distance, 3 how many miles will he run in one week? (Lesson 2-3) Evaluate 2ab + 2bc + 2ac for the values of the variables indicated. (Lesson 1-2) 4. a = 5, b = 4, c = 8 6. a = 5.4, b = 2.9, c = 7.1 7. a = 2.6, b = 6.4, c = 10.8 Find the value of each expression. Use π ≈ 3.14. Round to the nearest tenth. (Prior Grade) 10. π · 7 2 9. 2 · π · 3.2 11. π · (19 ÷ 2) = 60 Evaluate 6 2. Multiply 5 by 36. Multiply _1 by 180. 2 3 Example 2 Evaluate 2ab + 2bc + 2ac if a = 7, b = 4, and c = 2. 2ab + 2bc + 2ac = 2(7)(4) + 2(4)(2) + 2(7)(2) = 56 + 16 + 28 = 100 5. a = 2, b = 3, c = 9 8. π · 15 3 1 _ · 5 · 6 2 = _1 · 5 · 36 3 3 _ = 1 · 180 3 Substitute a = 7, b = 4, and c = 2. Multiply. Add. Example 3 Evaluate π · 16 2. Use π ≈ 3.14. Round to the nearest tenth. π · 16 2 ≈ 3.14 · 256 ≈ 803.8 Evaluate 16 2. Multiply 3.14 by 256. 12. PIZZA The distance, in inches, around a circular pizza with diameter 14 inches is given by the expression π · 14. Evaluate this expression. Round to the nearest tenth. (Prior Grade) Chapter 7 Get Ready for Chapter 7 351 7-1 Circumference and Area of Circles Main IDEA Find the circumference and area of circles. Standard 7MG2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Standard 7MG3.1 Identify and construct basic elements of geometric figures (e.g., altitudes, midpoints, diagonals, angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles) by using a compass and straightedge. Measure and record the distance d across the circular part of an object, such as a battery or a can, through its center. Place the object on a piece of paper. Mark the point where the object touches the paper on both the object and on the paper. Carefully roll the object so that it makes one complete rotation. Then mark the paper again. Finally, measure the distance C between the marks. in. 1 2 3 4 5 6 1. What distance does C represent? 2. Find the ratio _ for this object. C d NEW Vocabulary circle center radius chord diameter circumference pi 3. Repeat the steps above for at least two other circular objects and compare the ratios of C to d. What do you observe? 4. Graph the data you collected as ordered pairs, (d, C). Then describe the graph. A circle is a set of points in a plane circumference radius center that are the same distance from a (C) (r) given point in the plane, called the center. The segment from the center diameter (d) to any point on the circle is called the radius. A chord is any segment The diameter of a circle is with both endpoints on the circle. twice its radius or d ⫽ 2r. A diameter is a chord that passes through the center. It is the longest chord. The distance around the circle is called the circumference. The ratio of the circumference of a circle to its diameter is always 3.1415926…. _ It is represented by the Greek letter π (pi). The numbers 3.14 and 22 7 C are often used as approximations for π. So, _ = π. This can also be d written as C = πd or C ≈ 3.14d. 352 Chapter 7 Measurement: Area and Volume +%9 #/.#%04 Circumference of a Circle The circumference C of a circle is equal to its diameter d times π, or 2 times its radius r times π. Words Model C d r C = πd or C = 2πr Symbols Find the Circumferences of Circles Calculating with π Using 3.14 for π will result in a close approximation. Find the circumference of each circle. Round to the nearest tenth. 1 IN C = πd Circumference of a circle C=π·9 Replace d with 9. C = 9π This is the exact circumference. C ≈ 9 · 3.14 or 28.3 Replace π with 3.14 and multiply. The circumference is about 28.3 inches. 2 7.2 cm C = 2πr Circumference of a circle C ≈ 2 · 3.14 · 7.2 Replace π with 3.14 and r with 7.2. C ≈ 45.2 Multiply. The circumference is about 45.2 centimeters. a. b. c. IN FT M Personal Tutor at ca.gr7math.com A circle can be separated into congruent wedge-like pieces. Then the pieces can be rearranged to form a figure that resembles a parallelogram. 1 C 2 radius 1 C 2 Since the circle has an area that is relatively close to the area of the parallelogram-shaped figure, you can use the formula for the area of a parallelogram to find the formula for the area of a circle. A = bh 1 A= _ ·C r 2 1 A= _ · 2πr r 2 ( ( ) ) A = π · r · r or πr 2 Extra Examples at ca.gr7math.com Area of a parallelogram The base of the parallelogram is one-half the circumference and the height is the radius. Replace C with 2πr. Simplify. Lesson 7-1 Circumference and Area of Circles 353 +%9 #/.#%04 Area of a Circle Words The area A of a circle is equal to π times the square of the radius r. Symbols A = πr 2 Model Find the Areas of Circles Find the area of each circle. Round to the nearest tenth. Estimation To estimate the area of a circle, square the radius and then multiply by 3. A = πr 2 3 KM A ≈ 3.14 · 8 Area of a circle 2 Replace π with 3.14 and r with 8. A ≈ 3.14 · 64 Evaluate 8 2. A ≈ 201.0 Multiply. The area is about 201.0 square kilometers. A = πr 2 4 FT A ≈ 3.14 (7.5) Area of a circle 2 Replace π with 3.14 and r with half of 15 or 7.5. A ≈ 3.14 · 56.25 Evaluate 7.5 2. A ≈ 176.6 Multiply. The area is about 176.6 square feet. Find the area of each circle. Round to the nearest tenth. d. The radius is 11 inches. e. The diameter is 5 meters. 5 FOUNTAINS Refer to the information at the left. Suppose that you Real-World Link The Sonic Pool, at the Huntington Botanical Gardens in San Marino, California, is a circular bowl filled with lake water. The water is vibrated to create wave patterns that visitors can reach in and touch. walk around the edge of the Sonic Pool and estimate its circumference to be 16 feet. Based on your estimate, what is the approximate diameter of the fountain? C = πd Circumference of a circle 16 ≈ 3.14d Replace π with 3.14 and C with 16. 16 _ ≈d Divide each side by 3.14. 5.1 ≈ d Divide. 3.14 The diameter of the fountain is about 5 feet. Source: nedkahn.com f. HOME DECOR A catalog states that a circular area rug covers 19.5 square feet. What is the approximate diameter of the rug? 354 Chapter 7 Measurement: Area and Volume Vickie Kirby/Austin College Find the circumference of each circle. Round to the nearest tenth. Examples 1, 2 1. (p. 353) 2. 3. YD CM MI Find the area of each circle. Round to the nearest tenth. Examples 3, 4 4. (p. 354) 5. 6. FT 14.5 m IN Example 5 7. BRACELETS When Cammie finished making a friendship bracelet, it was (p. 354) 7.9 inches long. What was the diameter of the bracelet? (/-%7/2+ (%,0 Find the circumference of each circle. Round to the nearest tenth. For Exercises 8–11 12–15 16–19 8. See Examples 1, 2 3, 4 5 9. 10. 11. IN MI MM KM Find the area of each circle. Round to the nearest tenth. 12. 13. M 14. 15. FT MI IN 16. CARS If the tires on a car each have a diameter of 25 inches, how far will the car travel in 100 rotations of its tires? 17. MEASUREMENT A circular table top has a radius of 2_ feet. A decorative trim 1 4 is placed along the outside edge of the table. How long is the trim? 18. SAFETY A light in a parking lot illuminates a circular area 15 meters across. What is the area of the parking lot covered by the light? 19. ANIMALS A California ground squirrel usually stays within 150 yards of its burrow. Find the area of a California ground squirrel’s world. Find the circumference and area of each circle. Round to the nearest tenth. 20. The radius is 3.5 centimeters. 21. The diameter is 8.6 kilometers. 22. The diameter is 9 inches. 23. The radius is 0.6 mile. Lesson 7-1 Circumference and Area of Circles 355 24. Find the diameter of a circle if its area is 706.9 square millimeters. 25. LAWN CARE The pattern of water distribution from a sprinkler is commonly a circle or part of a circle. A certain sprinkler is set to cover part of a circle measuring 270°. Find the area of the grass watered if the sprinkler reaches a distance of 15 feet. FT _ Another approximate value for π is 22 . Use this value to find the 7 circumference and area of each circle. 26. The diameter is 7 feet. 27. The radius is 2_ inches. 1 3 28. PIZZA The pizzeria has a special that offers one large, two medium, or three small pizzas for $12. Which offer is the best buy? Explain your reasoning. Real-World Link Trees should be planted so that they have plenty of room to grow. The planting site should have an area of at least 2 to 3 times the diameter of the circle the spreading roots of the maturing tree are expected to occupy. IN IN IN 29. SPORTS Three tennis balls are packaged one on top of the other in a can. Which measure is greater, the can’s height or circumference? Explain. 30. TREES During a construction project, barriers are placed around trees. For each inch of trunk diameter, the protection 1 zone should have a radius of 1_ feet. 2 Find the area of this zone for a tree with a trunk circumference of 63 inches. Source: www.forestry. uga.edu d in. 1 12 d 31. GRAPHIC ARTS Michael is painting a sign for a new coffee shop. On the sign, he drew a circle with a radius of 2 feet. He then drew another circle with a radius 1.5 times larger. How much greater is the area of the larger circle? %842!02!#4)#% See pages 693, 714. 32. Self-Check Quiz at ca.gr7math.com H.O.T. Problems FIND THE DATA Refer to the California Data File on pages 16–19. Choose some data and write a real-world problem in which you would determine the area of a circle. 33. OPEN ENDED Draw and label a circle that has a circumference between 10 and 20 centimeters. Justify your answer. 34. NUMBER SENSE If the radius of a circle is halved, how will this affect its circumference and its area? What happens to the circumference and area if the radius is doubled or tripled? Explain your reasoning. (Hint: Find the circumference and area for each circle and organize the data in a table.) 356 Chapter 7 Measurement: Area and Volume (l)Jonathan Nourok/PhotoEdit, (r)Aaron Haupt CHALLENGE Find the area of each shaded region. 35. 36. 37. IN M CM M CM 38. */ -!4( Explain how to find the diameter of a circle if you (*/ 83 *5*/( know the measure of its area. 39. In the figure below, the radius of the inscribed circle is 8 inches. What is the perimeter of square WXYZ? X 40. Using the two circles shown below, circumference of circle x what is __ ? circumference of circle y Y 8 cm W A 16π in. B 64 in. C 32 in. Z 12 cm circle x 3π F _ 4 4π G _ 3 circle y 2 H _ J 3 _4 3 D 64π in. GEOMETRY For Exercises 41 and 42, use ABC with vertices A(-2, -2), B(-1, 2), and C(1, 0). 41. GEOMETRY Graph ABC and its image after it is translated 2 units right and 1 unit up. (Lesson 6-7) 42. GEOMETRY What are the coordinates of A’B’C’ when ABC is reflected over the x-axis? (Lesson 6-6) 43. ART At an auction in New York City, a 2.55-square inch portrait of George Washington sold for $1.2 million. About how much did the buyer pay per square inch for the portrait? (Lesson 4-1) 44. PREREQUISITE SKILL The price of calculators has been decreasing. A calculator sold for $125 in 1995. A similar calculator sold for $89 in 2005. Use the look for a pattern strategy to determine the price of a similar calculator in 2025 if the price decrease continues at the same rate. Lesson 7-1 Circumference and Area of Circles 357 Extend 7- 1 Main IDEA Find measures of arcs and inscribed angles. Standard 7MG3.1 Identify and construct basic elements of geometric figures (e.g., altitudes, midpoints, diagonals, angle bisectors, and perpendicular bisectors; central angles, radii, diameters, and chords of circles) by using a compass and straightedge. Standard 7MR2.4 Make and test conjectures using inductive and deductive reasoning. Geometry Lab Investigating Arcs and Angles In Lesson 6-1, you learned about angle relationships. Angles can also be placed in circles. A central angle is an angle that intersects a circle in two points and has its vertex at the center of the circle. It separates the circle into two parts, each of which is an arc. The measure of a central angle is equivalent to the measure of its corresponding arc. There are three types of arcs. A minor arc measures less than 180°. A major arc measures more than 180°. D AC A E 110 B A semicircle measures 180°. DFE JKL J N G C K 60 L F M JML NEW Vocabulary central angle arc minor arc major arc semicircle inscribed angle An inscribed angle is an angle that has its vertex on the circle, and its sides contain chords of the circle. B Measure of Inscribed Angles 1 Use a compass to draw a circle and label the center C. C A D Use a straightedge to draw chords BA and BD that do not go through the center of the circle. Use a straightedge to −− −− draw AC and CD. B C A Measure ∠ABD and ∠ACD. D ANALYZE THE RESULTS 1. What seems to be the relationship between m∠ABD and m∠ACD? 2. Repeat Steps 1–4 with several different inscribed angles. READING Math Arcs and Segments AC is−− read arc The symbol AC. The symbol AC is read segment AC. 358 3. MAKE A PREDICTION Draw a circle with a radius of 3 inches. Then draw a central angle that measures 60° and an inscribed angle that intercepts the same arc. Without measuring, predict the measure of the inscribed angle. Then check your prediction by measuring. Chapter 7 Measurement: Area and Volume Angles Inscribed in a Semicircle 2 Z Use a compass to draw a circle −− with center X and diameter YZ. X Draw and label any point R on YZ. Use a straightedge to −− −− draw RY and RZ. Vocabulary Link Inscribe Everyday Use to write, engrave, or print characters on Math Use to have its vertex (or vertices) on a circle and its sides containing chords of the circle. ANALYZE THE RESULTS −− −− R Y −− 4. What shape is formed by RY, RZ, and YZ? 5. Find m∠YRZ. What kind of triangle is triangle YRZ? −− −− 6. Draw and label another point T on YZ. Draw TY and TZ. Find m∠YTZ. 7. MAKE A CONJECTURE What is true about inscribed angles that intercept a semicircle? 8. Find the measures of the missing angles and arcs in the figure at the right. a. DB b. ∠a c. ECA d. ∠b f. BA h. CB e. ∠ECB g. DC A 20˚ B 130˚ a b E C D Chords and Diameters 3 Use a compass to draw a circle and label the center P. Draw a chord −− that is not a diameter. Label it EF. Construct a line segment through P −− that is perpendicular to EF with endpoints on the circle. −− Label this as diameter GH. F G P E H ANALYZE THE RESULTS 9. Compare the lengths of EG and FG. Then compare the lengths of EH and FH. −−− −− 10. What is the relationship between diameter GH and chord EF? 11. MAKE A CONJECTURE What is the relationship among a diameter, a chord, and its arc if the diameter is perpendicular to the chord? Extend 7-1 Geometry Lab: Investigating Arcs and Angles 359 7-2 Problem-Solving Investigation MAIN IDEA: Solve a simpler problem. Standard 7MR1.3 Determine when and how to break a problem into simpler parts. Standard 7MR2.2 Apply strategies and results from simpler problems to more complex problems. Standard 7AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation. e-Mail: SOLVE A SIMPLER PROBLEM YOUR MISSION: Solve a problem by solving a simpler problem. THE PROBLEM: What is the largest number of pieces that can be cut from one pizza using 5 straight cuts? ▲ GINA: I have a circular pizza. A “cut” doesn’t have to go through the center, just edge to edge. EXPLORE PLAN SOLVE You know that a “cut” does not have to go through the center. Also, the pieces do not necessarily have to be the same size. Solve a simpler problem using 1, 2, 3, and 4 cuts and then look for a pattern. £ÊVÕÌ ÓÊVÕÌÃ ÎÊVÕÌÃ {ÊVÕÌÃ Cuts 0 1 2 3 4 5 Pieces 1 2 4 7 11 16 +1 +2 +3 +4 +5 CHECK So, the largest number of pieces formed by 5 cuts is 16. Check your answer by making a diagram for 5 cuts. 1. Explain why it was helpful for Gina to solve a simpler problem. */ -!4( Write about a situation in which you might need to (*/ 83 *5*/( 2. solve a simpler problem in order to solve a more complicated problem. Then solve the problem. 360 Brent Turner Chapter 7 Measurement: Area and Volume Use the solve a simpler problem strategy to solve Exercises 3–6. 3. GEOMETRY How many squares of any size are in the figure at the right? READING For Exercises 9 and 10, use the following information. Carter Middle School has 487 fiction books and 675 nonfiction books. Of the nonfiction books, 84 are biographies. 9. Draw a Venn diagram of the situation. 10. How many books are not biographies? 4. TABLES The school cafeteria has 15 square tables that can be pushed together to form one long table for class parties. Each square table can seat only one person on each side. How many people can be seated at the combined tables? 5. PARTY SUPPLIES Paper cups come in packages of 40 or 75. Monica needs 350 paper cups for the school party. How many packages of each size should she buy? 11. STATISTICS The graph represents a survey of 400 students. Determine the difference in the number of students who prefer cola to lemon-lime soda. Soft Drink Preferences 37% Cola 15% Orange 20% 18% Lemon- Root Lime Beer 10% Cherry 6. GIFT WRAPPING During the holidays, Tyler and Abigail earn extra money by wrapping gifts at a department store. Tyler wraps 8 packages an hour and Abigail wraps 10 packages an hour. Working together, about how long will it take them to wrap 40 packages? Use any strategy to solve Exercises 7–11. Some strategies are shown below. G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s rn. • Look for a patte agram. • Use a Venn di r problem. • Solve a simple For Exercises 12–14, select the appropriate operation(s) to solve the problem. Justify your selection(s) and solve the problem. 12. TRAVEL When Mrs. Lopez started her trip from Jackson, Mississippi, to Atlanta, Georgia, her odometer read 35,400 miles. When she reached Atlanta, her odometer 1 read 35,782 miles. If the trip took 6_ hours, 2 what was her average speed? 13. SCHOOL SUPPLIES Ethan wishes to buy 7. CUBES Three different views of a cube are shown. If the fish is currently faceup, what figure is facedown? 4 pens, 1 ruler, and 8 folders at the school store. The prices are shown in the table below. If there is no tax, is $11 enough to pay for Ethan’s school supplies? Explain. Item Cost Pens $1.75 Ruler $1.09 Folder $0.55 14. HEALTH A human heart beats an average of 8. NUMBER SENSE Find the sum of all the whole numbers from 1 to 40, inclusive. 72 times in one minute. Estimate the number of times a human heart beats in one year. Lesson 7-2 Problem-Solving Investigation: Solve a Simpler Problem 361 Explore 7-3 Main IDEA Estimate areas of irregular figures. Standard 7MG2.2 Estimate and compute the area of more complex or irregular two- and three-dimensional figures by breaking the figures down into more basic geometric objects. Standard 7MR2.2 Apply strategies and results from simpler problems to more complex problems. Measurement Lab Area of Irregular Figures An irregular figure has sides that are not line segments. To estimate area of an irregular figure, separate the figure into simpler shapes. Then find the sum of these areas. Estimate the area of Idaho. triangle First, separate the figure into a triangle and a rectangle. Area of triangle 100 mi 481 mi 1 bh A=_ 2 1 =_ · 200 · 311 2 b = 300 –100 or 200 = 31,100 Simplify. I DAHO 170 mi 300 mi h = 481–170 or 311 rectangle Area of rectangle A = w = 300 · 170 or 51,000 = 300 and w = 170 The area of Idaho is about 31,100 + 51,000 or 82,100 square miles. Check for Reasonableness Solve the problem another way. How does it compare to the answer in the activity? ANALYZE THE RESULTS 1. In the figure at the right, the area of California is separated into polygons. Explain how polygons can be used to estimate the total land area. 210 mi 213.3 mi 546.7 mi 2. Estimate the area of each region. 3. Estimate the total area of California. 280 mi 4. RESEARCH Use the Internet or another source to find the actual total land area of California. How does it compare to your answer in Exercise 3? 160 mi 40 mi 5. RESEARCH Estimate the area of another state. Use the Internet or another source to compare your estimate with the actual area. 362 Chapter 7 Measurement: Area and Volume 160 mi 133.3 mi 7-3 Area of Complex Figures Main IDEA Find the area of complex figures. Standard 7MG2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Standard 7MG2.2 Estimate and compute the area of more complex or irregular two- and three-dimensional figures by breaking the figures down into more basic geometric objects. CARPETING When carpeting, you must calculate the amount of floor space. Sometimes the space is made up of several shapes. &AMILY 2OOM .OOK 1. Identify some of the polygons that make up the family room, nook, and foyer area shown in this floor plan. &OYER $INING 2. How can the polygons be used to find the total area that will be carpeted? A complex figure is made up of two or more shapes. HALF OF A CIRCLE OR SEMICIRCLE PARALLELOGRAM TRAPEZOID NEW Vocabulary complex figure RECTANGLE SQUARE TRIANGLE To find the area of a complex figure, separate the figure into shapes whose areas you know how to find. Then find the sum of these areas. The following is a review of area formulas. +%9 #/.#%04 Area Formulas Shape Words Formula Parallelogram The area A of a parallelogram is the product of any base b and its height h. A = bh Triangle The area A of a triangle is half the product of any base b and its height h. A= _1 bh READING in the Content Area Trapezoid The area A of a trapezoid is half the product of the height h and the sum of the bases, b 1 and b 2. A= _1 h(b1 + b2) For strategies in reading this lesson, visit ca.gr7math.com. Circle The area A of a circle is equal to π times the square of the radius r. A = πr 2 2 2 Lesson 7-3 Area of Complex Figures 363 Find the Area of a Complex Figure 1 Find the area of the complex figure. Semicircle Since a semicircle is half a circle, its area is _1 πr2. The figure can be separated into a semicircle and a triangle. 6m 11 m 2 Area of semicircle 1 A=_ πr 2 2 1 A≈_ · 3.14 · 3 2 2 Area of triangle 1 A=_ bh 2 1 A=_ · 6 · 11 2 A ≈ 14.1 A = 33 The area of the figure is about 14.1 + 33 or 47.1 square meters. Find the area of each figure. Round to the nearest tenth if necessary. a. b. 12 cm 12 cm 6 cm c. 20 in. 7m 20 in. 13 in. 15 m 18 cm 25 in. 2 GOLF The plan for one hole of a FT miniature golf course is shown. It is composed of a trapezoid and a parallelogram. How many square feet of turf will be needed for this plan? Real-World Link There are 336 dimples on a regulation golf ball. Source: mygolfrecord.com FT FT FT FT Area of trapezoid 1 A=_ h(b 1 + b 2) 2 1 A=_ (3)(2 + 3) 2 Area of parallelogram A = 7.5 A = 15 A = bh A = 6 · 2.5 So, 7.5 + 15 or 22.5 square feet of turf will be needed. 4 ft d. SHEDS Chloe’s father is building a shed. How many square feet of wood are needed to build the back of the shed shown at the right? Personal Tutor at ca.gr7math.com 364 Chapter 7 Measurement: Area and Volume Bill Bachmann/Photo Researchers 12 ft 15 ft Find the Area of a Shaded Region 1 in. 1 in. 3 In the figure at the right, four congruent triangles are cut from a rectangle. Find the area of the shaded region. Round to the nearest tenth if necessary. Congruent Triangles Congruent triangles have corresponding sides and angles that are congruent. 5 in. 12 in. Find the area of the rectangle and subtract the area of the four triangles. Area of rectangle Area of triangles A = w 1 A = 4 · (_ bh) A = 12 · 5 = 12, w = 5 A = 60 Simplify. 2 1 A=4·_·1·1 2 b = 1, h = 1 A=2 Simplify. The area of the shaded region is 60 – 2 or 58 square inches. 13 cm e. Two rectangles are cut from a 3 cm larger rectangle. Find the area of the shaded region. Round to the nearest tenth if necessary. 7cm 4 cm 1 cm 2 cm Example 1 (p. 364) Find the area of each figure. Round to the nearest tenth if necessary. 1. 2. 12 in. 3 yd 11 in. 8 yd 17 in. 10 yd 16 in. Examples 2, 3 (p. 364, 365) 3. WINDOWS The Lunas installed the window shown below. How many square feet is the window? 4. A triangle is cut from a rectangle. Find the area of the shaded region. 11 ft FT 6 ft FT 4 ft Lesson 7-3 Area of Complex Figures 365 (/-%7/2+ (%,0 Find the area of each figure. Round to the nearest tenth if necessary. For Exercises 5–10 11, 12 13–16 5. See Examples 1 2 3 CM 6. 7. YD CM YD CM YD YD CM 8. CM CM YD 9. M 10. 6.4 ft M 7 ft 3.6 ft CM CM CM 9 ft CM 11. CARPENTRY Scott is constructing 12. JEWELRY A necklace comes with a a deck like the one shown. What is the area of the deck? gold pendant. What is the area of the pendant in square centimeters? 5 ft CM 3.5 ft 12 ft CM CM CM Find the area of the shaded region. Round to the nearest tenth if necessary. 13. 14. 20 m 10 yd 25 m 8 yd 22 m 9 yd 15 yd 42 m 15. 16. 2 cm 5 ft 2 cm 12 ft 8 cm 25 ft 16 cm %842!02!#4)#% 17. PAINTING Suppose you are painting one side of a house. One gallon of paint covers See pages 694, 714. 350 square feet and costs $21.95. How much will it cost to buy enough paint Self-Check Quiz at if you apply one coat of paint? ca.gr7math.com 13 ft 18 ft 35 ft 366 Chapter 7 Measurement: Area and Volume H.O.T. Problems 18. CHALLENGE In the diagram at the right, a 16 ft 3-foot-wide wooden walkway surrounds a garden. What is the area of the walkway? 19. */ -!4( Explain at least two (*/ 83 *5*/( 15 ft 27 ft different ways of finding the area of a hexagon. Include a drawing with your answer. 20. What is the total area of the figure 21. A rectangular vegetable garden that is shown? CM 12 ft 32 feet long and 21 feet wide is on a rectangular lot that is 181 feet long and 48 feet wide. The rest of the lot is grass. Approximately how many square feet is grass? CM A 92.5 cm 2 B 64.3 cm 21 ft 2 48 ft 32 ft C 56.5 cm 2 181 ft D 36.0 cm 2 F 8,688 ft 2 H 8,016 ft 2 G 8,635 ft 2 J 282 ft 2 22. MODELS Suppose you had 100 cubes. Use the solve a simpler problem strategy to determine the largest cube you could build with the cubes. (Lesson 7-2) 23. MONUMENTS Stonehenge is a circular array of giant stones in England. The diameter of Stonehenge is 30.5 meters. Find the approximate distance around Stonehenge. (Lesson 7-1) 24. GEOMETRY Graph rectangle ABCD with vertices A(-1, 3), B(5, 3), C(5, -2), and D(-1, -2). Then graph its image after a translation 2 units right and 4 units down. (Lesson 6-7) PREREQUISITE SKILL Classify each polygon according to its number of sides. 25. 26. 27. 28. Lesson 7-3 Area of Complex Figures 367 7-4 Three-Dimensional Figures Main IDEA Identify and draw threedimensional figures. Standard 7MG3.6 Identify elements of threedimensional geometric objects (e.g., diagonals of rectangular solids) and describe how two or more objects are related in space (e.g., skew lines, and the possible ways three planes might intersect). CRYSTALS A two-dimensional figure has two dimensions, length and width. A three-dimensional figure, like the Amethyst crystal shown at the right, has three dimensions, length, width, and depth (or height). Amethyst 1. Name the two-dimensional top shapes that make up the sides of this crystal. 2. If you observed the crystal from directly above, what two-dimensional figure would you see? NEW Vocabulary coplanar parallel solid polyhedron edge face vertex diagonal skew lines prism base pyramid sides 3. How are two- and three-dimensional figures related? bottom The figure at the right shows rectangle ABCD. Lines AB and DC are coplanar because they lie in the same plane. They are also parallel because they will never intersect, no matter how far they are extended. Just as two lines in a plane can intersect or be parallel, there are different ways that planes may be related in space. ÌiÀÃiVÌÊÊ>Êi ÌiÀÃiVÌÊ>ÌÊ>Ê*Ì ! A B D C J ÊÌiÀÃiVÌ 4HESE ARE CALLED AL E P N S R Intersecting planes can also form three-dimensional figures or solids. A polyhedron is a solid with flat surfaces that are polygons. Some terms associated with three-dimensional figures are edge, face, vertex, and diagonal. Vocabulary Link Intersection Everyday Use the place where two roads cross Math Use any point or points that figures have in common. 368 An edge is where two planes intersect in a line. A face is a flat surface. A vertex is where three or more planes intersect at a point. Chapter 7 Measurement: Area and Volume Craig Kramer A diagonal is a line segment whose endpoints are vertices that are neither adjacent nor on the same face. Notice that in the figure at the right, −−− −− WX and KL do not intersect. These segments are not parallel because they do not lie in the same plane. Lines that do not intersect and are not coplanar are called skew lines. W X J K Z Y L M Identify Relationships 1 Name a plane that is parallel to G C D H plane ABC. Plane EFG is parallel to plane ABC. −−− 2 Identify a segment that is skew to CG. −− −− CG and EH are skew. B F E A 3 Identify a pair of points between which a diagonal can be drawn. A segment between points A and G forms a diagonal. a. Identify the intersection of planes ABC and CDH. Prisms and pyramids are common solids. Their names are based on the shape of their bases. A prism is a polyhedron with two parallel, congruent faces called bases. A pyramid is a polyhedron with one base that is a polygon and faces that are triangles. prism pyramid bases base Identify Prisms and Pyramids Common Error In the drawing of a rectangular prism, the bases do not have to be on the top and bottom. Any two parallel rectangles are bases. In a triangular pyramid, any face is a base. Identify each solid. Name the number and shapes of the faces. Then name the number of edges and vertices. 4 The figure has two parallel congruent bases that are triangles, so it is a triangular prism. The other three faces are rectangles. It has a total of 5 faces, 9 edges, and 6 vertices. 5 The figure has one base that is a pentagon, so it is a pentagonal pyramid. The other faces are triangles. It has a total of 6 faces, 10 edges, and 6 vertices. b. Extra Examples at ca.gr7math.com Stephen Frisch/Stock Boston c. d. Lesson 7-4 Three-Dimensional Figures 369 You can use three-dimensional drawings of objects to describe how different parts of the objects are related in space. Analyze Drawings 6 ARCHITECTURE The drawing shows the plans for a new office building. Draw and label the top, front, and side views. Real-World Link Architects use computer-aided design and drafting technology to produce their drawings. FRONT Ì«ÊÛiÜ vÀÌÊÛiÜ SIDE ÃìÊÛiÜ e. CABINETS Julian’s brother drew plans for a cabinet as shown. Draw and label the top, front, and side views. SIDE FRONT Personal Tutor at ca.gr7math.com Examples 1–3 (p. 369) Use the figure at the right to identify the following points, lines, or planes. B F C G 1. parallel planes 2. skew lines 3. two points that form a diagonal when connected 4. intersecting planes Examples 4, 5 (p. 369) D Identify each solid. Name the number and shapes of the faces. Then name the number of edges and vertices. 5. Example 6 (p. 370) 370 A 6. 8. PETS A pet lizard lives in an aquarium with a height of 2 units and a rectangular base 3 units long and 2 units wide. Draw and label the top, front, and side views. Chapter 7 Measurement: Area and Volume Stephen Frisch/Stock Boston 7. E H (/-%7/2+ (%,0 For Exercises 9–12 13–16 17–18 See Examples 1–3 4, 5 6 T Use the figure at the right to identify the following points, lines, or planes. X Y S 9. parallel planes 10. skew lines 11. two points that form a diagonal W U when connected. R V 12. intersecting planes Identify each solid. Name the number and shapes of the faces. Then name the number of edges and vertices. 13. 14. 15. 17. PEDESTALS The plans for a 16. 18. STEPS The Taros are planning sculpture pedestal are shown. Draw and label the top, front, and side views. to install the porch steps shown. Draw and label the top, front, and side views. -VÕ«ÌÕÀiÊ*iìÃÌ> *ÀV Ê-Ìi«Ã FRONT SIDE FRONT SIDE UNIT IN UNIT IN CRYSTALS For Exercises 19–21, complete parts a and b for each crystal. a. Identify the solid or solids that form the crystal. b. Draw and label the top and one side view of the crystal. 19. %842!02!#4)#% See pages 694, 714. Self-Check Quiz at ca.gr7math.com 20. Emerald 21. Quartz Fluorite 22. State whether the following conjecture is true or false. If false, provide a counterexample. Two planes in three-dimensional space can intersect at one point. Lesson 7-4 Three-Dimensional Figures (l)Biophoto Associates/Photo Researchers, (c)E.B. Turner, (r)Stephen Frisch/Stock Boston 371 H.O.T. Problems 23. OPEN ENDED Choose a real-world object such as a chair or a desk. Draw the top, front, and side views of your object. CHALLENGE Determine whether each statement is always, sometimes, or never true. Explain your reasoning. 24. A prism has two congruent bases. 25. A pyramid has five vertices. 26. */ -!4( Explain whether a top-front-side view diagram (*/ 83 *5*/( always provides enough information to draw a figure. If not, provide a counterexample. 27. Benita received the gift box shown. 28. Which of the following represents a side view of the figure below? Which drawing best represents the top view of the gift box? F A B G C H D J 29. Find the area of the figure. Round to the nearest tenth. (Lesson 7-3) IN IN 30. MANUFACTURING The label that goes around a jar of peanut butter 3 inch. If the diameter of the jar is 2 inches, what overlaps itself by _ 8 IN IN is the length of the label? (Lesson 7-1) PREREQUISITE SKILL Find the area of each triangle described. 31. base, 3 in.; height, 10 in. 372 32. base, 8 ft; height, 7 ft Chapter 7 Measurement: Area and Volume 33. base, 5 cm; height, 11 cm 7-5 Volume of Prisms and Cylinders Main IDEA Find the volumes of prisms and cylinders. Standard 7MG2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Standard 7MG2.2 Estimate and compute the area of more complex or irregular two- and three-dimensional figures by breaking the figures down into more basic geometric objects. The rectangular prism at the right has a volume of 12 cubic units. Model three other rectangular prisms with a volume of 12 cubic units. Copy and complete the following table. Prism Length (units) Width (units) Height (units) Area of Base (units 2) A 4 1 3 4 B C D 1. Describe how the volume V of each prism is related to its length , width w, and height h. 2. Describe how the area of the base B and the height h of each prism is related to its volume V. NEW Vocabulary volume cylinder complex solid Volume is the measure of the space occupied by a solid. Standard measures of volume are cubic units such as cubic inches (in 3) or cubic feet (ft 3). +%9 #/.#%04 Words Symbols Volume of a Prism The volume V of a prism is the area of the base B times the height h. Models V = Bh B h B h Find the Volumes of Prisms Volume The formula for the volume of a rectangular prism is often written as V = wh since the area of base B of a rectangular prism is always equal to w. 1 Find the volume of the rectangular prism. V = Bh Volume of a prism V = ( · w) h The base is a rectangle, so B = · w. V = (9 · 5) 6.5 = 9, w = 5, h = 6.5 V = 292.5 Simplify. 6.5 cm 5 cm 9 cm The volume is 292.5 cubic centimeters. Lesson 7-5 Volume of Prisms and Cylinders 373 2 Find the volume of the triangular prism. V = Bh Common Error Remember that the bases of a triangular prism are triangles. In Example 2, these bases are not on the top and bottom of the figure, but on its sides. IN IN Volume of a prism ) (_2 1 V = (_ · 6 · 7)10 2 The base is a triangle, so B = 1 · 6 · 7. V= 1 ·6·7 h _ IN 2 The height of the prism is 10. V = 210 Simplify. The volume is 210 cubic inches. Find the volume of each prism. a. b. c. FT CM MM MM MM FT FT CM CM A cylinder is a solid with bases that are congruent, parallel circles connected with a curved side. You can use the formula V = Bh to find the volume of a cylinder, where the base is a circle. +%9 #/.#%04 Volume of a Cylinder Words The volume V of a cylinder with radius r is the area of the base B times the height h. Symbols V = Bh, where B = π r 2 or V = π r 2h Model Find the Volume of a Cylinder 3 Find the volume of the cylinder. Estimation You can estimate the volume of the cylinder in Example 3 to be about 3 · 7 2 · 20 or 2,940 ft 3 to check the reasonableness of your result. Round to the nearest tenth. FT Since the diameter is 13 feet, the radius is 6.5 feet. V = π r 2h FT Volume of a cylinder 2 V ≈ 3.14 · 6.5 · 20 Replace π with 3.14, r with 6.5, and h with 20. V ≈ 2,653.3 Simplify. The volume is about 2,653.3 cubic feet. Find the volume of each cylinder. Round to the nearest tenth. d. radius, 2 in.; height 7 in. 374 Chapter 7 Measurement: Area and Volume e. diameter, 18 cm; height 5 cm Objects that are made up of more than one type of solid are called complex solids. To find the volume of a complex solid, separate the figure into solids whose volumes you know how to find. Find the Volume of a Complex Solid 4 DISPENSERS Find the volume of the soap 3 in. dispenser at the right. Estimation You can check the reasonableness of your result in Example 4 by estimating the volume. The volume should be between 5 · 7 · 5 or 175 in 3 and 5 · 7 · 8 or 280 in 3. 5 in. The dispenser is made of one rectangular prism and one triangular prism. Find the volume of each prism. Rectangular Prism Triangular Prism IN IN IN IN IN 5 in. 7 in. IN V = Bh V = Bh V = (5 · 7)5 or 175 V = 1 · 7 · 3 5 or 52.5 (_2 ) The volume of the dispenser is 175 + 52.5 or 227.5 cubic inches. MM f. CRAFTS Tanya uses cube beads similar to the one shown to make jewelry. Each bead has a circular hole through the middle. Find the volume of the bead. MM MM Personal Tutor at ca.gr7math.com Examples 1, 2 (pp. 373–374) Find the volume of each prism. Round to the nearest tenth if necessary. 1. 2. 6 ft 2 ft Example 3 (p. 374) 7m 14 m 11 m 3 ft Find the volume of each cylinder. Round to the nearest tenth. 3. 9 yd 4. CM 5 yd CM CM Example 4 (p. 375) 5. TOYS Gloria’s younger sister received the toy house shown as a gift. What is the volume of the toy house? Extra Examples at ca.gr7math.com CM CM CM Lesson 7-5 Volume of Prisms and Cylinders 375 (/-%7/2+ (%,0 Find the volume of each solid. Round to the nearest tenth if necessary. For Exercises 6–9, 12, 13 10, 11, 14, 15 16, 17 6. See Examples 1, 2 3 4 7. 8. IN IN YD MM IN MM 9. 10. MM 11. CM M YD YD M CM M M M 12. rectangular prism: length, 4 in.; width, 6 in.; height, 17 in. 13. triangular prism: base of triangle, 5 ft; altitude, 14 ft; height of prism, 8_ ft 1 2 14. cylinder: radius, 25 m; height, 20 m 15. cylinder: diameter, 7.2 cm; height, 5.8 cm 17. TOWELS An unused roll of 16. MAILBOXES The Francos have the mailbox shown below. Find the volume of the mailbox. 3.5 in. paper towels has the dimensions shown. What is the volume of the unused roll? CM CM 9 in. 15 in. 7 in. CM 18. Find the height of a rectangular prism with a length of 6.8 meters, a width of 1.5 meters, and a volume of 91.8 cubic meters. 19. Find the height of a cylinder with a radius of 4 inches and a volume of 301.6 cubic inches. 20. MEASUREMENT A bar of soap in the shape of a rectangular prism has a 1 inches by volume of 16 cubic inches. After several uses, it measures 2_ 1 inches. How much soap was used? 2 inches by 1_ 4 2 21. PACKAGING The Cooking Club is selling their own special blends of rice mixes. They can choose from the two containers below to package their product. Which container will hold more rice? Explain your reasoning. CM #OOKING #LUB 2ICE -IX #OOKING #LUB 2ICE -IX CM CM CM 376 Chapter 7 Measurement: Area and Volume CM 22. POOLS A wading pool is to be 20 feet long, 11 feet wide, and 1.5 feet deep. The excavated dirt is to be hauled away by wheelbarrow. If the wheelbarrow holds 9 cubic feet of dirt, how many wheelbarrows of dirt must be hauled away from the site? CONVERTING UNITS OF MEASURE For Exercises 23–25, use the cubes and the information below. YD YD YD FT FT FT The volume of the left cube is 1 cubic yard. The right cube is the same size, but the unit of measure has been changed. So, 1 cubic yard = (3)(3)(3) or 27 cubic feet. Use a similar process to convert each measurement. 23. 1 ft 3 = in 3 24. 1 cm 3 = mm 3 25. 1 m 3 = cm 3 26. GARDENING Candice is making a vegetable %842!02!#4)#% See pages 695, 714. garden with the dimensions shown. Each bag of planting soil she plans to use fills 0.5 cubic yard. How many bags of soil will she need to buy to fill her garden? FT FT FT M 27. GEOMETRY Explain how you would find Self-Check Quiz at ca.gr7math.com the volume of the hexagonal prism shown at the right. Then find its volume. M M M M M H.O.T. Problems CHALLENGE For Exercises 28–31, describe how the volume of each solid is affected after the indicated change in its dimension(s). 28. You double one dimension of a rectangular prism. 29. You double two dimensions of a rectangular prism. 30. You double all three dimensions of a rectangular prism. 31. You double the radius of a cylinder. 32. OPEN ENDED Find the volume of a can or other cylindrical object, making sure to include appropriate units. Explain your method. 33. FIND THE ERROR Erin and Dulce are finding the volume of the prism shown at the right. Who is correct? Explain your reasoning. 8 in. 10 in. 7 in. A = Bh A = (10 · 7) · 8 A = 560 in 3 A = Bh A = _21 · 7 · 8 · 10 ) ( A = 280 in Erin 3 Dulce Lesson 7-5 Volume of Prisms and Cylinders (l)Stewart Cohen/Getty Images, (r)Stewart Cohen/Getty Images 377 34. SELECT A TOOL Tyree needs to find the volume M of the figure at the right. Which of the following tools might Tyree use to find the volume of the figure? Justify your selection(s). Then, use the tool(s) to solve the problem. make a model 35. calculator M paper/pencil */ -!4( Write two formulas that you can use to find the (*/ 83 *5*/( volume of a rectangular prism. State the formula that you prefer to use and explain why. 36. A cylinder is 30 inches tall and has a 37. A cardboard box has the dimensions diameter of 12 inches. Which is the closest to the volume of the cylinder in cubic feet? shown below. Which is the closest to the volume of the box in cubic feet? A 1 ft 3 IN B 2 ft 3 C 3 ft 3 D 4 ft IN IN 3 F 8 ft 3 H 15.5 ft 3 G 10 ft 3 J 17 ft 3 38. How many edges does an octagonal pyramid have? (Lesson 7-4) 39. PAINTING You are painting a wall of this room red. Find the area of the red wall to the nearest square foot. (Lesson 7-3) 2 yd 4 yd 2 yd 40. MEASUREMENT The circumference of a circle is 16.5 feet. What is its area to the nearest tenth of a square foot? (Lesson 7-1) 4 yd 41. WOOL Texas ranchers produce about 20% of U.S. wool. If 27.5 million pounds of wool are produced each year, how many pounds of wool are not produced in Texas? (Lesson 5-7) Write each percent as a fraction or mixed number in simplest form. (Lesson 5-1) 42. 0.12% 43. 225% 44. 135% 45. _% 48. _ · 4 2 · 9 49. _ · 6 2 · 20 3 8 PREREQUISITE SKILL Multiply. 46. _ · 6 · 10 1 3 378 47. _ · 7 · 15 1 3 Chapter 7 Measurement: Area and Volume 1 3 1 3 CH APTER Mid-Chapter Quiz 7 Lessons 7-1 through 7-5 Find the circumference and area of each circle. Round to the nearest tenth. (Lesson 7-1) 1. 8. GEOMETRY Draw and label the top view, a side view, and the front view of the figure. (Lesson 7-4) 2. MI IN MEASUREMENT For Exercises 3 and 4, use the following information. Round to the nearest tenth. (Lesson 7-1) A shot-putter must stay inside the circle shown. 9. STANDARDS PRACTICE Juanita wants to sketch all of the faces of a triangular prism. What shapes will appear on her paper? (Lesson 7-4) A 2 squares, 2 triangles B 2 triangles, 3 rectangles FT C 3 triangles D 1 triangle, 3 rectangles 3. What is the area of the region in which the athlete is able to move? 4. What is the circumference of the circular region? Find the volume of each solid. Round to the nearest tenth if necessary. (Lesson 7-5) 10. CM 5. DANCE Balloons come in packages of 15 or 35. Julie needs 195 balloons for the spring dance. How many packages of each size should she buy? Use the solve a simpler problem strategy. (Lesson 7-2) CM 11. YD YD Find the area of each figure. Round to the nearest tenth if necessary. (Lesson 7-3) 12. 6. CM 7. CM CM STANDARDS PRACTICE Find the volume of a cube-shaped box with edges 15 inches in length. (Lesson 7-5) F 225 in 3 H 1,350 in 3 G 900 in 3 J 3,375 in 3 M 13. Find the width of a rectangular prism with a M M M length of 7.6 meters, a height of 8 meters, and a volume of 88.4 cubic meters. Round to the nearest tenth. (Lesson 7-5) 7-6 Volume of Pyramids and Cones Animation ca.gr7math.com Main IDEA Find the volumes of pyramids and cones. Standard 7MG2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Standard 7MG3.5 Construct two-dimensional patterns for threedimensional models, such as cylinders, prisms, and cones. In this Mini Lab, you will investigate the relationship between the volume of a pyramid and the volume of a prism with the same base area and height. NEW Vocabulary cone 1. Compare the base areas and the heights of the two solids. 2. Fill the pyramid with rice, sliding a ruler across the top to level the amount. Pour the rice into the cube. Repeat until the prism is filled. How many times did you fill the pyramid in order to fill the cube? 3. What fraction of the cube’s volume does one pyramid fill? The volume of a pyramid is one-third the volume of a prism with the same base area and height. Volume of a Pyramid Words The volume V of a pyramid is onethird the area of the base B times the height h. Symbols V= Model _1 Bh 3 The height of a pyramid or cone is the distance from the vertex, perpendicular to the base. 380 Chapter 7 Measurement: Area and Volume Find the Volume of a Pyramid 1 Find the volume of the pyramid. Round to the nearest tenth. Estimation You can estimate the volume of the pyramid in Example 1 to be ( ) 1 V=_ Bh 3 1 _ V= 1 _ · 8.1 · 6.4 11 3 2 ( ) Volume of a pyramid _ B = 1 · 8.1 · 6.4, h = 11 2 1 1 about _ _ · 8 · 6 (11) V = 95.04 or 88 m 3. Since 95.04 m 3 is close to 88 m 3, the answer is reasonable. The volume is about 95.0 cubic meters. 3 2 11 m 6.4 m Simplify. 8.1 m a. Find the volume of a pyramid that has a height of 5 yards and a square base with sides 2 yards long. 2 ARCHITECTURE The volume of the Pyramid Arena in Memphis, Tennessee, is about 38,520,000 cubic feet. If the height of the pyramid is 321 feet, find the area of the rectangular base. 1 V=_ Bh Real-World Link The Pyramid Arena’s structure is appropriate as the city of Memphis gets its name from an Egyptian city, known for its ancient pyramids. Volume of a pyramid 3 _ 38,520,000 = 1 · B · 321 Replace V with 38,520,000 and h with 321. 3 _ 38,520,000 = 321 B Multiply. 3 3 3 321 3 · 38,520,000 = ·_ B Multiply each side by 321 . 321 321 3 _ _ _ 360,000 ≈ B Simplify. The area of the base of the pyramid is about 360,000 square feet. Source: pyramidarena.org b. CRAFTS Nicco made a pyramid-shaped candle. The volume of the candle is 864 cubic centimeters and its base has an area of 144 square centimeters. How high is the candle? Personal Tutor at ca.gr7math.com A cone is a three-dimensional figure with one circular base. A curved surface connects the base and the vertex. The volumes of a cone and a cylinder are related in the same way as those of a pyramid and prism. +%9 #/.#%04 Words The volume V of a cone with radius r is one-third the area of the base B times the height h. Symbols V= Extra Examples at ca.gr7math.com 4x5 Coll-A Briere Productions/SuperStock Volume of a Cone Model _1 Bh or V = _1 πr2h 3 3 Lesson 7-6 Volume of Pyramids and Cones 381 Find the Volume of a Cone 3 Find the volume of the cone. 1 2 V=_ πr h 3 mm Volume of a cone 3 _ V ≈ 1 · 3.14 · 3 2 · 14 Replace π with 3.14, r with 3, and 3 14 mm h with 14. V ≈ 131.9 Simplify. The volume is about 131.9 cubic millimeters. Find the volume of each cone. Round to the nearest tenth. c. d. FT FT Example 1 (p. 381) CM CM Find the volume of each pyramid. Round to the nearest tenth. 1. 11 cm IN 2. 8 cm IN 14 cm IN 3. Find the volume of a pyramid that has a height of 125 centimeters and a square base with sides 95 centimeters long. 4. Find the volume of a pyramid that has a height of 17 feet and a square base with sides 22 feet long. Example 2 5. ARCHITECTURE The Louvre Pyramid in Paris has a square base with (p. 381) Example 3 (p. 382) sides 112 feet long. If the volume is 296,875 cubic feet, find the height of the pyramid. Find the volume of each cone. Round to the nearest tenth. 6. 7. IN 7m 5m 8. 7 yd 9. 12 cm 4 yd 15 cm 382 Chapter 7 Measurement: Area and Volume IN (/-%7/2+ (%,0 Find the volume of each pyramid. Round to the nearest tenth. For Exercises 10–13 18 14–17 10. See Examples 1 2 3 11. IN 12. CM YD IN IN CM YD CM YD 13. triangular pyramid: triangle base, 10 cm; triangle height, 7 cm; pyramid height, 15 cm Find the volume of each cone. Round to the nearest tenth. 14. 15. 16. MI FT MM FT MM MI 17. cone: diameter, 12 m; height, 5 m 18. VOLCANO A model of a volcano constructed for a science project is cone-shaped with a diameter of 10 inches. If the volume of the model is about 287 cubic inches, how tall is the model? Find the volume of each solid. Round to the nearest tenth if necessary. 19. 20. YD 2.5 m 21. MM MM 3m YD 2m YD YD MM 4m 22. FROZEN CUSTARD You are filling cone-shaped glasses with frozen custard. Each glass has the dimensions shown. One gallon of custard is equivalent to 4,000 cubic centimeters. About how many glasses can you completely fill using one gallon of custard? %842!02!#4)#% See pages 695, 714. Self-Check Quiz at ca.gr7math.com 23. IRRIGATION A water tank like the one at the right is used to water flowers at a park. Water can be pumped from the tank at a rate of 25 liters per minute. How long will it take to use all of the water in a full tank? Round to the nearest minute. (Hint: 1 liter = 1,000 cm 3) 8 cm 15 cm M M M Lesson 7-6 Volume of Pyramids and Cones 383 H.O.T. Problems 24. CHALLENGE How could you change the height of a cone so that its volume would remain the same when its radius was tripled? 25. OPEN ENDED Draw and label a rectangular pyramid with a volume of 48 cubic centimeters. 26. NUMBER SENSE Which would have a greater effect on the volume of a cone, doubling its radius or doubling its height? Explain your reasoning. */ -!4( Write about a real-world situation (*/ 83 *5*/( 27. that can be solved by finding the volume of a cone. 28. A rectangular pyramid has a base 18 inches by 30 inches and a height of 36 inches. Which is closest to the volume of the pyramid in cubic feet? 29. Find the volume of the cylinder. Round to the nearest tenth if necessary. 6ft A 2.5 ft 3 8ft B 3 ft 3 C 4 ft 3 D 5.5 ft 3 F 48 ft 3 H 288 ft 3 G 150.3 ft 3 J 30. MEASUREMENT Find the volume of the doghouse at the right. (Lesson 7-5) 904.3 ft 3 FT FT 31. Name the number and shapes of the faces of a trapezoidal prism. Then name the number of edges and vertices. (Lesson 7-4) FT 32. GEOMETRY Graph triangle ABC with vertices A(1, 2), B(4, -1), FT and C(2, -4). Then graph its image after a reflection over the y-axis, and write the coordinates of the image’s vertices. (Lesson 6-6) 33. SHOPPING Etu saved $90 when he purchased a DVD recorder on sale. If the sale price was 37.5% off the regular price, what was the regular price of the DVD recorder? (Lesson 5-7) PREREQUISITE SKILL Find the circumference of each circle. Round to the nearest tenth. (Lesson 7-1) 34. diameter, 9 in. 384 35. diameter, 5.5 ft Chapter 7 Measurement: Area and Volume 36. radius, 2 m 37. radius, 3.8 cm Explore 7-7 Main IDEA Find the surface area of cylinders using models and nets. Measurement Lab Surface Area of Cylinders Nets are two-dimensional patterns of three-dimensional figures. You can use a net to find the area of each surface of a three-dimensional figure such as a cylinder. Standard 7MG3.5 Construct twodimensional patterns for three-dimensional models, such as cylinders, prisms, and cones. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Use an empty cylinder-shaped container that has a lid. Measure and record the height of the container. Then label the lid and bottom face using a blue marker. Label the curved side using a red marker. Take off the lid of the container and make 2 cuts as shown. Next, cut off the sides of the lid. Finally, lay the lid, the curved side, and the bottom flat to form the net of the container. NEW Vocabulary TOP net CUT SIDE CUT CUT BOTTOM ANALYZE THE RESULTS 1. Classify the two-dimensional shapes that make up the net of the container. 2. Find the area of each shape. Then find the sum of these areas. 3. Find the diameter of the top of the container and use it to find the perimeter or circumference of that face. 4. Multiply the circumference by the height of the container. What does this product represent? 5. Add the product from Exercise 4 to the sum of the areas of the two circular bases. 6. Compare your answers from Exercises 2 and 5. 7. MAKE A CONJECTURE Write a method for finding the area of all the surfaces of a cylinder given the measures of its height and the diameter of one of its bases. Explore 7-7 Measurement Lab: Surface Area of Cylinders 385 7-7 Surface Area of Prisms and Cylinders CUT CUT Main IDEA Use an empty box with a tuck-in lid. Measure and record the height of the box and the perimeter of the top or bottom face. Find the lateral and total surface area of prisms and cylinders. Standard 7MG2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Standard 7MG3.5 Construct two-dimensional patterns for three-dimensional models, such as cylinders, prisms, and cones. CUT CUT Label the top, bottom, front, back, and side faces using a marker. Open the lid and make 5 cuts as shown. Then open the box and lay it flat to form a net of the box. Measure and record the dimensions of each face. CUT BACK SIDE BOTTOM SIDE FRONT TOP 1. Find the area of each face. Then find the sum of these areas. 2. Multiply the perimeter of a base by the height of the box. What does this product represent? 3. Add the product from Exercise 2 to the sum of the areas of NEW Vocabulary lateral face lateral surface area total surface area Vocabulary Link Lateral Everyday Use situated on, directed toward, or coming from the side Lateral face Math Use a face of a solid that is not a base the two bases. 4. Compare your answers from Exercises 1 and 3. In the Mini Lab, you found the area of each surface, or face, of a box. A lateral face of a solid is any flat surface that is not a base. The lateral surface area of a solid is the sum of the areas of its lateral faces. The total surface area of a solid is the sum of the areas of all its surfaces. +%9 #/.#%043 Lateral Surface Area of a Prism Words The lateral area L of a prism is the perimeter P of the base times the height h of the prism. Model Symbols L = Ph Total Surface Area of a Prism 386 Words The total surface area S of a prism is the lateral surface area L plus the area of the two bases 2B. Symbols S = L + 2B or S = Ph + 2B Chapter 7 Measurement: Area and Volume Model Surface Areas of a Prism 1 Find the lateral and total surface areas of Bases of Rectangular Prisms For the examples and exercises in this book, assume that the top and the bottom faces of a rectangular prism are its bases. the rectangular prism. 12 m The bases of this prism are rectangles that are 3 meters wide and 7 meters long. Begin by finding the perimeter and area of one base. Perimeter of Base Area of Base P = 2 + 2w B = w P = 2(7) + 2(3) or 20 B = 7(3) or 21 3m 7m Use this information to find the lateral and total surface areas. Lateral Surface Area Total Surface Area L = Ph S = L + 2B L = 20(12) or 240 S = 240 + 2(21) or 282 The lateral surface area is 240 square meters, and the total surface area of the prism is 282 square meters. 2 SKATEBOARDING A wedge skateboarding ramp is built in the shape of a triangular prism. You plan to paint all surfaces of the ramp. Find the total surface area to be painted. Estimate 55.3 in. 12 in. 32 in. 54 in. 2 S = (60 + 50 + 10)(30) + 60(10) or 4,200 in . The bases of the prism are triangles with side lengths of 12 inches, 54 inches, and 55.3 inches. Find the perimeter and area of one base. Perimeter of Base Real-World Link Other types of skateboarding ramps include angled boxes, lo-banks, quarterpipes, and micro halfpipes. Kits for building ramps can include isometric drawings of side and rear views. Area of Base 1 B=_ bh 2 1 B=_ (54)(12) or 324 2 P = 55.3 + 12 + 54 P = 121.3 Use this information to find the total surface area. S = Ph + 2B Total surface area of prism S = 121.3(32) + 2(324) P = 121.3, h = 32, and B = 324. S = 4,529.6 Simplify. The surface area is 4,529.6 square inches. Compare to the estimate. Find the lateral and total surface areas of each prism. a. b. 3ft 4ft 6 ft 9 yd 6 yd 21 yd 5ft Personal Tutor at ca.gr7math.com Lesson 7-7 Surface Area of Prisms and Cylinders Tony Freeman/PhotoEdit 387 You can find the total surface area of a cylinder by finding the area of its two bases and adding the area of the curved surface. The lateral area of a cylinder is the area of the curved surface. If you unfold a cylinder, its net is two circles and a rectangle. Cylinders The formulas for the lateral and total surface areas of cylinders are similar to those of prisms. Prism: L = Ph For cylinders, the base is a circle, so its perimeter is the circumference. Prism: S = L + 2B For cylinders, the base B is a circle with area πr 2. Model Net Area 2 circular bases 2 congruent circles with radius r 2(πr 2) or 2πr 2 1 curved surface 1 rectangle with width h and length 2πr 2πr · h or 2πrh Just as with prisms, you can use the measures of a cylinder to find the lateral and total surface areas of a cylinder. +%9 #/.#%043 Lateral Surface Area of a Cylinder Words The lateral area L of a cylinder with height h and radius r is the circumference of the base times the height. Symbols L = 2πrh Model Total Surface Area of a Cylinder Words The surface area S of a cylinder Model with height h and radius r is the lateral area plus the area of the two bases. Symbols S = L + 2πr 2 or S = 2πrh + 2πr 2 Surface Areas of Cylinders 3 Find the lateral area and the total surface area of the cylinder. Round to the nearest tenth. Lateral Surface Area Total Surface Area L = 2πrh S = L + 2πr 2 L ≈ 2(3.14)(2)(3) S ≈ 37.7 + 2(3.14)(2) 2 L ≈ 37.7 S ≈ 62.8 2 ft 3 ft The lateral area is about 37.7 square feet, and the surface area of the cylinder is about 62.8 square feet. 388 Chapter 7 Measurement: Area and Volume Extra Examples at ca.gr7math.com 4 LABELS Find the area of the label on the can IN of vegetables shown at the right. Since the label covers the lateral surface of the can, you only need to find the can’s lateral surface area. IN L = 2πrh Estimate L ≈ 2(3)(2)(5) L ≈ 60 in L = 2πrh π ≈ 3, r = 1.75 ≈ 2, h = 5 2 Lateral surface area of cylinder L ≈ 2(3.14)(1.75)(5) π = 3.14, r = 1.75, h = 5 L ≈ 55.0 Simplify. The area of the label is about 55 square inches. Compare to the estimate. Find the lateral and total surface areas of each cylinder. Round to the nearest tenth. c. d. 5 mm 7 cm 10 mm Examples 1, 2 (p. 387) 14.8 cm Find the lateral and total surface areas of each solid. Round to the nearest tenth if necessary. 1. 2. 6 in. 4 yd 5 yd Example 3 (p. 388) 3. 10 in. 3 yd 7 in. 8 in. 4. 8m YD 9.4 m YD Example 4 (p. 389) IN 5. CONTAINERS Frozen orange juice often comes in /RANGE *UICE cylindrical cardboard containers with metal lids. Find the area of the cardboard portion of the orange juice container shown. IN Lesson 7-7 Surface Area of Prisms and Cylinders 389 (/-%7/2+ (%,0 For Exercises 6, 7 8, 9, 13 10, 11 12, 13 See Examples 1 2 3 4 Find the lateral and total surface areas of each solid. Round to the nearest tenth if necessary. 6. 7. 8. FT IN CM FT FT IN CM CM FT IN 9. 10. M 11. M MM YD M MM YD M M 12. CAMPING A manufacturer makes nylon tents like the one shown. How much material is needed to make the tent? CM CM CM CM 25 ft 13. POOL A vinyl liner covers the inside walls and bottom of the swimming pool. Find the area of this liner to the nearest square foot. 4 ft 14. A rectangular prism has length 12 centimeters and width 4 centimeters. If its surface area is 467 square centimeters, what is the height of the prism? 15. MANUFACTURING Find the amount of metal needed to construct the mailbox at the right to the nearest tenth of a square inch. 2 in. 4 in. MAIL 4 in. 16. GARDENING The door of the %842!02!#4)#% See pages 695, 714. greenhouse below has an area of 4.5 square feet. How many square feet of plastic are needed to cover the roof and sides of the greenhouse? FT 390 a cylindrical pipe is shown. Find the total surface area of the pipe, including the interior. IN IN IN Self-Check Quiz at ca.gr7math.com 17. PLUMBING A hollow piece of FT FT FT FT Chapter 7 Measurement: Area and Volume 10 in. H.O.T. Problems 18. REASONING Determine whether the following statement is true or false. If false, give a counterexample. If two rectangular prisms have the same volume, then they also have the same surface area. 19. CHALLENGE Will the surface area of a cylinder increase more if you double the height or double the radius? Explain your reasoning. 20. NUMBER SENSE If you double the edge length of a cube, explain how this affects the surface area of the prism. 21. */ -!4( Explain the difference between lateral area and (*/ 83 *5*/( surface area. 22. Molly is painting the rectangular toy 23. A paint roller like the one shown is chest shown in the diagram below. used for painting. 16 in. IN 15 in. IN 24 in. If Molly paints only the outside of the toy chest, what is the total surface area, in square inches, she will paint? A 330 in 2 C 1,968 in 2 2 2 B 399 in D 5,760 in To the nearest tenth, how many square inches does a single rotation of the paint roller cover? F 18.0 H 56.5 G 28.3 J 113.1 Find the volume of each solid. Round to the nearest tenth if necessary. (Lesson 7-6) 24. rectangular pyramid: length, 14 m; width, 12 m; height, 7 m 25. cone: diameter, 22 cm; height, 24 cm 26. HEALTH The inside of a refrigerator in a medical laboratory measures 17 inches by 18 inches by 42 inches. You need at least 8 cubic feet to refrigerate some samples from the lab. Is the refrigerator large enough for the samples? Explain your reasoning. (Lesson 7-5) PREREQUISITE SKILL Multiply. (Lesson 2-3) 27. _ · 2.8 1 2 28. _ · 10 · 23 1 2 29. _ · 2.5 · 16 1 2 30. _ 3_ (20) 1 2 ( 12 ) Lesson 7-7 Surface Area of Prisms and Cylinders 391 Extend 7- 7 Main IDEA Make a net of a cone. Measurement Lab Net of a Cone A cone is a three-dimensional figure with one circular base. The lateral surface is part of a larger circle. So that the edges match, the circumference of the base is equal to part of the circumference of the larger circle. Standard 7MG3.5 Construct twodimensional patterns for three-dimensional models, such as cylinders, prisms, and cones. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Make a Net of a Cone Use a compass to draw two circles slightly touching, one with a radius of 17 centimeters and one with a radius of 8 centimeters. A 17 cm B 8 cm Think: What part of the circumference of A is equal to the circumference of B? Let x represent the part. x(34π) = 16π The circumference of A is 34π. The circumference of B is 16π. 34π x·_ =_ 16π 34π 34π Divide each side by 34π. x ≈ 0.47 Simplify. You need 0.47 of the circumference of A. Find the size of the central angle to be cut from A. 0.47 · 360° ≈ 170° Cut a central angle of 170° from circle A and make a cone. A A 17 cm B B 8 cm ANALYZE THE RESULTS Find the central angle of each cone and then make a net and the cone. A 1. A 2. 4 in. 25 cm B B 10 cm 392 Chapter 7 Measurement: Area and Volume 2 in. 7-8 Surface Area of Pyramids Main IDEA Find the lateral and total surface areas of pyramids. Standard 7MG2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. HISTORY In 1485, Leonardo da Vinci sketched a pyramid-shaped parachute in the margin of his notebook. In June 2000, using a parachute created with tools and materials available in medieval times, Adrian Nicholas proved da Vinci’s design worked by descending 7,000 feet. 1. How many cloth faces does this pyramid have? What shape are they? 2. How could you find the total area NEW Vocabulary regular pyramid slant height of the material used for the parachute? A regular pyramid is a pyramid with a base that is a regular polygon. The lateral faces of a regular pyramid are congruent isosceles triangles. At the top of the pyramid, these triangles meet at a common point called the vertex. The altitude or height of each lateral face is called the slant height of the pyramid. ìÊvÊ,i}Õ>ÀÊ*ÞÀ>` iÌÊvÊ,i}Õ>ÀÊ*ÞÀ>` VERTEX BASE LATERAL FACE LATERAL FACE SLANT HEIGHT BASE Look Back You can review area of triangles on page 693. SIDE LENGTH S OF REGULAR POLYGON SLANT HEIGHT Ű To find the lateral area L of a regular pyramid, look at its net. The lateral area of a pyramid is the sum of the areas of its lateral faces, which are all triangles. The net of a square pyramid is a square and four triangles as shown above. 1 L=4 _ s Area of the lateral faces 1 L=_ (4s) Commutative Property of Multiplication (2 ) 2 _ L = 1 P 2 The perimeter of the base P is 4s. The total surface area of a regular pyramid is the lateral surface area plus the area of the base. Lesson 7-8 Surface Area of Pyramids (t)Heathcliff O’Malley/The Daily Telegraph, (b)Biblioteca Ambrosiana, Milan/Art Resource, NY 393 +%9 #/.#%043 Lateral Surface Area of a Pyramid Words The lateral surface area L of a regular pyramid is half the perimeter P of the base times the slant height . Symbols L= Model _1 P 2 Total Surface Area of a Pyramid Words The total surface area S of a regular pyramid is the lateral area L plus the area of the base B. Symbols S = L + B or S = Model _1 P + B 2 Surface Areas of a Pyramid 1 Find the lateral and total surface areas of 10 ft 12 ft the triangular pyramid. 1 P L=_ 2 S=L+B 1 L=_ · 30 · 12 2 1 S = 180 + 43.5 B = _ · 10 · 8.7 2 L = 180 S = 223.5 8.7 ft 10 ft 10 ft The lateral and total surface areas are 180 and 223.5 square feet. a. Find the lateral and total surface areas of a pyramid with a slant height of 18 meters and a square base with 11-meter sides. 2 ARCHITECTURE Use the information at the left to find the lateral surface area of the Pyramid of the Sun if it has a slant height of 132.5 meters. Real-World Link The Pyramid of the Sun in Teotihuacán, Mexico, was built in the second century, A.D. It is about 71 meters tall, and its square base has side lengths of 223.5 meters. Source: infoplease.com 1 L=_ P 2 1 L=_ · 894 · 132.5 2 L = 59,227.5 Lateral surface area of a pyramid P = 223.5(4) or 894 and = 132.5 Simplify. The lateral area of the pyramid is 59,227.5 square meters. b. AWARDS A music award is a square pyramid with a 6-inch-long base and a 13-inch slant height. Find the award’s total surface area. Personal Tutor at ca.gr7math.com 394 Chapter 7 Measurement: Area and Volume Charles & Josette Lenars/CORBIS Example 1 (p. 394) Find the lateral and total surface areas of each regular pyramid. Round to the nearest tenth if necessary. 1. 2. 6 ft 15 m 12 m 10.2 m 4 ft 12 m 12 m 4 ft Example 2 3. HISTORY Refer to the lesson opener. Each face of the parachute has a base of (p. 394) about 12 yards and a height of about 17 yards. Find the amount of material needed to make the parachute. (/-%7/2+ (%,0 Find the lateral and total surface area of each regular pyramid. Round to the nearest tenth if necessary. For Exercises 4–9 10, 11 See Examples 1 2 4. 5. IN IN 7. IN MM 6. FT FT MM M FT M M CM 8. M M FT 9. MM MM MM CM CM FT FT 10. ARCHITECTURE The Transamerica Pyramid in San Francisco is shaped like a square pyramid. It has a slant height of 856.1 feet and each side of its base is 145 feet. Find the lateral area of the building. 11. ROOFS A pyramid-shaped roof has a slant height of 16 feet and its square base is 40 feet wide. How much roofing material is needed to cover the roof? 12. A square pyramid has a lateral area of 107.25 square centimeters and a slant height of 8.25 centimeters. Find the length of each side of its base. 13. GLASS The Luxor Hotel in Las Vegas, Nevada, %842!02!#4)#% See pages 696, 714. Self-Check Quiz at ca.gr7math.com is a pyramid-shaped building standing 350 feet tall and covered with glass. Its base is a square with each side 646 feet long. Find the surface area of the glass on the Luxor. (Hint: Use the Pythagorean Theorem to find the pyramid’s slant height .) FT FT FT FT Lesson 7-8 Surface Area of Pyramids Mike Yamashita/Woodfin Camp & Associates 395 H.O.T. Problems CHALLENGE For Exercises 14–16, use the drawings of the pyramid shown, in which the lateral faces are equilateral triangles. 14. Find the measure of the slant Side View height . 15. Use the slant height to find the height h of the pyramid. 16. Find the volume and surface area of the pyramid. 17. OPEN ENDED Draw a square pyramid, giving measures for its slant height and base side length. Then find its lateral area. Justify your answer. 18. */ -!4( Explain how the slant height and the height of a (*/ 83 *5*/( pyramid are different. 19. Which is the best estimate for the 20. The net of a paperweight is shown surface area of the pyramid? below. Which is closest to the lateral surface area of the paperweight? FT CM FT FT CM A 107 ft 2 B 180 ft 2 C 429 ft 2 D 608 ft 2 F 32 cm 2 H 127 cm 2 G 49 cm 2 J 176 cm 2 21. GEOMETRY Find the surface area of a cylinder that has a diameter of 22 feet and a height of 7.5 feet. (Lesson 7-7) 22. MOUNTAINS A student is creating a clay model of a mountain shaped like a cone. If the mountain is 4 feet tall and the radius of the base is 2 feet, what is the volume of clay needed to make the mountain? Round to the nearest tenth if necessary. (Lesson 7-6) PREREQUISITE SKILL Solve each proportion. (Lesson 4-2) 23. _ = _ 16 n 396 12 40 24. _3 = _x 5 8 Chapter 7 Measurement: Area and Volume 25. _ = _ a 13 7 39 26. _ = _ 10 26 30 w Explore 7-9 Main IDEA Investigate the relationships between the surface areas and volumes of similar solids. Spreadsheet Lab Similar Solids In this activity you will use a spreadsheet to investigate the relationship between surface areas and volumes of similar solids, solids that have the same shape and whose linear measures are proportional. Prism A Standard 7MG2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a threedimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor. Standard 7MR2.2 Apply strategies and results from simpler problems to more complex problems. 1 Find the surface area and volume of 3 cm the prism at the right. Then find the surface areas and volumes of similar prisms with scale factors of 2, 3, and 4. 2 cm 5 cm 3IMILAR 0RISMSXLS ! " # $ 0RISM 3CALE &ACTOR ,ENGTH 7IDTH ! " # $ 3HEET 3HEET 3HEET The spreadsheet evaluates the formula 2*C3*D3⫹2*C3*E3⫹2*D3*E3. % & ' (EIGHT 3URFACE 6OLUME !REA The spreadsheet evaluates the formula C5*D5*E5. ANALYZE THE RESULTS 1. What is the ratio of the surface area of prism B to the surface area of prism A? of prism C to prism A? of prism D to prism A? 2. How are the answers to Exercise 1 related to the scale factors? 3. What is the ratio of the volume of prism B to the volume of prism A? of prism C to prism A? of prism D to prism A? 4. How are the answers to Exercise 3 related to the scale factors? 5. MAKE A PREDICTION If the dimensions of prism E are 5 times that of prism A, predict the ratio of the surface areas of prism E to prism A. 6. Explain how you can use the ratio in Exercise 5 to predict the surface area of prism E. Find the surface area using the spreadsheet. 7. MAKE A PREDICTION If the dimensions of prism E are 5 times that of prism A, predict the ratio of the volumes of prism E to prism A. 8. Explain how you can use the ratio in Exercise 6 to predict the volume of prism E. Find the volume using the spreadsheet. Explore 7-9 Spreadsheet Lab: Similar Solids 397 Cylinder A 3 in. 2 Find the surface area and volume of the cylinder at the right. Then find the surface areas and volumes of similar cylinders with scale factors of 2, 3, and 4. 4 in. 3IMILAR #YLINDERSXLS ! Spreadsheet Notation in Microsoft ® Excel ®, the expression PI() gives the value for π. " # #YLINDER 3CALE &ACTOR ! " # $ 3HEET 3HEET 4HE SPREADSHEET EVALUATES THE FORMULA 0) #> 0) # $ Spreadsheet Notation The expression C5^2 squares the value in cell C5. $ 2ADIUS % (EIGHT 3URFACE !REA & ' 6OLUME 3HEET 4HE SPREADSHEET EVALUATES THE FORMULA 0) #> $ ANALYZE THE RESULTS 9. What is the ratio of the surface areas of cylinder B to cylinder A? of cylinder C to cylinder A? of cylinder D to cylinder A? 10. How are the answers to Exercise 9 related to the scale factors of each cylinder? 11. What is the ratio of the volume of cylinder B to the volume of cylinder A? of cylinder C to cylinder A? of cylinder D to cylinder A? 12. How are the answers to Exercise 11 related to the scale factors of each cylinder? 13. MAKE A PREDICTION If the dimensions of cylinder F are 6 times that of cylinder A, predict the ratio of the surface areas of cylinder F to cylinder A. 14. Explain how you can use the ratio to predict the surface area of cylinder F. Find the surface area using the spreadsheet. 15. MAKE A PREDICTION If the dimensions of cylinder F are 5 times that of cylinder A, predict the ratio of the volumes of cylinder F to cylinder A. 16. MAKE A CONJECTURE If two solids A and B are similar and the scale factor relating solid A to solid B is _a , write expressions for the ratios b of their surface areas and volumes. 398 Chapter 7 Measurement: Area and Volume 7-9 Similar Solids Main IDEA Find dimensions, surface area, and volume of similar solids. 1 The model car at the right is _ the 43 size of the original car. 1. If the model car is 4.2 inches long, Standard 7MG2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a threedimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor. 1.6 inches wide, and 1.3 inches tall, what are the dimensions of the original car? 2. Make a conjecture about the radius of the wheel of the original car compared to the model. The pyramids at the right have the same shape. The ratios of their corresponding linear measures, such as length, width, or height, 6 12 are _ or 3 and _ or 3. We say that 2 M M M M 4 3 is the scale factor. NEW Vocabulary similar solids REVIEW Vocabulary scale factor the ratio of corresponding measures of two similar figures (Lesson 4-5) These pyramids are called similar solids because they have the same shape, their corresponding linear measures are proportional, and their corresponding faces are similar polygons. If you know two solids are similar, you can use a proportion to find a missing measure. Find Missing Linear Measures 1 The cylinders at the right are similar. Find the height of cylinder A. Since the two cylinders are similar, the ratios of their corresponding linear measures are proportional. FT H FT ÞìÀÊ ÞìÀÊ radius cylinder A height cylinder A __ is proportional to __ Words radius cylinder B Variable _5 8 h _5 = _ Write the proportion. 5 · 12 = 8 · h Find the cross products. 5 · 12 8·h _ =_ Divide each side by 8. 12 8 7.5 = h height cylinder B Let h represent the height of cylinder A. Equation 8 FT 8 = h _ 12 Simplify. The height of cylinder A is 7.5 feet. Lesson 7-9 Similar Solids Susan Jones/Age Fotostock 399 Find the missing measure for each pair of similar solids. a. b. FT MM FT FT MM MM As you discovered in the Geometry Lab prior to this lesson, the surface areas and volumes of similar solids are proportional. +%9 #/.#%043 Ratios of Similar Solids Ratios of Surface Area Words Symbols If two solids are similar, the ratio of their surface areas is proportional to the square of the scale factor between them. Model Solid A surface area of solid A a 2 __ = (_) surface area of solid B Solid B b Ratios of Volumes Words Symbols If two solids are similar, the ratio of their volumes is proportional to the cube of the scale factor between them. volume of solid A a 3 __ = (_) volume of solid B b Find Surface Area of a Similar Solid 2 The pyramids at the right are similar. Scale Factor Remembering that area is expressed in square units can help you remember to square the scale factor when working with surface areas of similar solids. Find the total surface area of pyramid B. *ÞÀ>`Ê 3 IN *ÞÀ>`Ê The ratio of the measures of pyramid A 8 2 or _ . to pyramid B is _a = _ b 12 IN 3 surface area of pyramid A 2 ___ = _a () Write a proportion. () Substitute the known values. Let S = the surface area of pyramid B. surface area of pyramid B b 2 224 _ _ = 2 3 S 224 4 _=_ 9 S (_23 ) = _23 · _23 or _49 2 224 · 9 = 4S Find the cross products. 224 · 9 4S _ =_ Divide each side by 4. 4 IN 4 504 = S Simplify. The surface area of pyramid B is 504 square inches. 400 Chapter 7 Measurement: Area and Volume Extra Examples at ca.gr7math.com Find the missing measure for each pair of similar solids. Round to the nearest tenth if necessary. c. d. Find Volume of a Similar Solid 3 A cube has a volume of 27 cubic feet. Suppose the dimensions are doubled. What is the volume of the new cube? A 13.5 ft 3 B 54 ft 3 C 108 ft 3 D 216 ft 3 Read the Item You know that the cubes are similar, the ratio of the side lengths Scale Factors When the lengths of all dimensions of a solid are multiplied by a scale factor x, then the surface area is multiplied by x 2 and the volume is multiplied by x 3. _a is _1 , and the volume of the smaller cube is 27 cubic feet. b 2 Solve the Item Since the volumes of similar solids have a ratio of _a ( b )3 and _ba = _12 , replace a with 1 and b with 2 in _a . ( b )3 3 volume of smaller cube __ = _a () Write a proportion. () Substitute the known values. Let V represent the volume of the larger cube. volume of larger cube b 27 1 3 _ = _ 2 V 27 1 _ =_ 8 V (_12 ) = _12 · _12 · _12 3 27 · 8 = V · 1 216 = V Find the cross products. Multiply. So, the volume of the larger cube is 216 cubic feet. The answer is D. e. A triangular prism has a volume of 896 cubic meters. If the prism is reduced to one-fourth its original size, what is the volume of the new prism? F 14 m 3 H 64 m 3 G 56 m 3 J M M 224 m 3 Personal Tutor at ca.gr7math.com Lesson 7-9 Similar Solids 401 Examples 1, 2 (pp. 399–400) For Exercises 1 and 2, use the two similar pyramids shown. Round to the nearest tenth if necessary. 15 in. 7 in. 1. Find the missing side length. 2. Find the missing surface area. 10 in. ? Example 3 3. (p. 401) STANDARDS PRACTICE A cone has a volume of 134.4 cubic centimeters. Suppose that the dimensions are reduced to half their current value. What is the volume of the resulting cone? CM A 8.4 cm 3 (/-%7/2+ (%,0 For Exercises 4, 5, 10 6, 7 8, 9, 11, 23, 24 See Examples 1 2 B 16.8 cm 3 CM C 33.6 cm 3 Find the missing measure for each pair of similar solids. Round to the nearest tenth if necessary. 4. MM 5. FT FT MM MM 3 FT 6. 7. 6 in. 8. 27 m 9m 402 D 67.2 cm 3 Chapter 7 Measurement: Area and Volume 12 in. 9. 3 cm 4 cm 10. ARCHITECTURE The model of a high-rise apartment building is 25.2 inches tall. On the model, 2 inches represents 45 feet. What is the height of the building? 11. ART In art class, Rueben made two similar cylindrical containers. One was 4 inches tall, and the other was 8 inches tall. If the volume of the smaller container is 16.7 cubic inches, find the volume of the larger container. For Exercises 12–14, use the two similar prisms at the right. Prism A Prism B 12. Write the ratio of the surface areas and the ratio of the volumes. 13. Find the surface area of prism B. 14. Find the volume of prism A. 15. The surface areas of two similar solids are 36 square yards and 144 square yards. Find the ratio of their linear measures. 16. HOBBIES Darcy is building a doll house FT similar to her family’s house. If the doll house FT 1 will be _ the size of her actual house, what will 20 be the lateral surface area of her doll house, not including the roof? Round to the nearest tenth. %842!02!#4)#% 17. AQUARIUMS A zoo has three cylindrical 3 aquariums. The smallest is _ the size of See pages 696, 714. CM 4 1 the one shown, while the largest is 1_ times 2 Self-Check Quiz at CM larger. Determine the volumes of the three aquariums. Round to the nearest tenth. ca.gr7math.com H.O.T. Problems FT FT 18. CHALLENGE The ratio of the surface areas of two similar pyramids is _. 1 25 What is the ratio of the volumes of the pyramids? Explain your reasoning. 19. OPEN ENDED Draw and label two cones that are similar. Explain why they are similar. 20. SELECT A TECHNIQUE Ruby is packing two similar boxes. The smaller box is 9 inches long and 12 inches tall, and the larger box is 18 inches long and 24 inches tall. Which of the following techniques might Ruby use to determine how much greater the volume of the larger box is? Justify your selection(s). Then use the technique(s) to solve the problem. mental math number sense estimation 21. REASONING True or False? All spheres are similar. Explain your reasoning. 22. */ -!4( Refer to the application at the beginning of the (*/ 83 *5*/( lesson. Write a real-world problem involving a model car. Then solve your problem. Lesson 7-9 Similar Solids 403 23. The triangular prisms shown are 24. The dimensions of two cubes are similar. shown below. 7 in. 14 in. Find the volume of the smaller prism. A 211 in 3 B 844 in 3 C 3,376 in 3 D 6,752 in 3 The volume of the smaller cube is 125 cubic feet. Find the volume of the larger cube. F 375 ft 3 G 3,375 ft 3 H 5,125 ft 3 J 15,625 ft 3 25. HISTORY The great pyramid of Khufu in Egypt was originally 481 feet high, and had a square base measuring 756 feet on a side and slant height of about 611.8 feet. What was its lateral surface area? Round to the nearest tenth. (Lesson 7-8) 26. MEASUREMENT Find the lateral surface and total surface area of the rectangular prism at the right. (Lesson 7-7) 27. GEOMETRY Graph parallelogram QRST with vertices Q(-3, 3), R(2, 4), S(3, 2), and T(-2, 1). Then graph the image of the figure after a reflection over the x-axis, and write the coordinates of its vertices. (Lesson 6-6) CM CM 28. ALGEBRA Find the value of x in the two congruent triangles. (Lesson 6-4) 29. MONEY A $750 investment earned $540 in 6 years. Find the simple interest rate. (Lesson 5-9) Math and Architecture Under construction It’s time to complete your project. Use the information and data you have gathered about floor covering costs and loan rates to prepare a Web page or brochure. Be sure to include a labeled scale drawing with your project. Cross-Curricular Project at ca.gr7math.com 404 Chapter 7 Measurement: Area and Volume CM CH APTER 7 Study Guide and Review Download Vocabulary Review from ca.gr7math.com Key Vocabulary base (p. 369) Be sure the following Key Concepts are noted in your Foldable. lateral surface area center (p. 352) (p. 386) chord (p. 352) net (p. 385) circle (p. 352) pi (p. 352) circumference (p. 352) polyhedron (p. 368) complex figure (p. 363) prism (p. 368) Circles (Lesson 7-1) complex solid (p. 375) pyramid (p. 369) • Circumference: C = πd or C = 2πr cone (p. 381) radius (p. 352) • Area: A = πr 2 coplanar (p. 368) regular pyramid (p. 393) cylinder (p. 374) similar solids (p. 399) diameter (p. 352) slant height (p. 393) edge (p. 368) solid (p. 368) face (p. 368) total surface area (p. 386) lateral face (p. 386) vertex (p. 368) "REA 7OLUME Key Concepts Volume (Lessons 7-5 and 7-6) • Prism: V = Bh • Cylinder: V = Bh or V = π r 2h _ • Pyramid: V = 1 Bh 3 1 1 _ • Cone: V = Bh or V = _π r 2h 3 volume (p. 373) 3 Surface Area (Lessons 7-7 and 7-8) • Prism Lateral Surface Area: L = Ph Total Surface Area: S = L + 2B Vocabulary Check • Pyramid Lateral Surface Area: L = P Total Surface Area: S = L + B State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. • Cylinder Lateral Surface Area: L = 2πrh Total Surface Area: S = L + 2B 1. The flat surface of a prism is called a face. 2. Circumference is the distance around a Similar Solids (Lesson 7-9) _a • If two solids are similar with a scale factor of , a 2 b then the surface areas have a ratio of and b a 3 the volumes have a ratio of . (_b ) (_) circle. 3. The measure of the space occupied by a solid is called the total surface area. 4. A cylinder is a figure that has two parallel, congruent circular bases. 5. A solid is any two-dimensional figure. 6. The side of a prism is called a vertex. 7. The radius is the distance across a circle through its center. Vocabulary Review at ca.gr7math.com Chapter 7 Study Guide and Review 405 CH APTER 7 Study Guide and Review Lesson-by-Lesson Review 7-1 Circumference and Area of Circles (pp. 352–357) Find the circumference and area of each circle. Round to the nearest tenth. 8. radius: 18 in. 9. diameter: 6 cm 10. LANDSCAPING Bill is planting a circular flowerbed. What is the area of the flowerbed if the diameter is 30 feet? 7-2 11. GEOGRAPHY The total area of Arizona is 114,006 square miles. Of that, about 42% of the land is desert. About how many square miles of Arizona’s land is not covered by desert? A = πr 2 A ≈ 3.14 · 5 2 A ≈ 78.5 yd 2 Example 2 A total of 450 students were surveyed. If 60% of the students voted to hold a carnival, find the number of students who voted for the carnival. Find 10% of 450 and use the result to find 60% of 450. 10% of 450 = 45; so 60% is 6 × 45 or 270. So, 270 students voted for the carnival. Area of Complex Figures (pp. 363–367) Find the area of each figure. Round to the nearest tenth if necessary. CM 12. MM 13. CM CM MM Example 3 Find the area of the complex figure. M MM 14. BASKETBALL Travis is going to paint part of a basketball court as shown. What is the area of the court? Chapter 7 Measurement: Area and Volume Area of semicircle 1 A≈_ · 3.14 · 2 2 2 CM FT M M MM MM 406 The radius is 5 yards. C = 2πr C ≈ 2 · 3.14 · 5 C ≈ 31.4 yd YD PSI: Solve a Simpler Problem (pp. 360–361) Solve. Use the solve a simpler problem strategy. 7-3 Example 1 Find the circumference and area of the circle. Area of trapezoid 1 A=_ (6)(4 + 10) 2 A ≈ 6.3 A = 42 The area is about 6.3 + 42 or 48.3 square meters. Mixed Problem Solving For mixed problem-solving practice, see page 714. 7-4 Three-Dimensional Figures (pp. 368–372) Identify each solid. Name the number and shapes of the faces. Then name the number of edges and vertices. 15. 16. Example 4 Name the number and shapes of the faces of a rectangular prism. Then name the number of edges and vertices. RECTANGULAR FACES VERTICES EDGES 17. CRYSTALS Kelli found a crystal in the shape of a pentagonal pyramid. How many faces, edges, and vertices does the crystal have? 7-5 Volume of Prisms and Cylinders (pp. 373–378) Find the volume of each solid. 18. 19. YD MM MM YD YD YD MM The base of this prism is a triangle. FT FT V = Bh 1 · 13 · 10 18 V= _ 20. FOOD A can of green beans has a diameter of 10.5 centimeters and a height of 13 centimeters. Find its volume. 7-6 Example 5 Find the volume of the solid. FT ) (2 V = 1,170 ft 3 Volume of Pyramids and Cones (pp. 380–384) Find the volume of each solid. Round to the nearest tenth if necessary. Example 6 Find the volume of the pyramid. 21. The base B of the pyramid is a rectangle. 22. CM FT FT IN IN 1 V=_ Bh CM FT IN CM 23. cone: diameter, 9 yd; height, 21 yd 24. ICE CREAM A waffle cone is five inches 3 1 V=_ (12 · 6)8 3 V = 192 in 3 tall. The opening of the cone has a radius of 1.5 inches. What is the volume of ice cream that the cone can hold? Chapter 7 Study Guide and Review 407 CH APTER 7 Study Guide and Review 7-7 Surface Area of Prisms and Cylinders (pp. 386–391) Find the surface area of each solid. Round to the nearest tenth if necessary. 25. 26. IN Example 7 Find the surface area of the cylinder. MM MM M IN M M M S ≈ 2(3.14)(8) 2 + 2(3.14)(8)(11) r = 8 and h = 11 S ≈ 954.6 mm 7-8 Surface area of a cylinder S = 2πr 2 + 2πrh 2 Simplify. Surface Area of Pyramids (pp. 393–396) 27. ARCHITECTURE A hotel shaped like a square pyramid has a slant height of 92.5 meters and each side of its base is 183.5 meters long. What is the lateral surface area of the pyramid? Example 8 Find the total surface area of the square pyramid. 1 A=_ bh M Area of triangle 2 _ A = 1 (3)(7) or 10.5 2 M M The total lateral area is 4(10.5) or 42 square meters. The area of the base is 3(3) or 9 square meters. So, the total surface area is 42 + 9 or 51 square meters. 7-9 Similar Solids (pp. 399–404) 28. Cylinders A and B are ÞìÀÊ FT similar. If the total surface area of cylinder A is 84 square feet, ÞìÀÊ what is the total FT surface area of cylinder B? Example 9 Two similar cones are shown at the right. Find the volume of the smaller cone. 3 V ⫽ 184 cm 3 volume of smaller cone __ = _a volume of larger cone FT 9 cm V 1 _ =_ 184 27 (b) Write a proportion. _1 = (_1 )3. 27 3 Find the cross products. 184 27V _ =_ Divide each side by 27. 27 V = 6.8 cm 3 Simplify. Chapter 7 Measurement: Area and Volume V⫽? V · 27 = 184 · 1 27 408 3 cm CH APTER Practice Test 7 Find the circumference and area of each figure. Round to the nearest tenth if necessary. Find the volume of each solid. Round to the nearest tenth. 1. 8. 2. FT 9.4 cm 9. 5.2 in. 3 in. 3. STANDARDS PRACTICE A jogger ran around a circular track two times. If the track has a radius of 25 yards, about how far did the jogger run? A 314 yd B 157 yd 15 mm 9.4 mm 12 mm 10. FUEL The fuel tank is made up of a cylinder. What is the volume of the tank? Round to the nearest tenth. C 78.5 yd D 50 yd 8.4 m 21.2 m Find the area of each figure. Round to the nearest tenth if necessary. 4. FT 5. M M FT M M M Find the volume and the total surface area of each solid. Round to the nearest tenth if necessary. 11. 12. 3.3 m 6m 11 ft 10.4 ft M 6m FT 7m 7 ft 6. GEOMETRY Identify the solid. Name the number and shapes of its faces. Then name its number of edges and vertices. 13. 7. CAKE DECORATION Mrs. Lee designed the flashlight birthday cake shown below. If one container of frosting covers 250 square inches of cake, how many containers will she need to frost the top of this cake? Explain. IN STANDARDS PRACTICE Find the volume of the solid. FT F 2,160 ft 3 G 2,520 ft 3 H 3,600 ft 3 J 7,200 ft 3 FT FT FT For Exercises 14–16, use the two similar prisms. 14. Write the ratio of the surface areas. IN IN 7 ft 10 m Prism B Prism A 15. Find the total surface IN area of prism B. 16. Find the volume of prism A. Chapter Test at ca.gr7math.com Chapter 7 Practice Test 409 CH APTER 7 California Standards Practice Cumulative, Chapters 1–7 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1 3 What is the surface area of the shoe box? IN The figure shows a circle inside a square. IN IN A 200 in 2 C 400 in 2 B 224 in 2 D 448 in 2 Which procedure should be used to find the area of the shaded region? A Find the area of the square and then subtract the area of the circle. B Find the area of the circle and then subtract the area of the square. C Find the perimeter of the square and then subtract the circumference of the circle. D Find the circumference of the circle and then subtract the perimeter of the square. 2 Question 3 Most standardized tests will include any commonly used formulas at the front of the test booklet, but it will save you time to memorize many of these formulas. For example, you should memorize that the surface area of a prism is 2h + 2w + 2hw. 4 What is the area of the shaded region in the figure below? If LMN is translated 7 units up and 2 units to the right, what are the coordinates of point L? 4 3 2 1 ⫺8⫺7⫺6⫺5⫺4⫺3⫺2⫺1 O L 2cm 2cm y 3.5 cm 1 2 3 4x ⫺2 ⫺3 ⫺4 ⫺5 ⫺6 N⫺7 ⫺8 3.5 cm F 6.5 cm H 13 cm G 7 cm J 26 cm G (7, 2) Martin and his two brothers equally shared the cost of a new computer game with a list price of $35. They received a 25% discount on the video game and paid 5.5% sales tax on the discounted price. Find the approximate amount that each of the brothers paid toward the cost of the game. H (2, 7) A $14.77 C $9.23 J (4, -1) B $11.73 D $8.42 M F (-1, 4) 410 Chapter 7 Measurement: Area and Volume 5 More California Standards Practice For practice by standard, see pages CA1–CA39. 6 Suppose you know the side lengths of each figure below. Which one would contain enough information to let you find the length of diagonal d? F A stackable block shown below is made of wood. The height and width of each section is 6 cm. The length is 12 cm. 9 6 cm H 6 cm d d 6 cm G J d 12 cm d What is the volume, in cubic centimeters, of the wood used to create this block? 7 An isosceles triangle is removed from a rectangle as shown in the figure below. Find the area of the remaining part of the rectangle. A 2,592 C 432 B 1,296 D 30 Pre-AP 5 cm 5 cm Record your answers on a sheet of paper. Show your work. Front 10 The diagrams show 12 cm A 60 cm 2 C 47.5 cm 2 B 55 cm 2 D 35 cm 2 the design of the trash cans in the school cafeteria. Back 3 ft 4 3 ft 2 ft a. Find the volume 1 1 2 ft of each trash can to the nearest tenth. 8 Susan has two similar rectangular packages. The dimensions of the first box is three times that of the second package. How many times greater is the volume of the first package than of the second package? F 81 H 9 G 27 J 3 b. The tops and sides of the cans need to be painted. Find the surface area of each can to the nearest tenth. c. The paint used by the school covers 200 square feet per gallon. How many trash cans can be covered with 1 gallon of paint? NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 10 Go to Lesson... 7-1 6-7 7-7 7-3 5-8 3-5 7-3 7-9 7-5 7-7 For Help with Standards... MG2.2 MG3.2 MG2.1 MG2.2 NS1.7 MG3.3 MG2.2 MG2.3 MG2.3 MG2.1 California Standards Practice at ca.gr7math.com Chapters 1–7 California Standards Practice 411 Algebraic Thinking: Linear and Nonlinear Functions Focus Graph linear equations and understand the idea of slope and its relation to ratio. CHAPTER 8 Algebra: More Equations and Inequalities Express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs. Solve simple linear equations and inequalities over the rational numbers. CHAPTER 9 Algebra: Linear Functions Graph and interpret linear and some nonlinear functions. CHAPTER 10 Algebra: Nonlinear Functions and Polynomials Use exponents, powers, and roots and use exponents in working with fractions. Interpret and evaluate expressions involving integer powers and simple roots. 412 Michael Newman/PhotoEdit Math and Economics Getting Down to Business How would you like to run your own business? On this adventure, you’ll be creating your own company. Along the way, you’ll come up with a company name, slogan, and product to sell to your peers at school. You’ll research the cost of materials, create advertisements, and calculate potential profits. You’ll also survey your peers to find out what they would be willing to pay for your product, analyze the data, and adjust your projected profit model. You’re going to need your algebra tool kit to make this company work, so let’s get down to business! Log on to ca.gr7math.com to begin. Unit 4 Algebraic Thinking: Linear and Nonlinear Functions Michael Newman/PhotoEdit 413 Algebra: More Equations and Inequalities 8 • Standard 7AF2.0 Interpret and evaluate expressions involving integer powers and simple roots. • Standard 7AF4.0 Solve simple linear equations and inequalities over the rational numbers. Key Vocabulary equivalent expressions (p. 416) like terms (p. 417) two-step equation (p. 422) Real-World Link Beaches The California shoreline has been decreasing at an average rate of about 9 inches per year. You can write an equation to describe the change in the amount of shoreline for a given number of years. Algebra: More Equations and Inequalities Make this Foldable to help you organize your notes. Begin with a plain sheet of 11” × 17” paper. 1 Fold in half lengthwise. 2 Fold again from top to bottom. 3 Open and cut along the second fold to make two tabs. 4 Label each tab as shown. %QUATIONS )NEQUALIT IES 414 Chapter 8 Algebra: More Equations and Inequalities © Larry Dale Gordon/zefa/Corbis GET READY for Chapter 8 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Determine whether each statement is true or false. (Lesson 1-3) Example 1 1. 10 > 4 2. 3 < -3 Determine whether the statement -2 > 1 is true or false. 3. -8 < -7 4. -1 > 0 Plot the points on a number line. 5. WEATHER The temperature in Sioux City, Iowa, was -7°F while the temperature in Des Moines, Iowa, was -5°F. Which city was warmer? Explain. (Lesson 1-3) Write an algebraic equation for each verbal sentence. (Lesson 1-7) 6. Ten increased by a number is -8. 7. The difference of -5 and 3x is 32. 8. Twice a number decreased by Since -2 is to the left of 1, -2 < 1. The statement is false. Example 2 Write an algebraic equation for the verbal sentence twice a number increased by 3 is -5. Let x represent the number. twice a number increased by 3 is -5 4 is 26. +3 2x 9. MONEY Bianca has $1 less than twice as much as her brother. If her brother had $15, how much money did Bianca have? (Lesson 1-9) So, the equation is 2x + 3 = -5. Solve each equation. Check your solution. (Lessons 1-9 and 1-10) Example 3 10. n + 8 = -9 11. 4 = m + 19 12. -4 + a = 15 13. z - 6 = -10 14. 3c = -18 15. -42 = -6b 44 = k - 7 + 7= +7 ______ 51 = k 16. _ = -8 17. 12 = _ w 4 = -5 Solve 44 = k - 7. Write the equation. Add 7 to each side. Simplify. r -7 Chapter 8 Get Ready for Chapter 8 415 8-1 Simplifying Algebraic Expressions Main IDEA Use the Distributive Property to simplify algebraic expressions. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A. Standard 7AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, commutative) and justify the process used. Standard 7AF1.4 Use algebraic terminology (e.g. variable, equation, term, coefficient, inequality, expression, constant) correctly. NEW Vocabulary equivalent expressions term coefficient like terms constant simplest form simplifying the expression You can use algebra tiles to rewrite the algebraic expression 2(x + 3). Double this amount of tiles to represent 2(x + 3). Represent x + 3 using algebra tiles. 1 x 1 x 1 1 1 Rearrange the tiles by grouping together the ones with the same shape. 1 x 1 1 x x 1 1 1 1 1 1 1 1. Choose two positive and one negative value for x. Then evaluate 2(x + 3) and 2x + 6 for each of these values. What do you notice? 2. Use algebra tiles to rewrite the expression 3(x - 2). (Hint: Use one green x-tile and 2 red –1-tiles to represent x - 2.) In Chapter 1, you learned that expressions like 2(4 + 3) can be rewritten using the Distributive Property and then simplified. 2(4 + 3) = 2(4) + 2(3) Distributive Property = 8 + 6 or 14 Multiply. Then add. The Distributive Property can also be used to simplify an algebraic expression like 2(x + 3). 2(x + 3) = 2(x) + 2(3) Distributive Property = 2x + 6 Multiply. The expressions 2(x + 3) and 2x + 6 are equivalent expressions, because no matter what x is, these expressions have the same value. Write Expressions With Addition Use the Distributive Property to rewrite each expression. 2 (y + 2)5 1 4(x + 7) 4(x + 7) = 4(x) + 4(7) READING in the Content Area For strategies in reading this lesson, visit ca.gr7math.com. 416 = 4x + 28 (y + 2)5 = y · 5 + 2 · 5 Simplify. = 5y + 10 Commutative Property a. 6(a + 4) Chapter 8 Algebra: More Equations and Inequalities b. (n + 3)8 c. -2(x + 1) Write Expressions with Subtraction Look Back You can review multiplying integers in Lesson 1-6. Use the Distributive Property to rewrite each expression. 3 6(p - 5) 6(p - 5) = 6[p + (-5)] Rewrite p - 5 as p + (-5). = 6(p) + 6(-5) Distributive Property = 6p + (-30) Simplify. = 6p - 30 Definition of subtraction 4 -2(x - 8) -2(x - 8) = -2[x + (-8)] Rewrite x - 8 as x + (-8). = -2(x) + (-2)(-8) Distributive Property = -2x + 16 Simplify. d. 3(y - 10) e. -7(w - 4) f. (n - 2)(-9) When plus or minus signs separate an algebraic expression into parts, each part is called a term. The numerical factor of a term that contains a variable is called the coefficient of the variable. This expression has three terms. -2x + 16 + x 1 is the coefficient of x - 2 is the coefficient of x Vocabulary Link Constant Everyday Use unchanging Math Use a numeric term without a variable Like terms contain the same variables to the same powers. For example, 3x 2 and -7x 2 are like terms. So are 8xy 2 and 12xy 2. But 10x 2z and 22xz 2 are not like terms. A term without a variable is called a constant. Constant terms are also like terms. Identify Parts of an Expression 5 Identify the terms, like terms, coefficients, and constants in the expression 6n - 7n - 4 + n. 6n - 7n - 4 + n = 6n + (-7n) + (-4) + n Definition of subtraction = 6n + (-7n) + (-4) + 1n Identity Property; n = 1n • Terms: 6n, -7n, -4, n • Like terms: 6n, -7n, n • Coefficients: 6, -7, 1 • Constants: -4. Identify the terms, like terms, coefficients, and constants in each expression. g. 9y - 4 - 11y + 7 Extra Examples at ca.gr7math.com h. 3x + 2 - 10 - 3x Lesson 8-1 Simplifying Algebraic Expressions 417 An algebraic expression is in simplest form if it has no like terms and no parentheses. You can use the Distributive Property to combine like terms. This is called simplifying the expression. Simplify Algebraic Expressions 6 Simplify the expression 3y + y. Equivalent Expressions To check whether 3y + y and 4y are equivalent expressions, substitute any value for y and see whether the expressions have the same value. 3y and y are like terms. 3y + y = 3y + 1y Identity Property; y = 1y = (3 + 1)y or 4y Distributive Property; simplify. 7 Simplify the expression 7x - 2 - 7x + 6. 7x and -7x are like terms. -2 and 6 are also like terms. 7x - 2 - 7x + 6 = 7x + (-2) + (-7x) + 6 Definition of subtraction = 7x + (-7x) + (-2) + 6 Commutative Property = [7 + (-7)]x + (-2) + 6 Distributive Property = 0x + 4 Simplify. = 0 + 4 or 4 0x = 0 · x or 0 Simplify each expression. i. 4z - z j. 6 - 3n + 3n k. 2g - 3 + 11 - 8g 8 FOOD At a baseball game, you buy some hot dogs that cost $3 each and the same number of soft drinks for $2.50 each. Write an expression in simplest form that represents the total amount spent. Words Real-World Link In a recent year, Americans were expected to eat 26.3 million hot dogs in major league ballparks. This is enough to stretch from Dodger Stadium in Los Angeles to the Pirates’ PNC Stadium in Pittsburgh. $3 each some number and for of hot dogs $2.50 the same number each for of drinks Let x represent the number of hot dogs or drinks. Variable 3·x Expression + 2.50 · x Simplify the expression. 3x + 2.50x = (3 + 2.50)x = 5.50x Distributive Property Simplify. The expression $5.50x represents the total amount spent. Source: www.hot-dog.org l. MONEY You have saved some money. Your friend has saved $50 less than you. Write an expression in simplest form that represents the total amount of money you and your friend have saved. Personal Tutor at ca.gr7math.com 418 Chapter 8 Algebra: More Equations and Inequalities DiMaggio/Kalish/CORBIS Examples 1–4 (pp. 416–417) Example 5 (p. 417) Examples 6, 7 (p. 418) Example 8 (p. 418) (/-%7/2+ (%,0 For Exercises 16–27 28–33 34–39 40–43 See Examples 1–4 5 6, 7 8 Use the Distributive Property to rewrite each expression. 1. 5(x + 4) 2. 2(n + 7) 3. (y + 6)3 4. (a + 9)4 5. 2(p - 3) 6. 6(4 - k) 7. -6(g - 2) 8. -3(a + 9) Identify the terms, like terms, coefficients, and constants in each expression. 9. 5n - 2n - 3 + n 10. 8a + 4 - 6a - 5a 11. 7 - 3d - 8 + d Simplify each expression. 12. 8n + n 13. 7n + 5 - 7n 14. 4p - 7 + 6p + 10 15. MOVIES You buy 2 drinks that each cost x dollars and a large bag of popcorn for $3.50. Write an expression in simplest form that represents the total amount of money you spent. Use the Distributive Property to rewrite each expression. 16. 3(x + 8) 17. -8(a + 1) 18. (b + 8)5 19. (p + 7)(-2) 20. 4(x - 6) 21. 6(5 - q) 22. -8(c - 8) 23. -3(5 - b) 24. (d + 2)(-7) 25. -4(n - 3) 26. (10 - y)(-9) 27. (6 + z) 3 Identify the terms, like terms, coefficients, and constants in each expression. 28. 2 + 3a + 9a 29. 7 - 5x + 1 30. 4 + 5y - 6y + y 31. n + 4n - 7n - 1 32. -3d + 8 - d - 2 33. 9 - z + 3 - 2z Simplify each expression. 34. n + 5n 35. 12c - c 36. 5x + 4 + 9x 37. 2 + 3d + d 38. -3r + 7 - 3r - 12 39. -4j - 1 - 4j + 6 Write an expression in simplest form that represents the total amount in each situation. 40. SHOPPING You buy x shirts that each cost $15.99, the same number of jeans for $34.99 each, and a pair of sneakers for $58.99. 41. PHYSICAL EDUCATION Each lap around the school track is a distance of 1 laps on Wednesday, and 100 yards y yards. You ran 2 laps on Monday, 3_ 2 on Friday. 42. FUND-RAISING You have sold t tickets for a school fund-raiser. Your friend has sold 24 fewer than you. 43. BIRTHDAYS Today is your friend’s birthday. She is y years old. Her brother is 5 years younger. Lesson 8-1 Simplifying Algebraic Expressions 419 44. GOVERNMENT In 2005, in the Texas Legislature, there were 119 more members in the House of Representatives than in the Senate. If there were m members in the Senate, write an expression in simplest form to represent the total number of members in the Texas Legislature. 45. FIND THE DATA Refer to the California Data File on pages 16–19. Choose some data and write a real-world problem in which you would write and simplify an algebraic expression. Use the Distributive Property to rewrite each expression. 46. 3(2y + 1) 47. -4(3x + 5) 48. -6(12 - 8n) 49. 4(x - y) 50. -2(3a - 2b) 51. (-2 - n)(-7) 52. 5x(y - z) 53. -6a(2b + 5c) ALGEBRA Simplify each expression. 54. -_ a - _ + _ a - _ 2 5 1 4 7 10 1 5 55. 6p - 2r - 13p + r 56. -n + 8s - 15n - 22s 57. SCHOOL You are ordering T-shirts with your school’s mascot printed on them. Each T-shirt costs $4.75. The printer charges a set-up fee of $30 and $2.50 to print each shirt. Write two expressions that you could use to represent the total cost of printing n T-shirts. MEASUREMENT Write two equivalent expressions for the area of each figure. 58. 59. 10 12 60. x⫹4 x⫺7 x⫹5 16 61. SCHOOL You spent m minutes studying on Monday. On Tuesday, you studied 15 more minutes than you did on Monday. Wednesday, you studied 30 minutes less than you did on Tuesday. You studied twice as long on Thursday as you did on Monday. On Friday, you studied 20 minutes less than you did on Thursday. Write an expression in simplest form to represent the number of minutes you studied for these five days. %842!02!#4)#% See pages 696, 715. Self-Check Quiz at ca.gr7math.com H.O.T. Problems 62. OPEN ENDED Write an expression that has four terms and simplifies to 3n + 2. Identify the coefficient(s) and constant(s) in your expression. 63. Which One Doesn’t Belong? Identify the expression that is not equivalent to the other three. Explain your reasoning. x - 3 + 4x 5(x - 3) 6 + 5x - 9 5x - 3 64. CHALLENGE Simplify the expression 8x 2 - 2x + 12x - 3. Show that your answer is true for x = 2. 65. */ -!4( Is 2(x - 1) + 3(x - 1) = 5(x - 1) a true statement? (*/ 83 *5*/( If so, justify your answer using mathematical properties. If not, give a counterexample. 420 Chapter 8 Algebra: More Equations and Inequalities 66. Which property is used in the equation 67. Which expression is equivalent to 5a + 5b? below? 4x + 32 = 4(x + 8) F 5ab A Associative Property of Addition G 5(a + b) B Commutative Property of Addition H 5a + b C Distributive Property J a + 5b D Reflexive Property 68. MEASUREMENT The prisms at the right are similar. Find the volume of the smaller prism. Round to the nearest tenth. (Lesson 7-9) IN IN IN IN MEASUREMENT Find the lateral area and the surface area of each regular pyramid. Round to the nearest tenth if necessary. (Lesson 7-8) 69. 70. CM CM 71. FT CM IN CM FT CM IN FT IN 72. INTERNET The number of U.S. households with high-speed Internet access increased 66% from 2003 to 2004. If 63 million households had high-speed Internet access in 2004, how many households had highspeed Internet access in 2003? (Lesson 5-8) 73. Determine whether the set of numbers in the table is proportional. (Lesson 4-2) Time (hours) 1 2 3 4 5 6 Rental Charge $13 $23 $33 $43 $53 $63 Express each rate as a unit rate. Round to the nearest tenth if necessary. (Lesson 4-1) 74. $5 for 4 loaves of bread 75. 183.4 miles in 3.2 hours PREREQUISITE SKILL Solve each equation. Check your solution. (Lessons 1-9 and 1-10) 76. x + 8 = 2 77. y - 5 = -9 78. 32 = -4n 79. _ = -15 a 3 Lesson 8-1 Simplifying Algebraic Expressions 421 8-2 Solving Two-Step Equations Main IDEA Solve two-step equations. Standard 7AF4.1 Solve twostep linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. NEW Vocabulary two-step equation BOOK SALE Linda bought four books at a book sale benefiting a local charity. The handwritten receipt she received was missing the cost for the hardback books she purchased. I]Vc`Ndj[dg NdjgHjeedgi (]VgYWVX`h &eVeZgWVX`h & 1. Explain how you could use the work IdiVaeV^Y , backward strategy to find the cost of each hardback book. Then find the cost. The solution to this problem can also be found by solving the equation 3x + 1 = 7, where x is the cost per hardback book. This equation can be modeled using algebra tiles. 1 x x x 3x 1 1 1 1 1 1 1 1 7 A two-step equation contains two operations. In the equation 3x + 1 = 7, x is multiplied by 3 and then 1 is added. To solve two-step equations, undo each operation in reverse order. Solve Two-Step Equations 1 Solve 3x + 1 = 7. METHOD 1 Use a model. Remove one 1-tile from each mat. 1 x x x 3x 1 1 1 1 1 1 1 1 1 71 Separate the remaining tiles into 3 equal groups. x x x 3x 1 1 1 1 1 1 Aaron Haupt Chapter 8 Algebra: More Equations and Inequalities Use the Subtraction Property of Equality. 3x + 1 = 7 - 1 =-1 ____________ 3x = 6 6 Write the equation. Subtract 1 from each side. Use the Division Property of Equality. 3x = 6 3x 6 _ =_ Divide each side by 3. x=2 Simplify. 3 There are 2 1-tiles in each group, so the solution is 2. 422 Use symbols. METHOD 2 3 BrainPOP® ca.gr7math.com _ 2 Solve 25 = 1 n - 3. 4 METHOD 1 METHOD 2 Vertical method 1 25 = _ n-3 Write the equation. +3= Add 3 to each side. 4 +3 1 28 = _ n Simplify. 4 1 4 · 28 = 4 · _ n 4 112 = n Multiply each side by 4. Horizontal method _1 n - 3 = 25 4 1 _n - 3 + 3 = 25 + 3 4 _1 n = 28 4 _ 4 · 1 n = 4 · 28 4 n = 112 The solution is 112. Solve each equation. Check your solution. a. 3x + 2 = 20 b. 5 + 2n = -1 c. -1 = _ a + 9 1 2 Personal Tutor at ca.gr7math.com Some two-step equations have a term with a negative coefficient. Equations with Negative Coefficients 3 Solve 6 - 3x = 21. 6 - 3x = 21 Common Error A common mistake when solving the equation in Example 3 is to divide each side by 3 instead of -3. Remember that you are dividing by the coefficient of the variable, which in this instance is a negative number. 6 + (-3x) = 21 6 - 6 + (-3x) = 21 - 6 Write the equation. Rewrite the left side as addition. Subtract 6 from each side. -3x = 15 Simplify. -3x 15 _ =_ Divide each side by -3. -3 -3 x = -5 Simplify. The solution is -5. 6 - 3x = 21 Write the equation. 6 - 3(- 5) 21 Replace x with -5. 6 - (-15) 21 Multiply. Check 6 + 15 21 21 = 21 ✓ To subtract a negative number, add its opposite. The sentence is true. Solve each equation. Check your solution. d. 10 - _p = 52 2 3 Extra Examples at ca.gr7math.com e. -19 = -3x + 2 f. n _ - 2 = -18 -3 Lesson 8-2 Solving Two-Step Equations 423 Sometimes it is necessary to combine like terms before solving an equation. Combine Like Terms First 4 Solve -2y + y - 5 = 11. Check your solution. -2y + y - 5 = 11 Write the equation. -2y + 1y - 5 = 11 Identity Property; y = 1y -y - 5 = 11 Combine like terms; -2y + 1y = (-2 + 1)y or -y. -y - 5 + 5 = 11 + 5 Add 5 to each side. -y = 16 Simplify. -1y 16 _ =_ -y = -1y; divide each side by -1. -1 -1 y = -16 Simplify. The solution is –16. Check -2y + y - 5 = 11 Write the equation. -2(-16) + (-16) - 5 11 Replace y with -16. 32 + (-16) - 5 11 11 = 11 ✓ Multiply. The statement is true. Solve each equation. Check your solution. g. x + 4x = 45 Examples 1–3 (pp. 422–423) h. 10 = 2a + 13 - a i. -3 = 6 - 5w + _w 5 2 Solve each equation. Check your solution. 1. 6x + 5 = 29 2. -2 = 9m - 11 3. 10 = _ + 3 4. _x - 5 = 7 5. 3 - 5y = -37 6. _ - 4 = 3 2 3 c -2 a 4 Example 3 7. ELECTRONICS Mr. Sampson bought a home theater system. The total cost of (p. 423) the system was $816, and he pays $34 a month on the balance. The current balance owed is $272. Solve the equation 272 = 816 - 34m to determine the number of monthly payments Mr. Sampson has made. Example 4 (p. 424) Solve each equation. Check your solution. 8. 6k - 10k = 16 9. 5d + 4 - 6d = 11 10. 1 = 4_ - 2p + _p 1 2 10 3 11. MOVIES Cassidy went to the movies with some of her friends. The tickets cost $6.50 apiece, and each person received a $1.75 student discount. The total amount paid for all the tickets was $33.25. Solve the equation 33.25 = 6.50p - 1.75p to determine the number of people who went to the movies. 424 Chapter 8 Algebra: More Equations and Inequalities (/-%7/2+ (%,0 For Exercises 12–19, 24, 25 20–23 26–33 See Examples 1, 2 3 4 Solve each equation. Check your solution. 12. 2h + 9 = 21 13. 11 = 2b + 17 14. 5 = 4a - 7 15. -17 = 6p - 5 16. 2g - 3 = -19 17. 16 = 5x - 9 g 18. 13 = _ + 4 3 y 19. 5 + _ = -3 8 1 _ 22. - x - 7 = -11 2 21. 13 - 3d = -8 20. 3 - 8c = 35 23. 15 - _ = 28 w 4 24. SCHOOL TRIP At an amusement park, each student is given $19 for food. This covers the cost of 2 meals at x dollars each plus $7 worth of snacks. Solve 2x + 7 = 19 to find how much money the school expects each student will spend per meal. 25. SHOPPING Suppose you receive a $75 online gift to your favorite music site. You want to purchase some CDs that cost $14 each. There will be a $5 shipping and handling fee. Solve 14n + 5 = 75 to find the number of CDs you can purchase. Solve each equation. Check your solution. 26. 28 = 3m - 7m 27. y + 5y = 24 28. 3 - 6x + 8x = 9 29. -21 = 9a - 15 - 3a 30. 26 = g + 10 - 3g 31. 8x + 5 - x = -2 32. GAMES Brent had $26 when he went to the fair. After playing 5 games and then 2 more, he had $15.50 left. Solve 15.50 = 26 - 5p - 2p to find the price for each game. 33. SPORTS LaTasha paid $75 to join a summer golf program. The course where she plays charges $30 per round, but since she is a student, she receives a $10 discount per round. If LaTasha spent $375, use the equation 375 = 30g - 10g + 75 to find out how many rounds of golf LaTasha played. Solve each equation. Check your solution. 34. 4(x + 2) = 20 35. 6(w - 2) = 54 37. _ = 12 38. _ = -4 a-4 5 36. -4_ = _(t + 1) 6 2 5 5 6+z _ 39. = -2 10 n+3 8 14 ft 40. HOME IMPROVEMENT If Mr. Arenth wants to put new carpeting in the room shown, how many square feet should he order? 6c 8 ft 41. ANIMALS Solve 4x + 12 = 171. If x stands %842!02!#4)#% See pages 697, 715. for the number of animals in a pet store, can it be a solution? Explain. 5 3c ft 25 42. GEOMETRY Write an equation to Self-Check Quiz at ca.gr7math.com −− represent the length of AB. Then find the value of x. 13 x 2x A Lesson 8-2 Solving Two-Step Equations B 425 H.O.T. Problems 43. FIND THE ERROR Alexis and Tomás are solving the equation 2x + 7 = 16. Who is correct? Explain. 2x + 7 = 16 2x + 7 - 7 = 16 - 7 2x = 9 2x + 7 = 16 16 2x _ +7=_ 2 Alexis 9 2x _ =_ 2 2 x+7=8 x+7-7=8-7 x=1 2 x = 4.5 Tomás 44. CHALLENGE Solve (x + 5) 2 = 49. (Hint: There are two solutions.) */ -!4( Explain how you can use the work backward problem(*/ 83 *5*/( 45. solving strategy to solve a two-step equation. 46. What value of y makes the equation true? 47. What is the value of m if -6m + 4 = -32? _y - 7 = 3 F 6 4 2 G 4_ A 3 3 1 H 2_ 3 B 16 C 40 J D 84 -6 Use the Distributive Property to rewrite each expression. (Lesson 8-1) 48. 6(a + 6) 49. -3(x + 5) 50. (y - 8)4 52. MEASUREMENT The cylinders at the right are similar. 51. -8(p - 7) MM Find the surface area of the larger cylinder. Round to the nearest tenth. (Lesson 7-9) MM MM 53. MEASUREMENT If one leg of a right triangle is 5 feet and its hypotenuse is 13 feet, how long is the other leg? (Lesson 3-5) 54. Write 4.78 × 10 -4 in standard form. (Lesson 2-10) PREREQUISITE SKILL Write an algebraic equation for each verbal sentence. (Lesson 1-7) 55. A number increased by 5 is 17. 426 Chapter 8 Algebra: More Equations and Inequalities (l)RubberBall/Alamy Images, (r)CORBIS 56. The quotient of a number and 2 is -2. 8-3 Writing Two-Step Equations Main IDEA Write two-step equations that represent real-life situations. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A. HOME ENTERTAINMENT Your parents offer to loan you the money to buy a $600 sound system. You give them $125 as a down payment and agree to make monthly payments of $25 until you have repaid the loan. Payments Amount Paid 0 125 + 25(0) = $125 1 125 + 25(1) = $150 1. Let n represent the number of 2 125 + 25(2) = $175 3 125 + 25(3) = $200 payments. Write an expression that represents the amount of the loan paid after n payments. 2. Write and solve an equation to find the number of payments you will have to make in order to pay off your loan. 3. What type of equation did you write for Exercise 2? Explain your reasoning. In Chapter 1, you learned how to write verbal sentences as one-step equations. Some verbal sentences translate to two-step equations. Words The sum of 125 and 25 times a number is 600. Variable Let n represent the number. Equation 125 + 25n = 600 Translate Sentences into Equations Translate each sentence into an equation. Sentence Equation 1 Eight less than three times a number is -23. 3n - 8 = -23 2 Thirteen is 7 more than twice a number. 13 = 2n + 7 3 The quotient of a number and 4, decreased by 1, n _ -1=5 is equal to 5. 4 Translate each sentence into an equation. a. Fifteen equals three more than six times a number. b. If 10 is increased by the quotient of a number and 6, the result is 5. c. The difference between 12 and twice a number is 18. Extra Examples at ca.gr7math.com Lesson 8-3 Writing Two-Step Equations 427 4 FUND-RAISING Your Class Council needs $600. With only $210 in the treasury, they decide to raise the rest by selling donuts for a profit of $1.50 per dozen. How many dozen will they need to sell? Treasury amount Words plus 1.50 per dozen sold $600. Let d represent the number of dozens. Variable 210 Equation + 1.50 · 210 + 1.50d = 600 Real-World Career How Does a FundRaising Professional Use Math? Fund-raising professionals use equations to help set and meet fund-raising goals. equals 210 - 210 + 1.50d = 600 - 210 d = Write the equation. Subtract 210 from each side. 1.50d = 390 Simplify. 1.50d 390 _ =_ Divide each side by 1.50. 1.50 1.50 600 d = 260 They need to sell 260 dozen. 5 DINING You and your friend’s lunch totaled $19. Your lunch cost $3 For more information, go to ca.gr7math.com. more than your friend’s. How much was your friend’s lunch? Your friend’s lunch Words your lunch equals $19. Let f represent the cost of your friend’s lunch. Variable Look Back You can review writing equations in Lesson 1–7. plus Equation f f + f + 3 = 19 Write the equation. 2f + 3 = 19 Combine like terms. 2f + 3 - 3 = 19 - 3 + Simplify. 2f 16 _ =_ Divide each side by 2. 2 = 19 Subtract 3 from each side. 2f = 16 2 f+3 f=8 Your friend spent $8. d. METEOROLOGY Suppose the current temperature is 54°F. It is expected to rise 2°F each hour for the next several hours. In how many hours will the temperature be 78°F? e. MEASUREMENT The perimeter of a rectangle is 40 inches. The width is 8 inches shorter than the length. Write and solve an equation to find the dimensions of the rectangle. Personal Tutor at ca.gr7math.com 428 Chapter 8 Algebra: More Equations and Inequalities Jon Feingersch/CORBIS Examples 1–3 (p. 427) Translate each sentence into an equation. 1. One more than three times a number is 7. 2. Seven less than twice a number is -1. 3. The quotient of a number and 5, less 10, is 3. For Exercises 4 and 5, write and solve an equation to solve each problem. Example 4 (p. 428) unpaid overdue balance of $1.30, your new balance is $2.05. How much is the daily fine for an overdue book? Example 5 5. SHOPPING Marty paid $121 for shoes and clothes. He paid $45 more for (p. 428) clothes than he did for shoes. How much did Marty pay for the shoes? (/-%7/2+ (%,0 For Exercises 6–9 10–13 14, 15 4. BOOK FINES You return a book that is 5 days overdue. Including a previous See Examples 1–3 4 5 Translate each sentence into an equation. 6. Four less than five times a number is equal to 11. 7. Fifteen more than twice a number is 9. 8. Eight more than four times a number is -12. 9. Six less than seven times a number is equal to -20. For Exercises 10–15, write and solve an equation to solve each problem. 10. PERSONAL FITNESS Angelica joins a local gym called Fitness Solutions. If she sets aside $1,000 in her annual budget for gym costs, use the ad at the right to determine how many hours she can spend with a personal trainer. 11. VACATION While on vacation, you purchase 4 identical T-shirts for some friends and a watch for yourself, all for $75. You know that the watch cost $25. How much did each T-shirt cost? Annual Membership: $720 Personal Trainers Available ($35/h) 12. PHONE SERVICE A telephone company advertises long distance service for 7¢ per minute plus a monthly fee of $3.95. If your bill for one month was $12.63, find the number of minutes you used making long distance calls. 13. VIDEO GAMES You and two of your friends share the cost of renting a video game system for 5 nights. Each person also rents one video game for $6.33. If each person pays $11.33, what was the cost of renting the video game system? 14. MONUMENTS From ground level to the tip of the torch, the Statue of Liberty and its pedestal are 92.99 meters high. The pedestal is 0.89 meter higher than the statue. How high is the Statue of Liberty? Lesson 8-3 Writing Two-Step Equations 429 15. GEOMETRY Find the value of x in the x˚ parallelogram at the right. 134˚ 134˚ x˚ ANIMALS For Exercises 16 –18, use the information at the left. 16. The top speed of a peregrine falcon is 20 miles per hour less than three times the top speed of a cheetah. What is the cheetah’s top speed? 17. A sailfish can swim up to 1 mile per hour less than one fifth the top speed of a peregrine falcon. Find the top speed that a sailfish can swim. 18. The peregrine falcon can reach speeds about 14 miles per hour more than Real-World Link When diving, the peregrine falcon can reach speeds of up to 175 miles per hour. 7 times the speed of the fastest human. What is the approximate top speed of the fastest human? 19. BASKETBALL In a basketball game, 2 points are awarded for making a regular basket, and 1 point is awarded for making a foul shot. Emeril scored 21 points during one game. Three of those points were for foul shots. The rest were for regular goals. Find the number of regular baskets that Emeril made during the game. Source: Time for Kids Almanac 20. SKIING In aerial skiing competitions, the total judges’ score is multiplied by the jump’s degree of difficulty and then added to the skier’s current score to obtain their final score. After her second jump, Martin’s final score is 216.59. The degree of difficulty for Toshiro’s second jump is 4.45. What must the judges’ score for Toshiro’s jump be in order for her to tie Martin for first place? Skier Score Martin, S. 100.23 Toshiro, M. 105.34 Moseley, K. 93.99 Long, A. 87.50 Cruz, P. 80.63 Thompson, L. 75.23 21. ALGEBRA Three consecutive even integers can be represented by n, n + 2, and n + 4. If the sum of three consecutive even integers is 36, what are the integers? %842!02!#4)#% See pages 697, 715. JOBS For Exercises 22 and 23, use the following information. Hunter and Amado are each trying to save $600 for a summer trip. Hunter started with $150 and earns $7.50 per hour working at a grocery store. Amado has nothing saved, but he earns $12 per hour painting houses. 22. Make a conjecture about who will take longer to save enough money for the trip. Justify your reasoning. Self-Check Quiz at ca.gr7math.com H.O.T. Problems 23. Write and solve two equations to check your conjecture. 24. OPEN ENDED Write two different statements that translate into the same two-step equation. 25. CHALLENGE Student Council has $200 to divide among the top class finishers in a used toy drive. Second place will receive twice as much as third place. First place will receive $15 more than second place. Write and solve an equation to find how much each winning class will receive. 430 Chapter 8 Algebra: More Equations and Inequalities (l)Tim Fitzharris/Masterfile, (r)Cris Cole/Getty Images 26. SELECT A TECHNIQUE Sherrie bought 3 bottles of sports drink for $6.42. If the sales tax was $0.42, which technique would you use to determine the cost of each bottle of sports drink? Justify your selection. Then find the cost of each bottle of sports drink. mental math 27. estimation paper/pencil */ -!4( Write about a real-world situation that can be solved (*/ 83 *5*/( using a two-step equation. Then write the equation and solve the problem. 28. A company employs 72 workers. 29. Kimberly needs $45 to go to the It plans to increase the number of employees by 6 per month until it has twice its current workforce. Which equation can be used to determine m, the number of months it will take for the number of employees to double? amusement park. She has $13. She earns $8 per hour working at her job. The equation 8h + 13 = 45 shows this relationship. How many hours does Kimberly need to work to earn enough money to go to the park? A 6m + 72m = 144 F 8 B 2m + 72 = 144 G 7 C 2(6m + 72) = 144 H 6 D 6m + 72 = 144 J 4 Solve each equation. Check your solution. (Lesson 8-2) 30. 5x + 2 = 17 31. -7b + 13 = 27 32. -6 = _ + 1 33. -15 = -4p + 9 36. 7a - 7a - 9 37. 3 - 4y + 9y n 8 Simplify each expression. (Lesson 8-1) 34. 5x + 6 - x 35. 8 - 3n + 3n 38. GEOMETRY Copy the figure at the right onto graph paper. Then draw the image of the figure after it is translated 4 units left and 2 units up. (Lesson 6-7) 39. Find the percent of change from 32 feet to 79 feet. Round to the nearest tenth if necessary. Then state whether the percent of change is a percent of increase or a percent of decrease. (Lesson 5-8) PREREQUISITE SKILL Simplify each expression. (Lesson 8-1) 40. 2x - 8 + 2x 41. -5n + 7 + 5n 42. 8p -3 + 3 43. -6 - 15a + 6 Lesson 8-3 Writing Two-Step Equations 431 Explore 8-4 Main IDEA Algebra Lab Equations with Variables on Each Side You can use algebra tiles to solve equations that have variables on each side of the equation. Solve equations with variables on each side using algebra tiles. Standard 7AF4.1 Solve twostep linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Interactive Lab ca.gr7math.com 1 Use algebra tiles to solve 3x + 1 = x + 5. 1 x x x 1 3x 1 x x x x x 1 1 1 1 1 Remove the same number of 1-tiles from each side of the mat until the x-tiles are by themselves on one side. 1 x 1 1 1 1 2x 1 1 51 x 1 1 1 1 x 2x Remove the same number of x-tiles from each side of the mat until there are x-tiles on the only one side. xx5 1 x 1 Model the equation. 1 3x x 1 1 x5 1 x 1 Separate the tiles into two equal groups. 4 Therefore, x = 2. Since 3(2) + 1 = 2 + 5, the solution is correct. Use algebra tiles to solve each equation. a. x + 2 = 2x + 1 b. 2x + 7 = 3x + 4 c. 2x - 5 = x - 7 d. 8 + x = 3x e. 4x = x - 6 f. 2x - 8 = 4x - 2 ANALYZE THE RESULTS 1. Identify the property of equality that allows you to remove a 1-tile or -1-tile from each side of an equation mat. 2. Explain why you can remove an x-tile from each side of the mat. 432 Chapter 8 Algebra: More Equations and Inequalities 2 Use algebra tiles to solve x - 4 = 2x + 2. 1 1 x 1 1 1 x4 x x 1 1 xx4 x x 1 Remove the same number of x-tiles from each side of the mat until there is an x-tile by itself on one side. 1 2x x 2 1 1 1 1 1 1 Model the equation. 2x 2 1 1 x 1 x 1 1 1 1 To isolate the x-tile, it is not possible to remove the same number of 1-tiles from each side of the mat. Add two -1-tiles to each side of the mat. 4 (2) x 2 (2) 1 1 1 1 1 1 6 x 1 1 1 1 Remove the zero pairs from the right side. There are six -1-tiles on the left side of the mat. The x-tile is isolated on the right side of the mat. x Therefore, x = -6. Since -6 - 4 = 2(-6) + 2, the solution is correct. Use algebra tiles to solve each equation. g. x + 6 = 3x - 2 h. 3x + 3 = x - 5 i. 2x + 1 = x - 7 j. x - 4 = 2x + 5 k. 3x - 2 = 2x + 3 l. 2x + 5 = 4x - 1 ANALYZE THE RESULTS 3. Solve x + 4 = 3x - 4 by removing 1-tiles first. Then solve the equation by removing x-tiles first. Does it matter whether you remove x-tiles or 1-tiles first? Is one way more convenient? Explain. 4. MAKE A CONJECTURE In the set of algebra tiles, -x is represented by x . Explain how you could use -x-tiles and other algebra tiles to solve -3x + 4 = -2x - 1. Explore 8-4 Algebra Lab: Equations with Variables on Each Side 433 8-4 Solving Equations with Variables on Each Side Main IDEA Solve equations with variables on each side. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A. Standard 7AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. SPORTS You and your friend are having a race. You give your friend a 15-meter head start. During the race, you average 6 meters per second and your friend averages 5 meters per second. Time (s) Friend’s Distance (m) Your Distance (m) 0 15 + 5(0) = 15 6(0) = 0 1 15 + 5(1) = 20 6(1) = 6 2 15 + 5(2) = 25 6(2) = 12 3 15 + 5(3) = 30 6(3) = 18 1. Copy the table. Continue filling in rows to find how long it will take you to catch up to your friend. 2. Write an expression for your distance after x seconds. 3. Write an expression for your friend’s distance after x seconds. 4. What is true about the distances you and your friend have covered when you catch up to your friend? 5. Write an equation that could be used to find how long it will take for you to catch up to your friend. Some equations, like 15 + 5x = 6x, have variables on each side of the equals sign. To solve these equations, use the Addition or Subtraction Property of Equality to write an equivalent equation with the variables on one side of the equals sign. Then solve the equation. Equations with Variables on Each Side 1 Solve 15 + 5x = 6x. Check your solution. 15 + 5x = 6x Write the equation. 15 + 5x - 5x = 6x - 5x Subtract 5x from each side. 15 = x Simplify by combining like terms. Subtract 5x from the left side of the equation to isolate the variable. Subtract 5x from the right side of the equation to keep it balanced. To check your solution, replace x with 15 in the original equation. Check 15 + 5x = 6x Write the original equation. 15 + 5(15) 6(15) 90 = 90 Replace x with 15. ✓ The sentence is true. The solution is 15. 434 Chapter 8 Algebra: More Equations and Inequalities Westlight Stock/OZ Production/CORBIS Extra Examples at ca.gr7math.com 2 Solve 6n - 1 = 4n - 5. 6n - 1 = 4n - 5 Write the equation. 6n - 4n - 1 = 4n - 4n – 5 Subtract 4n from each side. 2n - 1 = -5 Simplify. 2n - 1 + 1 = -5 + 1 Add 1 to each side. 2n = -4 Simplify. n = -2 Mentally divide each side by 2. Solve each equation. Check your solution. a. 8a = 5a + 21 b. 3x - 7 = 8x + 23 c. 7g - 12 = 3 + _g 7 3 3 CELL PHONES A cellular phone provider charges $24.95 per month plus $0.10 per minute for calls. Another cellular provider charges $19.95 per month plus $0.20 per minute for calls. For how many minutes of calls is the monthly cost of both providers the same? Words $24.95 per month plus $0.10 per minute equals $19.95 per month plus $0.20 per minute Variable Let m represent the minutes. Equation 24.95 + 0.10m = 19.95 + 0.20m 24.95 + 0.10m = 19.95 + 0.20m 24.95 + 0.10m - 0.10m = 19.95 + 0.20m - 0.10m 24.95 = 19.95 + 0.10m 24.95 - 19.95 = 19.95 - 19.95 + 0.10m 5 = 0.10m 5 0.10m _ =_ 0.10 0.10 50 = m Write the equation. Subtract 0.10m from each side. Subtract 19.95 from each side. Divide each side by 0.10. Check for Reasonableness $25 + 50($0.10) = $30 $20 + 50($0.20) = $30 The monthly cost is the same for 50 minutes of calls. Real-World Link Congress established the first official United States flag on June 14, 1777. Source: firstgov.gov d. FLAGS The length of a flag is 0.3 foot less than twice its width. If 17.4 feet of gold fringe is used along the perimeter of the flag, find the dimensions of the flag. Personal Tutor at ca.gr7math.com Lesson 8-4 Solving Equations with Variables on Each Side MPI/Getty Images 435 Examples 1, 2 (pp. 434–435) Example 3 Solve each equation. Check your solution. 1. 5n + 9 = 2n 2. 3k + 14 = k 3. 10x = 3x - 28 4. 7y - 8 = 6y + 1 5. 2a + 21 = 8a - 9 6. -4p - 3 = 2 + p 7. CAR RENTAL EZ Car Rental charges $40 a day plus $0.25 per mile. Ace (p. 435) (/-%7/2+ (%,0 For Exercises 8–11 12–19 20–23 See Examples 1 2 3 Rent-A-Car charges $25 a day plus $0.45 per mile. What number of miles results in the same cost for one day? Solve each equation. Check your solution. 8. 7a + 10 = 2a 9. 11x = 24 + 8x 10. 9g - 14 = 2g 11. m - 18 = 3m 12. 5p + 2 = 4p - 1 13. 8y - 3 = 6y + 17 14. 15 - 3n = n - 1 15. 3 - 10b = 2b - 9 16. -6f + 13 = 2f - 11 17. 2z - 31 = -9z + 24 18. 2.5h - 15 = 4h 19. 21.6 - d = 5d Define a variable, write an equation, and solve to find each number. 20. Eighteen less than three times a number is twice the number. 21. Eleven more than four times a number equals the number less 7. For Exercises 22 and 23, write and solve an equation to solve each problem. 22. MOVIES For an annual membership fee of $30, you can join a movie club that will allow you to purchase tickets for $5.50 each at your local theater. If the theater in your area charges $8 for movie tickets, determine how many movie tickets you will have to buy through the movie club for the cost to equal that of buying tickets at the regular price. 23. FOOD DRIVES The seventh graders at your school have collected 345 cans for the canned food drive and are averaging 115 cans per day. The eighth graders have collected 255 cans, but vow to win the contest by collecting an average of 130 cans per day. If both grades continue collecting at these rates, after how many days will the number of cans they have collected be equal? MEASUREMENT Write an equation to find the value of x so that each pair of polygons has the same perimeter. Then solve. 24. %842!02!#4)#% x4 x1 x2 x3 12x 25. 12x 12x x5 12x x7 x 10 12x See pages 697, 715. 26. MEASUREMENT Write and solve an equation to Self-Check Quiz at ca.gr7math.com 436 find the perimeter and area of the square at the right. Chapter 8 Algebra: More Equations and Inequalities 6x 9 2x 8 4x 2 27. CRAFT FAIRS The Art Club is selling mugs at a local craft fair. They must pay $5 for a booth plus 10% of their sales. It costs $8 in materials to make each mug. The club sells each mug for $10. Write and solve an equation to find how many mugs they must sell to break even. H.O.T. Problems 28. OPEN ENDED Write an equation that has variables on each side with a solution of 5. 3x 3 29. CHALLENGE Find the area of the parallelogram at the right. 30. x3 */ -!4( Explain how to solve (*/ 83 *5*/( 5x 1 the equation 1 - 3x = 5x - 7. 31. Carpet cleaner A charges $28.25 plus $18 a room. Carpet cleaner B charges $19.85 plus $32 a room. Which equation can be used to find the number of rooms for which the total cost of both carpet cleaners is the same? 32. Find the value of x so that the polygons have the same perimeter. 2x x4 2x 2x 2x 2x x4 x1 A 28.25x + 18 = 19.85x + 32 2x B 28.25 + 32x = 19.85 + 18x F 4 H 2 C 28.25 + 18x = 19.85 + 32x G 3 J 1 D (28.25 + 18)x = (19.85 + 32)x 33. SHOPPING Marisa bought 4 paperback books, each at the same price. The tax on her purchase was $2.35, and the total was $34.15. Write and solve an equation to find the price of each book. (Lesson 8-3) ALGEBRA Solve each equation. (Lesson 8-2) 34. 9 + 5y = 19 35. -6 = 4 + 2x 36. 8 - k = 17 37. 2 = 18 - 4d 38. SAVINGS Shala’s savings account earned $4.57 in 6 months at a simple interest rate of 4.75%. How much was in her account at the beginning of that 6-month period? (Lesson 5-9) 39. PREREQUISITE SKILL Enrique has $37.50 to spend at the cinema. A drink costs $1.75, popcorn costs $2.25, and tickets cost $8.50. Use the work backward strategy to determine how many friends he can invite to go with him if he pays for himself and for his friends. (Lesson 1-8) Lesson 8-4 Solving Equations with Variables on Each Side 437 8-5 Problem-Solving Investigation MAIN IDEA: Guess and check to solve problems. Standard 7MR2.8 Make precise calculations and check the validity of the results from the context of the problem. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A. e-Mail: GUESS AND CHECK YOUR MISSION: Solve the problem by guessing and checking the solution. THE PROBLEM: Find the number of tickets collected at the Balloon Pop and the Bean-Bag Toss. Missy: We collected 150 tickets during the Fall Carnival. It took 3 tickets to play the Bean-Bag Toss and 2 tickets to play the Balloon Pop. Ten more games were played at the Bean-Bag Toss booth than at the Balloon Pop. EXPLORE PLAN SOLVE CHECK The Bean-Bag Toss was 3 tickets, and the Balloon Pop was 2 tickets. The number of games played at the Bean-Bag Toss was 10 more than at the Balloon Pop. Make a systematic guess and check to see if it is correct. Find the combination that gives 150 total tickets. In the list, p is the number of Balloon Pop games and t is the number of Bean-Bag Toss games. p t 2p + 3t Check 12 22 2(12) + 3(22) = 90 too low 30 40 2(30) + 3(40) = 180 too high 27 37 2(27) + 3(37) = 165 still too high 24 34 2(24) + 3(34) = 150 correct So, 2(24) or 48 tickets were from the Balloon Pop and 3(34) or 102 tickets were from the Bean-Bag Toss. Thirty-four Balloon Pop games is 10 more than 24 Bean-Bag Toss games. Since 48 tickets plus 102 tickets is 150 tickets, the guess is correct. 1. Explain why it is important to make a systematic, organized list of your guesses and their results when using the guess and check strategy. */ -!4( Write a problem that could be solved by guessing (*/ 83 *5*/( 2. and checking. Then write the steps you would take to find the solution. 438 Chapter 8 Algebra: More Equations and Inequalities John Evans 9. RECREATION During a routine, ballet dancers For Exercises 3–5, solve using the guess and check strategy. 3. NUMBER THEORY A number is squared, and the result is 576. Find the number. are evenly spaced in a circle. If the sixth person is directly opposite the sixteenth person, how many people are in the circle? ANALYZE TABLES For Exercises 10 and 11, use the following information. 4. MONEY MATTERS Dominic has exactly $2 in quarters, dimes, and nickels. If he has 13 coins, how many of each coin does he have? 5. GIFTS At a park souvenir shop, a mug costs The school cafeteria surveyed 34 students about their dessert preference. The results are listed below. $3, and a pin costs $2. Chase bought either a mug or a pin for each of his 11 friends. If he spent $30 on these gifts and bought at least one of each type of souvenir, how many of each did he buy? Number of Students Preference of Students 25 apples 20 oranges 15 bananas 2 all three 1 no fruit 15 apples or oranges 8 bananas or apples 3 oranges only Use any strategy to solve Exercises 6–9. Some strategies are shown below. G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s m. • Draw a diagra 10. How many students prefer only bananas? • Make a table. 11. How many do not prefer apples? eck. • Guess and ch 6. MEASUREMENT The length of the rectangle below is longer than its width w. List the possible whole number dimensions for the rectangle, and identify the dimensions that give the smallest perimeter. For Exercises 12–14, select the appropriate operation(s) to solve the problem. Justify your selection(s) and solve the problem. 12. TECHNOLOGY The average Internet user A ⫽ 84 in 2 1 spends 6_ hours online each week. What w 2 percent of the week does the average user spend online? ᐉ 7. DINING The cost of your meal is $8.25. If you want to leave a 15% tip, would it be more reasonable to expect the tip to be about $1.25 or about $1.50? 8. DESIGN Edu-Toys is designing a new package to hold a set of 30 alphabet blocks like the one shown. Give two possible dimensions for the box. 13. READING Terrence is reading a 255-page book for his book report. He needs to read twice as many pages as he has already read to finish the book. How many pages has he read so far? 14. NUMBER SENSE Find the product of 2 in. 2 in. 2 in. 1 1 1 1 1 1 1-_ ,1-_ ,1-_ ,1-_ , ..., 1 - _ ,1-_ , 2 2 1 and 1 - _ . 3 4 48 49 50 Lesson 8-5 Problem-Solving Investigation: Guess and Check 439 CH APTER 8 Mid-Chapter Quiz Lessons 8-1 through 8-5 Use the Distributive Property to rewrite each expression. (Lesson 8-1) Translate each sentence into an equation. Then find each number. (Lesson 8-3) 1. 3(x + 2) 2. -2(a - 3) 16. Nine more than the quotient of a number 3. 5(3c - 7) 4. -4(2n + 3) and 3 is 14. 17. The quotient of a number and -7, less 4, is -11. Simplify each expression. (Lesson 8-1) 5. 2a - 13a 6. 6b + 5 - 6b 7. 2m + 5 - 8m 8. 7x + 2 - 8x + 5 18. The difference between three times a number and 10 is 17. 19. The difference between twice a number 9. Identify the terms, like terms, coefficients, and constants in the expression 5 - 4x + x - 3. (Lesson 8-1) Solve each equation. Check your solution. (Lesson 8-2) 10. 3m + 5 = 14 11. -2k + 7 = -3 12. 11 = _a + 2 13. -15 = -7 - p 1 3 and 13 is -21. 20. MOVING A rental company charges $52 per day and $0.32 per mile to rent a moving van. Ms. Misel was charged $202.40 for a 3-day rental. How many miles did she drive? (Lesson 8-3) Solve each equation. Check your solution. STANDARDS PRACTICE A diagram of a room is shown below. 14. (Lesson 8-4) 21. 3x + 7 = 2x 22. 7p - 6 = 4p w 23. 3y - 5 = 5y + 7 24. 4m + 7 = -3m + 49 2w 3 If the perimeter of the room is 78 feet, find its width. (Lesson 8-2) A 12 ft 25. MEASUREMENT Write and solve an equation to find the value of x so that the polygons have the same perimeter. (Lesson 8-4) 2x B 15 ft C 25 ft x2 x2 4x 2x D 27 ft 4x 1 3x 2 15. EXERCISE Brandi rode her bike the same distance on Tuesday and Thursday, and 20 miles on Saturday for a total of 50 miles for the week. Solve the equation 2m + 20 = 50 to find the distance Brandi rode on Tuesday and Thursday. (Lesson 8-2) 440 Chapter 8 Algebra: More Equations and Inequalities 26. MONEY Marlisa has exactly $61 in one- dollar, five-dollar, and ten-dollar bills. If she has 14 bills in all, how many of each bill does she have? (Lesson 8-5) 8-6 Inequalities Main IDEA Write and graph inequalities. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A. SIGNS The top sign indicates that trucks more than 10 feet 6 inches tall cannot pass. The other sign indicates that a speed of 45 miles per hour or less is legal. 1. Name three truck heights that can safely pass on a road where the first sign is posted. Can a truck that is 10 feet 6 inches tall pass? Explain. 2. Name three speeds that are legal according to the second sign. Is a car traveling at 45 miles per hour driving at a legal speed? Explain. In Chapter 1, you learned that a mathematical sentence that contains > or < is called an inequality. When used to compare a variable and a number, inequalities can describe a range of values. Write Inequalities with < or > Write an inequality for each sentence. 1 SAFETY A package must weigh less than 80 pounds. Let w = package’s weight. w < 80 2 AGE You must be over 55 years old to join. Let a = person’s age. a > 55 a. ROLLER COASTERS Riders must be taller than 48 inches. b. SPORTS Members of a swim team must be under 15 years of age. READING Math Inequality Symbols ≤ less than or equal to ≥ greater than or equal to The symbols ≤ and ≥ combine < and > with part of the equals sign. Write Inequalities with ≤ or ≥ Write an inequality for each sentence. 3 VOTING You must be 18 years of age or older to vote. Let a = person’s age. a ≥ 18 4 DRIVING Your speed must be 65 miles per hour or less. Let s = car’s speed. s ≤ 65 c. CARS A toddler must weigh at least 40 pounds to use a booster seat. d. TRAVEL A fuel tank holds at most 16 gallons of gasoline. Extra Examples at ca.gr7math.com Doug Martin Lesson 8-6 Inequalities 441 Inequalities • is less than • is fewer than Words • is greater than • is more than • exceeds < Symbols • is less than or equal to • is no more than • is at most • is greater than or equal to • is no less than • is at least ≤ ≥ > Inequalities with variables are open sentences. When the variable is replaced with a number, the inequality becomes either true or false. Determine the Truth of an Inequality For the given value, state whether each inequality is true or false. 6 10 ≤ 7 - x, x = -3 5 a + 2 > 8, a = 5 Symbols Read 7 ≯ 8 as 7 is not greater than 8. a + 2 > 8 Write the inequality. 10 ≤ 7 - x Write the inequality. 5 + 2 8 Replace a with 5. 10 7 - (-3) Replace x with -3. 10 ≤ 10 Simplify. 7 ≯ 8 Simplify. Since 7 is not greater than 8, 7 > 8 is false. While 10 < 10 is false, 10 = 10 is true, so 10 ≤ 10 is true. For the given value, state whether each inequality is true or false. e. n - 6 < 15, n = 18 f. -3p ≥ 24, p = 8 g. -2 > 5y - 7, y = 1 Inequalities can be graphed on a number line. Since it is impossible to show all the values that make an inequality true, an open or closed circle is used to indicate where these values begin, and an arrow to the left or to the right is used to show that they continue in the indicated direction. Graph an Inequality Graph each inequality on a number line. 8 n≥3 7 n<3 Place an open circle at 3. Then draw a line and an arrow to the left. 1 2 3 4 5 The open circle means the number 3 is not included in the graph. Place a closed circle at 3. Then draw a line and an arrow to the right. 1 2 3 4 5 The closed circle means the number 3 is included in the graph. Graph each inequality on a number line. h. x > 2 i. x < 1 Personal Tutor at ca.gr7math.com 442 Chapter 8 Algebra: More Equations and Inequalities j. x ≤ 5 k. x ≥ -4 Examples 1–4 (p. 441) Write an inequality for each sentence. 1. RESTAURANTS Children under the age of 6 eat free. 2. TESTING A maximum of 45 minutes is given to complete section A. Examples 5, 6 (p. 442) Examples 7, 8 (p. 442) (/-%7/2+ (%,0 For Exercises 10–15 16–21 22–29 See Examples 1–4 5, 6 7, 8 For the given value, state whether each inequality is true or false. 3. x - 11 < 9, x = 20 4. 42 ≥ 6a, a = 8 5. n _ + 1 ≤ 6; n = 15 3 Graph each inequality on a number line. 6. n > 4 7. p ≤ 2 8. x ≥ 0 9. a < 7 Write an inequality for each sentence. 10. MOVIES Children under 13 are not permitted without an adult. 11. SHOPPING You must spend more than $100 to receive a discount. 12. ELEVATORS An elevator’s maximum load is 3,400 pounds. 13. FITNESS You must run at least 4 laps around the track. 14. GRADES A grade of no less than 70 is considered passing. 15. MONEY The cost can be no more than $25. For the given value, state whether each inequality is true or false. 16. 12 + a < 20, a = 9 17. 15 - k > 6, k = 8 18. -3y < 21; y = 8 19. 32 ≤ 2x, x = 16 20. _ ≥ 5, n = 12 21. _ > 9, x = -2 -18 x n 4 Graph each inequality on a number line. 22. x > 6 23. a > 0 24. y < 8 25. h < 2 26. w ≤ 3 27. p ≥ 7 28. 1 ≤ n 29. 4 ≥ d Jgfikj@eali`\j SPORTS For Exercises 30 –33, use the graph that shows the number of children ages 5 –14 treated recently in U.S. emergency rooms. "ICYCLING "ASKETBALL 30. In which sport(s) were more 31. In which sport(s) were at least 75,000 children injured? 32. Of the sports listed, which have fewer than 100,000 injuries? %842!02!#4)#% 33. Write an inequality comparing See pages 698, 715. the number treated for soccerrelated injuries with those Self-Check Quiz at treated for football-related ca.gr7math.com injuries. 3PORT than 150,000 children injured? &OOTBALL "ASEBALL 3OFTBALL 3OCCER 3KATEBOARDING .UMBER OF 4REATED )NJURIES THOUSANDS Source: Children’s Hospital of Pittsburgh Lesson 8-6 Inequalities 443 H.O.T. Problems 34. FIND THE ERROR Valerie and Diego are writing an inequality for the expression at least 2 hours of homework. Who is correct? Explain. h≤2 h≥2 Valerie Diego 35. CHALLENGE Determine whether the following statement is always, sometimes, or never true. Explain your reasoning. If x is a real number, then x ≥ x. */ -!4( If a < b and b < c, what is true about the relationship (*/ 83 *5*/( 36. between a and c? Explain your reasoning and give examples using both positive and negative values for a, b, and c. 37. Conner can spend no more than 38. What inequality is graphed below? 4 hours at the swimming pool today. Which graph represents the time that Conner can spend at the pool? F x > -3 A G x ≥ -3 H x < -3 B J x ≤ -3 C D 39. SOUVENIRS The Green Gables gift shop sells regular postcards in packages of 5 and large postcards in packages of 3. If Román bought 16 postcards, how many packages of each did he buy? (Lesson 8-5) 40. ALGEBRA Suppose you can rent a car for either $35 a day plus $0.40 a mile or for $20 a day plus $0.55 per mile. Write and solve an equation to find the number of miles that result in the same cost for one day. (Lesson 8-4) PREREQUISITE SKILL Solve each equation. (Lesson 1-8) 41. y + 15 = 31 444 42. n + 4 = -7 Chapter 8 Algebra: More Equations and Inequalities (l)Robin Lynne Gibson/Getty Images, (r)Richard Hutchings/Photo Researchers 43. a - 8 = 25 44. -12 = x - 3 8-7 Solving Inequalities by Adding or Subtracting Main IDEA v Solve inequalities by using the Addition or Subtraction Properties of Inequality. FAMILY The table shows the age of each member of Victoria’s family. Notice that Victoria is younger than her brother Greg, since 13 < 16. Will this be true 10 years from now? Mom 41 Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A. Standard 7AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. 1. Add 10 to each side of the inequality 13 < 16. Greg 16 Write the resulting inequality and decide whether it is true or false. Victoria 13 Family Member Age Dad 43 2. Was Victoria’s dad younger or older than Victoria’s mom 13 years ago? Explain your reasoning using an inequality. The examples above demonstrate properties of inequality. These properties are also true for a ≥ b and a ≤ b. +%9 #/.#%04 Words Symbols Properties of Inequality When you add or subtract the same number from each side of an inequality, the inequality remains true. For all numbers a, b, and c, 1. if a > b, then a + c > b + c and a - c > b - c. 2. if a < b, then a + c < b + c and a - c < b - c. 2 > -3 2 + 5 > -3 + 5 ✓ 7>2 Examples 3<8 3-4<8-4 -1 < 4 ✓ Solving an inequality means finding the values of the variable that make the inequality true. Solving Inequalities 1 Solve n - 8 < 15. Check your solution. n - 8 < 15 Write the inequality. n - 8 + 8 < 15 + 8 Add 8 to each side. n < 23 Check Simplify. n - 8 < 15 Write the inequality. 22 - 8 15 Replace n with a number less than 23, such as 22. 14 < 15 ✓ This statement is true. The solution is n < 23. Extra Examples at ca.gr7math.com John Evans Lesson 8-7 Solving Inequalities by Adding or Subtracting 445 2 Solve -4 ≥ a + 7. Check your solution. Equivalent Inequalities If -11 is greater than or equal to a, then a is less than or equal to -11. -4 ≥ a + 7 Write the inequality. -4 - 7 ≥ a + 7 - 7 Subtract 7 from each side. -11 ≥ a or a ≤ - 11 Check Simplify. Replace a in the original equality with -11 and then with a number less than -11. The solution is a ≤ -11. Solve each inequality. b. n + _ ≥ 4 1 2 a. t + 3 > 12 c. y - 1.5 < 2 3 A manatee can weigh up to 1,300 pounds. Suppose a manatee currently weighs 968 pounds. Which inequality indicates how much more weight this manatee might gain? A w < 332 B w > 332 C w ≤ 332 D w ≥ 332 Read the Item Real-World Link An average manatee eats about 73 pounds of seagrass and other vegetation each day. Source: Kids Discover The phrase up to means less than or equal to. Solve the Item Let w = weight gained by the manatee. Estimate 1,300 - 1,000 = 300 Manatee’s current weight plus weight gained is less than or equal to 1,300 pounds. 968 + w ≤ 1,300 968 + w ≤ 1,300 968 - 968 + w ≤ 1,300 - 968 w ≤ 332 Write the inequality. Subtract 968 from each side. Simplify. Check for Reasonableness 332 ≈ 300 ✓ The answer is C. d. A tornado is classified using the Fujita Tornado Damage Scale, the F-Scale. An F1 tornado has wind speeds that are at least 73 miles per hour. An F2 tornado has wind speeds that are at least 113 miles per hour. Which inequality indicates how much the winds of an F1 tornado need to increase so it becomes at least an F2 tornado? F x ≥ 40 G x < 40 Personal Tutor at ca.gr7math.com 446 Chapter 8 Algebra: More Equations and Inequalities Douglas Faulkner/Corbis H x ≤ 40 J x > 40 Examples 1–2 (pp. 445–446) Solve each inequality. Check your solution. 1. b + 5 > 9 2. 12 + n ≤ 4 3. x - 4 < 10 4. Write an inequality for three more than a number is at most 15. Then solve. Example 3 (p. 446) 5. STANDARDS PRACTICE A certain city receives an average of 37 inches of rain per year, and there has been 13 inches of rain so far this year. Which inequality indicates how much more rainfall the city can get and stay at or within the average? A r < 24 (/-%7/2+ (%,0 For Exercises 6–27 28, 29 See Examples 1, 2 3 B r > 24 C r ≤ 24 D r ≥ 24 Solve each inequality. Check your solution. 6. 5 + x ≤ 18 7. 10 + n ≥ -2 8. -4 < k + 6 9. 3 < y + 8 10. c + 10 < 9 11. g - 4 ≥ 13 12. -2 < b - 6 13. s - 12 ≤ -5 14. t - 3 < -9 15. -10 ≥ x + 6 16. a - 3 ≤ 5 17. -11 > g - 4 18. 2 + m ≥ 3.5 19. q + 0.8 ≤ -0.5 20. v - 6 > 2.7 21. p - 4.8 > -6 22. d - _ ≤ _ 23. 5 > f + 1 _ 2 3 1 2 1 4 Write an inequality and solve each problem. 24. Five more than a number is at least 13. 25. The difference between a number and 11 is less than 8. 26. Nine less than a number is more than 4. 27. The sum of a number and 17 is no more than 6. 28. HEALTH Suppose Mr. Herr has a temperature of 99.2°. Write and solve an inequality to find how much his temperature must increase before he is considered to have a high fever. Range of Human Temperature Below Normal Low-Grade Fever 98.6 High Fever 101 Body Temperature (F) 29. HEALTH Hypothermia occurs when a person’s body temperature falls below 95°F. Write and solve an inequality that describes how much lower the body temperature of a person with hypothermia will be than a person with a normal body temperature of 98.6°F. 30. INSECTS There are more than 350,000 species of beetles. Suppose your local science museum has a collection representing 320 of these species. Write and solve an inequality to find how many beetle species are not shown in this collection. Interpret the solution. Lesson 8-7 Solving Inequalities by Adding or Subtracting 447 %842!02!#4)#% 31. MEASUREMENT The base of the rectangle is greater than its height. Write and solve an inequality to find the possible values of x. Interpret the solution. See pages 698, 715. Self-Check Quiz at x 3 cm 15 cm ca.gr7math.com 32. H.O.T. Problems FIND THE DATA Refer to the California Data File on pages 16–19. Choose some data and write a real-world problem in which you would need to solve an inequality using addition or subtraction. CHALLENGE Determine whether each equation or inequality has no solution, one solution, or more than one solution. 33. y - y = 0 34. x + 4 = 9 35. x + 4 > 9 36. y > y + 1 37. OPEN ENDED Write an inequality that has a solution of n > 5 and can be solved by using the Addition or Subtraction Property of Equality. 38. */ -!4( Explain how solving an inequality by using (*/ 83 *5*/( subtraction is similar to solving an equation by using subtraction. 39. Adriana has $30 to spend on food and 40. If x - 6 > 17, then x could be which of rides at a carnival. She has already spent $12 on food. Which inequality represents how much money she can spend on rides? A m < 18 C m > 18 B m ≤ 18 D m ≥ 18 the following values? F 11 G 22 H 23 J 24 For the given value, state whether each inequality is true or false. (Lesson 8-5) 41. 18 - n > 4, n = 11 42. 13 + x < 21, x = 8 43. 34 ≤ 5p, p = 7 44. ALGEBRA A family membership to the zoo costs $75 per year and covers admission, but not the $3 parking fee. Regular admission is $7 per person. Write and solve an equation to determine how many trips to the zoo a family of four could make for the cost of a membership to equal regular admission. (Lesson 8-4) PREREQUISITE SKILL Solve each equation. (Lesson 1-9) 45. 3y = -15 448 46. -18 = -2a Chapter 8 Algebra: More Equations and Inequalities 47. _ = 12 w 4 8-8 Solving Inequalities by Multiplying or Dividing Main IDEA Solve inequalities by using the Multiplication or Division Properties of Inequality. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A. Standard 7AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. SHOPPING The table shows the prices of the same brand name of shoes at a sports store. Notice that walking shoes cost less than cross-training shoes since 80 < 150. Will this inequality be true if the store sells both pairs of shoes at half price? Shoe Style Regular Price ($) athletic sandal 60 walking 80 running 100 basketball 120 1. Divide each side of the inequality cross training 150 80 < 150 by 2. Write the resulting inequality and decide whether it is true or false. 2. Would the cost of three pairs of basketball shoes be greater or less than the cost of three pairs of running shoes all sold at the regular price? Explain your reasoning using an inequality. The examples above demonstrate additional properties of inequality. +%9 #/.#%04 Properties of Inequality Words When you multiply or divide each side of an inequality by a positive number, the inequality remains true. Symbols For all numbers a, b, and c, where c > 0, _ 1. if a > b, then ac > bc and _ c > c. a b _ 2. if a < b, then ac < bc and _ c < c. a Examples b 5<8 2 > -10 4(5) < 4(8) -10 2 _ >_ 20 < 32 1 > -5 2 2 These properties are also true for a ≥ b and a ≤ b. Solve Inequalities by Dividing Checking Solutions You can check this solution by substituting numbers greater than -6 into the inequality and testing it to verify that it holds true. 1 Solve 7y > -42. Check your solution. 7y > -42 Write the inequality. 7y -42 _ >_ Divide each side by 7. y > -6 Simplify. 7 7 The solution is y > -6. Lesson 8-8 Solving Inequalities by Multiplying or Dividing Doug Martin 449 Solve Inequalities by Multiplying 1 2 Solve _ x ≤ 8. Check your solution. 3 _1 x ≤ 8 3 1 3 _ x ≤ 3(8) 3 ( ) x ≤ 24 Write the inequality. Multiply each side by 3. Simplify. The solution is x ≤ 24. You can check this solution by substituting 24 and a number less than 24 into the inequality. Solve each inequality. Check your solution. b. _ < -16 n 4 a. 3a ≥ 45 c. 81 ≤ 9p What happens when each side of an inequality is multiplied or divided by a negative number? Graph 3 and 5 on a number line. 5432 1 0 1 2 3 4 5 Multiply each number by -1. 5432 1 0 1 2 3 4 5 Since 3 is to the left of 5, 3 < 5. Since -3 is to the right of -5, -3 > -5. Notice that the numbers being compared switched positions as a result of being multiplied by a negative number. In other words, their order reversed. These and other examples suggest the following properties. +%9 #/.#%04 Common Error Do not reverse the inequality symbol just because there is a negative sign in the inequality, as in 7y < -42. Only reverse the` inequality symbol when you multiply or divide each side by a negative number. Properties of Inequality Words When you multiply or divide each side of an inequality by a negative number, the direction of the inequality symbol must be reversed for the inequality to remain true. Symbols For all numbers a, b, and c, where c < 0, _ 1. if a > b, then ac < bc and _ c < c. a b _ 2. if a < b, then ac > bc and _ c > c. a Examples b 8>5 -1(8) < -1(5) -3 < 9 Reverse the inequality symbols. -3 9 _ >_ -3 -3 1 > -3 -8 < -5 These properties also hold true for a ≥ b and a ≤ b. 450 Chapter 8 Algebra: More Equations and Inequalities Extra Examples at ca.gr7math.com Multiply or Divide by a Negative Number a 3 Solve _ ≥ 8. Check your solution. -2 a _ ≥8 -2 _ -2 a ≤ -2(8) -2 ( ) a ≤ -16 Write the inequality. Multiply each side by -2 and reverse the inequality symbol. Check this result. 4 Solve -24 > -6n. Check your solution. -24 > -6n Write the inequality. -6n -24 _ <_ Divide each side by -6 and reverse the symbol. -6 -6 4 < n or n > 4 Check this result. Solve each inequality. d. _ < -14 c -7 e. -5d ≥ 30 f. -3 ≤ _ w -8 Some inequalities involve more than one operation. To solve, work backward as you did in solving two-step equations. 5 WORK Jason wants to earn at least $30 this week to go to the state fair. His dad will pay him $12 to mow the lawn. For washing their cars, his neighbors will pay him $8 per car. Suppose Jason mows the lawn. Write and solve an inequality to find how many cars he needs to wash to earn at least $30. Interpret the solution. Real-World Link If you are 14 or 15 and have a part-time job, you can work no more than 3 hours on a school day, 18 hours in a school week, 8 hours on a nonschool day, or 40 hours in a nonschool week. Source: www.youthrules. dol.gov The phrase at least means greater than or equal to. Let c = the number of cars he needs to wash. Then write an inequality. 12 + 8c ≥ 30 12 - 12 + 8c ≥ 30 - 12 Write the inequality. Subtract 12 from each side. 8c ≥ 18 Simplify. 8c 18 _ ≥_ Divide each side by 8. c ≥ 2.25 Simplify. 8 8 Since he will not get paid for washing one fourth of a car, Jason must wash at least 3 cars. g. DVDS Joan has a total of $250. DVDs cost $18.95 each. Write and solve an inequality to find how many DVDs she can buy and still have at least $50. Interpret the solution. Personal Tutor at ca.gr7math.com Lesson 8-8 Solving Inequalities by Multiplying or Dividing Aaron Haupt 451 Examples 1–2 (pp. 449–450) Examples 3–4 Solve each inequality. Check your solution. 1. 3x > 12 2. _ < _y 3. 8x ≤ -72 5. -4y > 32 6. -56 ≤ -7p 7. _ < -7 3 4 7 9 (p. 451) 4. _ ≥ -6 h 4 d _ 8. ≥ -3 -3 g -2 Example 5 9. RENTAL CARS A rental car company charges $45 plus an additional $0.19 per (p. 451) mile to rent a car. If Lawrence does not want to spend more than $100 for his rental car, write and solve an inequality to find how many miles he can drive and not spend more than $100. Interpret the solution. Solve each inequality. Check your solution. For Exercises 10–15 16–27 28–29 See Examples 1, 2 3, 4 5 10. 5x < 15 11. 9n ≤ 45 12. 14k ≥ -84 13. -12 > 3g 14. -100 ≤ 50p 15. 2y < -22 16. -4w ≥ 20 17. -3r > 9 18. -72 < -12h 19. -6c ≥ -6 20. _ > 4 21. _ ≥ 5 22. _ ≤ -3 x 9 t _ 25. ≤ -2 -5 v -4 n 23. _ < -14 7 y 26. -8 ≤ _ 0.2 a -3 m 24. _ < -7 -2 -1 27. _k > -10 2 28. BUS TRAVEL A city bus company charges $2.50 per trip. They also offer a monthly pass for $85.00. Write and solve an inequality to find how many times a person should use the bus so that the pass is less expensive than buying individual tickets. Interpret the solution. 29. BABY-SITTING You want to buy a pair of $42 inline skates with the money you make baby-sitting. If you charge $5.25 an hour, write and solve an inequality to find how many whole hours you must baby-sit to buy the skates. Interpret the solution. Solve each inequality. Check your solution. 30. 5y – 2 > 13 31. 8k + 3 ≤ -5 32. -3g + 8 ≥ -4 33. 7 + _ < 4 34. _ - 4 ≤ -5 35. 10 - 3x ≥ 25 + 2x n 3 w 8 Write an inequality for each sentence. Then solve the inequality. See pages 699, 715. Self-Check Quiz at ca.gr7math.com 452 36. Three times a number increased by four is less than -62. 37. The quotient of a number and -5 increased by one is at most 7. 38. The quotient of a number and 3 minus two is at least -12. 39. The product of -2 and a number minus six is greater than -18. Chapter 8 Algebra: More Equations and Inequalities H.O.T. Problems 40. OPEN-ENDED Write an inequality that can be solved using the Multiplication Property of Equality where the inequality symbol needs to be reversed. 41. FIND THE ERROR Olivia and Lakita each solved 8a ≤ -56. Who is correct? Explain. Olivia 8a ≤ -56 Lakita 8a ≤ -56 8a -56 _ ≥_ 8a -56 _ ≤_ a ≥ -7 a ≤ -7 8 8 8 8 42. CHALLENGE You have scores of 88, 92, 85, and 87 on four tests. What number of points must you get on your fifth test to have a test average of at least 90? 43. */ -!4( Explain when you should reverse the inequality when (*/ 83 *5*/( solving an inequality. 44. Which number is a possible base 45. As a salesperson, you are paid $60 per length of the triangle if its area is greater than 45 square yards? week plus $25 per sale. This week you want your pay to be at least $700. Which inequality can be used to find the number of sales you must make this week? F 60 + 25x ≥ 700 18 yd G 60x + 25 ≥ 700 x yd A 3 C 5 B 4 D 6 H 60 + 25 ≤ 700 J 60x + 25 ≤ 700 Solve each inequality. Check your solution. (Lesson 8-7) 46. y + 7 < 9 47. a - 5 ≤ 2 48. j - 8 ≥ -12 49. -14 > 8 + n Write an inequality for each sentence. (Lesson 8-6) 50. SPEED A minimum speed on a certain highway is 45 miles per hour. 51. BIRDS A hummingbird’s wings can beat up to 200 times per second. 52. MEASUREMENT Three boxes with height 12 inches, width 10 inches, and length 13 inches are stacked on top of each other. What is the volume of the space that they occupy? (Lesson 7-5) Lesson 8-8 Solving Inequalities by Multiplying or Dividing 453 CH APTER 8 Study Guide and Review Download Vocabulary Review from ca.gr7math.com Key Vocabulary Be sure the following Key Concepts are noted in your Foldable. coefficient (p. 417) %QUATIONS constant (p. 417) equivalent expressions (p. 416) )NEQUALIT IES like terms (p. 417) simplest form (p. 418) Key Concepts Algebraic Expressions (Lesson 8-1) • Like terms contain the same variables to the same powers. simplifying the expression (p. 418) term (p. 417) two-step equation (p. 422) • A constant is a term without a variable. • An algebraic expression is in simplest form if it has no like terms and no parentheses. Equations (Lessons 8-2, 8-3, and 8-4) • To solve a two-step equation, undo each operation in reverse order. • To solve equations with variables on each side of the equals sign, use the Addition or Subtraction Property of Equality to write an equivalent equation with the variables on one side of the equals sign. Then solve the equation. Inequalities (Lesson 8-6) • When used to compare a variable and a number, inequalities can describe a range of values. Inequality Properties (Lessons 8-7 and 8-8) • When you add or subtract the same number from each side of an inequality, the inequality remains true. • When you multiply or divide each side of an inequality by a positive number, the inequality remains true. • When you multiply or divide each side of an inequality by a negative number, the direction of the symbol must be reversed for the inequality to be true. 454 Chapter 8 Algebra: More Equations and Inequalities Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. 1. Like terms are terms that contain different variables. 2. A two-step equation is an equation that contains two operations. 3. A coefficient is a term without a variable. 4. The numerical factor of a term that contains a variable is called the constant of the variable. 5. When plus or minus signs separate an algebraic expression into parts, each part is called a term. 6. An algebraic expression is in simplest form if it has no like terms and no parentheses. 7. The expressions 4(y + 7) and 4y + 28 are two-step equations. 8. When you use the Distributive Property to combine like terms, you are simplifying the expression. Vocabulary Review at ca.gr7math.com Lesson-by-Lesson Review 8-1 Simplifying Algebraic Expressions (pp. 416–421) Use the Distributive Property to rewrite each expression. Example 1 Use the Distributive Property to rewrite -8(x - 9). 9. 4(a + 3) -8(x - 9) Write the expression. = -8[x + (-9)] Rewrite x - 9 as x + (-9) = -8(x) + (-8)(-9) Distributive Property = -8x + 72 Simplify. 10. (n - 5)(-7) Simplify each expression. 11. p + 6p 12. 6b - 3 + 7b + 5 13. SOCCER Pam scored n goals. Leo scored 5 fewer than Pam. Write an expression in simplest form to represent the total number of goals scored. 8-2 Solving Two-Step Equations (pp. 422–426) Solve each equation. Check your solution. 14. 2x + 5 = 17 15. 4 = -3y - 2 16. _ + 2 = 9 17. 39 = a + 6a + 11 c 5 18. ZOO Four adults spend $37 for admission and $3 for parking at the zoo. Solve the equation 4a + 3 = 40 to find the cost of admission per person. 8-3 Example 2 Solve 5h + 8 = -12. 5h + 8 = -12 Write the equation. 5h + 8 - 8 = -12 - 8 Subtract 8 from each side. -12 - 8 = -12 + (-8) or 20 Divide each side by 5. 5h = -20 5h -20 _ =_ 5 5 h = -4 The solution is -4. Simplify. Check this solution. Writing Two-Step Equations (pp. 427–431) 19. Six more than twice a number is -4. Example 3 Translate the following sentence into an equation. Then solve. 20. The quotient of a number and 8, less 2, 6 less than 4 times a number is equal to 10. Translate each sentence into an equation. is 5. 6 less than 4 times a number 4n - 6 21. MEDICINE Dr. Miles recommended that Jerome take 8 tablets on the first day and then 4 tablets each day until the prescription was used. The prescription contained 28 tablets. How many days will Jerome be taking tablets after the first day? Write an equation and then solve. 4n - 6 = 10 4n - 6 + 6 = 10 + 6 4n = 16 is 10. = 10 Write the equation. Add 6 to each side. Simplify. 16 4n _ =_ 4 4 Divide each side by 4. n=4 Simplify. Chapter 8 Study Guide and Review 455 CH APTER 8 Study Guide and Review 8-4 Solving Equations with Variables on Each Side (pp. 434–437) Solve each equation. Check your solution. 22. 11x = 20x + 18 23. 4n + 13 = n - 8 24. 7b - 3 = -2b + 24 25. 9 - 2y = 8y - 6 26. GEOGRAPHY The coastline of California is 46 miles longer than twice the length of Louisiana’s coastline. It is also 443 miles longer than Louisiana’s coastline. Find the lengths of the coastlines of California and Louisiana. 8-5 Solve -7x + 5 = x - 19. -7x + 5 = x - 19 Write the equation. -7x + 7x + 5 = x + 7x - 19 Add 7x to each side. 5 = 8x - 19 5 + 19 = 8x - 19 + 19 Add 19 to each side. 24 = 8x 8x 24 _ =_ Divide each side by 8. 8 8 3=x Simplify. The solution is 3. PSI: Guess and Check (pp. 438–439) Solve using the guess and check strategy. 27. FUND-RAISER The Science Club sold candy bars and pretzels to raise money. They raised a total of $62.75. If they made $0.25 on each candy bar and $0.30 on each pretzel, how many of each did they sell? 28. FOOD A store sells apples in 2-pound bags and oranges in 5-pound bags. How many bags of each should you buy if you need exactly 11 pounds of apples and oranges? 29. BONES Each hand in the human body has 27 bones. There are 6 more bones in the fingers than in the wrist. There are 3 fewer bones in the palm than in the wrist. How many bones are in each part of the hand? 456 Example 4 Chapter 8 Algebra: More Equations and Inequalities Example 5 The product of two consecutive even integers is 1,088. What are the integers? The product is close to 1,000. Make a guess. Try 24 and 26. 24 × 26 = 624 This product is too low. Adjust the guess upward. Try 30 and 32. 30 × 32 = 960 This product is still too low. Adjust the guess upward again. Try 34 and 36. 34 × 36 = 1,224 This product is too high. Try between 30 and 34. Try 32 and 34. 32 × 34 = 1,088 This is the correct product. The integers are 32 and 34. Mixed Problem Solving For mixed problem-solving practice, see page 715. 8-6 Inequalities (pp. 441–444) Write an inequality for each sentence. 30. SPORTS Participants must be at least 12 years old to play. 31. PARTY No more than 15 people at the party. Example 6 All movie tickets are $9 and less. Write an inequality for this situation. Let t = the cost of a ticket. t≤9 For the given value, state whether each inequality is true or false. Example 7 Graph the inequality a < -4 on a number line. 32. 19 - a < 20, a = 18 Place an open circle at -4. Then draw a line and an arrow to the left. 33. 9 + k > 16, k = 6 Graph the inequality on a number line. 34. t < 2 35. g ≥ 92 36. NUTRITION A food can be labeled low- fat only if it has no more than 3 grams of fat per serving. Write an inequality to describe low-fat foods. 8-7 Solving Inequalities by Adding or Subtracting (pp. 445-448) Solve each inequality. Check your solution. Example 8 solution. 37. b - 9 ≥ 8 38. 15 > 3 + n Write the inequality. 39. x + 4.8 ≤ 2 40. r + 5.7 ≤ 6.1 41. t + _ < 4 42. -1_ < k - 3 x-7<3 x-7+7<3+7 x < 10 Check Write the inequality. 1 2 2 5 43. MOVING A moving company is loading a 920-pound piano into a service elevator. The elevator can carry a maximum of 1,800 pounds. Write and solve an inequality to determine how much additional weight the elevator can carry. Solve x - 7 < 3. Check your x-7<3 9-73 2<3✓ Add 7 to each side. Simplify. Replace x with a number less than 10, such as 9. This statement is true. Chapter 8 Study Guide and Review 457 CH APTER 8 Study Guide and Review 8-8 Solving Inequalities by Multiplying and Dividing (pp. 449-453) Solve each inequality. 44. _ < 6 45. _ ≤ 3 46. 0.5x > 3.2 47. -56 ≥ 8y 48. 9 > _ 49. -_a ≤ 2 n 4 x -4 k 1.7 5 6 50. GOLF Aubrey wants to spend less than $38.50 on new golf balls. If each box costs $11, what is the maximum number of boxes of golf balls that she can buy? 51. JOBS Dakota earns $8 per hour working at a landscaping company and wants to earn at least $1,200 this summer. a. Write an inequality to represent this situation. b. Solve the inequality that you found in part a. c. What is the minimum number of hours Dakota will have to work? 458 Chapter 8 Algebra: More Equations and Inequalities Example 9 solution. Solve -2n ≥ 26. Check your -2n ≥ 26 Write the inequality. 26 -2n _ ≤_ -2 -2 n ≤ -13 Divide each side by -2 and reverse the symbol. Simplify. The solution is n ≤ -13. You can check this solution by substituting -13 and a number less than -13 into the inequality. CH APTER 8 Practice Test Use the Distributive Property to rewrite each expression. 1. -7(x - 10) 19. MUSICAL Joseph sold tickets to the school musical. He had 12 bills worth $175 for the tickets sold. If all the money was in $5 bills, $10 bills, and $20 bills, how many of each bill did he have? 2. 8(2y + 5) Simplify each expression. 3. 9a - a + 15 - 10a - 6 20. BANKING First Bank charges $4.50 per 4. 2x + 17x month for a basic checking account plus $0.15 for each check written. Citizen’s Bank charges a flat fee of $9. How many checks would you have to write each month in order for the cost to be the same at both banks? Solve each equation. Check your solution. 5. 3n + 18 = 6 6. _ - 11 = 5 7. -23 = 3p + 5 + p 8. 4x - 6 = 5x 9. -3a - 2 = 2a + 3 10. -2y + 5 = y - 1 k 2 11. FUND-RAISER The band buys coupon books for a one-time fee of $60 plus $5 per book. If they sell the books for $10 per book, write and solve an equation to determine how many books they will need to sell in order to break even. 21. STANDARDS PRACTICE The perimeter of the rectangle is 44 inches. 4x in. x 7 in. Translate each sentence into an equation. What is the area of the rectangle? 12. Three more than twice a number is 15. F 22 in 2 G 120 in 2 H 392 in 2 J 440 in 2 13. The quotient of a number and 6 plus 3 is 11. 14. The product of a number and 5 less 7 is 18. 15. STANDARDS PRACTICE In the inequality 3x + $5,000 ≤ $80,000, x represents the salary of an employee at a factory. Which phrase most accurately describes the employee’s salary? A Less than $25,000 B More than $25,000 C At least $25,000 D At most $25,000 For Exercises 22 and 23, write an inequality and then graph the inequality on a number line. 22. COMPUTERS A recordable DVD can hold at most 4.7 gigabytes of data. 23. GAMES Your score must be over 55,400 to have the new high score. Solve each equation. Check your solution. 16. x + 5 = 4x + 26 Solve each inequality. Check the solution. 17. 3d = 18 - 3d 24. -4 > _ 18. -2g + 15 = 45 - 8g 25. -2g + 15 > 45 Chapter Test at ca.gr7math.com c 9 Chapter 8 Practice Test 459 CH APTER 8 California Standards Practice Cumulative, Chapters 1–8 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1 5 Which property is used in the equation below? 5(x - 2) = 5x - 10 A Eddie scored the most touchdowns. A Associative Property of Addition B Orlando scored the most touchdowns. B Commutative Property of Addition C Dante scored exactly half of the total number of touchdowns. C Distributive Property D Dante scored the most touchdowns. D Reflexive Property 2 Orlando, Eddie, and Dante scored a total of 108 touchdowns this season. Eddie scored 8 more touchdowns than Dante, and Orlando scored twice as many touchdowns as Dante. Which is a reasonable conclusion about the number of touchdowns scored by the players? A farmer packs tomatoes in boxes that weigh 1.4 kilograms when empty. The average tomato weighs 0.2 kilogram and the total weight of a box filled with tomatoes is 11 kilograms. How many tomatoes are packed in each box? 6 The largest possible circle is to be cut from a 2 meter board. What will be the approximate area, in square meters, of the remaining board (shaded region)? (A = πr 2 and π ≈ 3.14) F 62 G 55 H 48 J 13.6 3 2 meters There are 4 children in the Owens family. F 8.56 1 times as tall as Kelly, and he Jamie is 1_ G 0.86 2 is 6 inches taller than Olivia. Sammy is 56 inches tall, which is 2 inches taller than Olivia. Find Jamie’s height. 4 A 52 inches C 58 inches B 56 inches D 60 inches H 2.28 J 3.14 The sum of a number n and 6 is 23. Which equation shows this relationship? A rectangular prism has a length of 7.5 inches, a width of 1.4 inches, and a volume of 86.4 cubic inches. What is the height of the rectangular prism? Round to the nearest tenth. F 23 + n = 6 A 0.1 G 6n = 23 B 8.2 H 6 + n = 23 C 462.9 J n – 6 = 23 D 907.2 460 Chapter 8 Algebra: More Equations and Inequalities 7 California Standards Practice at ca.gr7math.com More California Standards Practice For practice by standard, see pages CA1–CA39. 8 9 Which expression is equivalent to 2ab + 4ac? F 6abc H 2a(b + c) G ab + ac J 2a(b + 2c) 12 In the figure below, every angle is a right angle. 6 2 3 3 About how much paper is needed to make a label that covers only the sides of the soup can shown below? Use 3.14 for π and round to the nearest square inch. 3 2 2 4 4 3 2 6 2 in. What is the area in square units? 5 in. F 49 H 54 G 50 J 57 13 Which expression has the smallest value? A 31 C 63 B 62 D 72 A ⎪78⎥ C ⎪-22⎥ B ⎪14⎥ D ⎪-47⎥ Pre-AP Record your answers on a sheet of paper. Show your work. Question 9 When answering a test question involving a 3-dimensional shape, always study the shape and its labels carefully. Ask yourself, “ Am I finding surface area or volume?” 14 The table below gives prices for two different bowling alleys in your area. Bowling Alley X Y 10 What is the value of x if -5x - 4 = -34? F -7 H 6 G -6 J 7 Shoe Rental $2.50 $3.50 Cost per Game $4.00 $3.75 a. Write an equation to find the number of games g for which the total cost to bowl at each alley would be equal. 11 √ 625 = A 15 C 30 B 25 D 35 b. How many games will you have to bowl at each alley for the cost to be equal? NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Go to Lesson... 8-1 8-2 1-9 1-7 1-1 7-1 7-5 8-1 7-7 8-2 3-1 7-3 1-3 8-4 For Help with Standard... AF1.3 AF4.1 AF4.1 AF1.1 MR1.1 MG2.1 MG2.1 AF1.3 MG2.1 AF4.1 NS2.4 MG2.2 NS2.5 AF1.1 Chapters 1–8 California Standards Practice 461 Algebra: Linear Functions 9 • Standard 7AF3.0 Graph and interpret linear and some nonlinear functions. Key Vocabulary constant of variation (p. 483) line of fit (p. 505) linear function (p. 472) slope (p. 477) Real-World Link Roller Coasters If you ride the Boomerang roller coaster, located in Buena Park, California, you will travel 935 feet in 108 seconds. You can use the linear function 935 = 108r to model the average speed of this coaster. Algebra: Linear Functions Make this Foldable to help you organize your notes. Begin with seven 1 sheets of 8 ” × 11” paper. _ 2 1 Fold a sheet of paper in half lengthwise. Cut a 1” tab along the left edge through one thickness. 2 Glue the 1” tab down. Write the title of the lesson on the front tab. -INEAR 'UNCTIONS 462 Chapter 9 Algebra: Linear Functions Eric Gieszl/Ultimate Rollercoaster 3 Repeat Steps 1–2 for the remaining sheets of paper. Staple together to form a booklet. -INEAR 'UNCTIONS GET READY for Chapter 9 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Graph each point on the same coordinate grid. (Prior Grade) 1. A(-3, -4) 2. B(2, -1) 3. C(0, -2) 4. D(-4, 3) Example 1 Graph P(-1, 2), Q(3, -1), and R(-4, 0) on a coordinate grid. y P 5. WALKING From his cabin, Derek walked 4 miles south and 2 miles west, where he rested. If the origin represents the cabin, graph the point representing Derek’s resting point. (Prior Grade) R x O Q Start at the origin. The first number in each ordered pair is the x-coordinate. The second number in each ordered pair is the y-coordinate. Evaluate each expression if x = 6. Example 2 (Lesson 1-2) Evaluate 6x - 1 if x = 4. 6. 3x 7. 4x - 9 8. 2x + 8 9. 5 + x 6x - 1 = 6(4) - 1 = 24 - 1 = 23 Replace x with 4. Multiply 6 by 4. Subtract. 10. PROFIT The weekly profit of a certain company is 48x - 875, where x represents the number of units sold. Find the weekly profit, if the company sells 37 units. (Lesson 1-2) Solve each equation. (Lesson 1-9) Example 3 11. 14 = n + 9 12. z - 3 = 8 Solve 18 + m = 7. 13. -17 = b - 21 14. 23 + r = 15 18 + m = 7 -18__________ = -18 ____ m = -11 Write the equation. Subtract 18 from each side. Chapter 9 Get Ready for Chapter 9 463 Explore 9-1 Main IDEA Model functions using real-world situations. Preparation for Standard 7AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Algebra Lab Functions A relation expresses how objects in one group called inputs are assigned or related to objects in another group called outputs. Suppose three students select a favorite color from the colors blue, red, or green. The relation diagrams below show two of several possible results. )NPUT 2ELATION /UTPUT 2ELATION /UTPUT )NPUT BLUE RED GREEN !LONSO -ALINA #HEN !MBER 4IM +AELA BLUE RED GREEN A function is a relation in which exactly one output is assigned to each input. In the example above, the first relation is a function, since each person chose one favorite color. The second relation is not a function, since Alonso chose two colors as his favorite. Create a spinner like the one shown. ! Each of the four people in your group should spin the spinner once to simulate # selecting TV show A, B, C, or D as their favorite. Each person should keep spinning until they get a result different from anyone else. Record the results as Relation 1. " $ Next, have each person spin again, this time spinning until one or more persons are assigned the same letter. Record the results as Relation 2. Finally, have each person spin again, and allow one person to spin twice. Record the five results as Relation 3. ANALYZE THE RESULTS 1. Make a diagram like the one shown above for each relation. 2. Determine whether each relation is a function. Explain your reasoning in the context of selecting a favorite TV show. 3. MAKE A CONJECTURE Draw a diagram of each of the relations described below. Then determine if the relation is a function. Explain. a. one person spins four times, each time spinning a different letter b. each of four people spins the same letter 464 Chapter 9 Algebra: Linear Functions 9-1 Functions Main IDEA Complete function tables. Preparation for Standard 7AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. ENTERTAINMENT Suppose you can buy DVDs for $15 each. Cost ($) 1. Copy and complete the table at the right. 1 15 2. If 6 DVDs are purchased, what is the 2 30 3 total cost? 3. Explain how to find the total cost of 9 DVDs. 4 5 The total cost depends on, or is a function of, the number of DVDs purchased. A relationship that assigns exactly one output value for each input value is called a function. Functions are often written as equations. The input x is any real number. f(x) = 7x NEW Vocabulary function domain range function table DVDs f(x) is read the function of x, or more simply, f of x. It is the output. The operations performed in the function are sometimes called the rule. To find the value of a function for a certain number, substitute the number for the variable x. Find a Function Value Find each function value. 1 f(9) if f(x) = x - 5 f(x) = x - 5 Write the function. f(9) = 9 - 5 or 4 Substitute 9 for x into the function rule. So, f(9) = 4. 2 f(-3) if f(x) = 2x + 1 f(x) = 2x + 1 READING in the Content Area For strategies in reading this lesson, visit ca.gr7math.com. Write the function. f(-3) = 2(-3) + 1 Substitute -3 for x into the function rule. f(-3) = -6 + 1 or -5 Simplify. So, f(-3) = -5. Find each function value. a. f(2) if f(x) = x - 4 b. f(6) if f(x) = 2x - 8 Lesson 9-1 Functions Michael Newman/PhotoEdit 465 Input and Output The variable for the input is called the independent variable because it can be any number. The variable for the output is called the dependent variable because it depends on the input value. The set of input values in a function is called the domain. The set of output values is called the range. You can organize the input, rule, and output into a function table. Make a Function Table 3 Complete the function table for f(x) = x + 5. Then state the domain and range of the function. Input Rule Output x f(x) = x + 5 f(x) -2 -1 Substitute each value of x, or input, into the function rule. Then simplify to find the output. Input Rule Output The domain is {-2, -1, 0, 1}. x f(x) = x + 5 f(x) -2 -2 + 5 f(-2) = 3 -1 -1 + 5 f(-1) = 4 0 0+5 f(0) = 5 1 1+5 f(1) = 6 0 1 The range is {3, 4, 5, 6}. Copy and complete each function table. Then state the domain and range of the function. c. f(x) = x - 7 x x-7 d. f(x) = 4x f(x) 4x x e. f(x) = 2x + 3 f(x) x -3 -5 -1 -2 -3 2 -1 2 3 0 5 5 2x + 3 f(x) Sometimes functions are written using two variables. One variable, usually x, represents the input and the other, usually y, represents the output. The function in Example 3 can also be written as y = x + 5. Real-World Career How Does a Zookeeper Use Math? A zookeeper must order the appropriate amount of various foods that will keep each animal healthy. For more information, go to ca.gr7math.com. Functions with Two Variables 4 ZOOKEEPER The zoo needs 1.5 tons of specially mixed elephant chow to feed its elephants each week. Write a function to represent the amount of elephant chow c needed for w weeks. Then determine how much elephant chow the zoo will need to feed its elephants for 12 weeks. Words Amount of chow equals 1.5 times the number of weeks. Function c = 1.5 · w The function c = 1.5w represents the situation. 466 Chapter 9 Algebra: Linear Functions Robert Brenner/PhotoEdit To find the amount of chow needed for 12 weeks, substitute 12 for w. c = 1.5w Write the function. c = 1.5(12) or 18 Substitute 12 for w. The zoo needs 18 tons of elephant chow. f. HOME REPAIR An air conditioner repair service charges $60 for a service call plus $30 per hour for labor. Write a function to represent the charge c for a service call with h hours of labor. How much would the charge be if there are 3 hours of labor? Personal Tutor at ca.gr7math.com Examples 1, 2 (p. 465) Example 3 (p. 466) Find each function value. 1. f(4) if f(x) = x - 6 Copy and complete each function table. Then state the domain and range of the function. 3. f(x) = 8 - x x Example 4 (p. 466) (/-%7/2+ (%,0 For Exercises 7–12 13–18 19–20 See Examples 1, 2 3 4 2. f(-2) if f(x) = 4x + 1 8-x 4. f(x) = 5x + 1 f(x) x 5x + 1 5. f(x) = 3x - 2 f(x) x -3 -2 -5 -1 0 -2 2 1 2 4 3 5 3x - 2 f(x) 6. MEASUREMENT The perimeter of a square is 4 times the length of a side. Write a function to represent the perimeter p of a square with sides measuring s units. What is the perimeter of a square with a 14-inch side? Find each function value. 7. f(7) if f(x) = 5x 8. f(9) if f(x) = x + 13 9. f(4) if f(x) = 3x - 1 10. f(5) if f(x) = 2x + 5 11. f(-5) if f(x) = 4x - 1 12. f(-12) if f(x) = 2x + 15 Copy and complete each function table. Then state the domain and range of the function. 13. f(x) = 6x - 4 x 6x - 4 14. f(x) = 5 - 2x f(x) x 5 - 2x 15. f(x) = 7 + 3x f(x) x -5 -2 -3 -1 0 -2 2 3 1 7 5 6 Extra Examples at ca.gr7math.com 7 + 3x f(x) Lesson 9-1 Functions 467 Copy and complete each function table. Then state the domain and range of the function. 16. f(x) = x - 9 x-9 x 17. f(x) = 7x 7x x f(x) 18. f(x) = 4x + 3 f(x) 4x + 3 x -2 -5 -4 -1 -3 -2 7 2 3 12 6 5 f(x) 19. SPORTS Tyree’s bowling score is handicapped by 30 points, meaning that he receives an additional 30 points on his final score. Write a function that can be used to represent Tyree’s final score s given his base score b. What is his adjusted score if he bowled 185? 20. PARTY PLANNING Sherry is having a birthday party at the Swim Center. The cost of renting the pool is $45 plus $3.50 for each person. Write a function to represent the total cost c for p people. What is the total cost if 20 people attend the party? Find each function value. 21. f _ if f(x) = 2x + _ ( 56 ) 22. f _ if f(x) = 4x - _ ( 58 ) 1 3 1 4 23. BIKING After 1 hour, a cyclist had ridden 12 miles. If she then continued riding at an average rate of 8 miles per hour, how long did it take her to ride 60 miles? %842!02!#4)#% 24. SCUBA DIVING The table shows the water pressure encountered by a diver. Write a function to represent the pressure p encountered at a depth of d feet. What would the pressure be at a depth of 175 feet? Round to the nearest tenth. See pages 699, 716. Self-Check Quiz at ca.gr7math.com H.O.T. Problems Depth (ft) Pressure (lb/in 2) 0 14.7 33 29.4 66 44.1 99 58.8 132 73.5 25. OPEN ENDED If f(x) = 2x - 4, find a value of x that will make the function value a negative number. 26. CHALLENGE Write the function rule for each function table. a. 27. x f(x) -3 b. c. x y -9 -2 -1 -5 3 -1 7 3 x f(x) -30 -5 -1 -10 2 20 6 60 d. x y -3 -2 -5 1 3 1 1 3 7 3 5 5 11 5 9 */ -!4( For the function y = x + 4, find the input value when (*/ 83 *5*/( the output value is -5. Write a rule that can be used to find the input value when the output value is known. 468 Chapter 9 Algebra: Linear Functions 28. The equation c = 6.50t represents c, 29. Stephanie received a $25 gift certificate the total cost of t tickets for a movie. Which table contains values that satisfy this equation? to an online music store. If the cost of purchasing a song is $0.95, which table best describes b, the balance remaining after she buys s songs? A t c Cost of Movie Tickets 1 2 3 $6.50 $13.00 $19.50 t c Cost of Movie Tickets 1 2 3 $6.50 $12.00 $18.00 B C t c Cost of Movie Tickets 1 2 3 $13.00 $19.50 $26.00 t c Cost of Movie Tickets 1 2 3 $6.50 $8.50 $9.50 D 4 $26.00 F b 24.10 2 23.10 2 23.20 4 21.20 4 21.40 5 20.25 6 19.60 8 17.40 8 17.80 10 15.50 s b s b 0 25.00 5 20.05 3 22.00 10 15.10 6 19.00 15 10.15 9 16.00 20 5.20 12 13.00 25 0.25 4 $24.50 4 $32.50 G 4 $10.50 H s s b 1 J 30. MEASUREMENT The length of a rectangle is 6 inches. Its area is greater than 30 square inches. Write an inequality for the situation. Solve the inequality. Interpret the solution. (Lesson 8-8) Solve each inequality. (Lesson 8-6) 31. b + 15 > 32 32. y - 24 ≤ 12 33. 9 ≤ 16 + t 35. UTILITIES An airport has changed the booths used for public telephones. The old booths consisted of four sides of a rectangular prism. The new booths are half of a cylinder with an open top. How much less material is needed to construct a new booth than an old booth? (Lesson 7-7) 34. 18 ≥ a - 6 Old Design New Design 45 in. 45 in. 26 in. 13 in. 26 in. 36. MEASUREMENT A block of cheese in the shape of a rectangular prism has a volume of 305 cubic centimeters. After several slices are cut from the block, it measures 10.25 centimeters by 6.5 centimeters by 2 centimeters. How much cheese was used? (Lesson 7-5) 37. Find the distance between the points (-1, 1) and (3, -2). (Lesson 3-7) PREREQUISITE SKILL Graph each point on the same coordinate plane. 38. A(-4, 2) 39. B(3, -1) 40. C(0, -3) 41. D(1, 4) Lesson 9-1 Functions 469 Explore 9-2 Main IDEA Algebra Lab Graphing Relationships In this lab, you will investigate a relationship between the number of pennies in a cup and how far the cup will stretch a rubber band. Graph relationships. Standard 7AF1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. Using a pencil, punch a small hole in the bottom of a paper cup. Place a paper clip onto a rubber band. Push the other end of the rubber band through the hole in the cup. Attach a second paper clip to the other end of the rubber band. Place it horizontally across the bottom of the cup to keep it from coming through the hole. Copy the table at the right. Number of Pennies x Distance y Tape the top paper clip to the edge of a desk. Measure and record the distance from the bottom of the desk to the bottom of the cup. Drop one penny into the cup. Measure and record the new distance from the bottom of the desk to the bottom of the cup. Continue adding one penny at a time. Measure and record the distance after each addition up to 10 pennies. ANALYZE THE RESULTS 1. MAKE A CONJECTURE Examine the data. Do you think the number of pennies affects the distance? Explain. 2. Graph the ordered pairs formed by your data. Do the points resemble a straight line? 3. MAKE A PREDICTION What will the distance of the bottom of the cup from the bottom of the desk be if 15 pennies are placed in the cup? 4. Find the ratio of each distance to the number of pennies. What do you notice about these ratios? 470 Chapter 9 Algebra: Linear Functions 9-2 Representing Linear Functions Main IDEA Represent linear functions using function tables and graphs. ROLLER COASTERS The Millennium Force has a maximum speed of 1.5 miles per minute. If x represents the minutes traveled at this speed, the function rule for the distance traveled is y = 1.5x. Standard 7AF3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. Standard 7AF1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. 1. Copy and complete Input the function table. 2. Graph the ordered pairs (x, y) on a coordinate plane. What do you notice? Rule Output (Input, Output) x 1.5x y (x, y) 1 1.5(1) 1.5 (1, 1.5) 2 1.5(2) 3 4 Functions can be represented in words, in a table, with a graph, and as ordered pairs. Graph a Function 1 SCHOOL SUPPLIES The school store sells book covers for $2 each and notebooks for $1. Toni wants to buy some of each. The cost of x books covers and y notebooks is 2x + y. Toni has $5 to spend, so 2x + y = 5. Graph 2x + y = 5 to find how many covers and notebooks Toni can buy. 2x + y = 5 NEW Vocabulary 2x - 2x + y = 5 - 2x y = 5 - 2x linear function Write the equation. Subtract 2x from each side to solve for y. Simplify. The equation y = 5 - 2x represents a function. Choose values for x and substitute them to find y. Then graph the ordered pairs (x, y). x 5 - 2x y (x, y) 0 5 - 2(0) 5 (0, 5) 1 5 - 2(1) 3 (1, 3) 2 5 - 2(2) 1 (2, 1) 3 Check for Reasonableness Check solutions in the context of the original problem to be sure they make sense. 5 - 2(3) -1 (3, -1) y (0, 5) (1, 3) (2, 1) O x (3, ⫺1) She cannot buy negative amounts, so she can buy 0 covers and 5 notebooks, 1 cover and 3 notebooks, or 2 covers and 1 notebook. a. DECORATING A repeating pattern is made using 6 triangular tiles x and 1 hexagonal tile y. Graph the function 6x + y = 35 to find the number of each tile needed if 35 tiles are used. Lesson 9-2 Representing Linear Functions 471 Graph a Function 2 Graph y = x + 2. • Select any four values for the input x. Substitute these values for x to find the output y. • Graph each ordered pair. Draw a line that passes through each point. Solutions The solutions of an equation are ordered pairs that make an equation representing the function true. x x+2 y (x, y) 0 0+2 2 (0, 2) 1 1+2 3 (1, 3) 2 2+2 4 (2, 4) 3 3+2 5 (3, 5) y yx 2 (3, 5) (2, 4) (1, 3) (0, 2) x O The line is the complete graph of the function. The ordered pair corresponding to any point on the line is a solution of the equation y = x + 2. The point where the line crosses the x-axis is the solution to the equation 0 = x + 2. It appears that (-2, 0) is also a solution. Check this by substitution. Check y=x+2 Write the function. 0 -2 + 2 Replace x with -2 and y with 0. 0=0 ✓ Simplify. Graph each function. b. y = x - 5 c. y = -2x d. y = 2x + 1 Personal Tutor at ca.gr7math.com REVIEW Vocabulary linear relationship relationships that have straight-line graphs (Lesson 4-10) A function in which the graph of the solutions forms a line is called a linear function. Therefore, y = x + 2 is a linear equation. #/.#%04 3UMMARY Words The value of y is one less than the corresponding value of x. Equation y=x-1 Table 472 Representing Functions Chapter 9 Algebra: Linear Functions x y 0 -1 1 0 2 1 3 2 Ordered Pairs (0, -1), (1, 0), (2, 1), (3, 2) Graph y yx 1 O x 3 Which line graphed below best represents the table of values for the ordered pairs (x, y)? A x -2 -1 0 1 y -3 -1 1 3 C y y x O B D y y x O x O x O Read the Item You need to decide which of the four graphs represents the data. Solve the Item Eliminate the Possibilities By testing the ordered pair (0, 1) first, choices B and D can be eliminated. The values in the table represent the ordered pairs (-2, -3), (-1, -1), (0, 1), and (1, 3). Test the ordered pairs. Graph C is the only graph that contains all these ordered pairs. The answer is C. y e. The graph of the line y = 3x + 2 is drawn on the coordinate grid. Which table of ordered pairs contains only points on this line? F G x -1 0 2 3 y -5 -2 4 7 x -1 5 7 8 y -1 Extra Examples at ca.gr7math.com 1 -3 2 H J O x -6 -3 0 3 y 0 -1 2 3 x -3 -1 1 2 y -7 -1 5 8 Lesson 9-2 Representing Linear Functions x 473 Example 1 1. GARDENING Marigolds x come in containers with 4 flowers and daisies y (p. 471) come individually. Graph the function 4x + y = 15 to find the number of containers of marigolds and daisies you can get if you want 15 flowers. Example 2 (p. 472) Example 3 Graph each function. 2. y = x + 5 3. y = 3x - 2 4. y = -2x + 1 STANDARDS PRACTICE Which line graphed best represents the table of values for the ordered pairs (x, y)? 5. (p. 473) A 8 x -7 -2 2 9 y -6.5 -4 -2 1.5 y C 8 4 ⫺8 ⫺4 y 4 ⫺4 8x O O 4 8x 4 8x ⫺4 ⫺8 B 8 ⫺8 D y 8 4 ⫺8 ⫺4 O y 4 4 8x ⫺8 ⫺4 O ⫺4 ⫺8 (/-%7/2+ (%,0 For Exercises 6, 7 8–15 27, 28 See Examples 1 2 3 ⫺8 6. PETS Fancy goldfish x cost $3 each and common goldfish y cost $1 each. Graph the function 3x + y = 20 to determine how many of each type of goldfish Tasha can buy for $20. 7. CLOTHES A store sells T-shirts x in packs of 5 and regular shirts y individually. Graph the function 5x + y = 10 to determine the number of each type of shirt Bethany can have if she buys 10 shirts. Graph each function. 8. y = 4x 12. y = 3x - 7 474 Chapter 9 Algebra: Linear Functions 9. y = -3x 10. y = x - 3 11. y = x + 1 13. y = 2x + 3 1 14. y = _x + 1 3 15. y = _x - 3 1 2 16. MEASUREMENT The equation s = 180(n - 2) relates the sum of the measures of angles s formed by the sides of a polygon to the number of sides n. Find four ordered pairs (n, s) that are solutions of the equation. Then graph the equation. MEASUREMENT For Exercises 17–19, use the following information. The equation y = 1.09x describes the approximate number of meters y in x yards. 17. Would negative values of x have any meaning in this situation? Explain. 18. Graph the function. 19. About how many meters is a 40-yard race? MOUNTAIN CLIMBING For Exercises 20 and 21, use the following information and the table at the right. If the temperature is 80°F at sea level, the function t = 80 - 3.6h describes the temperature t at a height of h thousand feet above sea level. Various California Mountains Mountain Elevation (ft) Mount Whitney 14,494 Pyramid Peak 9,984 Adams Peak 8,197 Mount Palomar 6,138 20. Graph the temperature function. 21. What is the temperature at each peak on a day that is 80°F at sea level? %842!02!#4)#% See pages 699, 716. 22. MONEY Drake is saving money to buy a new computer for $1,200. He already has $450 and plans to save $30 a week. The function f(x) = 30x + 450 represents the amount Drake has saved after x weeks. Graph the function to determine the number of weeks it will take Drake to save enough money to buy the computer. Self-Check Quiz at ca.gr7math.com H.O.T. Problems 23. OPEN ENDED Draw a graph of a linear function. Name three solutions of the function. 24. Which One Doesn’t Belong? Identify the ordered pair that is not a solution of y = 2x - 3. Explain your reasoning. (-2, -7) (0, 3) (2, 1) (1, -1) 25. CHALLENGE Name the coordinates of four points that satisfy each function. Then give the function rule. a. b. y O y x O 26. x */ -!4( Explain how a function table can be used to graph a (*/ 83 *5*/( function. Lesson 9-2 Representing Linear Functions 475 28. The graph shows the line y = 5x - 1. 27. Which line graphed below best represents the table of values for the ordered pairs (x, y)? x -4 0 4 8 y -2 -1 0 1 y A ⫺2 2 ⫺4 2x ⫺2 ⫺2 B 2 ⫺2 y O x Which table of ordered pairs contains only points on this line? ⫺2 y D 2x O F y 2 x O C O y O 4x 2 G H J x -2 -1 0 1 y -9 -4 1 6 x -3 -2 -1 0 y -8 -7 -6 -5 x 0 1 2 3 y -1 0 1 2 x -1 0 1 2 y -6 -1 4 9 Find each function value. (Lesson 9-1) 29. f(6) if f(x) = 7x – 3 30. f(-5) if f(x) = 3x + 15 31. f(3) if f(x) = 2x - 7 32. BAND The school band makes $0.50 for every flower they sell. They want to make at least $500 on the flower sale. Write and solve an inequality to find how many flowers they can sell and meet their goal. (Lesson 8-8) 33. MEASUREMENT Find the volume of a cylinder with a diameter of 7 inches and a height of 9 inches. Round to the nearest tenth if necessary. (Lesson 7-5) PREREQUISITE SKILL Find the constant rate of change for each graph. (Lesson 4-9) 34. O 476 35. y x Chapter 9 Algebra: Linear Functions 36. y O x y O x 9-3 Slope Main IDEA Find the slope of a line. Standard 7AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. SAFETY A ladder truck uses a moveable ladder to reach upper levels of houses and buildings. 1. The rate of change of the ladder compares the height it is raised to the distance of its base from the building. Write this rate as a fraction in simplest form. 45 ft 2. Find the rate of change of a NEW Vocabulary slope rise run 30 ft ladder that has been raised 100 feet and whose base is 50 feet from the building. The term slope is used to describe the steepness of a straight line. Slope is the ratio of the rise, or vertical change, to the run, or horizontal change. In linear functions, no matter which two points you choose, the slope of the line is always constant. rise slope = _ run vertical change between any two points horizontal change between the same two points 1 EXERCISE Find the slope of the treadmill at the right. rise slope = _ run Slope Slope provides a way of describing how steep a line is numerically. 10 in. =_ 48 in. 5 =_ 24 Definition of slope IN rise = 10 in., run = 48 in. IN Simplify. 5 The slope of the treadmill is _ . 24 a. HIKING A hiking trail rises 6 feet for every horizontal change of 100 feet. What is the slope of the hiking trail? Since slope is a rate of change, it can be positive (slanting upward) or negative (slanting downward). Lesson 9-3 Slope Tom Carter/PhotoEdit 477 Find Slope Using a Graph 2 Find the slope of the line. Translating Rise and Run up positive down negative right positive left negative y run Choose two points on the line. The vertical change is 2 units while the horizontal change is 3 units. rise slope = _ run 2 =_ 3 rise B 2 O x A Definition of slope rise = 2, run = 3 3 2 The slope of the line is _ . 3 b. c. y O y x O x Slope can be found by finding the ratio of the change in y-values (rise) to the change in x-values (run) for any two points on a line. Find Slope Using a Table 3 The points given in the table lie on +2 a line. Find the slope of the line. Then graph the line. Choose two points from the table to find the changes in the x- and y-values. 1 3 5 7 y 12 9 6 3 -3 -3 -3 (1, 12) ⫺3 2 change in x (3, 9) 9 - 12 =_ (5, 6) 3-1 -3 3 =_ or -_ 2 2 3 _ The slope is - . 2 (7, 3) x O d. x -6 -2 2 6 y -2 -1 0 1 Personal Tutor at ca.gr7math.com 478 +2 x y change in y slope = _ Slope You can choose any two points to calculate slope. Whichever y-value you use first, be sure to use the corresponding x-value first. +2 Chapter 9 Algebra: Linear Functions e. x -4 0 4 8 y -1 -2 -3 -4 change in y change in x READING Math rise _. You can also find You have found slope by using _ run and Subscripts x 1 is read x sub one and x 2 is read x sub two. They are used to indicate two different x-coordinates. the slope of a line by using the coordinates of any two points on the line. One point can be represented by (x 1, y 1) and the other by (x 2, y 2). The small numbers slightly below x and y are called subscripts. +%9 #/.#%04 Slope Formula The slope m of a line passing through points (x 1, y 1) and (x 2, y 2) is the ratio of the difference in the y-coordinates to the corresponding difference in the x-coordinates. Words Model y (x1, y1) (x2, y2) y2 - y1 m=_ x - x , where x 2 ≠ x 1 Symbols 2 O x 1 Find Slope Using Coordinates Find the slope of the line that passes through each pair of points. 4 C(-1, -4), D(2, 2) y -y y 2 1 m=_ x -x Slope formula 2 - (-4) m=_ 2 - (-1) (x 1, y 1) = (-1, -4) (x 2, y 2) = (2, 2) 6 m=_ or 2 Simplify. 2 1 3 Check Using the Slope Formula • It does not matter which point you define as (x 1, y 1) and (x 2, y 2). y2 - y1 m=_ x2 - x1 • However, the coordinates of both points must be used in the same order. -5 To check Example 5, let (x 1, y 1) = (-4, 3) and (x 2, y 2) = (1, 2). Then find the slope. x O C(⫺1, ⫺4) When going from left to right, the graph of the line slants upward. This is correct for positive slope. 5 R(1, 2), S(-4, 3) S(⫺4, 3) Slope formula 3-2 m=_ (x 1, y 1) = (1, 2) (x 2, y 2) = (-4, 3) -4 - 1 1 1 m= _ or -_ Check D(2, 2) 5 y R(1, 2) x O Simplify. When going from left to right, the graph of the line slants downward. This is correct for negative slope. Find the slope of the line that passes through each pair of points. f. A(2, 2), B(5, 3) Extra Examples at ca.gr7math.com g. C(-2, 1), D(0, -3) h. J(-7, -4), K(-3, -2) Lesson 9-3 Slope 479 Example 1 1. BUILDINGS Find the slope of the (p. 477) roof of the storage shed. Example 2 Find the slope of each line. (p. 478) 2. 3 ft 15 ft 3. y y x O Example 3 (p. 479) (/-%7/2+ (%,0 For Exercises 9, 10 11–14 15, 16 17–22 O 4. The points given in the table lie on a line. Find the (p. 478) Examples 4, 5 x slope of the line. Then graph the line. 0 1 2 3 y 1 3 5 7 Find the slope of the line that passes through each pair of points. 5. A(-3, -2), B(5, 4) 6. C(-4, 2), D(1, 5) 7. E(-6, 5), F(3, -3) 8. G(1, 5), H(4, -3) 9. SKIING Find the slope of a ski 10. ROADS Find the slope of a road that rises 12 feet for every horizontal change of 100 feet. run that descends 15 feet for every horizontal change of 24 feet. See Examples 1 2 3 4, 5 12 ft 100 ft FT FT Find the slope of each line. 11. 12. y 13. 14. y O Chapter 9 Algebra: Linear Functions x y O x O 480 x x y O x The points given in each table lie on a line. Find the slope of the line. Then graph the line. 15. x 0 2 4 y 9 4 -1 -6 16. 6 x -3 3 9 15 y -3 1 5 9 Find the slope of the line that passes through each pair of points. 17. A(0, 1), B(2, 7) 18. C(2, 5), D(3, 1) 19. E(1, 2), F(4, 7) 20. G(-6, -1), H(4, 1) 21. J(-9, 3), K(2, 1) 22. M(-2, 3), N(7, -4) 23. AQUARIUMS The graph shows the depth y Depth (in.) of water in an aquarium over several days. Find the slope of the line and explain its meaning as a rate of change. 12 TRAVEL For Exercises 24–26, use the following information. After 2 hours, Kendra had traveled 110 miles. After 3 hours, she had traveled 165 miles. After 5 hours, she had traveled 275 miles. 8 4 0 4 8 12 x Day 24. Graph the information with the hours on the horizontal axis and miles traveled on the vertical axis. Draw a line through the points. 25. What is the slope of the graph? 26. What does the slope of the graph represent? HOUSING For Exercises 27–29, use the graph at the right. 53 (OME /WNERSHIP 0ERCENT OF &AMILIES 27. Find the slope of the line representing the change between each three-year period. 28. Does the graph show a constant rate Real-World Link After World War II, the rate of home ownership in the U.S. rose steadily for three decades, from 44% in the late 1940s to 65.6% in 1980. of change? Explain. 29. If the graph is extended in each direction, could you expect the slope to remain constant throughout the graph? Explain. 9EAR Source: U.S. Census Bureau 30. GEOMETRY Two lines that are parallel have the same slope. Determine whether quadrilateral ABCD is a parallelogram. Justify your reasoning. Source: www.census.gov y C D B %842!02!#4)#% See pages 700, 716. O A x 31. DISABILITIES Wheelchair ramps for access to public buildings are allowed Self-Check Quiz at ca.gr7math.com a maximum of one inch of vertical increase for every one foot of horizontal distance. Would a ramp that is 10 feet long and 8 inches tall meet this guideline? Explain your reasoning. Lesson 9-3 Slope Van D. Bucher/Photo Researchers 481 H.O.T. Problems 32. OPEN ENDED Write the coordinates of two points. Show that you can define either point as (x 1, y 1) and the slope of the line containing the points will be the same. 33. FIND THE ERROR Martin and Dylan are finding the slope of the line that passes through X(0, 2) and Y(2, 3). Who is correct? Explain. 3-2 m=_ 3-2 m=_ 0-2 1 m=_ or -_1 -2 2-0 m = _21 2 Martin Dylan 34. CHALLENGE Find the slope of the straight line that is the graph of the function expressing the circumference of a circle as a function of the radius. 35. */ -!4( For the slope of a linear function, explain why the (*/ 83 *5*/( vertical change (rise) and the horizontal change (run) is always the same. 36. Which line graphed below has a slope 37. What is the slope of the linear function of -2? A shown in the graph? C y O x y y x O O B D y 4 F -_ O x x y O 3 _ G -3 4 x 3 H _ 4 4 J _ 3 Graph each function. (Lesson 9-2) 38. y = 5x 39. y = x - 2 40. y = 2x - 1 41. y = 3x + 2 42. TEMPERATURE The function used to change a Celsius temperature C to a 9 C + 32. Change 25° Celsius to Fahrenheit temperature F is F = _ 5 Fahrenheit. (Lesson 9-1) PREREQUISITE SKILL Solve each equation. (Lesson 1-10) 43. 42 = -14x 482 44. 144 = 18a Chapter 9 Algebra: Linear Functions (l)First Light, (r)Yellow Dog Productions/Getty Images 45. _ = 7 n 3 46. -6 = _ t 9 9- 4 Direct Variation Main IDEA Use direct variation to solve problems. COMPUTERS Use the graph at the right that shows the output of a color printer. 24 Standard 7AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation. 1. What is the constant rate of change, 18 Pages y or slope, of the line? 2. Is the total number of pages printed 12 6 always proportional to the printing time? If so, what is the constant ratio? 2 0 3. Compare the constant rate of change 4 6 x 8 Time (minutes) to the constant ratio. In the example above, the number of minutes and the number of pages printed both vary, while the ratio of pages printed to minutes, 1.5 pages per minute, remains constant. direct variation constant of variation Find a Constant Ratio 1 FUND-RAISER The amount of y money Robin has raised for a bike-a-thon is shown in the graph at the right. Determine the amount that Robin raises for each mile she rides. Amount Raised ($) NEW Vocabulary When the ratio of two variable quantities is constant, their relationship is called a direct variation. The constant ratio is called the constant of variation. Since the graph of the data forms a line, the rate of change is constant. Use the graph to find the constant ratio. amount raised __ distance 15 7.5 _ or _ 2 1 30 20 10 2 0 4 6 x 8 Distance (miles) 30 7.5 _ or _ 4 40 45 7.5 _ or _ 6 1 1 60 7.5 _ or _ 8 1 Robin raises $7.50 for each mile she rides. a. SKYDIVING Two minutes after a skydiver opens his parachute, he has descended 1,900 feet. After 5 minutes, he has descended 4,750 feet. If the distance varies directly as the time, at what rate is the skydiver descending? Lesson 9-4 Direct Variation 483 +%9 #/.#%04 Direct Variation Model A direct variation is a relationship in which the ratio of y to x is a constant, k. We say y varies directly with x. Words 4 _y Symbols k = x or y = kx, where k ≠ 0 Example y = 3x 4 2 y 2 y 3x O 2 4x 2 4 Solve a Direct Variation 2 PETS Refer to the information at the left. Assume that the age of a dog varies directly as its equivalent age in human years. What is the human-year age of a dog that is 6 years old? METHOD 1 Use an equation. Write an equation of direct variation. Let x represent the dog’s actual age and let y represent the human-equivalent age. Real-World Link Most pets age at a different rate than their human companions. For example, a 3-yearold dog is often considered to be 21 in human years. y = kx 21 = k(3) 7=k y = 7x Direct variation y = 21, x = 3 Simplify. Substitute for k = 7. Use the equation to find y when x = 6. y = 7x y = 7(6) x=6 y = 42 Multiply. METHOD 2 Use a proportion. human equivalent age actual age x 21 _ =_ 21 · 6 = 3 · x 126 = 3x human equivalent age actual age Find the cross products. Multiply. 126 3x _ =_ Divide each side by 3. 42 = x Simplify. 3 3 6 3 A dog that is 6 years old is 42 years old in human-equivalent years. b. SHOPPING A grocery store sells 6 oranges for $2. How much would it cost to buy 10 oranges? Round to the nearest cent if necessary. Personal Tutor at ca.gr7math.com 484 Chapter 9 Algebra: Linear Functions Jane Burton/Photo Researchers Not all relationships with a constant rate of change are proportional. Likewise, not all linear functions are direct variations. Look Back To review proportional relationships, see Lessons 4-2 and 4-3. Identify Direct Variation Determine whether each linear function is a direct variation. If so, state the constant of variation. 3 Miles, x 25 50 75 100 Gallons, y 1 2 3 4 gallons _ 1 _ 2 1 _ or _ 25 miles 50 Compare the ratios to check for a common ratio. 3 1 _ or _ 25 75 4 1 _ or _ 25 25 100 Since the ratios are the same, the function is a direct variation. The 1 constant of variation is _ . 25 4 Hours, x 2 4 6 8 Earnings, y 36 52 68 84 earnings _ 36 18 _ or _ 2 hours 52 13 _ or _ 1 4 68 11.33 _ or _ 6 1 1 84 10.50 _ or _ 8 1 The ratios are not the same, so the function is not a direct variation. c. Days, x 5 10 15 20 Height, y 12.5 25 37.5 50 d. Time, x 4 6 8 10 Distance, y 12 16 20 24 #/.#%04 3UMMARY Table Direct Variations Notice that the graph of a direct variation, which is a proportional linear relationship, is a line that passes through the origin. x -2 -1 1 2 y -4 -2 2 4 _yx 2 2 2 2 Proportional Linear Function Graph Equation y y = 2x O x Nonproportional Linear Function Table x -2 -1 1 2 y -5 -3 1 3 _yx _5 3 1 _3 2 Extra Examples at ca.gr7math.com Graph Equation y y = 2x - 1 O x 2 Lesson 9-4 Direct Variation 485 (p. 483) Example 2 (p. 484) 1. MANUFACTURING The number of computers y built varies directly as the number of hours the production line operates. What is the ratio of computers built to hours of production? Computers Example 1 2. TRANSPORTATION A charter bus 1 hours. Assuming travels 210 miles in 3_ 60 40 20 2 0 2 (p. 485) (/-%7/2+ (%,0 For Exercises 4–5 6–11 12–15 See Examples 1 2 3, 4 3. Determine whether the linear function is a direct variation. If so, state the constant of variation. 4. GARDENING Janelle planted ornamental grass seeds. After the grass breaks the soil surface, its height varies directly with the number of days. What is the rate of growth? Hours, x 2 3 4 5 Miles, y 116 174 232 290 is directly proportional to the number of newspapers he delivers. How much does Dusty earn for each newspaper delivery? y 3 Earnings ($) Height (in.) x 5. JOBS The amount Dusty earns y 2 1 0 6 Hours that the distance traveled is directly proportional to the time traveled, how far will the bus travel in 6 hours? Examples 3, 4 4 2 4 Days 6 x 6 4 2 0 4 8 12 x Newspapers 6. SUBMARINES Ten minutes after a submarine is launched from a research ship, it is 25 meters below the surface. After 30 minutes, the submarine has descended 75 meters. At what rate is the submarine diving? 7. MOVIES The Stratton family rented 3 DVDs for $10.47. The next weekend, they rented 5 DVDs for $17.45. What is the rental fee for a DVD? 8. MEASUREMENT Morgan used 3 gallons of paint to cover 1,050 square feet and 5 gallons to paint an additional 1,750 square feet. How many gallons of paint would she need to cover 2,800 square feet? 9. MEASUREMENT The weight of an object on Mars varies directly with its weight on Earth. An object that weighs 70 pounds on Mars weighs 210 pounds on Earth. If an object weighs 160 pounds on Earth, how much would it weigh on Mars? 486 Chapter 9 Algebra: Linear Functions 10. ELECTRONICS The height of a wide-screen television screen is directly proportional to its width. A manufacturer makes a television screen that is 60 centimeters wide and 33.75 centimeters high. Find the height of a television screen that is 90 centimeters wide. 11. BAKING A cake recipe requires 2_ cups of flour for 12 servings. How much 3 4 Real-World Link The aspect ratio of a television screen describes the ratio of the width of the screen to the height. Standard screens have an aspect ratio of 4:3 while widescreen televisions have an aspect ratio of 16:9. Source: infoplease.com flour is required to make a cake that serves 30? Determine whether each linear function is a direct variation. If so, state the constant of variation. 12. 14. Pictures, x 5 6 7 8 Profit, y 20 24 28 32 Age, x 10 11 12 13 Grade, y 5 6 7 8 13. 15. Minutes, x 200 400 600 800 Cost, y 65 115 165 215 Price, x 10 15 20 25 Tax, y 0.70 1.05 1.40 1.75 ALGEBRA If y varies directly with x, write an equation for the direct variation. Then find each value. 16. If y = -12 when x = 9, find y when x = -4. 17. Find y when x = 10 if y = 8 when x = 20. 18. If y = -6 when x = -14, what is the value of x when y = -4? 19. Find x when y = 25, if y = 7 when x = 8. 20. Find y when x = 5, if y = 12.6 when x = 14. 21. MEASUREMENT The number of centimeters in a measure varies directly as the number of inches. Find the measure of an object in centimeters if it is 50 inches long. Inches, x %842!02!#4)#% Centimeters, y 6 9 12 15 15.24 22.86 30.48 38.10 See pages 700, 716. 22. MEASUREMENT The length of the rectangle shown Self-Check Quiz at varies directly as its width. What is the perimeter of a rectangle that is 10 meters long? ca.gr7math.com H.O.T. Problems ⫽4m w ⫽ 6.4 m 23. OPEN ENDED Identify values for x and y in a direct variation relationship where y = 9 when x = 16. 24. CHALLENGE The amount of stain needed to cover a wood surface is directly proportional to the area of the surface. If 3 pints are required to cover a square deck with a side of 7 feet, how many pints of stain are needed to paint a square deck with a side of 10 feet 6 inches? 25. */ -!4( Write a real-world problem involving a direct (*/ 83 *5*/( variation. Then solve your problem. Lesson 9-4 Direct Variation Samsung Electronics/Handout/Reuters/CORBIS 487 26. Students in a science class recorded 27. Nicole read 24 pages during a lengths of a stretched spring, as shown in the table below. 30-minute independent reading period. How many pages would she read in 45 minutes? Length of Stretched Spring Distance Stretched, x (centimeters) Mass, y (grams) 0 0 2 12 5 30 9 54 12 72 G 36 J 48 8 ounces of pineapple juice for every 12 ounces of orange juice. If she uses 32 ounces of orange juice, which proportion can she use to find x, the number of ounces of pineapple juice she should add to make the punch? 8 32 A _ =_ 8 x C _ =_ 12 x 8 32 _ _ B = 12 x x C y = -_ 6 x _ D y= 6 B y = 6x H 42 28. To make fruit punch, Kelli must add Which equation best represents the relationship between the distance stretched x and the mass of an object on the spring y? A y = -6x F 30 12 32 8 x D _ =_ 12 32 Find the slope of each line. (Lesson 9-3) 29. 30. y O x 31. y y x O O x 32. JOBS The function p = 7.5h describes the relationship between the number of hours h Callie works and the amount she is paid p. Graph the function. Then use your graph to determine how much Callie can expect to earn if she works 20 hours. (Lesson 9-2) 33. HEALTH Many health authorities recommend that a healthy diet contains no more than 30% of its Calories from fat. If Jennie consumes 1,500 Calories each day, what is the maximum number of Calories she should consume from fat? (Lesson 5-3) PREREQUISITE SKILL Solve each equation. (Lesson 1-9) 34. 7 + a = 15 488 35. 23 = d + 44 Chapter 9 Algebra: Linear Functions 36. 28 = n - 14 37. t - 22 = -31 CH APTER 9 Mid-Chapter Quiz Lessons 9-1 through 9-4 Find each function value. (Lesson 9-1) 1. f(9) if f(x) = 12x 2. f(6) if f(x) = x + 7 Find the slope of the line that passes through each pair of points. (Lesson 9-3) 3. f(8) if f(x) = 2x - 8 4. f(2) if f(x) = 6x + 1 11. A(2, 5), B(3, 1) 12. C(-1, 2), D(-5, 2) 5. SCIENCE Sonar units locate objects using the time it takes to reflect sound waves back from an object. The function f(x) = 727x, where f(x) is the distance to the object in meters and x is the time in seconds, can be used to locate objects under water. Find the distance to a wrecked ship if it takes 5.24 seconds for sound to reflect back. 13. E(5, 2), F(2, -3) STANDARDS PRACTICE Which graph has a negative slope? (Lesson 9-3) 14. y F y H x O O x (Lesson 9-1) 6. STANDARDS PRACTICE Which equation describes the function represented by the table? (Lesson 9-1) x y -2 -7 0 -3 2 1 4 5 A y = 2x – 3 C y=x+4 B y=x-3 D y = 2x + 3 7. PICNICS Shelby is hosting a picnic. The cost to rent the shelter is $25 plus $2 per person. Write a function using two variables to represent the situation. Find the total cost if 150 people attend. (Lesson 9-1) y G y J x O x O 15. BAKING Ernesto baked 3 cakes in 2_ hours. 1 2 Assuming that the number of cakes baked is directly proportional to the number of hours, how many cakes can he bake in 1 7_ hours? (Lesson 9-4) 2 16. JOBS The number of gallons of water Fina uses is directly proportional to the number of dogs she washes. How many gallons of water does she use for each dog she washes? (Lesson 9-4) Graph each function. (Lesson 9-2) 9. y = 2x - 5 10. WATER A store sells bottles of water x in packs of 6 and individual bottles of water y. Graph the function 6x + y = 17 to determine the number of each type of bottled water Sophia can have if she buys 17 bottles of water. (Lesson 9-2) 12 Gallons 8. y = x + 6 y 9 6 3 0 1 2 Dogs 3 4 x 9-5 Main IDEA Algebra Lab Proportional and Nonproportional Functions In this lab, you will use models to develop two different functions. Compare and contrast proportional and nonproportional linear functions. Using centimeter cubes, build the two tower patterns shown. Standard 7AF1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. A Pattern B Figures Figure Number 0 1 2 3 0 1 2 3 Let x represent the figure number and y represent the number of cubes in each tower. Copy and complete the table below for each pattern. Then graph and label each set of tower data on separate coordinate planes. x Process y y 0 1 2 3 4 Number of Cubes Explore 8 6 4 2 5 x 0 2 4 6 8 x Figure Number ANALYZE THE RESULTS 1. Compare and contrast the models of patterns A and B. 2. Compare and contrast the processes for patterns A and B. 3. Compare and contrast the graphs of patterns A and B. 4. Which pattern represents a proportional relationship or direct variation, and which represents a nonproportional relationship? Explain. How can you tell this from the process shown in the table? from the graph? 490 Chapter 9 Algebra: Linear Functions 9- 5 Slope-Intercept Form BrainPOP® ca.gr7math.com Main IDEA Graph linear equations using the slope and y-intercept. Graph each equation listed in the table at the right. Equation NEW Vocabulary slope-intercept form y-intercept y-intercept y = 3x + 2 _1 1. Use the graphs to find the Standard 7AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. Slope y = x + (-1) slope and y-intercept of each line. Copy and complete the table. 4 y = -2x + 3 2. Compare each equation with the value of its slope. What do you notice? Proportional linear functions can be written in the form y = kx, where k is the constant of variation, or slope of the line. Nonproportional linear functions can be written in the form y = mx + b. This is called the slope-intercept form. When an equation is written in this form, m is the slope and b is the y-intercept. The y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis. Find Slopes and y-intercepts of Graphs State the slope and the y-intercept of the graph of each equation. _ 3 2 y=_ x + (-4) Write the equation in the form y = mx + b. y = mx + b m= 1 y = 2x - 4 3 _2 , b = -4 3 2 The slope of the graph is _ , and the y-intercept is -4. 3 2 x+y=6 x+y=6 x-x+y=6-x y=6-x Write the original equation. Subtract x from each side. Simplify. y = -1x + 6 Write the equation in the form y = mx + b. y = mx + b Recall that -x means -1x. m = -1, b = 6 The slope of the graph is -1, and the y-intercept is 6. a. y = -5x + 3 Extra Examples at ca.gr7math.com b. y = _x - 6 1 4 c. y - x = 5 Lesson 9-5 Slope-Intercept Form 491 Graph Using Slope-Intercept Form _ 3 Graph y = - 3 x - 1 using the slope and y-intercept. 2 Step 1 Find the slope and y-intercept. 3 y = -_ x-1 2 Check for Accuracy To check your graph, substitute the x- and y-values of another point on your graph into the equation. For Example 3, test the point (2, -4). _ 2 _ -4 = - 3 (2) - 1 2 Step 2 Graph the y-intercept -1. y -3 3 Step 3 Write the slope -_ as _ . Use it to O 2 2 locate a second point on the line. m=_ -3 2 x down 3 units change in y: down 3 units change in x: right 2 units right 2 units Step 4 Draw a line through the two points. y = -3x - 1 2 _ slope = - 3 , y-intercept = -1 Graph each equation. e. y = _x - 1 1 2 Personal Tutor at ca.gr7math.com d. y = x + 3 -4 = -3 - 1 -4 = -4 ✓ f. y = -_x + 2 4 3 Graph an Equation to Solve Problems ADVERTISING It costs $15 to design a poster and $3 to print each poster. The cost y to print x posters is given by y = 3x + 15. 4 Graph the equation to find the number y of posters that can be printed for $45. 60 y = 3x + 15 slope = 3, y-intercept = 15 Plot the point (0, 15). Locate another point up 3 and right 1. Draw the line. The x-coordinate is 10 when the y-coordinate is 45, so the number of posters is 10. 40 ( 1,18) 20 (0,15) O 4 8 12 x 5 Describe what the slope and y-intercept represent. The slope 3 represents the cost in dollars per poster, and the y-intercept 15 is the one-time charge in dollars for preparing the design. 6 Is the total cost proportional to the number of posters? Explain. Compare the ratio of total cost to number of posters for two points. 18 _ = $18 per poster 1 45 _ = $4.50 per poster 10 The ratios are different. So, the total cost is not proportional to the number of posters. Real-World Link In the year 2000, over $236 billion was spent on advertising in the United States. Source: McCann-Erickson, Inc. TRANSPORTATION A taxi fare y can be determined by the equation y = 0.50x + 3.50, where x is the number of miles traveled. g. Graph the equation to find the cost of traveling 8 miles. h. What do the slope and y-intercept represent? i. Is the total fare proportional to the number of miles? Explain. 492 Chapter 9 Algebra: Linear Functions Juan Silva/Getty Images Examples 1, 2 (p. 491) Example 3 (p. 492) Examples 4–6 (p. 492) State the slope and the y-intercept for the graph of each equation. 2. y = -_x - _ 1 6 1. y = x + 2 1 2 3. 2x + y = 3 Graph each equation using the slope and the y-intercept. 4. y = _x - 2 5. y = -_x + 1 5 2 1 3 6. y = -2x + 5 MONEY MATTERS For Exercises 7–9, use the following information. Lydia borrowed $90 from her mother and plans to pay her mother $10 per week. The equation for the amount of money y Lydia owes her mother is y = 90 - 10x, where x is the number of weeks after the loan. 7. Graph the equation to find the amount Lydia owes her mother after 4 weeks. 8. What do the slope and x-intercept represent? 9. Is the amount owed proportional to the number of weeks? Explain. (/-%7/2+ (%,0 For Exercises 10–15 16–21 22–27 See Examples 1, 2 3 4–6 State the slope and the y-intercept for the graph of each equation. 10. y = 3x + 4 13. y = -_x - _ 3 7 1 7 11. y = -5x + 2 12. y = _x - 6 14. y - 2x = 8 15. 3x + y = -4 1 2 Graph each equation using the slope and the y-intercept. 16. y = _x - 5 1 3 3 _ 19. y = x - 4 2 17. y = -x + _ 18. y = -_x + 1 20. y + 2x = -3.5 21. 1.5 = y - 3x 3 2 4 3 BOATING For Exercises 22–24, use the following information. The Lakeside Marina charges a $35 rental fee for a boat, in addition to charging $15 an hour for usage. The total cost y of renting a boat for x hours can be represented by the equation y = 15x + 35. 22. Graph the equation to find the total cost for a 3-hour rental. 23. What do the slope and the y-intercept represent? 24. Is the total cost proportional to the number of hours? Explain. SPACE SCIENCE For Exercises 25–27, use the following information. From 4,074 meters above Earth, a space shuttle glides to the runway. Let y = 4,074 - 47x represent the altitude of the shuttle after x seconds. 25. Graph the equation to find the shuttle’s altitude after 50 seconds. 26. What do the slope and y-intercept represent? 27. Is the altitude propotional to the number of seconds? Explain. Lesson 9-5 Slope-Intercept Form 493 28. INSECTS The equation y = 15x + 37 can be used to approximate the temperature y in degrees Fahrenheit based on the number of chirps x a cricket makes in 15 seconds. Graph the equation to estimate the number of chirps a cricket will make in 15 seconds if the temperature is 80°F. GEOMETRY For Exercises 29–31, use the supplementary angles at the right. y˚ x˚ 29. Write the equation in slope-intercept form. x ⫹ y ⫽ 180˚ 30. Graph the equation. 31. Is the relationship between supplementary angles proportional? Explain. For Exercises 32–34, use the graph at the right. y 32. What is the slope of the line? 33. Identify the y-intercept of the graph. x O 34. What is an equation of the line in slope-intercept form? %842!02!#4)#% See pages 700, 716. WEATHER For Exercises 35–37, use the following information. The equation y = 1.5x + 2 can be used to find the total rainfall in y inches x hours after 12:00 P.M. during a tropical storm. 35. Graph the equation. Self-Check Quiz at ca.gr7math.com H.O.T. Problems 36. State the slope and y-intercept and describe what they represent. 37. Name the x-intercept and describe what it represents. 38. OPEN ENDED Draw the graph of a line that has a y-intercept but no x-intercept. What is the slope of the line? 39. FIND THE ERROR The table shows the results of a science experiment in which water is heated until it is boiling then removed from the heat source. Whose conclusion is correct? Explain. The temperature is proportional to the amount of time. Time (s) Temperature (°C) 0 100° 15 99° 30 98° 45 97° The temperature is not proportional to the amount of time. Aurelio Chantel 40. CHALLENGE A triangle’s original vertices are located at (3, 0), (4, -3), and (1, -4). The triangle is translated 1 unit to the right and 3 units up. It is then reflected across the graph of y = x + 1. Determine the new vertices of the triangle. 41. REASONING What is the slope and y-intercept of a vertical line? 494 Chapter 9 Algebra: Linear Functions (l)Rob Crandall/The Image Works, (r)David Young–Wolff/PhotoEdit */ -!4( Write a real-world problem that involves a linear (*/ 83 *5*/( 42. equation in slope-intercept form. Graph the equation. Explain the meaning of the slope and y-intercept. 43. Which best represents the graph of 44. Which statement could be true for the y = 3x + 4? 21 O 1 2 B C 4 y 3 2 1 321 O 1 2 1 2 3 4x 2 1 y 321 O 1 2 3 4 D 1 2 3 x 4 3 2 1 y Total Earnings A graph below? y 1 2 3 x 3000 2000 1000 x 4 3 2 1 y 4321 O 1 2 0 10 20 30 Sales (thousands of dollars) F Mr. Blackwell will earn $1,750 if his sales are $10,000. 1 2 x G Ms. Chu will not earn any money if she has no sales. H Mr. Montoya earns $250 for every $1,000 he sells. J Ms. James earns $2,500 if she sells $2,500 worth of merchandise. 45. BICYCLING Angel rides her bike 25 miles in 2_ hours. How long will it take her to ride 60 miles? (Lesson 9-4) 1 2 Find the slope of the line that passes through each pair of points. (Lesson 9-3) 46. M(4, 3), N(-2, 1) 47. S(-5, 4), T(-7, 1) 48. X(-9, 5), Y(-2, 5) 49. MEASUREMENT The function y = 0.39x approximates the number of centimeters y in x inches. Make a function table. Then graph the function. (Lesson 9-2) 50. MEASUREMENT When filled to capacity, a cylindrical silo can hold 8,042 cubic feet of grain. The circumference C of the silo is approximately 50.3 feet. Find the height h of the silo to the nearest foot. (Lesson 7-5) PREREQUISITE SKILL Solve each equation. Check your solution. (Lesson 8-2) 51. 3a - 12 = -3 52. -2 = -n + 4 53. -_p - 7 = -3 1 3 54. 4 - _x = 20 1 5 Lesson 9-5 Slope-Intercept Form 495 Extend 9-5 Main IDEA Use technology to investigate situations to determine if they display linear behavior. Standard 7AF1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. Standard 7MR3.3 Develop generalizations of the results obtained and the strategies used and apply them to new problem situations. Graphing Calculator Lab Modeling Linear Behavior Many situations in the real world exhibit linear behavior or behavior in which equal changes in one quantity produce approximately equal changes in another quantity. In this activity, you will examine a situation using a data collection device and a graphing calculator to determine if this situation displays linear behavior. 1 Connect a motion detector to your calculator. Start the data collection program by pressing APPS (CBL/CBR) ENTER , and then select RANGER, Applications, Meters, Dist Match. Place the detector on a desk or table so that it can read the motion of a walker. Mark the floor at a distance of 1 and 6 meters from the detector. Have a partner stand at the 1-meter mark. When you press the button to begin collecting data, have your partner begin to walk away from the detector at a slow but steady pace. Stop collecting data when your partner passes the 6-meter mark. Press ENTER to display a graph of the data. The x-values represent equal intervals of time in seconds. The y-values represent the distances from the detector in meters. ANALYZE THE RESULTS 1. Describe the DISTANCE graph of the collected data. Does the relationship between time and distance appear to be linear? Explain. 2. Use the TRACE feature on your calculator to find the y-intercept on the graph. Interpret its meaning. 496 Chapter 9 Algebra: Linear Functions Horizons Companies Other Calculator Keystrokes at ca.gr7math.com 3. Press STAT 1 and record the time data from List L1 List L2 L1 and the distance data from L2 in a table like the one shown. Then use these data to calculate _ for several pairs of the rate of change distance time points. What do you notice? 4. Does your answer to Exercise 3 support your conclusion about the graph in Exercise 1? Explain. 5. MAKE A PREDICTION Predict how your graph and answers to Exercise 3 would change if the person in the activity were to a. move at a steady but quick pace away from the detector. b. move at a steady but slow pace toward the detector. 6. COLLECT THE DATA Repeat the activity and answer Questions 1 through 3 again for each of the situations described in Exercise 5. 7. MAKE A CONJECTURE How could you change the situation to be one that does not display linear behavior? 8. COLLECT THE DATA Repeat the activity and answer Questions 1 through 3 again for the situation you described in Exercise 7. Families of graphs are graphs that are related in some manner. In this activity, you will study families of linear graphs. 2 Window Pressing ZOOM 6 changes the viewing window for a graph to be [-10, 10] scl:1 by [-10, 10] scl:1 Clear any existing equations from CLEAR . the Y= list by pressing Enter each of the following equations: y = -2x + 4, y = -2x + 1, and y = -2x – 3. Press ZOOM 6 to graph the equations. ANALYZE THE RESULTS 9. Compare the three equations and their graphs. 10. MAKE A CONJECTURE Consider equations of the form y = ax + b, where the value of a is constant but the value of b varies. What do you think is true for the graphs of these equations? 11. Use your calculator to graph y = 2x + 3, y = -x + 3, and y = -3x + 3. Compare the three equations and their graphs. 12. MAKE A CONJECTURE Consider equations of the form y = ax + b, where the value of a varies but the value of b remains constant. What do you think is true for the graphs of these equations? Extend 9-5 Graphing Calculator Lab: Modeling Linear Behavior 497 9- 6 Writing Systems of Equations and Inequalities Main IDEA Write systems of equations and inequalities. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A). Two Internet sites sell the same product for the same price, but their shipping charges differ as shown in the table. Internet Site Base Charge Charge per Pound A $3.00 $1.00 B $1.00 $2.00 The shipping charges can be represented by the following equations and tables where x represents the number of pounds, and y represents the shipping charge. Internet Site A y=x+3 NEW Vocabulary system of equations system of inequalities Internet Site B y = 2x + 1 x y=x+3 y (x, y) x y = 2x + 1 y (x, y) 0 0+3 3 (0, 3) 0 2(0) + 1 1 (0, 1) 1 1+3 4 (1, 4) 1 2(1) + 1 3 (1, 3) 2 2+3 5 (2, 5) 2 2(2) + 1 5 (2, 5) 3 3+3 6 (3, 6) 3 2(3) + 1 7 (3, 7) For Questions 1–3, refer to the tables above. 1. For what number of pounds are the shipping charges the same? 2. For what number of pounds are the shipping charges for Internet Site A less than the ones for Internet Site B? 3. For what number of pounds are the shipping charges for Internet Site A greater than Internet Site B? For Questions 4 and 5, refer to the graphs of the equations at the right. 4. At what point do the two lines intersect? 5. What does this ordered pair represent? x3 5 4 3 2 1 54321 O 1 2 3 4 5 y 2x 1 1 2 3 4 5x Together, the equations y = x + 3 and y = 2x + 1 are called a system of equations. There are two equations and two different unknowns, x and y. The solution of a system of equations is an ordered pair that satisfies each equation. 498 Chapter 9 Algebra: Linear Functions Writing Systems of Equations System of Equations When writing the system, it is important to keep like variables lined up in relation to each other. 1 MOVIES Seven adults and children went to the movies. The number of adults was one more than the number of children. Write a system of equations that represents the number of adults and children. Let a = the number of adults, and let c = the number of children. number of adults plus number of children equals total number of people. a + c = 7 number of adults equals number of children plus one. a = c + 1 So, the system of equations is a + c = 7 and a = c + 1. a. MONEY Jerry has a total of five nickels and dimes in his pocket. The value of the coins is 35 cents. Write a system of equations that represents the number of coins Jerry has. A system of inequalities is similar to a system of equations except that it contains the symbol <, ≤, >, or ≥. Writing Systems of Inequalities 2 ADVERTISING Jeremy wants to advertise in the classified section of two newspapers. He wants to spend less than $35 per newspaper. Newspaper A charges a fee of $5 per day, plus $2.25 per line. Newspaper B charges a fee of $4 per day plus $3.50 per line. Write a system of inequalities to represent what Jeremy will spend for advertising. Let x = the number of lines used, and let y = the number of days the ad runs. Cost per line times number of lines plus cost per day times number of days is less than $35. 2.25x 3.50x + + 5y 4y < < 35 35 So, the system of inequalities is. 2.25x + 5y < 35 3.50x + 4y < 35. b. The number of dimes and quarters is more than 15, but the value of the coins when added together is less than $5. Write a system of inequalities that represents the number of coins. (Hint: Remember that the value of a dime is $0.10 and the value of a quarter is $0.25.) Lesson 9-6 Writing Systems of Equations and Inequalities 499 Example 1 (p. 499) 1. AGE The sum of Sally’s age plus twice Jerry’s age is 48. The difference of Sally’s age minus Jerry’s age is 3. Write a system of equations that represents their ages. 2. BASKETBALL In 2006, Vince Carter and Jason Kidd cost the New Jersey Nets a total of approximately $33 million. If Jason Kidd makes $3 million dollars more than Vince Carter, write a system of equations that represents their salaries. Example 2 (p. 499) 3. CELL PHONES Shantel is considering two rate plans that a cell phone company offers. Plan A offers a standard basic charge plus $0.15 per minute used. Plan B offers the same basic charge plus $15 with unlimited minutes. She needs to spend less than $39 per month. Write a system of inequalities to represent the plans. 4. LANDSCAPING A homeowner is going to seed a new lawn and cover it with straw. The lawn is 3,500 square feet. He can only fit a total of 4 bags of seed or bales of straw in his vehicle at a time. Write a system of inequalities to represent the situation. (/-%7/2+ (%,0 For Exercises 5–7 8–10 See Examples 1 2 Units Coverage (sq. ft) Seed (S) 1 bag 2,000 Straw (R) 1 bale 436 Write a system of equations that represents each situation. 5. PETS A pet store currently has a total of 45 cats and dogs. There are 7 more cats than dogs. 6. PARKS The city park has ordered a total of 22 maple and oak trees to be planted. The total cost for the trees is $620, with maples costing $25 each and oaks costing $32 each. 7. TRACK There are 63 athletes on the high school track teams. There are 7 more girls than boys on the team. Write a system of inequalities that represents each situation. 8. ELECTRONICS A store is ordering two types of stereos. They want to make a total profit of more than $4,800. Model A stereo sells for a profit of $35, and Model B stereo sells for a profit of $75. The store plans on selling at least 110 stereos. %842!02!#4)#% See page 701, 716. 9. SCHOOL SUPPLIES The teacher tries to keep at least 50 pens and pencils in the classroom for students. He likes there to be at least ten more pencils than pens. 10. APPLIANCES A delivery truck can fit no more than 20 washers and dryers at Self-Check Quiz at ca.gr7math.com 500 a time. Washers weigh 175 pounds and dryers weigh 155 pounds. The truck’s maximum capacity is 3,300 pounds. Chapter 9 Algebra: Linear Functions H.O.T. Problems 11. CHALLENGE The solution of a system of 5 4 3 2 1 inequalities is the set of all ordered pairs that satisfies both inequalities. a. Write a system of inequalities for the graph at the right. b. List three solutions of the system. */ -!4( Write a real-world (*/ 83 *5*/( 12. problem that could be represented by a system of equations or inequalities. Explain how the system would be helpful in the situation. 13. Claire baked 36 cookies. There are 8 more chocolate chip cookies than peanut butter. Which system can be used to find the number of each type of cookie? A c + p = 36 p=c+8 C c+p=8 p = c + 36 B c + p = 36 c=p+8 D c+p=8 c = p + 36 y 1 O 5543221 1 2 3 4 55 1 2 3 4 5x 14. Which inequality represents the statement “A number n decreased by 11 is greater than or equal to 53”? F 11 - n ≤ 53 G 11 - n ≥ 53 H n - 11 ≥ 53 J n - 11 ≤ 53 READING For Exercises 15–17, use the following information. (Lesson 9-5) Eric has read 30 pages of a novel. He plans to read 50 pages every evening until he is finished. The equation y = 30 + 50x can be used to represent the number of pages y Eric has read after x days. 15. Graph the equation. 16. Use the graph to find the number of pages Eric will have read after 6 days. 17. What do the slope and y-intercept represent? 18. TRAVEL One and a half hours after leaving its main station, a commuter train has traveled 202.5 miles. At this rate, how far will the train travel after 5 hours? (Lesson 9-4) For the given value, state whether each inequality is true or false. (Lesson 8-6) 19. 18 - n > 4, n = 11 20. 13 + x < 21, x = 8 21. 34 ≤ 5p, p = 7 22. _ ≥ 3, a = -12 a -4 23. PREREQUISITE SKILL A display of video game boxes is stacked in the shape of a pyramid. There are 5 boxes in the top row, 7 boxes in the second row, 9 boxes in the third row, and so on. The display contains 10 rows of boxes. How many boxes are in the display? Use the look for a pattern strategy. (Lesson 2-8) Lesson 9-6 Writing Systems of Equations and Inequalities 501 9- 7 Problem-Solving Investigation MAIN IDEA: Solve problems by using a graph. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Standard 7SDAP1.2 Represent two numerical variables on a scatterplot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables (e.g., between time spent on homework and grade level). e-MAIL: USE A GRAPH YOUR MISSION: Use a graph to solve the problem. THE PROBLEM: Are the highest-rated bikes the most expensive? You want to know whether the highest-rated bikes are the most expensive. -OUNTAIN "IKES 0RICE EXPLORE ▲ JULIA: I am planning to buy a mountain bike. I found a graph with the ratings and the prices of 8 different bikes. 2ATING PLAN SOLVE CHECK Study the graph. The graph shows that the highest-rated bike is not the most expensive bike. Also, the prices of the two bikes with the second-highest rating vary considerably. Look at the graph. The dot farthest to the right is not the highest on the graph. 1. Explain why the bike represented by (48, 300) might be the best bike to buy. 2. Find a graph in a newspaper or magazine, or on the Internet. Write a sentence explaining the information contained in the graph. 502 Chapter 9 Algebra: Linear Functions For Exercises 3 and 4, solve by using a graph. 3. STATISTICS Teenagers were asked which they Students per Computer in U.S. Public Schools Year Students Year Students 1991 20 1996 10 7.8 1992 18 1997 -OST 4IME 3PENT ON %LECTRONIC %NTERTAINMENT 1993 16 1998 6.1 1994 14 1999 5.7 Number of Teenagers spent more time using: their computer, their video game system, or both equally. The graph shows the results of the survey. How many teenagers were surveyed? EDUCATION For Exercises 6 and 7, use the table below. 1995 10.5 2000 5.4 70 60 50 40 30 20 10 0 Source: National Center for Education Statistics 6. Make a graph of the data. Video Games Both Equally 7. Describe how the number of students per Type of Entertainment computer changed from 1991 to 2000. 8. ALGEBRA 4. STATISTICS A zoologist studied extinction times in years of birds on an island. Make a graph of the data. Does the bird with the greatest average number of nests have the greatest extinction time? Average Number of Nests Bird Extinction Time (yr) Cuckoo 1.4 2.5 Magpie 4.5 10.0 Swallow 3.8 2.6 Robin 3.3 4.0 Stonechat 3.6 2.4 Blackbird 4.7 3.3 Tree Sparrow 2.2 1.9 Cost of production The blue line 120 shows the cost 80 of producing Amount from 40 T-shirts. The sales green line 0 4 8 12 16 shows the amount Number of T-shirts of money received from the sale of the T-shirts. How many shirts must be sold to make a profit? Money (dollars) Computer For Exercises 9 and 10, select the appropriate operation(s) to solve the problem. Justify your selection(s) and solve the problem. Use any strategy to solve Exercises 5–8. Some strategies are shown below. 9. COLLEGE Caton’s big brother has a full scholarship for tuition, books, and room and board for four years of college. The total scholarship is $87,500. Room and board cost $9,500 per year. His books cost about $750 per year. What is the cost of his yearly tuition? G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s rn. • Look for a patte • Use a graph. 10. STATISTICS The results of a survey showed 5. MONEY Ming is towing a boat on the back of his minivan. Normally he gets 26 miles per gallon, but pulling the boat decreases his fuel consumption by 10 miles per gallon. If gasoline costs $2.75 per gallon, about how much more does a 520-mile trip cost to pull the boat? that 34% of eighth graders wanted to take an extra language class. The school’s policy says that there must be at least 32 students interested in the class. If 105 eighth graders were surveyed, is this enough students for an extra language class? Lesson 9-7 Problem-Solving Investigation: Use a Graph 503 9-8 Scatter Plots Main IDEA Construct and interpret scatter plots. Standard 7SDAP1.2 Represent two numerical variables on a scatterplot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables (e.g., between time spent on homework and grade level). Measure a partner’s height in inches. Then ask your partner to stand with his or her arms extended parallel to the floor. Measure the distance from the end of the longest finger on one hand to the longest finger on the other hand. Write these measures as the ordered pair (height, arm span) on the board. 1. Graph each of the ordered pairs listed on the board. 2. Examine the graph. Do you think there is a relationship between height and arm span? Explain. NEW Vocabulary scatter plot line of fit A scatter plot is a graph that shows the relationship, if any relationship exists, between two sets of data. In this type of graph, two sets of data are graphed as ordered pairs on a coordinate plane. Scatter plots often show a pattern, trend, or relationship between the variables. #/.#%04 3UMMARY Positive Relationship y O As x increases, y increases. Types of Relationships Negative Relationship y x O No Relationship y x As x increases, y decreases. x O No obvious pattern 1 Explain whether the scatter plot of the data for the hours traveled in a car and the distance traveled shows a positive, negative, or no relationship. As the number of hours you travel increases, the distance traveled increases. Therefore, the scatter plot shows a positive relationship. 504 Laura Sifferlin Chapter 9 Algebra: Linear Functions Distance Traveled (mi) Identify a Relationship Hours Traveled (h) Weight at Birth 2 Explain whether the scatter plot of the data for the month of birth and birth weight show a positive, negative, or no relationship. 10 8 6 4 2 0 Birth weight does not depend on the month of birth. Therefore, the scatter plot shows no relationship. y y il y e y t r r r r ar ar rch pr a un Jul gus be obe be be m m nu ru a A M J Au ptem Oct ve ece Ja Feb M No D Se Month Temperature (˚F) a. Explain whether the scatter plot of the data for time and temperature shows a positive, negative, or no relationship. y 60 40 20 10 0 20 30 x Time (min) Real-World Link The Great Lakes (Superior, Michigan, Huron, Erie, and Ontario) and their connecting waterways form the largest inland water transportation system in the world. A line of fit is a line that is very close to most of the data points. Line of Fit LAKES The water temperatures at various depths in a lake are given. Source: The World Book Water Depth 0 10 20 25 30 35 40 50 Temperature (°F) 75 72 71 64 61 58 53 53 3 Make a scatter plot using 80 Temperature (°F) the data. Then draw a line that best seems to represent the data. Graph each of the data points. Draw a line that fits the data. (25, 64) (35, 58) 60 50 0 10 4 Write an equation for this line of fit. Estimation Drawing a line of fit using the method in this lesson is an estimation. Therefore, it is possible to draw different lines to approximate the same data. 70 20 30 40 50 60 Water Depth (feet) The line passes through points at (25, 64) and (35, 58). Use these points to find the slope, or rate of change, of the line. y -y 2 1 m=_ x -x 2 1 58 - 64 m=_ 35 - 25 3 _ m = -6 or -_ 10 5 Definition of slope (x 1, y 1) = (25, 64), (x 2, y 2) = (35, 58) _ The slope is - 3 , and the y-intercept is 79. 5 The y-intercept is 79 because the line of fit crosses the y-axis at about the point (0, 79). y = mx + b _ y = - 3 x + 79 5 Slope-intercept form 3 The equation for the line of fit is y = -_ x + 79. 5 Extra Examples at ca.gr7math.com Phil Schermeister/CORBIS Lesson 9-8 Scatter Plots 505 5 Use the equation to predict the temperature at a depth of 55 feet. 3 x + 79 y = -_ Equation for the line of fit 5 3 y = -_ (55) + 79 or 46 5 The temperature will be about 46°F. EDUCATION The approximate numbers of high school graduates in Texas over a 10-year period are shown in the table. Graduating Class Number of Graduates Graduating Class Number of Graduates 1994 163,000 1999 203,000 1995 169,000 2000 213,000 1996 172,000 2001 215,000 1997 182,000 2002 225,000 1998 197,000 2003 238,000 Source: Texas Education Agency b. Make a scatter plot of the data. Then draw a line that represents the data. c. Write an equation for a line of fit. d. Use the equation to predict the number of graduates for the graduating class of 2015. Personal Tutor at ca.gr7math.com (pp. 504–505) Explain whether the scatter plot of the data for each of the following shows a positive, negative, or no relationship. y Units Produced 1. 2. 60 40 20 0 10 20 30 40 Fuel Remaining (gal) Examples 1, 2 x y 12 8 4 100 0 Time (hr) Examples 3–5 (pp. 505–506) 300 x 200 Distance Traveled (mi) EDUCATION For Exercises 3–5, use the table. Enrollment in U.S. Public and Private Schools (millions) Year Students Year Students Year Students 3. Draw a scatter plot for 1900 15.5 1940 25.4 1980 41.7 the data and draw a line of fit. 1910 17.8 1950 25.1 1990 40.5 1920 21.6 1960 35.2 2000 46.9 4. Write an equation for 1930 25.7 1970 45.6 the line of fit. 5. Estimate the enrollment in public and private schools in 2010. 506 Chapter 9 Algebra: Linear Functions Extra Examples at ca.gr7math.com 6. 7. y y 40 40 Mileage (mpg) See Examples 1, 2 3–5 30 20 10 30 20 10 2 0 4 6 x 8 0 2 4 Experience (weeks) 8 x 9. y 90 12 11.6 11.2 80 70 60 10.8 0 6 Engine Size (L) Test Score 8. 100-Meter Dash Speed (s) For Exercises 6–9 10–15 Explain whether the scatter plot of the data for each of the following shows a positive, negative, or no relationship. Keyboard Speed (wpm) (/-%7/2+ (%,0 7 8 9 x 10 0 10 20 30 40 Study Time (minutes) Shoe Size FOOD For Exercises 10–12, use the table at the right. Nutritional Information of Commercial Muffins 10. Draw a scatter plot for the Muffin (brand) Fat (grams) Calories A 2 250 B 3 300 C 4 260 D 14 410 E 15 390 F 10 300 G 18 430 H 23 480 I 20 490 data. Then draw a line of fit. 11. Write an equation for the line of fit. 12. Use your equation to estimate the number of fat grams in a muffin with 350 Calories. LIFE EXPECTANCY For Exercises 13–15, use the following table. Year Born 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 1999 2000 Life Expectancy 47.3 50.0 54.1 70.8 73.7 59.7 62.9 68.2 69.7 75.4 76.7 77.1 Source: U.S. Census Bureau 13. Draw a scatter plot for the data. Then draw a line that seems to best fit the data. 14. Write an equation for your line of fit. 15. Use the equation to predict the life expectancy for a person born in 2020. Lesson 9-8 Scatter Plots CORBIS 507 Explain whether a scatter plot of the data for each of the following would show a positive, negative, or no relationship. 16. length of a side of a square and perimeter of the square 17. grade in school and number of pets 18. length of time for a shower and amount of water used 19. outside temperature and amount of heating bill BASEBALL For Exercises 20–22, use the table at the right. Home Runs Runs Batted In A. Jones 51 128 to show the relationship between home runs and runs batted in. A. Rodriguez 48 130 D. Ortiz 47 148 D. Lee 46 107 21. Explain whether you can draw M. Ramirez 45 144 a line of fit to approximate the data. M. Teixeira 43 144 A. Pujols 41 117 A. Dunn 40 101 P. Konerko 40 100 R. Sexton 39 121 Player 20. Make a scatter plot of the data 22. Could you predict the number of runs batted in for a player if you are given the number of home runs hit by that player? Explain. Source: mlb.com 23. SCHOOL Determine the relationship a scatter plot of the data might show. Explain. %842!02!#4)#% Week 1 2 3 4 5 6 7 8 9 Quiz Score 91 91 84 85 90 87 86 97 97 See pages 701, 716. 24. Self-Check Quiz at ca.gr7math.com H.O.T. Problems FIND THE DATA Refer to the California Data File on pages 16–19. Choose some data and make a scatter plot with a line of fit. Use your graph to make predictions about unlisted data. 25. OPEN ENDED Give an example of data that y could be represented by the scatter plot at the right. 26. NUMBER SENSE Suppose a scatter plot shows that as the values of x decrease, the values of y decrease. Does the scatter plot show a positive, negative, or no relationship? O 27. CHALLENGE Determine whether the following statement is always, sometimes, or never true. Justify your answer. A scatter plot that shows a positive relationship suggests that the relationship is proportional. 28. */ -!4( Explain why a scatter plot of skateboard sales and (*/ 83 *5*/( swimsuit sales for each month of the year might show a positive relationship. Does this mean that one factor caused the other? Explain. 508 Chapter 9 Algebra: Linear Functions x 29. A car owner tracked the value of a car 30. The scatter plot shows the cost of fruit using a scatter plot. Franco bought from a produce stand in relation to the weight of the fruit. ,IFETIME 6ALUE OF #AR #OST OF &RUIT #OST 6ALUE THOUSANDS OF DOLLARS &RUIT LB !GE YEARS Which description best represents the relationship of the car’s value? Based on the information in the graph, which statement is a valid conclusion? A negative trend F As Franco bought more pieces of fruit, the cost of the fruit increased. B no trend G As Franco bought fewer pieces of fruit, the cost of the fruit decreased. C positive trend D cannot be determined H As Franco bought fewer pounds of fruit, the number of pieces of fruit decreased. J As Franco bought more pounds of fruit, the cost of the fruit increased. CITIES For Exercises 31–33, use the table. (Lesson 9-7) Largest U.S. Cities City 31. Make a graph of the data. 32. Describe how the population of Detroit, Michigan, changed from 1950 to 2000. 33. Which city had the greatest percent increase from 1950 to 2000? 34. SPORTS There are a total of 36 baseballs and softballs in a bin. There are 5 more softballs than baseballs. Write a system of equations that represents the situation. (Lesson 9-6) Solve each equation. Check your solution. (Lesson 8-5) 2000 1950 New York, NY 8,010,000 7,890,000 Los Angeles, CA 3,690,000 1,970,000 Chicago, IL 2,900,000 3,620,000 Houston, TX 1,950,000 600,000 Philadelphia, PA 1,520,000 2,070,000 Phoenix, AZ 1,320,000 110,000 San Diego, CA 1,220,000 330,000 Dallas, TX 1,190,000 430,000 San Antonio, TX 1,150,000 410,000 Detroit, MI 950,000 1,850,000 Source: U.S. Census Bureau 35. 2x + 16 = 6x 36. 4a - 9 = 7a + 6 37. 5y - 1 = 3y + 11 38. n + 0.8 = -n + 1 Lesson 9-8 Scatter Plots 509 Extend 9-8 Main IDEA Graphing Calculator Lab Scatter Plots A graphing calculator is useful for creating and analyzing scatter plots of large sets of data. Create scatter plots and calculate lines of fit using technology. Standard 7SDAP1.2 Represent two numerical variables on a scatter plot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables (e.g., between time spent on homework and grade level). Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 1 LEISURE The tables give the weekly number of hours spent watching television and weekly number of hours spent exercising for each person in a survey. Make a scatter plot of the data. Weekly Television (h) 17 20 11 10 15 38 5 25 Weekly Exercise (h) 5 4.5 7.5 8 6.5 1 7.5 3 Weekly Television (h) 25 32 5 17 40 28 20 30 Weekly Exercise (h) 2.5 3.5 6 7 0.5 5 4 1.5 Clear the existing data by pressing STAT CLEAR ENTER . ENTER Next enter the data. Input the number of weekly hours spent watching television in L1 and press ENTER . Then enter the weekly hours spent exercising in L2. Turn on the statistical plot by pressing 2nd [STAT PLOT] ENTER ENTER Select the scatter plot and confirm L1 as the Xlist, L2 as the Ylist, and the square as the mark. Graph the data by pressing ZOOM 9. Use the Trace feature and the left and right arrow keys to move from one point to another. ANALYZE THE RESULTS 1. Describe how the data are related. Explain your reasoning. 510 Chapter 9 Algebra: Linear Functions Other Calculator Keystrokes at ca.gr7math.com 2. WEATHER Use a graphing calculator to make a scatter plot of the following weather data. Store the data in L3 and L4 and use Plot 2 to create the graph. Then determine whether the data have a positive, negative, or no relationship. Explain your reasoning. Average Monthly Temperature (°F) 77 42 45 55 57 63 76 65 Average Monthly Rainfall (in.) 6.0 4.8 7 3.2 6.8 4.8 5.7 7.2 Average Monthly Temperature (°F) 67 73 51 81 84 86 64 43 Average Monthly Rainfall (in.) 2.6 5.5 5.9 6.3 7.9 4.2 6.3 4.5 2 LEISURE Find and graph a line of fit for the data in Activity 1. Access the CALC menu by pressing STAT . Select 4 to find a line of fit in the form y = ax + b. Press 2nd [L1] , 2nd [L2] ENTER to find a line of fit for the data in lists L1 and L2. Graph the line of fit in Y1 by pressing and then VARS 5 to access the Statistics… menu. Use the and ENTER keys to select EQ and then press 1 to select RegEQ, the line of fit equation. Finally, press '2!0( . ANALYZE THE RESULTS 3. MAKE A PREDICTION Use the TRACE feature to predict the average number of hours of exercise someone who watches 35 hours of television would get. 4. COLLECT THE DATA Collect a set of data that can be represented in a scatter plot. Enter the data in a graphing calculator. Determine whether the data have a positive, negative, or no relationship. Then use the calculator to find a line of fit and to make a prediction. Extend 9-8 Graphing Calculator Lab: Scatter Plots 511 CH APTER 9 Study Guide and Review Download Vocabulary Review from ca.gr7math.com Key Vocabulary Be sure the following Key Concepts are noted in your Foldable. constant of variation -INEAR 'UNCTIONS Key Concepts Functions (Lessons 9-1 and 9-2) • A function is a relationship in which one value is dependent upon another. rise (p. 477) (p. 483) run (p. 477) direct variation (p. 483) scatter plot (p. 504) domain (p. 466) slope (p. 477) function (p. 465) slope-intercept form (p. 491) function table (p. 466) system of equations linear function (p. 472) line of fit (p. 505) (p. 498) y-intercept (p. 491) range (p. 466) • Functions can be represented by words, equations, tables, ordered pairs, and graphs. Slope (Lesson 9-3) • The slope m of a line passing through points (x 1, y 1) and (x 2, y 2) is the ratio of the difference in the y-coordinates to the corresponding difference in the x-coordinates. Vocabulary Check Direct Variation (Lesson 9-4) 1. The (domain, range) is the set of input • A direct variation is a relationship in which the ratio of y to x is a constant, k. Slope-Intercept Form (Lesson 9-5) Choose the correct term or number to complete each sentence. values of a function. 2. The range is the set of (input, output) values of a function. • An equation written in slope-intercept form is written as y = mx + b. 3. A relationship where one thing depends • When an equation is written in slope-intercept form, m is the slope and b is the y-intercept. 4. A (scatter plot, function table) is a graph Systems of Equations (Lesson 9-6) • Two equations together are called a system of equations. on another is called a (function, slope). that shows the relationship between two sets of data. 5. The (x-intercept, y-intercept) has the coordinates (0, b). (y - y x - x ) _ 6. The slope formula is _ x2 - x1 , y2 - y1 . 2 Scatter Plots (Lesson 9-8) • In a positive relationship, x increases and y increases. • In a negative relationship, x increases and y decreases. • In a no relationship, no obvious pattern exists between x and y. 1 2 1 7. A line that is very close to most of the data points in a scatter plot is called a (line of fit, y-intercept). 8. The (rise, run) is the vertical change between two points on a line. 9. A(n) (dependent, independent) variable is the variable for the output of a function. 512 Chapter 9 Algebra: Linear Functions Vocabulary Review at ca.gr7math.com Lesson-by-Lesson Review 9-1 Functions (pp. 465–469) Example 1 Complete the function table for f(x) = 2x - 1. Then state the domain and range of the function. Find each function value. 10. f(3) if f(x) = 3x + 1 11. f(-11) if f(x) = -2x 12. f(2) if f(x) = _x - 4 1 2 x 2x - 1 f(x) -2 2(-2) - 1 -5 13. FITNESS Wilson’s Fitness Club charges 0 2(0) - 1 -1 $20 for a membership fee plus $28 a month. Write a function to represent the cost c for n months. How much would it cost if you belonged to the club for 9 months? 1 2(1) - 1 1 5 2(5) - 1 9 Domain: {-2, 0, 1, 5} Range: {-5, -1, 1, 9} 14. Complete the function table for f(x) = 3x + 2. Then state the domain and the range of the function. x 3x + 2 y -2 0 1 5 9-2 Representing Linear Functions (pp. 471–476) Graph each function. 15. y = -2x + 1 1 16. y = _x - 2 2 17. MEASUREMENT The function y = 4x represents the perimeter y of a square with sides x units long. Graph the function. Example 2 Graph y = 3 - x. x 3-x y (x, y) -1 3 - (-1) 4 (-1, 4) 0 3-0 3 (0, 3) 2 3-2 1 (2, 1) 3 3-3 0 (3, 0) y y 3 x O x 18. CANDY A regular fruit smoothie x costs $1.50, and a large fruit smoothie y costs $3. Graph the function 1.5x + 3y = 12 to determine how many of each type of fruit smoothie Lisa can buy with $12. Chapter 9 Study Guide and Review 513 CH APTER 9 Study Guide and Review 9-3 Slope (pp. 477–482) Find the slope of each line that passes through each pair of points. 19. A(-2, 3), B(-1, 5) Example 3 Find the slope of the line that passes through A(-3, 2) and B(5, -1). y -y 1 2 m=_ x -x 20. G(6, 2), H(1, 5) 2 21. Q(2, 1), R(3, -5) 1 -1 - 2 m=_ 22. SLIDES Find the slope of a slide that descends 8 feet for every horizontal change of 14 feet. 5 - (-3) 3 _ m = -3 or -_ 8 8 Definition of slope (x 1, y 1) = (-3, 2), (x 2, y 2) = (5, -1) Simplify. 14 ft 8 ft 23. ANIMALS A lizard is crawling up a hill that rises 5 feet for every horizontal change of 30 feet. Find the slope of the hill. 9-4 Direct Variation (pp. 483–488) 24. TIME It takes Gabriella 4 hours to knit 6 scarves. Assuming that the number of scarves made varies directly as the time spent knitting, how many scarves will she make in 6 hours? 25. MONEY Josiah spent $15.60 on 3 comic books. The next time, he spent $10.40 on 2 comic books. What is the cost for each comic book? 26. FRUIT The cost of peaches varies directly with the number of pounds bought. If 3 pounds of peaches cost $4.50, find the cost of 5.5 pounds. 514 Chapter 9 Algebra: Linear Functions Example 4 Mrs. Dimas paid $6.48 for 8 apples. The next weekend, she paid $9.72 for 12 apples. What is the cost of each apple? $0.81 $6.48 _ or _ 8 apples 1 apple $9.72 $0.81 _ or _ 12 apples So, each apple costs $0.81. 1 apple Mixed Problem Solving For mixed problem-solving practice, see page 716. 9-5 Slope-Intercept Form (pp. 491–495) State the slope and y-intercept for the graph of each equation. 27. y = 2x + 5 28. y = -_x + 6 29. y - 4x = 7 30. 3x + y = -2 1 5 31. MONEY Malik had $100 in his savings account. He plans to add $25 each week. The equation for the amount of money y Malik has in his savings account is y = 100 + 25x, where x is the number of weeks. Graph the equation. Example 5 State the slope and y-intercept of the graph of y = - 1 x + 3. _ 1 y = -_ x+3 2 2 Write the equation. y = mx + b 1 , and the The slope of the graph is -_ 2 y-intercept is 3. 32. BIRDS The altitude in feet y of an albatross that is slowly landing can be given by y = 400 - 100x, where x represents the time in minutes. State the slope and y-intercept of the graph of the equation and describe what they represent. 9-6 Writing Systems of Equations and Inequalities (pp. 498–501) 33. FOOD Twenty-five teenagers were surveyed. There were six more who preferred pizza than preferred steak. Write a system of equations that represents this situation. 34. CELL PHONES Sheryl is considering two different cell phone plans. The first plan costs $19 per month plus $0.10 per minute. The second plan costs $0.15 per minute with no monthly base fee. Write a system of equations that represents the total cost of these two plans. Example 6 There are seven more red fruit pops in a bag than orange ones. There are 53 fruit pops in the bag. Write a system of equations to represent the number of fruit pops. Let r = number of red fruit pops and g = number of orange fruit pops. r + g = 53 There is a total of 53 fruit pops. r=g+7 There are 7 more red fruit pops than orange. 35. RATES A video rental store offers two plans. Plan 1 charges a basic fee plus $1.25 per day. Plan 2 charges twice the basic fee and $0.50 per day. If a customer wants to spend less than $7.50, write a system of inequalities representing each situation. Chapter 9 Study Guide and Review 515 CH APTER 9 Study Guide and Review 9-7 PSI: Use a Graph (pp. 502–503) 36. BASKETBALL The graph shows the number of points scored in the first seven basketball games. What is the average number of points scored so far this season? Example 7 The graph shows the heights of maple trees. Find the average height of the trees. Round to the nearest tenth. 4REES "ASKETBALL (EIGHT FEET .UMBER OF 0OINTS -APLE 4REES 'AME Add the heights: 21 + 24 + 26 + 18 + 29 + 30 + 23 + 28 or 199 199 Divide: _ or 24.9 8 So, the average height is 24.9 feet. Scatter Plots (pp. 504–509) Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. 37. day of the week and temperature 38. child’s age and grade level in school 39. temperature outside and amount of clothing 40. ATTENDANCE Use the table to draw a scatter plot and a line of fit for the data. 516 Volleyball Game 1 Number of Students 28 30 37 35 36 39 40 2 3 4 Chapter 9 Algebra: Linear Functions 5 6 7 Example 8 Determine whether the graph at the right shows a positive, negative, or no relationship. Birth Weight (pounds) 9-8 10 9 8 7 6 5 0 10 20 30 Day of the Month Born Since there is no obvious pattern, there is no relationship. CH APTER 9 Practice Test Find each function value. STANDARDS PRACTICE Which is the graph of y = -3x? 15. 1. f(3) if f(x) = -2x + 6 y F 2. f(0) if f(x) = 3x - 1 y H 3. f(-2) if f(x) = _ + 5 x 2 directly proportional to the time she works. If she earns $187.50 after working 25 hours, how much will she earn working 30 hours? x O O 4. JOBS The amount Jerri earns working is y G O O x y J 5. RAIN By 6 P.M., 3 inches of rain had fallen. For the next 3 hours, 0.5 inch of rain fell per hour. How many inches fell by 9 P.M.? x O x O Graph each function. 6. y = -2x + 5 7. y = _x - 1 1 3 Find the slope of the line that passes through each pair of points. 8. A(-2, 5), B(-2, 1) 10. 9. E(2, -1), F(5, -3) STANDARDS PRACTICE Rico planted 18 flowers in 30 minutes. At the same rate, how many flowers would he plant in 55 minutes? A 30 B 33 C 36 D 38 CHILD CARE For Exercises 11–13, use the following information. The cost per child at a day care center is $35 a day plus a registration fee of $50. The cost c for d days of child care is c = 35d + 50. 11. Graph the equation to find the cost for 5 days. 12. What do the slope and y-intercept represent? 13. Is the cost proportional to the number of days? Explain. 14. MONEY Robert has 26 coins that are all nickels and dimes. The value of the coins is $1.85. Write a system of equations that represents this situation. Chapter Test at ca.gr7math.com SALES For Exercises 16 and 17, use the table. New Customers Month Customers Month Customers Jan 542 Jul 631 Feb 601 Aug 620 Mar 589 Sep 723 Apr 610 Oct 754 May 648 Nov 885 June 670 Dec 1,027 16. Make a graph of the data. 17. Describe how the number of new customers changed from January to December. TRAVEL For Exercises 18–20, use the table. Distance (mi) 50 100 150 200 250 Gas (gal) 2 6 8 15 18 18. Draw a scatter plot for the data and draw a line of fit. 19. Write an equation for the line of fit. 20. Use your equation to estimate the amount of gas needed to travel 375 miles. Chapter 9 Practice Test 517 CH APTER 9 California Standards Practice Cumulative, Chapters 1–9 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1 4 Which statement is true about the slope of line RT ? A pattern of equations is shown below. Which statement best describes this pattern of equations? R 54321 O 1 2 3 4 5 80% of 62.5 is 50 40% of 125 is 50 20% of 250 is 50 10% of 500 is 50 C When the percent is increased by 2 and the other number remains the same, the answer is 50. The area of a square is 20 square inches. Which best represents the length of a side of the square? 1 2 3 4 5x S T G The slope between point R and point S is greater than the slope between point S and point T. B When the percent is halved and the other number is halved, the answer is 50. 2 y F The slope is the same between any two points. A When the percent is halved and the other number is doubled, the answer is 50. D When the percent remains the same and the other number is increased by 2, the answer is 50. 5 4 3 2 1 H The slope between point R and point T is greater than the slope between point S and point T. J The slope is positive. 5 The graph of the line y = -2x + 1 is shown below. Which table of ordered pairs contains only points on this line? y F 4.5 inches x O G 5 inches H 10 inches J 11 inches 3 Beth’s monthly charge for Internet access c can be found using the equation c = 12 + 2.50h, where h represents the number of hours of usage during a month. What is the total charge for a month in which Beth used the Internet for 9 hours? A $39.95 C $27.00 B $34.50 D $22.50 518 Chapter 9 Algebra: Linear Functions A x y -2 5 -1 3 0 -1 B x y -2 3 -1 1 0 -1 C x y -1 -3 0 -1 1 1 D x y -1 3 0 1 1 -1 California Standards Practice at ca.gr7math.com More California Standards Practice For practice by objective, see pages CA1-CA39. 6 A truck used 6.3 gallons of gasoline to travel 107 miles. How many gallons of gasoline would it need to travel an additional 250 miles? F 8.4 gal H 18.9 gal G 14.7 gal J 21.0 gal 8 4 The slope of the line shown below is _ . 5 y n 15 x 0 Question 6 When working with units of measurement, remember to write the units to ensure that the numbers are compared correctly. F 4 G 8 H 12 Which of the following conclusions about the number of rebounds per game and the height of a player is best supported by the scatter plot below? Number of Rebounds (per game) 7 What is the value of n? J 16 Pre-AP Record your answers on a sheet of paper. Show your work. 12 11 10 9 8 7 6 5 9 5‘8” 5‘10” 6‘ Study the data in the table. 6‘2” 6‘4” Height Date Number of Customers Scoops Sold June 1 75 100 June 2 125 230 460 June 3 350 A The number of rebounds increases as the player’s height decreases. June 4 275 370 June 5 175 300 B The number of rebounds is unchanged as the player’s height increases. June 6 225 345 June 7 210 325 C The number of rebounds increases as the player’s height increases. a. What type of display would be most D There is no relationship between the number of rebounds and the player’s height. b. Graph the data. appropriate for this data? c. Describe the relationship of the data. NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 Go to Lesson... 2-8 3-2 9-1 9-3 9-5 9-4 9-8 9-3 9-8 For Help with Standard... MR2.4 NS2.4 MR2.5 AF3.3 AF3.3 AF4.2 SDAP1.2 AF3.3 SDAP1.2 Chapter 9 California Standards Practice 519 Algebra: Nonlinear Functions and Polynomials 10 • Standard 7AF2.0 Interpret and evaluate expressions involving integer powers and simple roots. • Standard 7NS2.0 Use exponents, powers, and roots and use exponents in working with fractions. Key Vocabulary cube root (p. 554) nonlinear function (p. 522) quadratic function (p. 528) Real-World Link Fountains Many real-world situations, such as this fountain at Paramount’s Great America theme park in Santa Clara California, cannot be modeled by linear functions. These can be modeled using nonlinear functions. Algebra: Nonlinear Functions and Polynomials Make this Foldable to help you organize your notes. Begin with eight sheets of grid paper. 1 Cut off one section of the grid paper along both the long and short edges. 2 Cut off two sections from the second sheet, three sections from the third sheet, and so on to the 8th sheet. 3 Stack the sheets from narrowest to widest. 4 Label each of the right tabs with a lesson number. 520 Chapter 10 Algebra: Nonlinear Functions and Polynomials Richard Cummins/SuperStock GET READY for Chapter 10 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Graph each equation. (Lesson 11-2) Example 1 1. y = x – 4 Graph y = x + 1. 2. y = 2x First, make a table of values. Then, graph the ordered pairs and connect the points. 3. y = x + 2 x y (x, y) y = 2.54x describes about how many centimeters y are in x inches. Graph the function. 0 1 (0, 1) 1 2 (1, 2) 2 3 (2, 3) (Lessons 11-2) 3 4 (3, 4) 4. MEASUREMENT The equation Write each expression using a positive exponent. (Lesson 2-9) 5. a -9 7. x -5 6. 6 -4 8. 5 -2 y yx1 O Example 2 Write n -3 using a positive exponent. 1 n -3 = _ 3 n definition of negative exponent Write each expression using exponents. (Lesson 2-9) Example 3 9. 6 · 6 · 6 · 6 5 is multiplied by itself 3 times and 4 is multiplied by itself 2 times. So, 5 · 4 · 5 · 4 · 5 = 5 3 · 4 2. 10. 3 · 7 · 7 · 3 · 7 11. FUND-RAISER The students at x Write 5 · 4 · 5 · 4 · 5 using exponents. Hampton Middle School raised 8 · 8 · 2 · 8 · 2 dollars to help build a new community center. How much money did they raise? (Lesson 2-9) Chapter 10 Get Ready for Chapter 10 521 Linear and Nonlinear Functions 10-1 Main IDEA ROCKETRY The tables show the flight data for a model rocket launch. The first table gives the rocket’s height at each second of its ascent, or upward flight. The second table gives its height as it descends back to Earth using a parachute. Determine whether a function is linear or nonlinear. Preparation for AF1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. NEW Vocabulary nonlinear function Ascent Time (s) Descent Height (m) Time (s) Height (m) 0 0 7 140 1 38 8 130 2 74 9 120 3 106 10 110 4 128 11 100 5 138 12 90 6 142 13 80 1. During its ascent, did the rocket travel the same distance each second? Justify your answer. 2. During its descent, did the rocket travel the same distance each second? Justify your answer. 3. Graph the ordered pairs (time, height) for the rocket’s ascent and descent on separate axes. Connect the points with a straight line or smooth curve. Then compare the graphs. REVIEW Vocabulary constant rate of change occurs when the rate of change between any two data points is proportional. (Lesson 4-10) In Lesson 9-2, you learned that linear functions have graphs that are straight lines. These graphs represent constant rates of change. Nonlinear functions are functions that do not have constant rates of change. Therefore, their graphs are not straight lines. Identify Functions Using Tables Determine whether each table represents a linear or nonlinear function. Explain. 1 +2 +2 +2 x y 2 50 4 35 6 20 8 5 2 -15 +3 -15 +3 -15 +3 As x increases by 2, y decreases by 15 each time. The rate of change is constant, so this function is linear. 522 Doug Martin Chapter 10 Algebra: Nonlinear Functions and Polynomials x y 1 1 4 16 7 49 10 100 +15 +33 +51 As x increases by 3, y increases by a greater amount each time. The rate of change is not constant, so this function is nonlinear. Determine whether each table represents a linear or nonlinear function. Explain. a. x 0 5 10 15 y 20 16 12 8 b. x 0 2 4 6 y 0 2 8 18 Identify Functions Using Graphs Determine whether each graph represents a linear or nonlinear function. Explain. y 3 y 4 x y2 1 y 0.5x 2 x O x O The graph is a curve, not a straight line. So, it represents a nonlinear function. This graph is also a curve. So, it represents a nonlinear function. Determine whether each graph represents a linear or nonlinear function. Explain. c. d. y O e. y x O y x O x Recall that the equation for a linear function can be written in the form y = mx + b, where m represents the constant rate of change. Identifying Linear Equations Always examine an equation after it has been solved for y to see that the power of x is 1 or 0. Then check to see that x does not appear in the denominator. Identify Functions Using Equations Determine whether each equation represents a linear or nonlinear function. Explain. _ 6 y = 6x 5 y=x+4 Since the equation can be written as y = 1x + 4, this function is linear. f. y = 2x 3 + 1 Extra Examples at ca.gr7math.com g. y = 3x The equation cannot be written in the form y = mx + b. So, this function is nonlinear. h. y = _ x 5 Lesson 10-1 Linear and Nonlinear Functions 523 7 BASKETBALL Use the table to determine whether Round(s) of play Teams 1 32 Examine the differences between the number of teams for each round. 2 16 3 8 16 - 32 = -16 4 - 8 = -4 4 4 5 2 the number of teams is a linear function of the number of rounds of play. 8 - 16 = -8 2 - 4 = -2 While there is a pattern in the differences, they are not the same. Therefore, this function is nonlinear. Graph the data to verify the ordered pairs do not lie on a straight line. Real-World Link The NCAA women’s basketball tournament begins with 64 teams and consists of 6 rounds of play. y 32 24 Teams Check 16 8 0 2 4 6 8 x Rounds of Play i. TICKETS Tickets to the school dance cost $5 per student. Are the ticket sales a linear function of the number of tickets sold? Explain. Number of Tickets Sold 1 2 Ticket Sales $5 $10 $15 3 Personal Tutor at ca.gr7math.com Determine whether each table, graph, or equation represents a linear or nonlinear function. Explain. Examples 1–6 1. (pp. 522–523) x 0 1 2 3 y 1 3 6 10 3. x 0 3 6 9 y -3 9 21 33 4. y O 2. y x O 5. y = _ x 3 Example 7 (p. 524) 524 6. y = 2x 2 7. MEASUREMENT The table shows the measures of the sides of several rectangles. Are the widths of the rectangles a linear function of the lengths? Explain. Chapter 10 Algebra: Nonlinear Functions and Polynomials Elise Amendola/AP/Wide World Photos x Length (in.) 1 4 8 10 Width (in.) 64 16 8 6.4 (/-%7/2+ (%,0 For Exercises 8–13 14–19 20–25 26–29 See Examples 1, 2 3, 4 5, 6 7 Determine whether each table, graph, or equation or represents a linear or nonlinear function. Explain. 8. 10. 12. x 3 6 9 12 y 12 10 8 6 x 5 10 15 20 y 13 28 43 58 x 2 4 6 8 y 10 12 16 24 y 14. 9. 11. 13. 15. x 1 2 3 4 y 1 4 9 16 x 1 3 y -2 -18 -50 -98 x 4 8 12 16 y 3 0 -3 -6 5 7 16. y y x O 17. 18. y O x 19. y y x O x O 20. y = x 3 - 1 21. y = 4x 2 + 9 22. y = 0.6x 23. y = _ 24. y = _ x 25. y = _ x +5 3x 2 x O x O 8 4 26. TRAVEL The Guzman family drove from Sacramento to Yreka. Use the table to determine whether the distance driven is a linear function of the hours traveled. Explain. Time (h) 1 2 3 4 Distance (mi) 65 130 195 260 27. BUILDINGS The table shows the height of several buildings in Chicago, Illinois. Use the table to determine whether the height of the building is a linear function of the number of stories. Explain. Stories Height (ft) Harris Bank III 35 510 One Financial Place 40 515 Kluczynski Federal Building 45 545 Mid Continental Plaza 50 582 North Harbor Tower 55 556 Building Source: The World Almanac Lesson 10-1 Linear and Nonlinear Functions 525 MEASUREMENT For Exercises 28 and 29, use the following information. Recall that the circumference of a circle is equal to pi times its diameter and that the area of a circle is equal to pi times the square of its radius. 28. Is the circumference of a circle a linear or nonlinear function of its diameter? Explain your reasoning. 29. Is the area of a circle a linear or nonlinear function of its radius? Explain your reasoning. For Exercises 30–34, determine whether each equation or table represents a linear or nonlinear function. Explain. 30. y - x = 1 33. 32. y = 2 x 31. xy = -9 x 0.5 1 1.5 2 y 15 8 1 -6 34. x -4 0 4 y 2 1 -1 -4 35. FOOTBALL The graphic shows 36. MEASUREMENT Make a graph :fcc\^\9fnc>Xd\j 8m\iX^\8kk\e[XeZ\ 6ISITORS the decrease in the average attendance at college bowl games from 1983 to 2003. Would you describe the decline as linear or nonlinear? Explain. 8 showing the area of a square as a function of its perimeter. Explain whether the function is linear. 9EAR Source: USA Today %842!02!#4)#% 37. GRAPHING Water is poured at a constant rate into the vase at the right. Draw a graph of See pages 702, 717. the water level as a function of time. Is the water level a linear or nonlinear function Self-Check Quiz at of time? Explain. ca.gr7math.com H.O.T. Problems 38. CHALLENGE True or false? All graphs of straight lines are linear functions. Explain your reasoning or provide a counterexample. 39. Which One Doesn’t Belong? Identify the function that is not linear. Explain your reasoning. y = 2x y = x2 y -2 = x x-y=2 40. OPEN ENDED Give an example of a nonlinear function using a table of values. 41. */ -!4( Describe two methods for determining whether a (*/ 83 *5*/( function is linear given its equation. 526 Chapter 10 Algebra: Nonlinear Functions and Polynomials 42. Which equation describes the data in 43. Which equation represents a nonlinear the table? function? x -7 -5 -3 0 4 y 50 1 17 26 10 A 5x + 1 = y C x2 + 1 = y B xy = 68 D -2x 2 + 8 = y F y = 3x + 1 x G y=_ 3 H 2xy = 10 J y = 3(x - 5) STATISTICS Determine whether a scatter plot of the data for the following might show a positive, negative, or no relationship. (Lesson 9-8) 44. grade on a test and amount of time spent studying 45. age and number of siblings 46. number of Calories burned and length of time exercising IC !R AB ISH AN I ND SH 3P 51. 8.1a + 2.3 = 5.1a - 3.1 GLI 50. 13.4w + 17 = 5w - 4 %N ND 49. 7k + 12 = 8 - 9k -A 48. 1 - 3c = 9c + 7 ARI (Lesson 8-4) (I Solve each equation. Check your solution. ,ANGUAGES 3POKEN BY .ATIVE 3PEAKERS N languages spoken by at least 100 million native speakers worldwide. What conclusions can you make about the number of Mandarin native speakers and the number of English native speakers? (Lesson 9-7) .ATIVE 3PEAKERS MILLIONS 47. LANGUAGES The graph shows the top five ,ANGUAGES 52. 4.1x - 23 = -3.9x - 1 53. 3.2n + 3 = -4.8n - 29 Source: The World Almanac For Kids 54. PARKS A circular fountain in a park has a diameter of 8 feet. The park director wants to build a fountain that has an area four times that of the current fountain. What will be the diameter of the new fountain? (Lesson 7-1) 55. MEASUREMENT The cylindrical air duct of a large furnace has a diameter of 30 inches and a height of 120 feet. If it takes 15 minutes for the contents of the duct to be expelled into the air, what is the volume of the substances being expelled each hour? (Lesson 7-5) PREREQUISITE SKILL Graph each equation. (Lesson 9-2) 56. y = 2x 57. y = x + 3 58. y = 3x - 2 59. y = _x + 1 1 3 Lesson 10-1 Linear and Nonlinear Functions 527 10-2 Graphing Quadratic Functions Main IDEA Graph quadratic functions. Standard 7AF1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. Standard 7AF3.1 Graph functions of the form y = nx 2 and y = nx 3 and use in solving problems. You know that the area A of a square is equal to the length of a side s squared, A = s 2. Copy and complete the table. s s2 (s, A) 0 0 (0, 0) 1 1 (1, 1) 2 3 Graph the ordered pairs from the table. Connect them with a smooth curve. 4 5 6 1. Is the relationship between the side length and the area of a square linear or nonlinear? Explain. 2. Describe the shape of the graph. NEW Vocabulary quadratic function A quadratic function, like A = s 2, is a function in which the greatest power of the variable is 2. Its graph is U-shaped, opening upward or downward. The graph opens upward if the coefficient of the variable that is squared is positive, downward if it is negative. Graph Quadratic Functions Quadratic Fuctions The graph of a quadratic function is called a parabola. 1 Graph y = x 2. To graph a quadratic function, make a table of values, plot the ordered pairs, and connect the points with a smooth curve. y x x2 y (x, y) -2 (-2) 2 = 4 4 (-2, 4) -1 2 (-1) = 1 1 (-1, 1) 0 (0) 2 = 0 0 (0, 0) 1 2 (1) = 1 1 (1, 1) 2 (2) 2 = 4 4 (2, 4) y x2 x O 2 Graph y = -2x 2. x -2x 2 -2 -2(-2) 2 = -8 (x, y) 4 -8 (-2, -8) 2 -2 (-1, -2) 0 -2(0) 2 = 0 0 (0, 0) ⫺4 1 -2(1) 2 = -2 -2 (1, -2) ⫺8 2 2 -8 (2, -8) -1 -2(-1) = -2 528 y -2(2) = -8 Chapter 10 Algebra: Nonlinear Functions and Polynomials y O ⫺8 ⫺4 4 8x y 2x 2 ⫺12 Extra Examples at ca.gr7math.com READING in the Content Area For more strategies in reading this lesson, visit ca.gr7math.com. 3 Graph y = x 2 + 2. x2 + 2 y (x, y) -2 (-2) 2 + 2 = 6 6 (-2, 6) 2 3 (-1, 3) 2 2 (0, 2) 2 3 (1, 3) 2 6 (2, 6) x -1 0 1 2 (-1) + 2 = 3 (0) + 2 = 2 (1) + 2 = 3 (2) + 2 = 6 y y x2 2 x O 4 Graph y = -x 2 + 4. x -x 2 + 4 y (x, y) -2 -(-2) 2 + 4 = 0 0 (-2, 0) -1 -(-1) 2 + 4 = 3 3 (-1, 3) 0 2 4 (0, 4) 2 -(0) + 4 = 4 1 -(1) + 4 = 3 3 (1, 3) 2 -(2) 2 + 4 = 0 0 (2, 0) y y x2 4 x O Graph each function. a. y = 6x 2 b. y = x 2 - 2 c. y = -2x 2 - 1 5 MONUMENTS The function h = 0.66d 2 represents the distance d in miles you can see from a height of h feet. Graph this function. Then use your graph and the information at the left to estimate how far you could see from the top of the Eiffel Tower. Distance cannot be negative, so use only positive values of d. Source: structurae.de h = 0.66d 2 0 2 0.66(0) = 0 (0, 0) 1,000 10 2 0.66(10) = 66 (10, 66) 800 20 2 0.66(20) = 264 (20, 264) 25 2 0.66(25) = 412.5 (25, 412.5) 30 0.66(30) 2 = 594 (30, 594) 400 35 0.66(35) 2 = 808.5 (35, 808.5) 200 40 2 (40, 1,056) 0.66(40) = 1,056 (d, h) Height (ft) Real-World Link The Eiffel Tower in Paris, France, opened in 1889 as part of the World Exposition. It is about 986 feet tall. h d 600 0 10 20 30 40 d Distance (mi) At a height of 986 feet, you could see approximately 39 miles. d. TOWERS The outdoor observation deck of the Space Needle in Seattle, Washington, is 520 feet above ground level. Estimate how far you could see from the observation deck. Personal Tutor at ca.gr7math.com Lesson 10-2 Graphing Quadratic Functions Lance Nelson/CORBIS 529 Examples 1–4 (pp. 528–529) Example 5 (p. 529) (/-%7/2+ (%,0 For Exercises 8–11 12–19 20, 21 See Examples 1, 2 3, 4 5 Graph each function. 1. y = 3x 2 2. y = -5x 2 3. y = -4x 2 4. y = -x 2 + 1 5. y = x 2 - 3 6. y = -x 2 + 2 7. CARS The function d = 0.006s 2 represents the braking distance d in meters of a car traveling at a speed s in kilometers per second. Graph this function. Then use your graph to estimate the speed of the car if its braking distance is 12 meters. Graph each function. 8. y = 4x 2 9. y = 5x 2 10. y = -3x 2 11. y = -6x 2 12. y = x 2 + 6 13. y = x 2 - 4 14. y = -x 2 + 2 15. y = -x 2 - 5 16. y = 2x 2 - 1 17. y = 2x 2 + 3 18. y = -4x 2 - 1 19. y = -3x 2 + 2 20. RACING The function d = _at 2 represents the distance d that a race car will 1 2 travel over an amount of time t given the rate of acceleration a. Suppose a car is accelerating at a rate of 5 feet per second every second. Graph this function. Then use your graph to find the time it would take the car to travel 125 feet. 21. WATERFALLS The function d = -16t 2 + 182 models the distance d in feet a drop of water falls t seconds after it begins its descent from the top of the 182-foot high American Falls in New York. Graph this function. Then use your graph to estimate the time it will take the drop of water to reach the river at the base of the falls. Graph each function. 22. y = 0.5x 2 + 1 23. y = 1.5x 2 24. y = 4.5x 2 - 6 25. y = _x 2 - 2 26. y = _x 2 27. y = -_x 2 + 1 1 3 1 2 1 4 MEASUREMENT For Exercises 28 and 29, write a function for each of the following. Then graph the function in the first quadrant. %842!02!#4)#% See pages 702, 717. 28. The surface area of a cube is a function of the edge length a. Use your graph to estimate the edge length of a cube with a surface area of 54 square centimeters. 29. The volume V of a rectangular prism with a square base and a fixed height Self-Check Quiz at ca.gr7math.com 530 of 5 inches is a function of the base edge length s. Use your graph to estimate the base edge length of a prism whose volume is 180 cubic inches. Chapter 10 Algebra: Nonlinear Functions and Polynomials H.O.T. Problems CHALLENGE The graphs of quadratic functions may have exactly one highest point, called a maximum, or exactly one lowest point, called a minimum. Graph each quadratic equation. Determine whether each graph has a maximum or a minimum. If so, give the coordinates of each point. 30. y = 2x 2 + 1 31. y = -x 2 + 5 32. y = x 2 - 3 33. OPEN ENDED Write and graph a quadratic function that opens upward and has its minimum at (0, -3.5). */ -!4( Write a quadratic function of the form y = ax 2 + c and (*/ 83 *5*/( 34. explain how to graph it. 35. Which graph represents the function y = -0.5x 2 - 2? y A O y B y C y D x O x x O O x Determine whether each equation represents a linear or nonlinear function. (Lesson 10-1) 36. y = x - 5 37. y = 3x 3 + 2 38. x + y = -6 39. y = -2x 2 STATISTICS For Exercises 40–42, use the information at the right. (Lesson 9-8) Year Population 40. Draw a scatter plot of the data and draw a line of fit. 2000 172 41. Does the scatter plot show a positive, negative, or no relationship? 42. Use your graph to estimate the population of the whooping crane at the refuge in 2005. Whooping Cranes 2001 171 2002 181 2003 194 2004 197 43. SAVINGS Anna’s parents put $750 into a college savings account. After 6 years, the investment had earned $540. Write an equation that you could use to find the simple interest rate. Then find the simple interest rate. (Lesson 5-9) 44. PREREQUISITE SKILL A section of a theater is arranged so that each row has the same number of seats. You are seated in the 5th row from the front and the 3rd row from the back. If your seat is 6th from the left and 2nd from the right, how many seats are in this section of the theater? Use the draw a diagram strategy. (Lesson 4-4) Lesson 10-2 Graphing Quadratic Functions 531 10-3 Problem-Solving Investigation MAIN IDEA: Solve problems by making a model. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Standard 7AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or inequalities that represents a verbal description (e.g. three less than a number, half as large as area A. e-Mail: MAKE A MODEL YOUR MISSION: Make a model to solve the problem. THE PROBLEM: Determine if there are enough tables to make a 10-by-10 square arrangement. EXPLORE PLAN ▲ Tonya: We have 35 square tables. We need to arrange them into a square that is open in the middle and has 10 tables on each side. You know Tonya has 35 square tables. Start by making models of a 4-by-4 square and of a 5-by-5 square. Then look for a pattern. SOLVE {LÞ{ ÃµÕ>Ài CHECK ÓÊ}ÀÕ«ÃÊvÊ{Ê>` ÓÊ}ÀÕ«ÃÊvÊÓ xLÞx ÃµÕ>Ài ÓÊ}ÀÕ«ÃÊvÊxÊ>`Ê ÓÊ}ÀÕ«ÃÊvÊÎ For a 10-by-10 square, Tonya needs 2 · 10 + 2 · 8 or 36 tables. She has 35 tables, so she needs one more. You can estimate that Tonya needs 4 × 10 or 40 tables. But each of the corner tables is counted twice. So, she needs 40 - 4 or 36 tables. 1. Draw a diagram showing another way the students could have grouped the tiles to solve this problem. Use a 4-by-4 square. */ -!4( Write a problem that can be solved by making a (*/ 83 *5*/( 2. model. Describe the model. Then solve the problem. 532 Laura Sifferlin Chapter 10 Algebra: Nonlinear Functions and Polynomials For Exercises 3–5, solve by making a model. 3. STICKERS In how many different ways can three rectangular stickers be torn from a sheet of 3 × 3 stickers so that all three stickers are still attached? Draw each arrangement. 4. MEASUREMENT A 10-inch by 12-inch piece of cardboard has a 2-inch square cut out of each corner. Then the sides are folded up and taped together to make an open box. Find the volume of the box. 8. PETS Mrs. Harper owns both cats and canaries. Altogether, her pets have thirty heads and eighty legs. How many cats does she have? GEOMETRY For Exercises 9 and 10, use the figure at the right. 9. How many cubes would it take to build this tower? 10. How many cubes would it take to build a similar tower that is 12 cubes high? 5. GEOMETRY A computer game requires players to stack arrangements of five squares arranged to form a single shape. One arrangement is shown at the right. How many different arrangements are there if touching squares must border on a full side? 11. CARS Yesterday you noted that the mileage on the family car read 60,094.8 miles. Today it reads 60,099.1 miles. Was the car driven about 4 or 40 miles? Use any strategy to solve Exercises 6–11. Some strategies are shown below. For Exercises 12 and 13, select the appropriate operation(s) to solve the problem. Justify your selections(s) and solve the problem. G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s 12. SCIENCE The light in the circuit will turn on m. • Draw a diagra if one or more switches are closed. How many combinations of open and closed switches will result in the light being on? eck. • Guess and Ch l. • Make a mode 6. CAMP The camp counselor lists 21 chores on separate pieces of paper and places them in a basket. The counselor takes one piece of paper, and each camper takes one as the basket is passed around the circle. There is one piece of paper left when the basket returns to the counselor. How many people could be in the circle if the basket goes around the circle more than once? 7. PARKING Parking space numbers consist of 3 digits. They are typed on a slip of paper and given to students at orientation. Tara accidentally read her number upside-down. The number she read was 795 more than her actual parking space number. What is Tara’s parking space number? a b c d e 13. HOBBIES Lorena says to Angela, “If you give me one of your baseball cards, I will have twice as many baseball cards as you have.” Angela answers, “If you give me one of your cards, we will have the same number of cards.” How many cards does each girl have? Lesson 10-3 Problem-Solving Investigation: Make a Model 533 10-4 Graphing Cubic Functions Main IDEA Standard 7AF3.1 Graph functions of the form y = nx 2 and y = nx 3 and use in solving problems. Standard 7AF3.2 Plot the values from the volumes of three-dimensional shapes for various values of the edge lengths (e.g., cubes with varying edge lengths or a triangle prism with a fixed height and an equilateral triangle base of varying lengths). MEASUREMENT You can find the area A of a square by squaring the length of a side s. This relationship can be represented in different ways. Words and Equation Area A s2 = s Table length of a side squared. equals s Graph s2 (s, A) 0 2 0 =0 (0, 0) 1 2 1 =1 (1, 1) 2 22 = 4 (2, 4) s A A s2 Area Graph cubic functions. s O Side 1. The volume V of a cube is found by cubing the length of a side s. Write a formula to represent the volume of a cube as a function of side length. s s s 2. Graph the volume as a function of side length. (Hint: Use values of s such as 0, 0.5, 1, 1.5, 2, and so on.) 3. Would it be reasonable to use negative numbers for x-values in this situation? Explain. You can graph cubic functions such as the formula for the volume of a cube by making a table of values. Graph a Cubic Function 1 Graph y = x 3. Graphing It is often helpful to substitute decimal values of x in order to graph points that are closer together. x y = x3 (x, y) -1.5 (-1.5) 3 ≈ -3.4 (-1.5, -3.4) -1 3 (-1) = -1 (-1, -1) 0 3 (0) = 0 (0, 0) 1 (1) 3 = 1 (1, 1) 1.5 (1.5) 3 ≈ 3.4 (1.5, 3.4) O Graph each function. a. y = x 3 - 1 b. y = -4x 3 Personal Tutor at ca.gr7math.com 534 y Chapter 10 Algebra: Nonlinear Functions and Polynomials c. y = x 3 + 4 x 2 PACKAGING A packaging company wants to manufacture a cardboard box with a square base of side length x inches and a height of (x – 3) inches as shown. Real-World Link Packaging is the nation’s third largest industry, with over $130 billion in sales each year. Source: San Jose State University (x 3) in. x in. x in. Write the function for the volume V of the box. Graph the function. Then estimate the dimensions of the box that would give a volume of approximately 8 cubic inches. V = lwh Volume of a rectangular prism V = x · x · (x – 3) Replace l with x, w with x, and h with (x – 3). V = x 2(x – 3) x · x = x2 V = x 3 – 3x 2 Distributive Property and Commutative Property The function for the volume V of the box is V = x 3 – 3x 2. Make a table of values to graph this function. You do not need to include negative values of x since the side length of the box cannot be negative. x 3 2 (x, V) 0 (0) – 3(0) = 0 (0, 0) 0.5 (0.5) 3 – 3(0.5) 2 ≈ –0.6 (0.5, –0.6) 1 (1) 3 – 3(1) 2 = –2 (1, –2) 1.5 (1.5) 3 – 3(1.5) 2 ≈ –3.4 (1.5, –3.4) 2 2.5 Analyze the Graph Notice that the graph is below the x-axis for values of x < 3. This means that the “volume” of the box is negative for x < 3. To have a box with a positive height and a positive volume, x must be greater than 3. V = x 3 – 3x 2 3 2 (2) – 3(2) = –4 3 2 (2.5) – 3(2.5) ≈ –3.1 3 2 (2, –4) (2.5, –3.1) 3 (3) – 3(3) = 0 (3, 0) 3.5 (3.5) 3 – 3(3.5) 2 ≈ 6.1 (3.5, 6.1) 4 (4) 3 – 3(4) 2 = 16 (4, 16) 20 18 16 14 12 10 8 6 4 2 2 4 y x 3 3x 2 1 2 3 4 5 6 7 8 9 10 Looking at the graph, we see that the volume of the box is approximately 8 cubic inches when x is about 3.6 inches. The dimensions of the box when the volume is about 8 cubic inches are 3.6 inches, 3.6 inches, and 3.6 – 3 or 0.6 inch. d. PACKAGING A packaging company wants to manufacture a cardboard box with a square base of side length x feet and a height of (x – 2) feet. Write the function for the volume V of the box. Graph the function. Then estimate the dimensions of the box that would give a volume of about 1 cubic foot. Extra Examples at ca.gr7math.com Getty Images Lesson 10-4 Graphing Cubic Functions 535 Example 1 (p. 534) Example 2 (p. 535) (/-%7/2+ (%,0 For Exercises 6–17 18, 19 See Examples 1 2 Graph each function. 1. y = -x 3 2. y = 0.5x 3 3. y = x 3 – 2 4. y = 2x 3 + 1 5. MEASUREMENT A rectangular prism with a square base of side length x centimeters has a height of (x + 1) centimeters. Write the function for the volume V of the prism. Graph the function. Then estimate the dimensions of the box that would give a volume of approximately 9 cubic centimeters. Graph each function. 6. y = -2x 3 7. y = -3x 3 8. y = 0.2x 3 9. y = 0.1x 3 10. y = x 3 + 1 11. y = 2x 3 + 1 12. y = x 3 – 3 13. y = 2x 3 – 2 14. y = _ x 3 15. y = _ x 3 + 2 16. y = -x 3 – 2 17. y = -x 3 + 1 1 4 1 3 18. MEASUREMENT Jorge built a scale model of the Great %842!02!#4)#% Pyramid. The base of the model is a square with side length s and the model’s height is (s – 1) feet. Write the function for the volume V of the model. Graph this function. Then estimate the length of one side of the square base of the model if the model’s volume is approximately 8 cubic feet. See pages 703, 717. 19. MEASUREMENT The formula for the volume V of a tennis ball is given by Self-Check Quiz at this function. Use 3.14 for π. Then estimate the length of the radius if the volume of the tennis ball is approximately 11 cubic inches. ca.gr7math.com 4 3 the equation V = _ πr where r represents the radius of the ball. Graph 3 Graph each pair of equations on the same coordinate plane. Describe their similarities and differences. 20. y = x 3 y = 3x 21. y = x 3 3 3 y=x –3 22. y = 0.5x 3 y = 2x 23. y = 2x 3 3 y = -2x 3 FARMING For Exercises 24 and 25, use the following information. A grain silo consists of a cylindrical main section and a hemispherical roof. The cylindrical main section has a radius of r units and a height h equivalent to the radius. The volume V of a cylinder is given by the equation V = πr 2h. 24. Write the function for the volume V of the cylindrical main section of the grain silo in terms of its radius r. 25. Graph this function. Use 3.14 for π. Then estimate the radius and height in meters of the cylindrical main section of the grain silo if the volume is approximately 15.5 cubic meters. 536 Chapter 10 Algebra: Nonlinear Functions and Polynomials r H.O.T. Problems 26. OPEN ENDED Write the equation of a cubic function whose graph in the first quadrant shows faster growth than the function y = x 3. CHALLENGE The zeros of a cubic function are the x-coordinates of the points at which the function crosses the x-axis. Find the zeros of each function below. 27. y = x 3 29. 28. y = x 3 + 1 */ -!4( The volume V of a cube with side length s is given by (*/ 83 *5*/( the equation V = s 3. Explain why negative values are not necessary when creating a table or a graph of this function. 30. Which equation could represent the graph shown below? 31. Which equation could represent the graph shown below? y O y x O A y = x3 F y = x3 – 2 B y = -x 3 G y = x3 + 2 C y = 2x 3 H y = -2x 3 D y = -2x 3 J x y = 2x 3 + 1 32. MANUFACTURING A company packages six small books for a children’s collection in a decorated 4-inch cube. They are shipped to bookstores in cartons. Twenty cubes fit in a carton with no extra space. What are the dimensions of the carton? Use the make a model strategy. (Lesson 10-3) Graph each function. (Lesson 10-2) 33. y = -2x 2 34. y = x 2 + 3 35. y = -3x 2 + 1 Estimate each square root to the nearest whole number. 37. √ 54 38. - √ 126 39. √ 8.67 36. y = 4x 2 + 3 (Lesson 3-2) 40. - √ 19.85 PREREQUISITE SKILL Write each expression using exponents. (Lesson 2-9) 41. 3 · 3 · 3 · 3 · 3 42. 5 · 4 · 5 · 5 · 4 43. 7 · (7 · 7) 44. (2 · 2) · (2 · 2 · 2) Lesson 10-4 Graphing Cubic Functions 537 Extend 10-4 Main IDEA Use a graphing calculator to graph families of nonlinear functions. Standard 7AF3.1 Graph functions of the form y = nx 2 and y = nx 3 and use in solving problems. Standard 7MR3.3 Develop generalizations of the results obtained and the strategies used and apply them to new problem situations. Graphing Calculator Lab Families of Nonlinear Functions Families of nonlinear functions share a common characteristic based on a parent function. The parent function of a family of quadratic functions is y = x 2. You can use a graphing calculator to investigate families of quadratic functions. Graph y = x 2, y = x 2 + 5, and y = x 2 - 3 on the same screen. Clear any existing equations from the Y= list by pressing CLEAR . Enter each equation. Press X,T,␪,n ENTER , X,T,␪,n 5 ENTER , and X,T,␪,n 3 ENTER . Graph the equations in the standard viewing window. Press ZOOM 6. ANALYZE THE RESULTS 1. Compare and contrast the three equations you graphed. 2. Describe how the graphs of the three equations are related. 3. MAKE A CONJECTURE How does changing the value of c in the equation y = x 2 + c affect the graph? 4. Use a graphing calculator to graph y = 0.5x 2, y = x 2, and y = 2x 2. 5. Compare and contrast the three equations you graphed in Exercise 4. 6. Describe how the graphs of the three equations are related. 7. MAKE A CONJECTURE How does changing the value of a in the equation y = ax 2 affect the graph? 8. Use a graphing calculator to graph y = 0.5x 3, y = x 3, and y = 2x 3. 9. Compare and contrast the three equations you graphed in Exercise 8 to the equations you graphed in Exercise 4. 538 Chapter 10 Nonlinear Functions and Polynomials Other Calculator Keystrokes at ca.gr7math.com 10-5 Multiplying Monomials Main IDEA Multiply monomials. Standard 7NS2.3 Multiply, divide, and simplify rational numbers by using exponent rules. Standard 7AF2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. Standard 7AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. SCIENCE The pH of a solution describes its acidity. Neutral water has a pH of 7. Lemon juice has a pH of 2. Each one-unit decrease in the pH means that the solution is 10 times more acidic. So, a pH of 8 is 10 times more acidic than a pH of 9. pH Times More Acidic Than a pH of 9 Written Using Powers 8 10 10 1 7 10 × 10 = 100 10 1 × 10 1 = 10 2 6 10 × 10 × 10 = 1,000 10 1 × 10 2 = 10 3 5 10 × 10 × 10 × 10 = 10,000 10 1 × 10 3 = 10 4 4 10 × 10 × 10 × 10 × 10 = 100,000 10 1 × 10 4 = 10 5 1. Examine the exponents of the factors and the exponents of the products in the last column. What do you observe? A monomial is a number, a variable, or a product of a number and one or more variables. Exponents are used to show repeated multiplication. You can use this fact to find a rule for multiplying monomials. 2 factors 4 factors NEW Vocabulary 3 2 · 3 4 = (3 · 3) · (3 · 3 · 3 · 3) or 3 6 monomial 6 factors Notice that the sum of the original exponents is the exponent in the final product. This relationship is stated in the following rule. +%9 #/.#%04 Product of Powers To multiply powers with the same base, add their exponents. Words Examples Numbers 4 3 2 ·2 =2 4+3 Algebra or 2 7 m a · an = am + n Multiply Powers 1 Find 5 2 · 5. Express using exponents. Common Error When multiplying powers, do not multiply the bases. 4 5 · 4 2 = 4 7, not 16 7. 52 · 5 = 52 · 51 5 = 51 Check 5 2 · 5 = (5 · 5) · 5 = 52 + 1 The common base is 5. =5·5·5 = 53 Add the exponents. = 53 Lesson 10-5 Multiplying Monomials CORBIS 539 2 Find -3x 2(4x 5). Express using exponents. -3x 2(4x 5) = (-3 · 4)(x 2 · x 5) = (-12)(x = -12x 2+5 Commutative and Associative Properties ) The common base is x. 7 Add the exponents. Multiply. Express using exponents. a. 9 3 · 9 2 b. 2 (_35 ) (_35 ) 9 c. -2m(-8m 5) 3 The population of Groveton is 6 5. The population of Putnam is 6 3 times as great. How many people are in Putnam? Real-World Link A census is taken every ten years by the U.S. Census Bureau to determine population. The government uses the data from the census to make many decisions. To find out the number of people, multiply 6 5 by 6 3. Source: census.gov d. RIVERS The Guadalupe River is 2 8 miles long. The Amazon River is 6 5 · 6 3 = 6 5+3 or 6 8 Product of Powers The population of Putnam is 6 8 or 1,679,616 people. almost 2 4 times as long. Find the length of the Amazon River. Personal Tutor at ca.gr7math.com In Lesson 2-9, you learned to evaluate negative exponents. Remember that any nonzero number to the negative n power is the multiplicative inverse of that number to the n th power. The Product of Powers rule can be used to multiply powers with negative exponents. Multiply Negative Powers 4 Find x 4 · x -2. Express using exponents. METHOD 1 METHOD 2 x 4 · x -2 = x 4 + (-2) The common x 4 · x -2 base is x. =x 2 Add the exponents. 1 _ 1 1 _ -2 =x·x·x·x·_ x · x x = x2 = x2 Simplify. Simplify. Express using positive exponents. e. 3 8 · 3 -2 540 f. n 9 · n -4 Chapter 10 Algebra: Nonlinear Functions and Polynomials Prisma/SuperStock g. 5 -1 · 5 -2 Extra Examples at ca.gr7math.com Examples 1–4 (pp. 539–540) Example 3 Simplify. Express using exponents. 1. 4 5 · 4 3 2. n 2 · n 9 3. -2a(3a 4) 4. 5 2x 2y 4 · 5 3xy 3 5. r 7 · r -3 6. 6m · 4m 2 7. AGE Angelina is 2 3 years old. Her grandfather is 2 3 times her age. How old (p. 540) (/-%7/2+ (%,0 For Exercises 8–25 26–28 See Examples 1, 2, 4 3 is her grandfather? Simplify. Express using exponents. 8. 6 8 · 6 5 9. 2 9 · 2 10. n · n 7 11. b 13 · b 12. 2g · 7g 6 13. (3x 8)(5x) 14. -4a 5(6a 5) 15. (8w 4)(-w 7) 16. (-p)(-9p 2) 17. -5y 3(-8y 6) 18. 4m -2n 5(3m 4n -2) 19. (-7a 4bc 3)(5ab 4c 2) 20. x 6 · x -3 21. y -1 · y 4 22. z -2 · z -3 23. m 2n -1 · m -3n 3 24. 3f -4 · 5f 2 25. -3ab · 4a -3b 3 26. INSECTS The number of ants in a nest was 5 3. After the eggs hatched, the number of ants increased 5 2 times. How many ants are there after the eggs hatch? 27. COMPUTERS The processing speed of a certain computer is 10 11 instructions per second. Another computer has a processing speed that is 10 3 times as fast. How many instructions per second can the faster computer process? 28. LIFE SCIENCE A cell culture contains 2 6 cells. By the end of the day, there are 2 10 times as many cells in the culture. How many cells are there in the culture by the end of the day? Simplify. Express using exponents. %842!02!#4)#% See pages 703, 717. 29. xy 2(x 3y) 3 32. (_23 ) (_23 ) 35. (_14 ) (_14 ) Self-Check Quiz at ca.gr7math.com 4 30. 2 6 · 2 · 2 3 -4 31. 4a 2b 3(7ab 2) -5 13 3 -2 33. (_78 ) (_78 ) 36. (_25 ) (_25 ) 4 -7 -2 -3 (_25 ) (_25 ) (_25 ) 2 37. (_) (_72 ) 7 34. 6 Lesson 10-5 Multiplying Monomials 541 38. CHALLENGE What is twice 2 30? Write using exponents. H.O.T. Problems 39. OPEN ENDED Write a multiplication expression whose product is 4 15. */ -!4( Determine whether the following statement is true or (*/ 83 *5*/( 40. false. Explain your reasoning or give a counterexample. If you change the order in which you multiply two monomials, the product will be different. 41. Which expression is equivalent to 42. Which expression describes the area in 8x 2y · 8yz 2? 2 square feet of the rectangle below? 2 2 A 64x y z F 11x 10 B 64x 2 yz 2 G 30x 10 C 16x 2 y 2z 2 H 11x 16 D 384x 2 y 2z 2 J 2 5x ft 8 6x ft 30x 16 Graph each function. (Lessons 10-2 and 10-4) 43. y = -x 3 44. y = 0.5x 3 45. y = x 3 - 2 46. y = 5x 2 47. y = x 2 + 5 48. y = x 2 – 4 49. BIOLOGY The table shows how long it took for the first 400 bacteria cells to grow in a petri dish. Is the growth of the bacteria a linear function of time? Explain. (Lesson 10-1) Time (min) 46 53 57 60 Number of cells 100 200 300 400 Express each number in scientific notation. (Lesson 2-10) 50. The flow rate of some Antarctic glaciers is 0.00031 mile per hour. 51. A human blinks about 6.25 million times a year. ALGEBRA Solve each equation. Check your solution. (Lesson 2-7) 53. 1_ + p = -6_ 3 4 52. k - 4.1 = -9.38 54. _ = 0.845 c 10 1 2 Find each sum or difference. Write in simplest form. (Lesson 2-6) 55. _ - _ 7 8 3 10 PREREQUISITE SKILL 59. 3 · 3 · 3 · 3 542 56. -_ + _ 1 5 5 12 57. 9_ + _ 2 3 1 6 58. -2_ - 1_ 3 4 1 8 Write each expression using exponents. (Lesson 2-9) 60. 5 · 4 · 4 · 5 · 4 61. 8 · (8 · 8) Chapter 10 Algebra: Nonlinear Functions and Polynomials 62. (5 · 5 · 5) · (5 · 5) CH APTER Mid-Chapter Quiz 10 Lessons 10-1 through 10-5 Determine whether each equation or table represents a linear or nonlinear function. Explain. (Lesson 10-1) STANDARDS PRACTICE Which graph shows y = x 2 + 1? (Lesson 10-2) 11. A C y y 1. 3y = x 2. y = 5x 3 + 2 3. 4. x 1 3 5 7 y -5 -6 -7 -8 x -1 0 1 2 y 1 0 1 4 x O x O B D y O y x O x 5. LONG DISTANCE The graph shows the amount of data transferred as a function of time. Is this a linear or nonlinear function? Explain your reasoning. (Lesson 10-1) 12. MEASUREMENT Brenda has a photograph that is 10 inches by 13 inches. She decides $ATA 4RANSFER 1 to frame it, using a frame that is 2_ inches 4 wide on each side. Find the total area of the framed photograph. Use the make a model strategy. (Lesson 10-3) 'IGABYTES Graph each function. (Lesson 10-4) 13. y = -2x 3 4IME MIN Graph each function. (Lesson 10-2) 6. y = 2x 2 2 14. y = 3x 3 15. y = 2x 3 16. y = 0.1x 3 Simplify. Express using exponents. (Lesson 10-4) 7. y = -x + 3 17. 10 4 · 10 7 8. y = 4x 2 - 1 18. 3 -3 · 3 5 · 3 2 9. y = -3x 2 + 1 19. 2 3a 7 · 2a -3 20. (3 2xy 4z 2)(3 5x 3y -2z 3) 10. AMUSEMENT PARK RIDES Your height h feet above the ground t seconds after being released at the top of a free-fall ride is given by the function h = -16t 2 + 200. Graph this function. After about how many seconds will the ride be 60 feet above the ground? (Lesson 10-2) 21. STANDARDS PRACTICE Which expression below has the same value as 5m 2? (Lesson 10-5) F 5m H 5·5·m·m G 5·m·m J 5·m·m·m 10-6 Dividing Monomials Main IDEA Divide monomials. Standard 7NS2.3 Multiply, divide, and simplify rational numbers by using exponent rules. Standard 7AF2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. Standard 7AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. NUMBER SENSE Refer to the table shown that relates division sentences using the numbers 2, 4, 8, and 16, and the same sentences written using powers of 2. 1. Examine the exponents of the divisors Division Sentence Written Using Powers of 2 4÷2=2 22 ÷ 21 = 21 8÷2=4 23 ÷ 21 = 22 8÷4=2 23 ÷ 22 = 21 16 ÷ 2 = 8 24 ÷ 21 = 23 16 ÷ 4 = 4 24 ÷ 22 = 22 16 ÷ 8 = 2 24 ÷ 23 = 21 and dividends. Compare them to the exponents of the quotients. What do you notice? 2. MAKE A CONJECTURE Write the quotient of 2 5 and 2 2 using powers of 2. As you know, exponents are used to show repeated multiplication. You can use this fact to find a rule for dividing powers with the same base. 7 factors Notice that the difference of the original exponents is the exponent in the final quotient. This relationship is stated in the following rule. 57 5·5·5·5·5·5·5 _ = __ or 5 3 54 4 factors +%9 #/.#%04 Words 5·5·5·5 Quotient of Powers To divide powers with the same base, subtract their exponents. Examples Numbers Algebra 37 _ = 3 7 – 3 or 3 4 am _ = a m – n, where a ≠ 0 an 33 Divide Powers Simplify. Express using exponents. 4 n9 2 _ 4 48 _ = 48 – 2 n9 _ = n9 – 4 48 1 _ 2 Common Error When dividing powers, do not divide 42 n =4 6 The common base is 4. Simplify. n4 =n 5 The common base is n. Simplify. 48 the bases. _2 = 4 6, 4 6 not 1 . Simplify. Express using exponents. 57 5 a. _ 4 544 x 10 x b. _ 3 Chapter 10 Algebra: Nonlinear Functions and Polynomials 12w 5 2w c. _ The Quotient of Powers rule can also be used to divide powers with negative exponents. It is customary to write final answers using positive exponents. Look Back To review adding and subtracting integers, see Lessons 1-4 and 1-5. Use Negative Exponents Simplify. Express using positive exponents. 69 3 _ -3 6 69 _ = 6 9 – (-3) Quotient of Powers 6 -3 = 6 9 + 3 or 6 12 Simplify. w -1 4 _ -4 w w -1 _ = w -1 – (-4) Quotient of Powers w -4 = w -1 + 4 or w 3 Simplify. Simplify. Express using positive exponents. 11 -8 11 d. _ 2 e. _ -7 b -4 b f. 6h 5 _ 3h -5 22 · 45 · 52 5 _ = 5 4 2 2 ·4 ·5 A 2 Remember that the Quotient of Powers Rule allows you to Read the Item 52 _ = 5 2 - 2 = 5 0 = 1. Solve the Item 52 5 simplify _2 . 52 1 C _ B 1 D 0 2 You are asked to divide one monomial by another. ( )( )( ) 22 · 45 · 52 52 22 _ 45 _ _ = _ 25 · 44 · 52 25 44 52 Group by common base. = 2 -3 · 4 1 · 5 0 Subtract the exponents. 1 =_ ·4·1 3 2 -3 = _3 4 1 =_ or _ Simplify. 2 8 2 1 2 The answer is C. Extra Examples at ca.gr7math.com Lesson 10-6 Dividing Monomials 545 _1 4 _1 -12 -3 (_1 ) (6) × (6) __ g. Simplify . 6 1 5 F _ 1 G _ (6) H 64 6 J 65 Personal Tutor at ca.gr7math.com 6 SOUND The loudness of a conversation is 10 6 times as intense as the loudness of a pin dropping, while the loudness of a jet engine is 10 12 times as intense. How many times more intense is the loudness of a jet engine than the loudness of a conversation? Real-World Link The decibel measure of the loudness of a sound is the exponent of its relative intensity multiplied by 10. A jet engine has a loudness of 120 decibels. To find how many times more intense, divide 10 12 by 10 6. 10 12 _ = 10 12 – 6 or 10 6 10 6 Quotient of Powers The loudness of a jet engine is 10 6 or 1,000,000 times as intense as the loudness of a conversation. h. SOUND The loudness of a vacuum cleaner is 10 4 times as intense as the loudness of a mosquito buzzing, while the loudness of a jack hammer is 10 9 times as intense. How many times more intense is the loudness of a jack hammer than that of a vacuum cleaner? Personal Tutor at ca.gr7math.com Examples 1–4 (pp. 544-545) Simplify. Express using positive exponents. 1. _ 76 7 2. _ 13 9c 7 3c 6. _ 6 5. _ 2 Example 5 (p. 545) 2 (p. 546) 546 3. _ 5 24k 9 6k 7. _ 2 8. _ -4 1 C _ 1 2 D _ 15 -6 15 z z 35p 1 5p 4 2 ·3 ·4 2·3 ·4 B 2 2 (2) 10. ASTRONOMY Venus is approximately 10 8 kilometers from the Sun. Saturn is more than 10 9 kilometers from the Sun. About how many times farther away from the Sun is Saturn than Venus? Chapter 10 Algebra: Nonlinear Functions and Polynomials Mug Shots/Corbis y 4. _ 2 9. Simplify _ . 3 5 A 22 Example 6 3 y8 29 2 (/-%7/2+ (%,0 For Exercises 11–26 27–30 31–34 See Examples 1–4 5 6 Simplify. Express using positive exponents. 11. _ 4 8 15 8 12. _ 29 2 13. _ 7 43 4 14. _ 5 15. _ 6 h7 h 16. _ 6 g 18 g 17. _ 11 x8 x 18. _ 8 19. _ 5 36d 10 6d 20. _ 16t 4 8t 21. _ 5 20m 7 5m 22. _ 5 22 -9 22 24. _ -5 3 -1 3 25. _ -2 42w -6 7w 26. _ -10 x 6y 14 _ 63 · 66 · 64 28. _ = 62 · 63 · 64 _1 × _1 (__ (5) 5) 29. (_15 ) 23. _ 4 27. x 4y 9 13 2 13 n n 75r 6 25r 12y -6 2y -6 2 2 3x 4 3 x 30. _ 4 -2 31. POPULATION The continent of North America contains approximately 10 7 square miles of land. If the population doubles, there will be about 10 9 people on the continent. At that point, on average, how many people will occupy each square mile of land? 32. FOOD An apple is 10 3 times as acidic as milk, while a lemon is 10 4 times as acidic. How many times more acidic is a lemon than an apple? 33. ANIMALS A common flea 2 -4 inch long can jump about 2 3 inches high. About how many times its body size can a flea jump? 34. MEDICINE The mass of a molecule of penicillin is 10 -18 kilograms and the mass of a molecule of insulin is 10 -23 kilograms. How many times greater is a molecule of penicillin than a molecule of insulin? Find each missing exponent. 17 17 35. _ = 17 8 4 2 36. _ =k k6 k 5 5 37. _ = 53 -9 ANALYZE TABLES For Exercises 39 and 40, use the information below and in the table. %842!02!#4)#% See pages 703, 717. Self-Check Quiz at ca.gr7math.com For each increase of one on the Richter scale, an earthquake’s vibrations, or seismic waves, are 10 times greater. p -1 10 38. _ =p p Earthquake Richter Scale Magnitude San Francisco, 1906 8.3 Adana, Turkey, 1998 6.3 Source: usgs.gov 39. How many times greater are the seismic waves of an earthquake with a magnitude of 6 than an aftershock with a magnitude of 3? 40. How many times greater were the seismic waves of the 1906 San Francisco earthquake than the 1998 Adana earthquake? Lesson 10-6 Dividing Monomials 547 3 100 3 41. NUMBER SENSE Is _ greater than, less than, or equal to 3? Explain your 99 H.O.T. Problems reasoning. 42. OPEN ENDED Write a division expression with a quotient of 4 15. 43. CHALLENGE What is half of 2 30? Write using exponents. 44. */ -!4( Explain why the Quotient of Powers Rule cannot (*/ 83 *5*/( 5 x be used to simplify the expression _ . 2 y 45. Which expression below is equivalent 8 47. One meter is 10 3 times longer than one millimeter. One kilometer is 10 6 times longer than one millimeter. How many times longer is one kilometer than one meter? 9m to _ ? 2 3m A 6m 4 C 3m 4 B 6m 6 D 3m 6 A 10 9 46. The area of a rectangle is 2 6 square B 10 6 feet. If the length is 2 3 feet, find the width of the rectangle. F 2 feet H 2 3 feet G 2 2 feet J C 10 3 D 10 2 9 feet Simplify. Express using positive exponents. (Lesson 10-5) 48. 6 4 · 6 7 49. 18 3 · 18 -5 50. (-3x 11)(-6x 3) 51. (-9a 4)(2a -7) 54. y = -2x 3 55. y = -0.1x 3 Graph each function. (Lesson 10-4) 52. y = x 3 + 2 53. y = _ x 3 1 3 State the slope and the y-intercept for the graph of each equation. (Lesson 9-5) 56. y = x – 3 57. y = _x + 7 2 3 58. 3x + 4y = 12 59. x + 2y = 10 60. COIN COLLECTING Jada has 156 coins in her collection. This is 12 more than 8 times the number of nickels in the collection. How many nickels does Jada have in her collection? (Lesson 8-3) Simplify. Express using positive exponents. (Lesson 10-5) 61. 5n · 3n 4 548 62. (-x)(-8x 3) 63. (-5b 7)(-2b 4) Chapter 10 Algebra: Nonlinear Functions and Polynomials 64. (-4w)(6w -2) 10-7 Powers of Monomials Main IDEA Find powers of monomials. MEASUREMENT Suppose the side length of a cube is 2 2 centimeters. Standard 7AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. 1. Write a multiplication expression for the volume of the cube. 2 2 cm 2. Simplify the expression. Write as a single power of 2. 3. Using 2 2 as the base, write the multiplication expression 2 2 · 2 2 · 2 2 using an exponent. 3 4. Explain why (2 2) = 2 6. You can use the rule for finding the product of powers to discover the rule for finding the power of a power. 5 factors (6 4) 5 = (6 4) (6 4) (6 4) (6 4) (6 4) Apply the rule for the product of powers. = 64 + 4 + 4 + 4 + 4 = 6 20 Notice that the product of the original exponents, 4 and 5, is the final power 20. This relationship is stated in the following rule. +%9 #/.#%04 Power of a Power To find the power of a power, multiply the exponents. Words Examples Numbers 2 3 (5 ) = 5 2·3 Algebra or 5 6 m n (a ) = a m · n Find the Power of a Power Common Error When finding the power of a power, do not add the exponents. 1 Simplify (8 4) 3. (8 4) 3 = 8 4 · 3 =8 12 2 Simplify (k 7) 5. Power of a Power (k 7) 5 = k 7 · 5 = k 35 Simplify. Power of a Power Simplify. (8 4) 3 = 8 12, not 8 7. Simplify. Express using exponents. a. (2 5) 2 Extra Examples at ca.gr7math.com b. (w 4) 6 c. [(3 2) 3] 2 Lesson 10-7 Powers of Monomials 549 Extend the power of a power rule to find the power of a product. 5 factors (3a 4) 5 = (3a 4) (3a 4) (3a 4) (3a 4) (3a 4) 4 4 4 Associative and 4 =3·3·3·3·3·a ·a ·a ·a ·a 4 = 3 5 · (a 4) 5 = 243 · a 20 or 243a 20 Commutative Properties of Multiplication Write using powers. Apply the rule for power of a power. This example suggests the following rule. +%9 #/.#%04 Words Power of a Product To find the power of a product, find the power of each factor and multiply. Examples Numbers 2 3 3 Algebra 2 3 (6x ) = (6) • (x ) or 216x 6 (ab) m = a mb m Power of a Product 3 Simplify (4p 3) 4. 4 Simplify (-2m 7n 6) 5. (4p 3) 4 = 4 4 · p 3 · 4 Alternative Method (4p 3) 4 can also be expressed as (4p 3)(4p 3)(4p 3)(4p 3) or (4 · 4 · 4 · 4) (p · p · p)(p · p · p) (p · p · p)(p · p · p) which is 256p 12. = 256p 12 (-2m 7n 6) 5 = (-2) 5m 7 · 5n 6 · 5 = -32m 35n 30 Simplify. Simplify. Simplify. d. (8b 9) 2 e. (6x 5y 11) 4 f. (-5w 2z 8) 3 5 GEOMETRY Express the area of the square as a monomial. A = s2 Area of a square 4 A = (7a b) 2 Replace s with 7a 4b. A = 7 2(a 4) 2(b 1) 2 Power of a Product A = 49a 8b 2 7a 4b Simplify. The area of the square is 49a 8b 2 square units. g. GEOMETRY Find the volume of a cube with sides of length 8x 3y 5. Express as a monomial. Personal Tutor at ca.gr7math.com 550 Chapter 10 Algebra: Nonlinear Functions and Polynomials Examples 1–4 Simplify. (pp. 549-550) 1. (3 2) Example 5 4. (7w 7) 5 2. (h 6) 4 3 3. [(2 3) 2] 3 12 5. (5g 8k ) 4 6. (-6r 5s 9) 2 (p. 550) 7. MEASUREMENT Express the volume of the cube at the right as a monomial. (/-%7/2+ (%,0 For Exercises 8–27 28–31 See Examples 1–4 3 3c 3d 2 Simplify. 8. (4 2) 3 9. (2 2) 7 10. (5 3) 3 11. (3 4) 2 12. (d 7) 6 13. (m 8) 5 14. (h 4) 9 15. (z 11) 5 16. [(3 2) 2] 2 17. [(4 3) 2] 2 18. [(5 2) 2] 2 19. [(2 3) 3] 2 20. (5j 6) 4 21. (8v 9) 5 22. (11c 4) 3 23. (14y) 4 24. (6a 2b 6) 3 25. (2m 5n 11) 6 26. (-3w 3z 8) 5 27. (-5r 4s 12) 4 GEOMETRY Express the area of each square below as a monomial. 28. 29. 8g 3h 12d 6e 7 GEOMETRY Express the volume of each cube below as a monomial. 30. 31. 5r 2s 3 7m 6n 9 Simplify. 32. (0.5k 5) 2 33. (0.3p 7) 3 34. (_w 5z ) 2 35. (_a -6b 9) 2 36. (3x -2) 4(5x 6) 2 37. (-2v 7) 3(-4v -2) 4 3 5 1 4 3 38. PHYSICS A ball is dropped from the top of a building. The expression 4.9x 2 %842!02!#4)#% gives the distance in meters the ball has fallen after x seconds. Write and simplify an expression that gives the distance in meters the ball has fallen after x 2 seconds. Then write and simplify an expression that gives the distance the ball has fallen after x 3 seconds. See pages 703, 717. 39. BACTERIA A certain culture of bacteria doubles in population every hour. At Self-Check Quiz at 1 P.M., there are 5 cells. The expression 5(2 x)gives the number of bacteria that are present x hours after 1 P.M. Simplify the expressions [5(2 x)] 2 and [5(2 x)] 3 and describe what they each represent. ca.gr7math.com Lesson 10-7 Powers of Monomials 551 MEASUREMENT For Exercises 40-42, use the table that gives the area and volume of a square and cube, respectively, with side lengths shown. Side Length (units) Area of Square (units 2) Volume of Cube (units 3) x x2 x3 40. Copy and complete the table. 2x 41. Describe how the area and volume are 3x each affected if the side length is doubled. Then describe how they are each affected if the side length is tripled. x2 x3 42. Describe how the area and volume are each affected if the side length is squared. Describe how they are each affected if the side length is cubed. 43. OPEN ENDED A googol is 10 100. Use the Power of a Power rule to write three H.O.T. Problems different expressions that are equivalent to a googol where each expression uses exponents. CHALLENGE Solve each equation for x. 44. (7 x) 3 = 7 15 46. 45. (-2m 3n 4) x = -8m 9n 12 */ -!4( Compare and contrast how you would correctly (*/ 83 *5*/( simplify the expressions (2a 3)(4a 6) and (2a 3) 6. 47. Which expression is equivalent to 49. Which of the following has the same (10 4) 8? value as 64m 6? A 10 2 C 10 12 4 32 B 10 D 10 A the area in square units of a square whose side length is 8m 2 B the expression (32m 3) 2 48. Which expression has the same value C the expression (8m 3) 2 as 81h 8k 6? F (9h 6k 4) 2 H (6h 5k 3) 3 G (9h 4k 3) 2 J D the volume in cubic units of a cube whose side length is 4m 3 (3h 2k) 6 Simplify. Express using positive exponents. (Lesson 10-6) 15 7 15 50. _ 4 y 10 51. _ 2 y 24g 3 18m 9 6m 52. _ 4 53. _ 8 3g 2 54. MEASUREMENT Find the area of a rectangle with a length of 9xy and a width of 4x 2y. (Lesson 10-5) Find each square root. (Lesson 3-1) 55. √ 49 552 56. √ 121 57. √ 225 Chapter 10 Algebra: Nonlinear Functions and Polynomials 58. √ 400 10-8 Roots of Monomials Main IDEA Find roots of monomials. Standard 7AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. NEW Vocabulary cube root REVIEW Vocabulary square root: a number whose square is that number (Lesson 3-1) NUMBER THEORY The square root of a number is a number whose square is that number. Some perfect squares can be factored into the product of two other perfect squares. 1. Find two factors of 100 that are also perfect squares. 2. Find the square roots of 4 and 25. Then find their product. 3. How does the product relate to 100? 4. Repeat Questions 1–3 using 144. The pattern you discovered about the factors of a perfect square is true for any number. +%9 #/.#%04 Words Product Property of Square Roots For any numbers a and b, where a ≥ 0 and b ≥ 0, the square root of the product ab is equal to the product of each square root. Examples Numbers √ 9 · 16 = √ 9 · √ 16 Algebra √ ab = √ a · √ b = 3 · 4 or 12 The square root of a monomial is a monomial whose square is that monomial. You can use the product property of square roots to find the square roots of monomials. √ x 2 = √ x · x = ⎪x⎥ Since x represents an unknown value, absolute value is used to indicate the positive value of x. √ x 4 = √ x2 · x2 = x2 Absolute value is not necessary since the value of x 2 will never be negative. Simplify Square Roots 1 Simplify √ 4y 2 . Absolute Value Use absolute value to indicate the positive value of y and q 3. 2 Simplify √ 36q 6 . 4y 2 = √4 · √ y2 √ 36q 6 = √ 36 · √ q6 √ = √ 6 · 6 · √ q3 · q3 = 6 ⎪q 3 ⎥ 2 · 2 · √ y·y = √ = 2⎪y⎥ Simplify. a. √ v2 Extra Examples at ca.gr7math.com b. √ c 6d 8 c. √ 121x 4z 10 Lesson 10-8 Roots of Monomials 553 READING Math Cube Root Symbol The cube root of a is shown by 3 the symbol √ a. The process of simplifying expressions involving square roots can be extended to cube roots. The cube root of a monomial is a monomial whose cube is that monomial. 3 3 √ 8 = √ 2·2·2=2 3 3 √ a3 = √ a ·a·a=a +%9 #/.#%04 Product Property of Cube Roots For any numbers a and b, the cube root of the product ab is equal to the product of each cube root. Words Examples Numbers 3 Algebra 3 3 = √ √ab a · √b 3 3 3 √ 216 = √ 8 · √ 27 = 2 · 3 or 6 Simplify Cube Roots 3 3 Simplify √ c3. 3 √ c3 = c (c) 3 = c 3 3 4 Simplify √ 64g 6 . 3 3 64g 6 = √ 64 · √ g6 √ 3 Product Property of Cube Roots 3 3 = √ 4 · 4 · 4 · √ g2 · g2 · g2 = 4 · g 2 or 4g 2 Absolute Value Because a cube root can be negative, absolute value is not necessary when simplifying cube roots. Simplify. Simplify. 3 d. √ s3 e. 3 27y 3 √ f. 3 √ 216k 12 5 GEOMETRY Express the length of one side of the square whose area is 81y 2z 6 square units in simplified form. A = s 2 Area of a square 81y 2z 6 = s 2 Replace A with 81y 2z 6. 81y 2z 6 = s Definition of square root. √ √ 81 · √y2 · √z6 = s Product Property of Square Roots. 9⎪yz 3⎥ = s Simplify. Add absolute value. The length of one side of the square is 9⎪yz 3⎥ units. g. GEOMETRY Find the length of one side of a cube whose volume is 125a 15 cubic units. Personal Tutor at ca.gr7math.com 554 Chapter 10 Algebra: Nonlinear Functions and Polynomials Examples 1–2 (p. 553) Example 3–4 Simplify. 1. √ d2 5. 3 √ m3 2. √ 25a 2 6. 8p 3 √ (p. 554) Example 5 3 3. 49x 6y 2 √ 7. √ 125r 6s 9 3 4. √ 121h 8k 10 8. 64 x 12y 3 √ 3 9. GEOMETRY Express the length of one side of the square whose area is 256u 2v 6 square units as a monomial. (p. 554) 10. GEOMETRY Express the length of one side of a cube whose volume is 27b 3c 12 cubic units as a monomial. (/-%7/2+ (%,0 For Exercises 11–18 14–26 27–34 See Examples 1–2 3 5 Simplify. 2 11. √n 12. y4 √ 13. g 8k 14 √ 14. √ 64a 2 15. √ 36z 12 16. √ 144k 4m 6 17. 9p 8q 4 √ 18. 225x 4y 6 √ 19. 3 √ h3 20. √ v3 21. √ 27b 3 22. √ 64k 3 23. √ 125d 9e 3 24. 8q 9r 18 √ 25. √ 343m 3n 21 26. √ 216x 12w 15 3 3 3 3 3 3 3 GEOMETRY Express the length of one side of each square whose area is given as a monomial. 27. 28. 29. 30. A 36m 6n 8 A 400x 2y 10 A 121a 2b 2 A 49p 4q 6 GEOMETRY Express the length of one side of each cube whose volume is given as a monomial. 31. 32. 33. 34. V 125k 9m 18 V 27g 24h 3 V 64w 3z 3 V 343c 6d 12 Simplify. %842!02!#4)#% See pages 704, 717. Self-Check Quiz at ca.gr7math.com √ 0.25x 2 3 36. 0.008p 9 √ Simplify each expression if . √_ab = _ √b 35. x √_ 16 2 38. 39. 37. 8 3 6 w x √_ 27 40. 121 √_ h k 3 √a 81 √_ m 4 8 6 Lesson 10-8 Roots of Monomials 555 H.O.T. Problems 41. OPEN ENDED Write a monomial and its square root. CHALLENGE Solve each equation for x. 42. √ 25a x = 5 ⎪a 3⎥ 43. 3 √ 64a 3b x = 4ab 7 44. √ 81a 4b x = 9a 2 ⎪b 5⎥ */ -!4( Explain why absolute value is necessary when (*/ 83 *5*/( 45. simplifying the expression √ y 2 and not necessary when simplifying √ y4. 46. Which expression is equivalent 48. Which of the following has the same 3 2 value as √ 27m 3n 6 ? to √144g ? A 12g C 12g 2 B 12⎪g⎥ D 12⎪g ⎥ A the length of the side of a square whose area is 27m 3n 6 2 B the expression 9mn 3 47. Which expression has the same value C the expression 3mn 2 as √ 400h 2k 4 ? F 20hk 2 H 20h 2k 4 G 20 ⎪h⎥ k 2 J D the length of the side of a cube whose volume is 3mn 2 200 ⎪h⎥ k 2 Simplify. (Lesson 10-7) 49. (6 3) 5 50. (n 7) 2 51. (2a 3b 2) 4 52. (-4p 11q) 3 Simplify. Express using positive exponents. (Lesson 10-6) 95 9 53. _ 3 54. _ 6 k 15 k 24y 4 55. _ 2 4y 45g 3 56. _ 7 3g 57. RETAIL Find the discount to the nearest cent for a flat-screen television that costs $999 and is on sale at 15% off. (Lesson 5-8) Math and Economics Getting Down to Business It’s time to complete your project. Use the information and data you have gathered about the cost of materials and the feedback from your peers to prepare a video or brochure. Be sure to include a scatter plot with your project. Cross-Curricular Project at ca.gr7math.com 556 Chapter 10 Algebra: Nonlinear Functions and Polynomials CH APTER 10 Study Guide and Review Download Vocabulary Review from ca.gr7math.com Key Vocabulary cube root (p. 554) monomial (p. 539) Be sure the following Key Concepts are noted in your Foldable. Key Concepts Functions (Lessons 10-1, 10-2, and 10-3) • Linear functions have constant rates of change. nonlinear function (p. 522) quadratic function (p. 528) Vocabulary Check • Nonlinear functions do not have constant rates of change. State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. • Quadratic functions are functions in which the greatest power of the variable is 2. 1. The expression y = x 2 - 3x is an example • Cubic functions are functions in which the greatest power of the variable is 3. Monomials (Lessons 10-5 through 10-8) • To multiply powers with the same base, add their exponents. • To divide powers with the same base, subtract their exponents. of a monomial. 2. A nonlinear function has a constant rate of change. 3. A quadratic function is a function whose greatest power is 2. 4. The product of 3x and x 2 + 3x will have 3 terms. • To find the power of a power, multiply the exponents. 5. A quadratic function is a nonlinear • To find the power of a product, find the power of each factor and multiply. 6. The graph of a linear function is a curve. function. 7. To divide powers with the same base, subtract the exponents. 8. The Quotient of Powers states when dividing powers with the same base, subtract their exponents. 9. The graph of a cubic function is a straight line. Vocabulary Review at ca.gr7math.com Chapter 10 Study Guide and Review 557 CH APTER 10 Study Guide and Review Lesson-by-Lesson Review 10-1 Linear and Nonlinear Functions (pp. 522–527) Determine whether each equation or table represents a linear or nonlinear function. Explain. 10. y - 4x = 1 12. 10-2 11. y = x 2 + 3 Time (h) 2 3 4 5 Number of Pages 98 147 199 248 Example 1 Determine whether the table represents a linear or nonlinear function. y -2 -3 -1 -1 0 1 1 3 As x increases by 1, y increases by 2. The rate of change is constant, so this function is linear. Graphing Quadratic Functions (pp. 528–531) 13. -4x 2 14. y = x 2 + 4 2 15. y = -2x + 1 Graph y = -x 2 - 1. Example 2 Graph each function. 2 16. y = 3x - 1 17. SCIENCE A ball is dropped from the Make a table of values. Then plot and connect the ordered pairs with a smooth curve. top of a 36-foot tall building. The quadratic equation d = -16t 2 + 36 models the distance d in feet the ball is from the ground at time t seconds. Graph the function. Then use your graph to find how long it takes for the ball to reach the ground. 10-3 x x y = -x 2 - 1 (x, y) -2 -(-2) 2 - 1 (-2, -5) -1 -(-1) 2 - 1 (-1, -2) 0 -(0) 2 - 1 (0, -1) 1 2 -(1) - 1 (1, -2) 2 2 (2, -5) -(2) - 1 y O x y x 2 1 PSI: Make a Model (pp. 532-533) Solve the problem by using the make a model strategy. 18. MEASUREMENT Sydney has a postcard that measures 5 inches by 3 inches. She decides to frame it, using a frame that 3 is 1_ inches wide. What is the 4 Example 3 DISPLAYS Cans of oil are displayed in the shape of a pyramid. The top layer has 2 cans in it. One more can is added to each layer, and there are 4 layers in the pyramid. How many cans are there in the display? perimeter of the framed postcard? 19. MAGAZINES A book store arranges it best-seller magazines in the front window. In how many different ways can five best-seller magazines be arranged in a row? So, based on the model there are 14 cans. 558 Chapter 10 Algebra: Nonlinear Functions and Polynomials Mixed Problem Solving For mixed problem-solving practice, see page 717. 10-4 Graphing Cubic Functions (pp. 534-537) 20. y = 2x 3 – 4 21. y = 0.25x 3 - 2 x y = -x 3 (x, y) 22. y = 2x 3 + 4 23. y = 0.25x 3 + 2 -2 -(-2) 3 (-2, 8) -1 -(-1) 3 (-1, 1) 0 -(0) 3 (0, 0) 1 -(1) 3 (1, -1) 2 -(2) 3 (2, -8) 24. MEASUREMENT A rectangular prism with a square base of side length x inches has a height of (x - 1) inches. Write the function for the volume V of the prism. Graph the function. Then estimate the dimensions of the box that would give a volume of approximately 18 cubic inches. 10-5 y y x 3 x Multiplying Monomials (pp. 539-542) Simplify. Express using exponents. 25. 4 · 4 5 26. x 6 · x 2 27. -9y 2(-4y 9) 28. _3 -4 · _3 2 (7) (7) 29. LIFE SCIENCE The number of bacteria after t cycles of reproduction is 2 t. Suppose a bacteria reproduces every 30 minutes. If there are 1,000 bacteria in a dish now, how many will there be in 1 hour? 10-6 Graph y = -x 3. Example 4 Graph each function. Example 5 Find 4 · 4 3. Express using exponents. 4 · 43 = 41 · 43 4 = 41 = 41 + 3 The common base is 4. 4 =4 Add the exponents. Example 6 Find 3a 3 · 4a 7. 3a 3 · 4a 7 = (3 · 4)a 3 + 7 Commutative and Associative Properties = 12a 10 Dividing Monomials (pp. 544-548) Simplify. Express using exponents. 59 30. _ 52 n5 31. _ n 32. _ 8 21c -7c 33. 68 6 Simplify_ . Express using exponents. 3 -1 3 11 Example 7 (_47 ) × (_47 ) __ _4 7 34. MEASUREMENT The area of the family room is 3 4 square feet. The area of the kitchen is 4 3 square feet. What is the difference in area between the two rooms? 68 _ = 68 - 3 63 The common base is 6. 5 =6 Example 8 Simplify. -8 s . Express using exponents. Simplify _ -4 s s -8 =_ = s -8 - (-4) s –4 1 = s -8 + 4 or _ 4 s Quotient of Powers Simplify. Chapter 10 Study Guide and Review 559 CH APTER 10 Study Guide and Review 10-7 Powers of Monomials (pp. 549-552) Example 9 Simplify. 2 3 35. (9 ) 36. (d 6f 3) 4 Simplify (7 3) 5. 37. (5y 5) 4 38. (6z 4x 3) 5 39. (_n -1) 2 40. [(p 2) 3] 2 41. (5 -1) 2 42. (-3k 2) 2(4k -3) 2 (7 3) 5 = 7 3 · 5 Power of a Power = 7 15 Simplify. Example 10 3 4 43. GEOMETRY Find the volume of a cube with sides of length 5s 2t 4 as a monomial. Simplify (2x 2y 3) 3. (2x 2y 3) 3 = 2 3 · x 2 · 3 y 3 · 3 Power of a Product = 8x 6y 9 Simplify. 44. GEOMETRY Find the area of a square with sides of length 6a 3b 5 as a monomial. 10-8 Roots of Monomials (pp. 553-556) Simplify. 45. √ a2 47. 49. 36x 2y 6 √ 3 p6 √ 3 51. √ 64c 6d 21 46. √ 49n 4 48. 81q 14 √ 50. Example 11 16f 8g 6 . Simplify √ 16f 8g 6 = √ 16 · √ f 8 · √ g 6 Product √ Property of Square Roots 3 √ 8m 18 3 52. √ 125r 9s 15 = 4 · f 4 · ⎪g 3⎥ or 4f 4 ⎪g 3⎥ 53. GEOMETRY Express the length of one side of the square whose area is 64b 16 square units as a monomial. 54. GEOMETRY Express the length of one side of a cube whose volume is 216a 9c 3 cubic units as a monomial. 560 Example 12 3 Simplify √ x9. 3 √ x9 = x3 Chapter 10 Algebra: Nonlinear Functions and Polynomials (x 3) 3 = x 9 Use absolute value to indicate the positive value of g 3. CH APTER Practice Test 10 Determine whether each graph, equation, or table represents a linear or nonlinear function. Explain. 1. 12. CRAFTS Martina is making cube-shaped gift boxes from decorative cardboard. Each side of the cube is to be 6 inches long, and there 1 is a _ -inch overlap on each side. How much 2. 2x = y y 2 cardboard does Martina need to make each box? x O Simplify. Express using exponents. 3. x -3 -1 1 3 y 2 18 26 10 13. 15 3 15 5 3 15 3 14. -5m 6(-9m 8) -40w 8 8w 15. _ 7 16. _ Graph each function. 4. y = _x 2 1 2 5. y = -2x 2 + 3 6. BUSINESS The function p = 60 + 2d 2 models the profit made by a manufacturer of digital audio players. Graph this function. Then use your graph to estimate the profit earned after making 20 players. Simplify. 2 17. √m 18. √ 144a 2b 6 19. 64x 3y 15 √ 20. STANDARDS PRACTICE Simplify the algebraic expression (3x 3y 2)(7x 3y). 7. A 21x 9y 2 B 21x 6y 2 C 21x 6y 3 D 21x 6y 6 Graph each function. 3 STANDARDS PRACTICE Which (12x 4)(4x 3) expression is equivalent to _ ? 5 F 12x 7 H 6x 4 G 12x 2 J 8x 6x 2 21. MEASUREMENT Find the area of the rectangle at the right. 4s 2t 2 3st 3 3 8. y = x + 4 9. y = x 3 - 4 Simplify. 10. y = _x 3 22. [(x 2) 4] 3 11. MEASUREMENT A neighborhood group 24. (3 -3) 2 1 3 23. (-2b 3) 2(4b 2) 2 would like Jacob to fertilize their lawns. The average area of each lawn is 6 4 square feet. If there are 6 2 lawns in this neighborhood, how many total square feet of lawn does Jacob need to fertilize? Chapter Test at ca.gr7math.com 25. GEOMETRY Express the length of one side of a square with an area of 121x 4y 10 square units in simplified form. Chapter 10 Practice Test 561 CH APTER 10 California Standards Practice Cumulative, Chapters 1–10 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 1 3 The equation c = 0.8t represents c, the cost of t tickets on a ferry. Which table contains values that satisfy this equation? A A car used 4.2 gallons of gasoline to travel 126 miles. How many gallons of gasoline would it need to travel 195 miles? t c Cost of Ferry Tickets 1 2 3 $0.80 $1.00 $1.20 4 $1.40 t c Cost of Ferry Tickets 1 2 3 $0.80 $1.60 $2.40 4 $3.20 t c Cost of Ferry Tickets 1 2 3 $0.75 $1.50 $2.25 4 $3.00 t c Cost of Ferry Tickets 1 2 3 $1.80 $2.60 $3.40 4 $4.20 B A 2.7 B 5.0 C 6.5 C D 7.6 2 The scatter plot below shows the cost of computer repairs in relation to the number of hours the repair takes. Based on the information in the scatter plot, which statement is a valid conclusion? Total Cost ($) Cost of Computer Repairs 55 50 45 40 35 30 25 20 15 10 5 y D 4 Shanelle purchased a new computer for $1,099 and a computer desk for $699 including tax. She plans to pay the total amount in 24 equal monthly payments. What is a reasonable amount for each monthly payment? F $50 G $75 1 2 3 4 5 6 7 x Number of Hours H $150 J $1,800 F As the length of time increases, the cost of the repair increases. G As the length of time increases, the cost of the repair stays the same. H As the length of time decreases, the cost of the repair increases. J As the length of time increases, the cost of the repair decreases. 562 Question 4 You can often use estimation to eliminate incorrect answers. In this question, Shanelle’s total spent can be estimated by adding $1,100 and $700, then dividing by 24. The sum of $1,100 and $700 is $1,800 before dividing by 24, so choice J can be eliminated. Chapter 10 Algebra: Nonlinear Functions and Polynomials More California Standards Practice For practice by standard, see pages CA1–CA39. 5 Which of the following is the graph of 2 x 2? y=_ 3 y y C A O O x y B D O 8 9 x The area of a rectangle is 30m 11 square feet. If the length of the rectangle is 6m 4 feet, what is the width of the rectangle? F 5m 7 ft H 36m 15 ft G 24m 7 ft J 180m 15 ft Which expression is equivalent to 5 4 × 5 6? A 5 10 C 25 10 B 5 24 D 25 24 y x Pre-AP O 6 Record your answers on a sheet of paper. Show your work. x 10 An electronics store is having a sale on Simplify the expression shown below. certain models of televisions. Mr. Castillo would like to buy a television that is on sale. This television normally costs $679. (3m 3n 2)(6m 4n) F 18m 12n 2 H 18 m 7n 3 G 18 m 7n 2 J 18 m 7n 6 Last Year’s Models 7 40% off What is the height h of the gutter in the figure below? Wednesday Only Television Sale! Take an additional 10% off a. What price, not including tax, will Mr. Castillo pay if he buys the television on Saturday? 20 ft h b. What price, not including tax, will Mr. Castillo pay if he buys the television on Wednesday? 12 ft A 10 ft C 16 ft B 14 ft D 18 ft c. How much money will Mr. Castillo save if he buys the television on Saturday? NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 10 Go to Lesson... 4-3 9-8 9-1 5-5 10-2 10-4 10-4 10-5 10-5 5-8 For Help with Standard... AF4.2 PS1.2 MR2.5 MR3.1 AF3.1 AF2.2 MG3.3 NS2.3 NS2.3 NS1.7 California Standards Practice at ca.gr7math.com Chapter 10 California Standards Practice 563 Statistics, Data Analysis, and Probability Focus Use statistical procedures and probability to describe data and make predictions. CHAPTER 11 Statistics Collect, organize, and represent data sets that have one or more variables and identify relationships among variables within a data set by hand and through the use of an electronic spreadsheet. CHAPTER 12 Probability Determine theoretical and experimental probabilities and use these to make predictions about events. 564 Lawrence Lawry/Getty Images Math and Science It’s all in the Genes Mirror, mirror on the wall... why do I look like my parents at all? You’ve been selected to join a team of genetic researchers to answer this very question. You’ll research basic genetic lingo and learn how to use a Punnett square. Then you’ll gather information about the genetic traits of your classmates. You’ll also make predictions based on an analysis of your findings. So grab your lab coat and your probability and statistics tool kits to begin this adventure. Log on to ca.gr7math.com to begin. Unit 5 Statistics, Data Analysis, and Probability Lawrence Lawry/Getty Images 565 11 Statistics • Standard 7SDAP1.0 Collect, organize, and represent data sets that have one or more variables and identify relationships among variables within a data set by hand and through the use of an electronic spreadsheet. Key Vocabulary circle graph (p. 576) histogram (p. 570) measures of central tendency (p. 585) measures of variation (p. 593) Real-World Link Population Statistics and statistical displays are frequently used to describe the populations of a country, state, or city. Statistics Make this Foldable to help you organize your notes. Begin with five pieces of 1 8 ” × 11” paper. _ 2 1 Place 5 sheets of paper _3 inch apart. 4 3 Crease and staple along the fold. 566 Chapter 11 Statistics Robert Brenner/PhotoEdit 2 Roll up bottom edges. All tabs should be the same size. 4 Label the tabs with the topics from the chapter. Label the last tab Vocabulary. a Table 11-1 Make rams 11-2 Histog Graphs 11-3 Circle en of Central Tend 11-4 Measures n s of Variatio 11-5 Measure s -Whisker Plot 11-6 Box-and f Plots -Lea -and Stem 11-7 y Appropriate Displa 11-8 Choose an lary Vocabu GET READY for Chapter 11 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at ca.gr7math.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Graph each set of points on a number line. (Lesson 1-3) Example 1 Graph the set {2, 4, 5, 9} on a number line. 1. {7, 8, 10, 15, 16} 2. {15, 20, 21, 25, 30} Add or subtract. (Lessons 1-4 and 1-5) Example 2 3. -4 + (-8) 4. -5 + 2 Find 6 + (-4). 5. 7 + (-3) 6. 1 - (-5) 6 + (-4) = 2 7. GOLF Gary’s golf scores relative to par on two holes were 3 and -2. Find his total score relative to par for the two holes. (Lesson 1-4) Order each set of rational numbers from least to greatest. (Lesson 2-2) 8. 0.23, 2.03, 0.32 9. 5.4, 5.64, 5.46, 5.6 10. 0.01, 1.01, 0.10, 1.10 11. LUNCH Horace’s lunch cost $3.71, Susan’s cost $3.17, and Paul’s cost $3.07. Write these costs in order from least to greatest. (Lesson 2-2) 6 and -4 have opposite signs. Subtract the absolute values, 6 and 4. The difference, 2, has the sign of the number with the larger absolute value, 6. Example 3 Order 6.08, 0.68, and 8.60 from least to greatest. Line up the decimal points. Compare the digits in each 6.08 place value position. 0.68 8.60 The order from least to greatest is 0.68, 6.08, and 8.60. Solve each problem. (Lesson 5-7) Example 4 12. Find 52% of 360. 72% of 360 = 0.72 × 360 13. What is 36% of 360? What is 72% of 360? = 259.2 Use the percent equation. Multiply. 14. Find 14% of 360. Chapter 11 Get Ready for Chapter 11 567 11-1 Problem-Solving Investigation MAIN IDEA: Solve problems by making a table. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Standard 7SDAP1.1 Know various forms of display for data sets, including stem-and-leaf plot or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data. MAKE A TABLE e-Mail: YOUR MISSION: Make a table to solve the problem. THE PROBLEM: How many acids, bases, and neutral substances were tested in science class? ▲ DARREN: In science class, I learned that substances with pH values less than 7 are acids, those with pH values greater than 7 are bases, and substances with pH values equal to 7 are neutral. I tested several solutions and listed the pH values. EXPLORE You have a list of the pH values. You need to know how many substances have a pH value of less than 7, greater than 7, and equal to 7. PLAN Make a table to show the frequency, or number, of pH values in each interval. SOLVE CHECK 7 8 4 3 pH number 8 7 9 7 Less than 7 9 5 2 3 7 Equal to 7 4 4 6 8 5 9 9 8 6 Greater than 7 7 Tally Frequency He tested 9 acids, 7 bases, and 4 neutral substances. He tested 9 + 4 + 7, or 20, substances. There are 20 values listed, so the table seems reasonable. 1. Tell an advantage and disadvantage of listing the values in a table. 2. Describe two types of information you have seen recorded in a table. */ -!4( Write a problem that can be answered using a table. (*/ 83 *5*/( 3. Then solve the problem by making a table. 568 Laura Sifferlin Chapter 11 Statistics Solve Exercises 4 and 5. Use the make a table strategy. 4. LUNCH The list shows lunch prices of various items at a local diner. Organize the data in a table using intervals $2.00–$2.99, $3.00–3.99, $4.00–$4.99, and so on. What is the most common interval of lunch prices? $3.00 $3.75 $4.25 $4.25 $4.50 $4.75 $4.75 $5.00 $5.00 $5.00 $5.00 $5.50 $5.50 $5.75 $5.80 $6.00 $6.00 $6.00 $6.50 $6.75 $7.00 $8.50 $10.00 $10.00 8. SPORTS In a recent survey of 120 students, 9. PHOTOGRAPHY How many ways are there 5. CARS Dexter’s brother wants to buy a used car. The list shows the model year of the cars listed in the classified ads. Which year is listed most frequently? 1998 2000 1999 1999 2001 2001 2002 1998 2000 2000 1997 2001 1998 1999 2001 2001 1999 2000 2000 1997 1999 1998 2002 1997 2000 1999 2000 2001 1999 1999 G STRATEGIES PROBLEM-SOLVIN tep plan. • Use the four-s asoning. • Use logical re r problem. • Solve a simple to arrange five French club members for a yearbook photo if the president and vice president must be seated in front with the other three members behind them? For Exercises 10 and 11, select the appropriate operation(s) to solve the problem. Justify your selection(s) and solve the problem. Use any strategy to solve Exercises 6–9. Some strategies are shown below. 10. GEOGRAPHY Name three countries that have a combined area of forests that is about equal to the area of forest in Russia. ,ARGEST !REAS OF &OREST #H IN A ! 3 5 IL "R AZ NA #A NE DO )N 2U DA SS Sample Tree Diameters from Cumberland National Forest IA diameters below are from 4 to 9.9 inches? SIA 6. FORESTS About what percent of the tree !REA IN MILLIONS MI • Make a table. Tally States was published in 1845. If 12 issues were published each year, including 1845, how many issues would be published through 2010? 50 said they play basketball and 60 said they play soccer. Of those, 20 play both sports. How many students do not play either basketball or soccer? $2.50 Diameter (in.) 7. MAGAZINES The first magazine in the United #OUNTRIES Frequency 2.0–3.9 6 4.0–5.9 30 6.0–7.9 28 8.0–9.9 24 10.0–11.9 19 12.0–13.9 4 Source: Top Ten Things 11. BASKETBALL The average salary of an NBA player is $4.5 million per season. The average salary of a WNBA player is $43,000 per season. About what percent of the average NBA player’s salary is the average WNBA player’s salary? Lesson 11-1 Problem-Solving Investigation: Make a Table 569 11-2 Histograms Main IDEA Display and interpret data in a histogram. Standard 7SDAP1.1 Know various forms of display for data sets, including stem-and-leaf plot or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data. BASKETBALL Kylie researched the average ticket prices to NBA basketball games for 30 teams. The frequency table shows the results. Price Interval ($) Frequency 20.00–29.99 1 30.00–39.99 11 40.00–49.99 10 50.00–59.99 5 60.00–69.99 1 70.00–79.99 2 1. What do you notice about the price intervals in the table? Tally 2. How many tickets were at least $20.00 but less than $50.00? NEW Vocabulary histogram Data from a frequency table can be displayed as a histogram. A histogram is a type of bar graph used to display numerical data that have been organized into equal intervals. Construct a Histogram REVIEW Vocabulary bar graph: a graphic form using bars to make comparisons of statistics (page 662) 1 FOOD Choose intervals and make a frequency table of the data shown. Then construct a histogram to represent the data. The least value in the data is 110 and the greatest is 380. An interval size of 50 Calories would yield the frequency table at the right. READING in the Content Area For strategies in reading this lesson, visit ca.gr7math.com. 570 Chapter 11 Statistics To construct a histogram, follow these steps. Step 1 Calories of Soup-in-a-Cup 380 225 110 176 218 241 280 160 155 180 195 166 178 207 240 239 144 212 235 221 Calories Tally Frequency 100–149 2 150–199 7 200–249 9 250–299 1 300–349 0 350–399 1 Draw and label a horizontal and vertical axis. Include a title. Show the intervals from the frequency table on the horizontal axis. Step 3 For each Calorie interval, draw a bar whose height is given by its frequency. Because all of the intervals are equal, all of the bars have the same width. Calories of Soup-in-a-Cup 10 8 6 4 2 35 0 39 – 9 30 0 34 – 9 25 0 29 – 9 20 0 24 – 9 15 0 19 – 9 0 10 0 14 – 9 There is no space between bars. Number of Soups Gaps Intervals with a frequency of 0 have a bar height of 0. This is referred to as a gap. Step 2 Calories a. SCHOOL The list at the right Test Scores gives a set of test scores. Choose intervals, make a frequency table, and construct a histogram to represent the data. 94 85 73 93 75 77 89 80 89 83 79 81 87 85 90 83 88 86 83 91 93 93 92 90 91 88 96 97 98 82 90 100 Personal Tutor at ca.gr7math.com Analyze and Interpret Data 2 HISTORY How many 3 HISTORY How old was the n n n n n n Two presidents were 40–44 years old, and six presidents were 45–49 years old. Therefore, 2 + 6 or 8 presidents were younger than 50 when they were first inaugurated. !GE OF 0RESIDENTS AT &IRST )NAUGURATION .UMBER OF 0RESIDENTS presidents were younger than 50 years old when they were first inaugurated? !GE YR Source: The World Almanac oldest president at his first inauguration? This cannot be determined from the data as presented in this graph. The histogram only tells us that the oldest president was either 65, 66, 67, 68, or 69 years old. b. How many presidents were 60 years old or older when they were first inaugurated? c. Based on the data above, at what age is a president most likely to be first inaugurated? Extra Examples at ca.gr7math.com Lesson 11-2 Histograms 571 100 118 117 116 118 121 114 114 105 109 107 112 114 115 118 117 118 125 106 110 122 108 110 121 113 120 119 111 104 111 120 113 120 117 105 110 118 112 114 115 Source: National Climatic Data Center AUTO RACING For Exercises 2–4, use the histogram at the right. Winning Speeds at Indianapolis 500 * 32 28 Number of Years 2. How many races had winning average speeds that were at least 150 miles per hour? 3. At which range of speeds is a car finishing in first place most likely to be? Explain your reasoning. 24 20 16 12 8 4 For Exercises 5, 6 7–14 See Examples 1 2, 3 19 9 Speed (miles per hour) Source: indy500.com (/-%7/2+ (%,0 17 5– 17 4 15 0– 74 50 – recorded at the Indianapolis 500? 14 9 0 4. What is the fastest winning speed 12 5– (p. 571) 112 100 128 120 134 118 106 110 109 112 –1 24 Examples 2, 3 State Record High Temperatures (°F) 10 0 record high temperatures in degrees Fahrenheit for each state in the United States. Choose intervals and make a frequency table. Then construct a histogram to represent the data. 99 1. WEATHER The list gives the (p. 570–571) 75 – Example 1 * through 2005 For each problem, choose intervals and make a frequency table. Then construct a histogram to represent the data. 5. 6. Calories of Frozen Fruit Bars Average Speed (mph), Selected Animals 25 35 200 280 80 80 90 70 61 50 50 50 45 8 40 45 50 50 60 90 70 43 42 40 40 40 35 0.17 40 100 120 45 60 350 35 32 32 30 30 30 1.17 30 25 20 9 18 12 200 Source: World Almanac for Kids, 2005 BASKETBALL For Exercises 7–10, use the histogram. 18,000–19,999 seats? 9. How likely is it that any given NBA court will seat more than 21,000 people? 10. Which court has the fewest seats? 572 Chapter 11 Statistics n n n 8. How many courts have ."! #OURT 3EATING n n n n n fewer than 19,000 seats? .UMBER OF #OURTS 7. How many courts have .UMBER OF 3EATS Source: The World Almanac LIBRARIES For Exercises 11–14, use the histogram at the right. .UMBER OF 0UBLIC ,IBRARIES IN %ACH 3TATE 12 8 4 a typical U.S. state have? Explain your reasoning. 9 ,1 9 99 9 Number of Public Libraries 1, 00 80 0– 1 0– 79 9 0– 0– 60 39 59 9 9 19 20 0– 13. How many public libraries does 9 0 public libraries? 0– 12. Which state has the fewest 16 40 600 public libraries? Number of States 11. How many states have at least 20 Source: Public Libraries Survey 14. How many states have between 400 and 800 public libraries? 3CORES OF 7INNING 4EAMS THROUGH /RANGE "OWL #OTTON "OWL 7INNING 4EAM 3CORE n n n n n n n n n n n n n .UMBER OF 3CORES n Source: World Almanac 2005 FOOTBALL For Exercises 15–18, use the histograms shown. .UMBER OF 3CORES Real-World Link There are 16,486 public libraries in the United States. The state with the most public libraries is New York (1,088) followed by California (1,074), and then Texas (848). 7INNING 4EAM 3CORE Source: The World Almanac 15. Which bowl had the highest winning team score? 16. Determine which bowl game has had a winning team score of 30–39 points more often. 17. Determine which bowl game has had a winning team score of at least 40 points more often. 18. What was the lowest winning team score in each bowl game? 19. COLLECT THE DATA Conduct a survey of your classmates to determine the number of hours each person spends on the Internet during a typical week. Then choose intervals, make a frequency table, and construct a histogram to represent the data. 20. RESEARCH Use the Internet or other resource to find the populations of each county, census division, or parish in your state. Make a histogram using your data. How does your county, census division, or parish compare with others in your state? %842!02!#4)#% See pages 704, 718. 21. OPEN ENDED Construct a histogram that has a vertical line of symmetry Self-Check Quiz at ca.gr7math.com and two gaps. Then construct a histogram that has a vertical line of symmetry and one gap. Lesson 11-2 Histograms Rafael Macia/Photo Researchers 573 H.O.T. Problems 22. CHALLENGE Describe how the histogram at the right would change if larger intervals, such as 0–9 and 10–19, were used. Describe how it would change if smaller intervals, such as 0–2, 3–5, 6–8, and so on, were used. Frequency n */ -!4( Describe when a (*/ 83 *5*/( 23. n n n Age (yr) histogram might be more useful than a table with individual data. Then describe when a table with individual data might be more useful than a histogram. 24. Which statement can be concluded from the histogram? 25. A group of mothers reported when their children got their first tooth. #HILDS !GE AT &IRST 4OOTH .UMBER OF #HILDREN .UMBER OF 3UPER "OWLS 7INNING 3CORES AT THE &IRST 3UPER "OWLS n n n n !GE MONTHS n n n n n n 7INNING 3CORES A The lowest winning score was 10. B The highest winning score was 59. C Most of the winning teams scored between 10 and 29 points. D Most of the winning teams scored between 20 and 39 points. What fraction of the number of children reported got their first tooth when they were six months old or older? 9 F _ 4 H _ 20 _ G 7 20 26. THEME PARKS The list gives the annual attendance in millions of persons for various theme parks in the United States. Use the make a table strategy to organize the data into intervals. (Lesson 11-1) 5 1 J _ 5 14.0 12.7 8.6 7.9 7.3 6.9 6.1 5.3 5.2 4.6 4.3 4.3 4.0 3.5 3.3 3.3 3.2 3.2 3.1 3.0 2.6 2.6 2.6 2.5 27. GEOMETRY Express the length of one side of a cube whose Source: World Almanac, 2005 volume is 64x 3 cubic units as a monomial. (Lesson 10-8) PREREQUISITE SKILL Solve each problem. (Lessons 5-3 and 5-7) 28. Find 26% of 360. 29. What is 53% of 360? 574 Chapter 11 Statistics 30. Find 73% of 360. Extend 11-2 Main IDEA Graphing Calculator Lab Histograms You can make a histogram using a graphing calculator. Use a graphing calculator to make histograms. Standard 7SDAP1.1 Know various forms of display for data sets, including stem-and-leaf plot or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data. Standard 7MR2.5 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Mr. Yamaguchi’s second period class has listed the distance each student lives from the school. Make a histogram. Distance from School (miles) 4 2 6 1 10 3 19 5 20 1 1 9 22 15 2 4 12 8 1 4 16 3 6 7 Clear any existing data in list L1 by pressing STAT ENTER CLEAR ENTER . Then enter the data in L1. Input each number and press ENTER . Turn on the statistical plot by pressing 2nd [STAT PLOT] ENTER ENTER . Select the histogram and L1 as the Xlist by ENTER 2nd L1 ENTER . pressing Press WINDOW . To set the viewing window to be [0, 25] scl: 5 by [0, 12] scl: 1, press WINDOW 0 ENTER 2.5 ENTER 5 ENTER 0 ENTER 12 ENTER 1 ENTER Press '2!0( '2!0( . to create the histogram. ANALYZE THE RESULTS 1. Press 42!#% . Find the frequency of each interval using the right arrow key. 2. Explain why the x-values for this data set were chosen as 0 to 25. 3. COLLECT THE DATA Use the graphing calculator to make a histogram of your classmates’ heights in inches. Other Keystrokes at ca.gr7math.com Extend 11-2 Graphing Calculator Lab: Histograms 575 11-3 Circle Graphs Main IDEA Construct and interpret circle graphs. Standard 7SDAP1.1 Know various forms of display for data sets, including stem-and-leaf plot or box-and-whisker plot; use the forms to display a single set of data or to compare two sets of data. Interactive Lab ca.gr7math.com MOVIES The graphic shows the results of a recent survey of 1,100 U.S. movie-goers. 9\_Xm`fij8d\i`ZXej=è[Dfjk 8eefpè^XkXDfm`\K_\Xk\i 3OMEONE TALKING ON A CELL PHONE DURING A MOVIE ! CELL PHONE RINGING DURING A MOVIE 3OMEONE TALKING TO THEIR SEATMATE DURING A MOVIE 3OMEONE WHO SAVES SEATS IN A CROWDED THEATER 3OMEONE LOUDLY EATING POPCORN OR SOME OTHER SNACK DURING A MOVIE 1. What percent of U.S. movie-goers found a ringing cell phone the most annoying behavior at a movie theater? 2. What percent of U.S. NEW Vocabulary circle graph movie-goers were annoyed with some kind of noise disturbance? Source: Braun Research 3. Which behavior was reported as the most annoying? 4. Are all the behaviors surveyed accounted for in the graphic? Explain. A circle graph

California Maths Grade 7

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