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
•
•
•
•
•
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
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
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
Explore 9-1
9-1
Algebra: Linear Functions
Algebra Lab: Functions . . . . . . . . . . . . . . . . . . . .464
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
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 . . . . . . . . . . . . . . . . . . 518
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
Extend 11-6
Graphing Calculator Lab:
Box-and-Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . .605
11-7 Stem-and-Leaf Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . .606
11-8 Select an Appropriate Display . . . . . . . . . . . . . . . . . . . . . 611
Prerequisite Skills
• Get Ready for Chapter 11 567
• Get Ready for the Next Lesson 574, 582,
590, 598, 604, 610
Reading and Writing Mathematics
• Reading in the Content Area 570
• Vocabulary Link 593
• Writing in Math 568, 574, 581, 590, 598,
604, 610, 614
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
Standard
Text of Standard
Primary
Citations
Supporting
Citations
3.3*
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.
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
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
ounces
160 students × 2 _ × 8 _ = 2,560 ounces
student
serving
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
$9.99
15.50
+
12.99
_______
$38.48
$5.49
×
2
______
$10.98
Step 2 $38.48
32.99
_______
$5.49
hat
scarf
gloves
total cost
cost of box set
difference
total cost
difference
number of box sets
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)
3 • 6
Keystrokes:
µ
14 µ 7
b.
ENTER
6×8
_
9+3
6 • 8
Keystrokes:
µ
9
3
ENTER
Evaluate each expression.
1. (15 - 5) + (9 × 2)
4.
2. (12 + 20) - (4 × 6)
3 × 10
_
9×8
_
5.
2+4
3. 6 × (9 ÷ 3) - 1
6.
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)
10. -6 + 8 - 10 + 15
11. -3(-8) - (-20)
9. 5 + 9 - 3 + (-17)
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.
or
t
al Tu
…
n
o
r
s
r
o
ok F with Pe
Lo
ne
Onli amples
h
t
a
s
Ex
•M
boxe
n
Extra
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
_1
4
STATE REPTILE
c lukewarm water
• 6 Tbsp. cornmeal
•
• _12 c cold water
• 1 c mashed potatoes
• 1 c boiling water
• 2 c all-purpose flour
• 1 tsp salt
• 2 c finely ground leached
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
3
$1.95
56”maximum
3
$1.95
56”maximum
3
$1.95
Ride
Tickets
Price
Bulgy
Red Baron
Speed Boats
Speedway
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
Space Race
e
Logger ’s Reveng
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
5
5
Rock-O-Plane
4
Sea Serpent
4
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.
Add.
Example 1 Find 14.63 + 2.9.
(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
91
4
20
3. Estimate the total number of paintings the artist
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?
ˆ}ÕÀiÊ£
ˆ}ÕÀiÊÓ
Figure Number
1
2
Perimeter
4
8
ˆ}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
g2
- 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.
d.
3p - 6
_
8-p
e.
4q
_
q + 2(p + 1)
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.
Example 3
(p. 30)
Example 4
(p. 31)
Example 5
(p. 31)
bc
_
6.
2
c2
_
b-5
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
17.
x2
-3
_
2z + 1
14. 12 + z - x
18.
wz2
_
y+6
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
in a community with a population of c people by
c
evaluating the expression _
n · p. The variable n is the
number of people per household, and p is the percent
of households owning that pet.
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.
National Percent of
Households Owning
Pets
Dogs
0.316
Cats
0.273
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.
39. Which of these gases freezes at the coldest
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.
Gas
hydrogen
-435°
krypton
-251°
oxygen
-369°
helium
-458°
argon
-309°
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
Freezing Point (°F)
at Sea Level
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.
Car Number
Laps Behind Winner
3
-1
8
-12
15
-3
24
0
48
-8
following statements is false?
F ⎪a⎥ > 2
G ⎪a⎥ = ⎪b⎥
H ⎪b⎥ < 2
J
⎪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.
51. r - 4n
52. 2m 2 - p + 3
53.
(Lesson 1-2)
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
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?
Change
Monday
-$2
Tuesday
+$1
Wednesday
+$3
Thursday
-$1
Friday
-$4
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.
45. -6 ● -11
46. 5 ● -5
(Lesson 1-3)
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.
29. How far below the surface
Erie
elevation is the deepest part of
Lake Huron?
30. Find the difference between the
deepest part of Lake Erie and the
deepest part of Lake Superior.
Deepest
Point (m)
Great Lakes
Surface
Elevation (m)
-64
174
Huron
-229
176
Michigan
-281
176
Ontario
-244
75
Superior
-406
183
Source: National Ocean Service
Evaluate each expression if a = -3, b = 14, and c = -8.
31. b - 20
35. b - a
32. c - 15
36. c - b
33. a - c
37. (b -
34. a - b
a)2
38. a - c -b2
+c
ANALYZE TABLES For Exercises 39 and 40, use the table.
39. The wind makes the air outside feel
See pages 677, 708.
Wind (miles per hour)
Temperature (°F)
%842!02!#4)#%
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?
40. How much warmer does 20°F feel than
-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
⫺5
54.
G -3 units
J
(Lesson 1-3)
⎪13 - (-7)⎥
53. ⎪20⎥ - ⎪-5⎥
55.
2
H 3 units
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)
52. ⎪-14⎥ + ⎪3⎥
0
F -7 units
51. BASEBALL The table shows the income of
Evaluate each expression.
B
7 units
Team
Income
(thousands)
Atlanta Braves
-$14,360
Chicago Cubs
$4,797
Florida Marlins
New York Yankees
-$27,741
$40,359
Source: mlb.com
⎪-12 + (-25)⎥
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
Dead Sea
Israel
–400
G 8,850, 6,194, -400, -156, -86
H -400, -156, -86, 6,194, 8,850
J
M
-156, -86, -400, 6,194, 8,850
Add or subtract.
A 2 × (36 ÷ 4) + (68 ÷ 4)
(Lessons 1-4, 1-5)
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
–86
F -86, -156, -400, 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
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.
_
The signs are the same.
6 Find -30 .
-15
-30
_
=2
-15
The quotient is positive.
Divide.
j. -28 ÷ (-7)
k.
36
_
l.
-2
-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
___
=_
5
5
= -6
Find the sum of the set of numbers.
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)
(p. 53)
Example 8
(p. 54)
54
5. (-3)
2
6. -4(5)(-7)
Divide.
7. -16 ÷ 4
22
10. _
11
Example 7
3. -3(7)
2. 3(-6)
8. 21 ÷ (-3)
9. -72 ÷ (-8)
-25
11. _
-5
12.
-96
_
12
Evaluate each expression if a = -5, b = 8, and c = -12.
13. 4a + 9
14.
b-c
_
15. 3b - a 2
a
16. FOOTBALL During a scoring drive, a
football team gained or lost yards on
each play as shown. What was the
average number of yards per play?
Chapter 1 Algebra: Integers
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)
35.
-84
_
4
32. -60 ÷ 3
36.
33. 45 ÷ 9
28
_
37.
-7
34. -34 ÷ (-2)
-72
_
38.
-6
64
_
8
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
42. 9 - wz
43.
w-x
_
44.
z
8y
_
x2 - 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.
-96
_
+c
55. -c 2 - 25
b-a
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
Subtract.
(Lesson 1-5)
68. 12 - 18
Add.
2°
69. -5 - (-14)
70. -3 - 20
71. 7 - (-15)
(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.
Self-Check Quiz at
ca.gr7math.com
H.O.T. Problems
23. MAPS The scale on a map indicates that 1 inch on the map represents an
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.
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?
ˆ}ÕÀiÊ£
ˆ}ÕÀiÊÓ
ˆ}ÕÀiÊÎ
B a = 87 - m
C a = 87 · m
D a = 87 ÷ m
Multiply or divide.
31. -9(10)
F P = 3x
H P=x+2
G P = 3x - 2
J
P=x-2
(Lesson 1-6)
32. -5(-14)
33. 34 ÷ (-17)
34.
-105
_
-5
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
Top
Speed
(mph)
Flight
Length
(mi)
Operating
Cost per
Hour
B747-400
534
3,960
$8,443
B727-200
430
644
$4,075
Airplane
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.
36. 3(9)
(Lesson 1-6)
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
Day
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.
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
12
12
_
=_
4
4
3=3
70
Chapter 1 Algebra: Integers
Photowood/CORBIS
c. -12 = -6k
Algebra
5x = -60
-60
5x
_
=_
5
5
x = -12
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)
-20 = -20
_x (2) = 8(2)
2
x = 16
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.
d.
y
_
= -8
-4
e.
m
_
= -9
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
Solve each equation. Check your solution.
1. 5b = 40
2. -7k = 14
p
4. _ = 9
9
5.
For
Exercises
8–13
14–19
20, 21
See
Examples
1
2
3
a
_
= -3
6. 22 = _
m
-2
12
7. LAWN SERVICE Josh charges $15 to mow an average size lawn in his
(p. 71)
(/-%7/2+ (%,0
3. -18 = -3n
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
11. 36 = -18y
14.
m
_
= 10
17.
q
20 = _
7
-5
9. 9b = 72
10. 34 = -2x
12. -32 = 8d
13. -35 = 5n
15.
u
_
=6
18.
c
-8 = _
9
12
16.
h
_
= 33
19.
r
_
= -3
-3
24
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)#%
See pages 679, 708.
Self-Check Quiz at
ca.gr7math.com
72
28.
10
_
= -5
x
29.
-126
_
= -21
a
30. -17 = _
g
136
31. PHYSICAL SCIENCE The amount of work, measured in foot-pounds, is equal
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?
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.
185 ft
1,250 ft
(Lesson 1-7)
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)
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
Order of Operations
(Lesson 1-2)
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
Variables, Expressions, and Properties
(pp. 29–34)
Evaluate each expression if a = 6, b = 2,
and c = 1.
13. a(b + 4)
14. 3b 2
15. 3a + 2b + c
16.
2
(a + 2)
_
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
The answer of $75 is close to
the estimate of $80, so the
answer is reasonable.
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.
Adding Integers
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.
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.
Subtracting Integers
29. -2 - (-5)
-16 + (-11)
= -27
Example 6
-7 + 20
= 13
Example 7
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?
Chapter 1 Algebra: Integers
Find -16 + (-11).
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⎥.
(pp. 46–49)
Subtract.
76
1
(pp. 41–45)
Add.
1-5
0
22. ⎪-12⎥ - ⎪4⎥
21. ⎪-5⎥
1-4
⫺8 ⫺7 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
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
Writing Equations
Find -48 ÷ (-12).
-48 ÷ (-12) = 4
The dividend and the
divisor have the same
sign. The quotient is
positive.
(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.
PSI: Work Backward
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
(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.
53.
_s = 42
7
52. -4x = 52
54.
y
_
= -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
Add, subtract, multiply, or divide.
9. -27 + 8
11.
-5
13. 7(-10)(-4)
15. 8(-9)
2. (2c + b) ÷ b - 3
1
H _
7
J
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?
the number of minutes of phone usage. Find
the charge if a person uses 622 minutes.
Solve each equation. Check your solution.
20. x + 15 = - 3
6. ⎪13⎥ ● - 13
22.
STANDARDS PRACTICE Evaluate the
following expression:
7.
7
its diameter. Given d, the diameter, write
an equation that could be used to find r,
the radius.
2
5. -8 ● -11
7
18. MEASUREMENT A circle’s radius is half
4. CELL PHONES The monthly charge in dollars
Replace each ● with <, >, or = to make a true
sentence.
⎪12 - 7⎥ - ⎪3 - 6⎥
A -8
B -2
16. 13 - 61
1
G -_
3. 4a - 5a - 12
x - 500
by the expression 40 + _
where x is
14. -9 + (-11)
F -7
2
for a specific cell phone company is given
12. -4 - (-35)
STANDARDS PRACTICE What is the
absolute value of -7?
17.
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.
-70
_
10. -105 ÷ 15
n
_
= 16
-2
21. -7 = a - 11
23. -96 = 8y
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?
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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.
Vocabulary Link
Terminating
Everyday Use bringing to
an end
Math Use a decimal that
ends
_3
b.
4
-3
_
c. 4_
13
25
5
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
Write each fraction or mixed number as a decimal.
(p. 85)
1.
_4
2.
5
5
4. _
9
Example 3
(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_
16
5
5. 4_
6
7. BASEBALL In a recent season, Ichiro Suzuki of the Seattle Mariners had
(p. 86)
Examples 4, 5
9
_
−−
−
12. -3.8
13. 2.15
Write each fraction or mixed number as a decimal.
14.
_3
15.
4
18. -_
7
16
22.
4
_
33
_2
16.
5
19. -_
5
32
6
23. -_
11
7
_
17.
80
1
20. 2_
8
33
_
40
5
21. 5_
16
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.
Weight
(ounces)
Animal
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.
IN
1
45.
46.
CM
CM
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.)
See pages 679, 709.
Self-Check Quiz at
ca.gr7math.com
47. WEATHER Carla recorded the rainfall totals for
%842!02!#4)#%
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
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.
Above/Below
Average (in.)
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 _
?
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?
5
A 2.4
C 2.55
B 2.45
D 2.6
55. Janet wants to buy a pair of jeans that
54. Felisa made 0.9 of her free throws in
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.
57. t + 17 = -5
58. a - 5 = 14
(Lesson 1-9)
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.
7
_3 ● _
4
12
b.
_5 ● _7
6
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.
_1 ● 0.3
e. 0.22 ● _
f. 2_ ● 2.42
5
12
11
50
3
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
12
16
1.
5
_1 ● _
2
2.
12
5. -_ ● -_
10
18
(p. 93)
k. -_ ● -_
7
10
4
5
Replace each ● with <, >, or = to make a true sentence.
(pp. 91–93)
Example 5
j. -3.15 ● -3.17
9
3
_
●_
25
3.
10
4
5
7
10
4. 3_ ● 3.625
5
8
11
−
6. -_ ● -_
16
18
3
_
● 0.25
−−
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 ● _7
3
_3 ● _5
5
8
12. 0.5 ● _
14. 6_ ● 6.5
15. 2_ ● 2.7
11.
9
13. 0.75 ● _
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. -_ ● -_
25. -_ ● -_
26. -_ ● -_
27. -_ ● -_
28. -1_ ● -1_
29. -5_ ● -5_
3
11
3
5
1
11
4
10
3
8
7
15
7
10
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 -_
5
5
C -_
F Point P
7
G Point Q
7
D -_
8
H Point R
J
S
1
2
3
4
1
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.
y
44. _ = 22
7
45. 4p = -60
47. 81 = -3d
48.
(Lesson 1-10)
46. 20 = _
t
15
_a = -108
49. -4n = -96
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.
51. -4(-7)
(Lesson 1-6)
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.
_3 · _1
2
1 3
c. _ · _
4 5
4
b.
_2 · _2
d.
_2 · _4
5
3
3
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 = _
b
d
bd
where b and d ≠ 0
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 · _2
b.
3
4
5 _
_
· 3
12
c.
20
7 _
_
· 7
10
16
Multiply Negative Fractions
_ _
2 Find - 5 · 3 . Write in simplest form.
8
6
Negative Fractions
-5
5
__
, and _ are
-5,
6
6
1
5 _
-5 _
· 3 =_
·3
-_
8
6
-6
all equivalent fractions.
Divide 6 and 3 by their GCF, 3.
8
6
2
-5 · 1
=_
Multiply the numerators.
Multiply the denominators.
5
= -_
The fractions have different signs,
so the product is negative.
2·8
16
Multiply. Write in simplest form.
d.
_8 · -_3
9
e. -_ · _
3
5
4
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.
_5 · 1_3
7
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
Real-World Link
A 757 aircraft has an
average cruising speed
of 540 miles per hour,
a capacity of 242
passengers, and a
_1
wingspan of 165 feet.
3
Source: Continental
Traveler
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.
j. CARPENTRY A piece of lumber is 4_ feet long. If you need a piece of
1
4
2
lumber that is _
this size, how long a piece do you need?
3
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.
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
The problem asks for the distance. When you
divide the common units, the answer is expressed in miles.
Check for Reasonableness
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.
_3 · _5
5
2.
7
4. -_ · _
_4 · _3
5
3.
8
6
12
2
6. -_ -_
3
13
3
7
_
_
9. -6 · 1
9
4
3
2 _
8
9
1
2
_
_
8. 2 · 1
2
5
5. -_ ·
1 4
8 9
1
1
_
7. 1 · 5_
3
2
_6 · _7
()
7
( )( )
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.
5 4
9 2
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
12.
1 _
_
·4
12
7
13.
3 _
_
·1
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.
_1 · -_3 · _4
3
5
8
( )
33.
35. 2_ · 1_ · 2_
2
7
38.
5
9
_1 · _2 · _3
2
5
36. 3_ · 1_ · 5
2
5
1
3
_1 · 0.25
37. 10 · 3.78 · _
1
2
1
5
−
39. -_ · 0.3
2
9
5
(-_25 ) · _16 · (-_25 )
34.
4
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.
%842!02!#4)#%
See pages 680, 709.
Self-Check Quiz at
ca.gr7math.com
44. abd 2
45. b 2c 2
46.
_1 a 2d
47. -3ac(-bd)
2
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
1
7
1
*/ -!4( Explain why the product of _
(*/
and _
is less than _
.
83 *5*/(
2
8
2
53. What number will make _ · _ = _ · n
3
4
true?
7
8
7
8
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
Replace each ● with <, >, or = to make a true sentence.
−−
1
4
2
55. _ ● _
56. _ ● 0.28
2
3 2
H _
in
4
5 2
G _
in
8
7
8
_1 in 2
6
J
(Lesson 2-2)
7
−
57. -_ ● -0.4
4
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.
62. 51 ÷ (-17)
(Lesson 1-6)
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 ÷ _1
e. -_ ÷ _
2
4
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.
The problem asks for the number of days.
When you divide the common units, the answer is expressed in days.
Check for Reasonableness
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
Write the multiplicative inverse of each number.
(p. 102)
1.
Example 2
3
4
4.
_2 ÷ _3
5.
6.
9
_3 ÷ -_
8
10
7. -_ ÷ -_
8.
_4 ÷ 8
9.
(p. 103)
3
4
( )
5
(
5
6
_5 ÷ _1
8
2
( 78 )
7
16
10. -5_ ÷ -4_
Examples 4, 5
3. -2_
2. -12
7
Divide. Write in simplest form.
(p. 103)
Example 3
_5
2
3
9
_
÷3
10
11. -3_ ÷ 6_
)
5
6
7
12
12. BIOLOGY The 300 million-year-old fossil of a cockroach was recently found
(p. 104)
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.
_2 ÷ _3
5
4
23. -_ ÷ _
4
5
27.
3
4
_2 ÷ 4
5
31. 3_ ÷ 2_
3
4
1
2
20.
_3 ÷ _2
21.
24.
3
2
_
÷ -_
10
3
25. -_ ÷ -_
28.
9
_
÷3
29.
8
3
( )
16
32. 7_ ÷ 2_
1
2
1
10
_2 ÷ _5
3
22.
6
( 23 )
5
9
_4 ÷ 6
33. -12_ ÷ 4_
1
4
2
3
5
10
26. -_ ÷ -_
7
12
30.
5
1
_2 ÷ _
( 56 )
_6 ÷ 4
7
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>Ãà œ««iÀ
>À}iÃÌÊ}À>Ãà œ««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.
48.
43 _
641
_
· 641 ÷ _
76
594
783 _
72
_
· 241 ÷ _
47.
594
241
783
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
Multiply. Write in simplest form.
51.
_1 · _3
2
52.
4
(Lesson 2-3)
7 _
_
·4
12
x = 15(20)
53. 1_ · 4_
2
3
7
1
5
54.
_2 · 3_1
3
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.
57. 10 candy bars short of his goal
PREREQUISITE SKILL Add or subtract.
59. -7 + 15
60. -9 + (-4)
(Lesson 1-3)
58. 7 bonus points
(Lessons 1-4 and 1-5)
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.
1
1_
3
_1
2
1. What is the sum of
_1
3
_1
3
_1
the whole-number
parts of the amounts?
2. How many _ cups
1
3
are there?
3
3
3. Can you combine these
Bread
cups of whole wheat
flour (sifted)
cups of white flour
(sifted)
cup oatmeal
cup apricots (diced)
cup hazelnuts
(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
8
( )
Add the numerators.
The denominators are the same.
-2
1
=_
or -_
8
Simplify.
4
Add. Write in simplest form.
a.
108
_5 + _7
9
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
( )
8
8
7
7
-_
-_
= -_
+ -_
9
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. -_ - _
3
5
4
5
e.
_3 - _5
8
f.
8
( )
_5 - -_4
7
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
(
7
4
7
4
5_
+ 8_
= (5 + 8) + _
+_
9
9
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 whole numbers
and fractions separately.
Add the numerators.
9
11
2
= 13_
or 14_
9
)
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
241 - 235
=_
4
6
1
=_
or 1_
4
2
Estimate
60 - 59 = 1
Write the mixed numbers
as improper fractions.
Subtract the numerators.
The denominators are the same.
1
Jasmine is 1_
inches taller than Amber.
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
Add or subtract. Write in simplest form.
(pp. 108–109)
1.
( )
_2 + -_4
5
5
3
4
3. -_ +
1
4
4
9
4. -_ - _
5.
7. 5_ - 2_
8. -1_ + -2_
9
10
7
10
4
9
Example 4
(-_79 )
5
2
6. -_ - (-_)
6
6
2. -_ + _
_3 - _7
8
8
3
7
2
9
( 27 )
9. 10 - 3_
5
16
10. CLOTHING Hat sizes are determined by the distance across a person’s head.
3
inches than
How much wider is a person’s head who wears a hat size of 7_
(p. 109)
4
1
inches?
someone who wears a hat size of 6_
4
(/-%7/2+ (%,0
For
Exercises
11–14
15–18
19–26
27, 28
See
Examples
1
2
3
4
Add or subtract. Write in simplest form.
( 27 )
11. -_ + _
12. -_ + -_
15. -_ - _
16.
19. 3_ + 7_
20. 9_ + 4_
3
7
4
9
1
9
3
5
4
5
5
8
5
8
16
7
9
3
4
7
12
1
7
_
-_
12
12
(
3
4
)
( )
14.
_8 + -_5
9
9
18.
_2 - _8
9
9
(
)
21. 8_ + -2_
22. 8_ + -5 _
25. 7 - 5_
26. 9 - 6_
1
10
24. -3_ - 7_
5
6
5
12
17.
16
5
9
23. -1_ - 3_
5
6
15
9
_
-_
13. -_ + _
9
10
5
12
11
12
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
Divide. Write in simplest form.
46.
_3 ÷ _6
5
47.
7
J
10
_2 in.
5
(Lesson 2-4)
_7 ÷ 2_4
8
48. -3_ ÷ 2_
1
4
5
1
2
49. Find the product of -_ and -_. (Lesson 2-3)
6
7
7
8
50. NUTRITION There is 2.3 times the
Fruit
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)
Evaluate each expression.
51. ⎪-20⎥ - ⎪17⎥
Vitamin C
(mg in 5.5 oz)
Orange
52
Strawberries
63
Kiwifruit
103.5
Source: Food and Drug Administration
(Lesson 1-3)
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
Replace each ● with <, >, or = to make a true
sentence. (Lesson 2-2)
4.
_1 ● _1
3
4
33
8.
Austin, TX
120
5. -_ ● -_
Denver, CO
115
Phoenix, AZ
215
7. -7.833… ● -7.8
Sacramento, CA
195
Santa Fe, NM
175
3
10
2
5
−−
4
6. 0.12 ● _
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)
Days
15
Divide. Write in simplest form.
(Lesson 2-4)
12.
3
4
2
_1 ÷ -_3
2
4
( )
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
6
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_
.
2
3
1
1
4
A 190_
, 191_
, 191_
, 191_
2
6
4
1
1
G Divide 25_
by 1_
.
15
2
3
1
4
1
B 191_
, 191_
, 191_
, 190_
4
6
15
6
15
2
4
J
3
1
4
1
D 191_
, 191_
, 190_
, 191_
6
15
2
2
1
1
to 1_
.
H Add 25_
2
3
1
1
4
C 190_
, 191_
, 191_
, 191_
2
2
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 + -_4
5
5
( )
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?
3.
e
ok
Bro
?
1
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.
_ ( _)
3
4
_1 + -_2 = _1 · _3 + -_2 · _4
( 3) 4 3 ( 3) 4
4
1 1 + -2
The LCD is 3 · 4 or 12.
3
8
=_
+ -_
( )
Rename using the LCD.
12
12
3
+
(-8)
5
= _ or -_
12
12
2 - 8 - (- 7 )
63
99
_
Add the numerators. Then simplify.
_
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.
3
1
_
+_
14
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.
d. -_ + -_
( 18 )
5
12
e. -3_ + 8_
1
2
f. 2_ - 6_
3
4
1
3
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.
_3 + -_1
4
6
2. -_ + _
4.
_7 - _3
5.
( )
8
5
8
4
7. -3_ + 1_
Example 4
( 23 )
14
12
6. _ - (-_)
15
21
7
7
9. -4_ - (-3_)
72
12
4
9
7
2
_
-_
9
13
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.
7
_1 + -_
4
12
12. -_ + _
15.
_1 - _7
16.
)
(
3
8
19. 3_ + -8_
(
1
5
1
2
)
23. -4_ - 5_
3
4
14. -_ + -_
2
_4 - -_
5
15
( )
1
2
20. 1_ + (-6_)
6
3
( 12 )
3
2
17. -_ - (-_)
11
9
3
1
21. 8_ - (-6_)
7
2
24. -8_ - 4_
25. -15_ + 11_
26. -22_ + 15_
3
8
5
8
13. -_ + -_
5
6
1
3
( 38 )
7
12
18. -_ - (-_)
15
25
3
1
22. 7_ - (-1_)
8
4
6
7
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
Youth
Camp
x
Glacier
Ridge
mi
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.
_3 of _2
3
4
3
2
40. _ less than _
3
4
39.
_3 more than _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 _
-_
?
the total amount of snack mix?
F
2
A 5_
c
3
G
7
B 5_
c
12
H
2
C 6_
c
3
J
7
D 6_
c
4
_3 × _5 – _2 × _6
5
3
6
4
( ) ( )
(_34 × _66 ) – (_23 × _55 )
(_34 × _33 ) – (_23 × _44 )
(_34 × _44 ) – (_23 × _33 )
3
12
Add or subtract. Write in simplest form.
44. -_ + _
7
11
5
11
(Lesson 2-5)
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.
118
51. d - 13 = -44
52. -18t = 270
53. -34 = y + 22
54. -5 = _
Chapter 2 Algebra: Rational Numbers
a
16
(Lessons 1-9 and 1-10)
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 .
2
2
4
4
4
_3 from each side.
4
_1
2
4
1
-_
=t
Simplify.
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
58.4
-7.3m
_
=_
-7.3
-7.3
-8 = m
Write the equation.
Divide each side by -7.3.
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
For
Exercises
10–15
16–21
22, 23
See
Examples
1, 2
3, 4
5
9
16
7
10
9. SPACE The planet Jupiter takes 11.9 Earth years to make one revolution
(p. 120)
(/-%7/2+ (%,0
5
8
3
8
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 + _
5
1
9
3
4
16. -_b = 16
7
14. g - (-1.5) = 2.35
15. -1.3 = n - (-6.12)
17. -_p = -8
18. -1.92 = -0.32s
19. -8.4 = 1.2t
20.
2
9
t
_
= -4.5
1
2
21.
3.2
2
3
h
_
= -2.2
-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
25.
1
-7.5r = -3_
26.
−
1
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)
1
2 _x
1
-_1 _x
(2 )
( )
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
B 24
1
B 2_
in.
2
C 30
1
C 3_
in.
4
D 54
3
D 3_
in.
4
Add or subtract. Write in simplest form.
43.
_1 + _1
6
7
44.
_7 - _1
8
(Lesson 2-6)
45. -5_ - 6_
1
2
6
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.
Add.
(Lesson 1-9)
(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
Write Expressions Using Powers
NEW Vocabulary
power
base
exponent
The exponent tells how many times
the base is used as a factor.
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 2
(
1
= (_
2)
3
)
· 34
Associative Property
Definition of exponents
2 a·b·b·a·b
a·b·b·a·b=a·a·b·b·b
= (a · a) · (b · b · b)
=
a2
·
b3
Commutative Property
Associative Property
Definition of exponents
Write each expression using exponents.
a.
_2 · 7 · _2 · _2 · 7 · _2
3
3
3
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
10 3
1,000
2
100
10 1
10
10 0
1
10
Negative powers
are the result of
repeated division.
Standard
Form
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 1
7 -3 = 1 · · or 3
7
7
7
7
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 .
(_23 )
4
2 _
2 _
=_
· 2 ·_
·2
3 3
16
_
=
81
3
Write the power as a product.
3
Multiply.
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.
_1 · p · k · _1 · p · p · k
2
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
15. x · _ · y · y · _ · 5 · y · 5 · x · _ · y · y
1
6
14. 2 · 7 · a · 9 · b · a · 7 · b · 9 · b · a
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.
28. How many stars can be seen with
unaided eyes in an urban area?
29. How many stars can be seen with
unaided eyes in a rural area?
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
With Small Telescope
2 · 10 6 stars
Source: Kids Discover
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
38.
3 3 · 10 2
_
2
3 · 10
4
Chapter 2 Algebra: Rational Numbers
39.
42 · 35 · 24
_
3
5
4 ·3 ·2
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
Boston
Number
of Arrivals
7.21 × 10
5
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
Canada
Number
of Visitors
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.
Element
Great Lakes are listed in the table.
Order the lakes according to their
area from least to greatest.
Mass per Atom
Great
Lake
9.91 × 10 3
Carbon
1.995 × 10 -23 g
Erie
Gold
3.272 × 10 -22 g
Huron
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
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
Chapter 2 Algebra: Rational Numbers
2.3 × 10 4
Source: World Book
Arrange these numbers in increasing order.
132
Area
(mi 2)
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.
40.
1
1
t + 3_
= 2_
3
2
41.
2
-_
y = 14
3
42.
(Lesson 2-7)
p
_
= 2.4
1.3
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
Rational Numbers
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)
(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. -_
1
3
13.
3
-2_
14.
10
_5
5
8
_
Write 3 as a decimal.
5
_3 means 3 ÷ 5.
5
9
Write each decimal as a fraction or mixed
number in simplest form.
15. 0.3
16. -7.14
17. 4.3
18. -5.7
−
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
Replace each ● with <, >, or = to make a
true sentence.
−−
8
8
2
21. _ ● _
22. -0.24 ● -_
3
9
55
1
23. -_ ● -_
2
110
33
24.
_5 ● _3
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?
(pp. 91–95)
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
Example 5
form.
Multiply. Write in simplest form.
27.
_3 · 1_2
5
3
5 3
29. _ · _
6 5
28.
(pp. 96–101)
2
2
-_
· -_
3
3
( )
_ _
Find 2 · 5 . Write in simplest
3
Multiply the numerators.
Multiply the denominators.
2·5
_2 · _5 = _
3
1 10
30. _ · _
2 11
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 ÷ _1
9
3
34. -4_ ÷ (-2)
2
5
7
2
_
÷ -_
12
3
( )
1
2
35. 6_ ÷ (-1_)
6
3
33.
_ _
Example 6 Find - 5 ÷ 3 . Write in
5
6
simplest form.
Divide. Write in simplest form.
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
Adding and Subtracting Like Fractions
5
6
_
+_
11
11
1
7
39. _ - _
8
8
38.
1
3
_
+ -_
28
28
(
)
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
18
(pp. 108–112)
Add or subtract. Write in simplest form.
37.
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)
42. -_ + _
2
3
3
5
43.
5
7
_
- -_
12
15
2
5 - 1_
( )
44. -4_ - 6_
45.
46. 7_ + 3_
47. 5_ - 12_
3
4
1
2
4
5
2
3
3
5
5
_ _
Example 8 Find 3 + 1 . Write in
3
4
simplest form.
Add or subtract. Write in 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
51.
_3 n = _7
4
50.
52.
_x = -2.2
4
r
-7.2 = _
1.6
3
53. AGE Trevor is _ of Maria’s age. Trevor
8
8
is 15. Write and solve a multiplication
equation to find Maria’s age.
2-8
PSI: Look for a Pattern
5
1
t+_
=_
3
6
_
_ _
Solve t + 1 = 5 .
3
6
Write the equation.
_
5
1
1
t+_
- 1 =_
- 1 Subtract _
from each side.
3
3
3
6
_
t=1
2
3
Simplify.
(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
3
5
() ()
3
1
64. (_) · (_)
2
4
62.
2
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
Scientific Notation
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
(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
20.
STANDARDS PRACTICE The following
table gives the approximate diameter,
in miles, for several planets.
(-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
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
G
H
J
Mercury, Neptune, Saturn, Earth
Mercury, Earth, Neptune, Saturn
Mercury, Neptune, Earth, Saturn
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?
A 8 hours
C 12 hours
B 9 hours
D 18 hours
4
Which number equals (3) -3?
1
F -_
27
G -9
1
H _
27
J 9
5
3 _
Which fraction is equivalent to _
+ 3?
6
A _
15
9
B _
10
2
The weight of a paper clip is 9.0 × 10 -4
kilograms. Which of the following represents
this weight in standard notation?
6
10
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
3
5
9
C _
50
9
D _
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_
2
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
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
A the brand of the computer
Oxygen
15.9994
Titanium
47.867
Mercury
200.59
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)
(
11.50 + 8.75 + 6.50
C t = (3 + 2 + 2) ____
3
Pre-AP
)
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
of blocks needed to fill the container.
1
H _
64
b. Write and simplify an expression to solve
1
J _
12
the problem.
c. How many blocks will it take?
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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
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,i>Ê ՓLiÀÃ
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*ÞÌ >}œÀi>˜
/ iœÀi“
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
5. 2 + 4
2
7. 10 2 + 8 2
2
6. 3 + 3
2
8. 7 2 + 5 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.
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
Real-World Link
The Great Pyramid
of Khufu is the
largest of the
ancient pyramids.
Source: infoplease.com
s = ± √
562,500
Write the equation.
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
15.
121
_
324
19. - √
2.56
12. - √
81
13. - √
484
14. ± √
36
16. -_
17. ±_
18. -_
20. ± √
1.44
21. √
0.25
22. ± √
0.0196
64
16
9
225
25
49
ALGEBRA Solve each equation. Check your solution(s).
23. v 2 = 81
24. b 2 = 100
27. w 2 = _
28.
36
100
9
_
= c2
64
25. 144 = s 2
26. 225 = y 2
29. 0.0169 = d 2
30. a 2 = 1.21
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.
( )
25
_
b. 2
( √
36 )
2
c.
81
2
( √
1.99 )
d.
( √x) 2
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.
Write each expression using exponents.
46. 6 · 6 · 6
(Lesson 2-10)
(Lesson 2-9)
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.
is closer to √
So, √
83 is between 9 and 10. Since √83
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
4 2 < 23.5 < 5 2
√
4 2 < √
23.5 < √
52
4 < √
23.5 < 5
Write an inequality.
16 = 4 2 and 25 = 5 2
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.
1
5_
5
19.
7
21_
10
20.
3
17_
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.
Multiply or divide.
38. (-5)(-13)
(Lesson 2-10)
(Lesson 1-6)
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
-«œÀÌÃ
ÕÈV
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
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?
8
runny nose and sneezing
15
watery eyes and sneezing
12
runny nose, watery eyes, and sneezing
For Exercises 10 and 11, select the appropriate
operation(s) to solve the problem. Justify your
selection(s) and solve the problem.
5
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«œÌi˜ÕÃi
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
1
g. √
11 ● 3_
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.
day, the number of miles
a person can see to the
horizon is about 1.23 times
the square root of his or
her distance from the
ground, in feet. Suppose
Domingo is at the top of
Bank of America Tower
and Jewel is at the top of
Two California Plaza.
How much farther can
Jewel see than Domingo?
1.6
1
i. √
6.25 ● 2_
2
Los Angeles Skyscrapers
1,018
1200
858
1000
800
699
750
600
400
200
0
Bank of
Two
America
California
Tower
Plaza
Source: National Park Service
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
For
Exercises
11–18
19–22
23–28
29–30
See
Examples
1–3
4
5,6
7
Name all sets of numbers to which each real number belongs.
_2
11. 14
12.
15. 4.83
16. 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
26.
2
2_
● √
5.76
5
24. √
12 ● 3.5
27.
−
1
5_
● 5.16
6
1
25. 6_ ● √
40
3
−
28. √
6.2 ● 2.4
29. LAW ENFORCEMENT Police can use the
to estimate the
formula s = 5.5 √0.75d
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.
45. t 2 = 25
(Lesson 3-1)
46. y 2 = _
1
49
47. 0.64 = a 2
PREREQUISITE SKILL Evaluate each expression.
48. 3 2 + 5 2
49. 6 2 + 4 2
(Lesson 2-9)
50. 9 2 + 11 2
51. 4 2 + 7 2
Lesson 3-4 The Real Number System
159
CH
APTER
Mid-Chapter Quiz
3
Lessons 3-1 through 3-4
Find each square root.
18.
(Lesson 3-1)
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
13. √
17
14. √
21
15. √
75
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
Name all sets of numbers to which each real
number belongs. (Lesson 3-4)
_2
22. √
25
23. - √
15
24. √3
25. 10
26. - √4
21.
3
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
Chapter 3 Real Numbers and the Pythagorean Theorem
−
● √
10
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
2
8 +
b2
= 24
Pythagorean Theorem
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
(p. 164)
Determine whether each triangle with sides of given lengths is a right
triangle. Justify your answer.
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
NORT H
CA ROL I NA
24 mi
Raleigh
401
40
54
1
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
x 2 = 225
Subtract 400 from each side.
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
the lake?
9. How high is the wire
10. How high is the
wheel chair ramp?
attached to the pole?
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.
( 34 )
25. -3_ + -5_
2
3
26. -1_ - 7_
1
8
3
4
(Lesson 2-6)
27.
_3 - 4_1
5
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.
30. T(5, 2)
31. A(-1, 3)
32. M(-5, 0)
(Page 688)
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.
5 units
3
units
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 .
0 1 2 3 4 5 6 7
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
)
(
Necklace
1
2 , 2_
(
)
2
2
1 _
dig. She finds a ring at _
, 1 1 and a
1
necklace at 2, 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
x
O
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,
1
So, the ordered pair for point P is 3_
,2 .
2
1
2
3
x
⫺1
which is 2.
(
O
P
)
⫺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_
(
1
4
1
2
)
f. S(-1.5, 3)
g. T -_, -3_
(
1
2
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
c = 3 + 3.5
2
7ASHINGTON
$#
!NNAPOLIS
4RENTON
$OVER
2ICHMOND
Pythagorean Theorem
2
c = 21.25
√
c 2 = ± √
21.25
c ≈ ±4.6
7EST
0OINT (ARTFORD
(ARRISBURG
is located at -1 1 , -1 1 .
"OSTON
Replace a with 3 and b with 3.5.
3 2 + 3.5 2 = 9 + 12.25 or 21.25
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
⫺1
Graph and label each point.
(_14 , 3_12 )
B
1
5. J
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
(p. 175)
(/-%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
13. PARKS On a park map, the ranger station is
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
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?
y
R
1
T
Q
W
⫺1
O
1
S
U
⫺1
V
x
P
Graph and label each point.
(_34 , 2_14 )
1 4
25. H(-2_, 3_)
4 5
22. E
23. F
(_25 , 1_12 )
24. G -3, 4_
26. J(4.3, -3.1)
27. K(-3.75, -0.5)
(
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 ՓLiÀÃ
>˜`ÊÌ i
*ÞÌ >}œÀi>˜
/ iœÀi“
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)
Example 1
Find each square root.
9. √
81
11. - √
64
10. ± √
225
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
Estimating Square Roots
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.
PSI: Use a Venn Diagram
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.
(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
Since 6 = 36, √
36 = 6.
(pp. 148–151)
Estimate to the nearest whole number.
3-3
.
Find √36
2
Chapter 3 Real Numbers and the Pythagorean Theorem
Example 5 The Venn diagram shows
the number of dog and cat owners.
œ}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.
30 in.
25 ft
20 ft
w
41. How long is
42. How far is the
the walkway?
ᐉ
h
plane from the
airport?
Use the Pythagorean Theorem to write the
equation 53 2 = h 2 + 25 2. Then solve the
equation.
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.
2. - √
0.25
3.
36
±_
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
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
√
18
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
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
9
D _
11
in a positive number?
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
7
B _
6
14 Which of the following expressions results
represents √8?
G H
6
C _
7
8
10 Which point on the number line best
F
5
1
A _
5
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.
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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.
10
1. _
24
36
3. _
81
(Prior Grade)
88
2. _
104
49
4. _
91
81
÷ 27
the $45 that he saved. Write a
fraction in simplest form that
represents the portion of his
savings he spent. (Prior Grade)
6-2
6. _
5+5
3-1
_
8.
1+9
7.
_
Simplify 54 .
5. MONEY Devon spent $18 of
Evaluate each expression.
Example 1
(Prior Grade)
7-4
_
8-4
5+7
_
9.
8-6
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
Solve each equation.
(Lessons 1-10)
Simplify.
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)
(/-%7/2+ (%,0
For
Exercises
8–11
12–15
16–21
22–23
See
Examples
1
2
3
4
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.
Express each ratio in simplest form.
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
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.
36.
19
_
5
37.
_3
8
38.
12.4
_
4
(Lesson 2-1)
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
CUP 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?
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
(pp. 194–195)
1. ELEPHANTS An adult elephant drinks about 225 liters of water each day.
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
%842!02!#4)#%
See pages 685, 711.
POSTAGE For Exercises 15 and 16, use the table below that shows the price to
mail a first-class letter for various weights.
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)
Cost ($)
1
2
3
0.39 0.63 0.87
4
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.
24. The product of -9 and a number is 45.
(Lesson 1-10)
25. A number divided by 4 is -16.
PREREQUISITE SKILL Solve each equation. Check your solution.
26. 5 · x = 6 · 10
27. 8 · 3 = 4 · y
28. 2 · d = 3 · 5
(Lesson 1-10)
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.
9
_x = _
4
10
b.
5
2
_
=_
c.
y
34
n
_7 = _
3
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
c = 2.79(11)
c = 2.79g
Write the equation.
c = 2.79(20)
Replace g with the number of gallons.
c = 30.69
c = 55.80
Multiply.
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
Solve each proportion.
(p. 199)
1.
1.5
10
_
=_
6
p
2.
3.2
n
_
=_
9
36
3.
5
41
_
=_
x
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
For
Exercises
6–13
14–19
20–25
See
Examples
1
2
3
Solve each proportion.
6.
32
_k = _
7.
18
x
_
=_
8.
44
11
_
=_
10.
6
d
_
=_
11.
2.5
h
_
=_
12.
3.5
a
_
=_
7
56
25
30
13
6
39
9
p
9.
0.4
2
_
=_
13 .
48
72
_
=_
5
8
3.2
w
0.7
9
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.
18
_2 = _
3
x+5
30.
x-4
7
_
=_
10
31.
5
4.5
3
_
=_
17 - x
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 previously-
viewed 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
Congruence The symbol is read is congruent to. Arcs
are used to show congruent
angles.
The similar triangles in the Mini Lab suggest the following.
+%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.
MN
JK
3
1
_
=_
or _
6
NP
10
2
K
7
M
N
10
6
6
Q
Next, check to see if corresponding
sides are proportional.
HJ
7
_
=_
3
L
Since the two polygons are rectangles,
all of their angles are right angles.
Therefore, all corresponding angles
are congruent.
Common Error
Do not assume that
two polygons are
similar just because
their corresponding
angles are congruent.
Their corresponding
sides must also
be proportional.
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
14
1.5
M
D
6
3.5
K
1.5
3.5
L
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
2
a corresponding length
long as
m=
2
_
is
on polygon ABCD.
−−
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)
(/-%7/2+ (%,0
For
Exercises
5–8
9–12
18, 19
See
Examples
1
2
3
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
8
Z
C
16
Determine whether each pair of polygons is similar. Explain.
5.
6.
3
3
3
3
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.
6 in.
B
S
W
1
A 13_
inches
C 24 inches
2
inches
B 22_
3
1
D 25_
inches
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.
22. (-2.5, 1.5), (1.5, -3.5)
23.
(-2, -1_12 ), (4, 3_12 )
24.
(Lesson 3-6)
(-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.
F 12
(Lesson 4-1)
6. 40 laps in 6 races
A 25
J
48
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
B 30
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.
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)
9. DISHES Jack washed 60 plates in
18.
Solve each proportion.
(Lesson 4-3)
33
11
_
=_
2
r
15
x
11. _ = _
36
24
5
4.5
12. _ = _
9
a
212
10.
H 24
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)
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
Asia
Europe
North America
Name
Speed
Dodonpa
172 kph
Stealth
128 kph
Kingda Ka
128 mph
Tallest Roller Coasters
Continent
Name
Height
Steel Dragon 2000
97 m
Europe
Silver Star
73 m
North America
Kingda Ka
456 ft
Asia
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
proportion to find the missing measure.
3 in.
(Lesson 4-5)
8 in.
4.5 in.
Solve each proportion.
y
5
60. _ = _
4
12
(Lesson 4-3)
61.
120
24
_
=_
b
60
62.
0.6
1.5
_
=_
m in.
n
5
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.
64.
(p. 674)
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
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
grams
=D2/B2*B5
grams
grams
Sheet 1
E
Amount
Sheet 2
1
Customary Unit
ounces
ounces
ounces
Sheet 3
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.
%842!02!#4)#%
43. MICROWAVES The inside of a microwave oven has a volume of 1.2 cubic
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.
See pages 687, 711.
44. MEASUREMENT The density of gold is 19.29 grams per cubic centimeter. To
Self-Check Quiz at
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)
5
3
_
÷_
12
53. -_ · _
7
48
20
PREREQUISITE SKILL Solve.
56.
3 cm
x cm
_
=_
5 ft
9 ft
9
14
2.4
2
3.2
3.2
3
Find each product or quotient. Write in simplest form.
52.
1.5
2
54. 2_ · 1_
3
4
4.8
(Lessons 2-3 and 2-4)
55. -3_ ÷ -_
2
3
1
5
( 23 )
(Lesson 4-3)
57.
5 in.
4 in.
_
=_
5 mi
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
1:30
1 inch represents an actual distance of 4 feet.
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.
Let a represent the actual distance in kilometers.
Let m represent the map distance in centimeters.
Variables
=
a
Equation
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
(p. 226)
(/-%7/2+ (%,0
For
Exercises
6–11
12–13
14–15
16–17
See
Examples
1
2
3
4
5. DECORATING Before redecorating, Nichelle makes a scale drawing of her
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.
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
9. bedroom 2
7. master bath
10. master bedroom
8. porch
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.)
ca.gr7math.com
H.O.T. Problems
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
Self-Check Quiz at
b. a tennis ball (8.25 in.)
how long would it take the Kowalski family to drive from Champaign to
Carbondale if they drove 60 miles per hour?
21. OPEN ENDED Choose a large or small rectangular item such as a calculator,
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.
3
27. 4ft =
?
yd
3
2
28. 160 cm =
?
m
2
2
29. 6 m =
(Lesson 4-7)
?
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?
Estimate each square root to the nearest whole number.
32. √
11
33.
√
48
35.
10 - 8
36.
85 - 67
_
2001 - 1995
(Lesson 3-2)
34. - √
118
PREREQUISITE SKILL Evaluate each expression.
45 - 33
_
(Lesson 4-6)
(Lesson 1-2)
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
10
38
Entries
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.
+?
Year
Entries
The change or difference in the
number of years is 2007–2004.
2004
2007
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
=_
Subtract to find the change in the
number of entries and years.
17 entries
≈_
Express this rate as a unit rate.
3 years
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)
Height (in.)
8
11
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.
Check
32
28
24
0
’98
’00
’02
’04
’06
Year
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
Graph
Doug Martin
Chapter 4 Proportions and Similarity
y
y slants
upward
O
232
Rates of Change
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
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
Time
Flyers Folded
2. Find the rate of temperature change between
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.
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.
<Xk`e^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
Distance (ft)
0
16
32
48
64
Rate of Change
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
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
4
Cost of Play Tickets
t
1
2
3
4
c
$2.50
$5.00
$7.50
$10.00
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.
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?
ALGEBRA Solve each equation. Check your solution(s).
2
32. p = 0.49
33.
1
t =_
144
2
(Lesson 4-8)
(Lesson 3-1)
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
(p. 236)
corresponding parts (p. 206)
Key Concepts
Proportions
cross products (p. 198)
(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.
(Lesson 4-5)
polygon (p. 206)
proportion (p. 198)
• The cross products of a proportion are equal.
Similar Polygons
nonproportional (p. 194)
ratio (p. 190)
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.
8 liters
10 milliliters
= __
8,000 milliliters
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
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?
Solving Proportions
1
The ratio in simplest form is _
or 1:800.
800
3
15.
6
n
_7 = _
4
2
(pp. 194–197)
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.
(pp. 198–202)
Solve each proportion.
_
13. _
r =8
Divide numerator and
denominator by 10.
800
Proportional and Nonproportional Relationships
11. INTERNET A high-speed Internet
4-3
1
=_
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?
8 liters = 8 × 1,000 or
8,000 milliliters.
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
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?
(pp. 213–218)
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
28. Which is greater: a 5,000-meter race or
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.
6
Age (yr)
1.00
$ per week
8
10
12
15
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
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
(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 = 12
3
9
6.
20
_5 = _
3
y
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 $1.99 per package,
what other information is needed to find
the cost of the pencils?
she in feet and inches?
(1 meter ≈ 39 inches)
9
F 3 feet 3 inches
H 4 feet 9 inches
G 4 feet 0 inches
J 4 feet 10.5 inches
A the cost of erasers
B the number of students in the whole
school
During a 3-hour period, 2,292 people rode
the roller coaster at an amusement park.
Which proportion can be used to find x,
the number of people who rode the coaster
during a 12-hour period if the rate is the
same?
3
x
A _
=_
2,292
12
3
12
C _
=_
2,292
x
3
12
=_
B _
x
12
D _
=_
2,292
x
3
C the number of students in her class
D the name of the store where she is buying
the pencils
9
2
12 Which fraction is between _
and _
?
2,292
3
1
F _
5
3
H _
4
1
G _
11
J _
2
10
9
10 A park is shaped like a rectangle with the
Pre-AP
dimensions shown below. Which of the
following is closest to the length of a
diagonal of the park?
Record your answers on a sheet of paper.
Show your work.
13 The table shows how
Time
(h)
2
4
6
8
much Susan earns
when she works at a
fast-food restaurant.
a. Graph the data
YD
from the table and
connect the points
with a line.
Wages
($)
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?
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AF4.2 6NS1.3 MG1.1 MG1.2 MG1.1 NS2.4
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Chapters 1–4 California Standards Practice
AF3.4
249
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
644 · _
1
1. _ · 303
2.
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. _
6.
5
3
7. _
4
_7
_
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
14.
3
x =_
_
10
5
15.
14
_4 = _
9
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)
_ _
6
5
w
_
=_
6
12
12
6 · w = 12 · 5
6w = 60
6w
60
_
=_
6
6
w = 10
Write the proportion.
Find cross products.
Simplify.
Divide each side by 6.
Simplify.
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
Utah
27 out of 100
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
Numbers
Algebra
Percent
A percent is a ratio that compares a number to 100.
Percent
Ratio
27%
27 out of 100
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.
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Îä¯ *>Ã̈V
Ó{¯
"Ì iÀÊ
/À>Ã
Îx¯
œœ`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.
4. 7 : 10
5.
9
_
6. 1 out of 500
20
Write each percent as a fraction in simplest form.
7. 19%
8. 50%
9. 18%
Write each ratio or fraction as a percent.
17
_
10. 23 out of 100
11. 110 per 100
12.
14. 8 out of 25
15. 54 out of 300
16. 2 : 5
13.
50
7
_
20
17. 9 : 10
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.
42.
_3
5
43.
_3
4
44.
_5
8
(Lesson 2-1)
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.
= 20%
Add the percent
symbol.
1.66 = 1.66
Multiply by 100.
= 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
100
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
_____
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
_2 = 0.66−6
Percents
In real-world
−
situations, 66.6% will
usually be rounded
to 67% or 66.7%.
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.
11
_
8.
25
13
_
9.
40
_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.
Chapter 5 Percent
David Forbert/SuperStock
13
17
_
, 60%, 0.062, _
25
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.
17
_
34.
20
1
37. _
40
12
_
35.
25
1
38. _
125
_8
36.
5
4
39. _
9
_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.
7
_3 , 0.8, 8%, _
4
10
41 4
50. 84%, 0.88, _, _
50 5
48. 0.2, _, 2%, _
49.
51. 31%, _, _, 0.305
52. 6.5%, _, 0.556, _
1
4
3
20
3 3
10 13
1
2
_
, 7%, 0.09, _
20
25
1
6
3
5
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.
3
1
_
●_
%
250
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.
56.
_7 ● 1_1 %
4
4
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3ERVING 3IZE CAN M,
“œÕ˜ÌÊ*iÀÊ-iÀۈ˜}
>œÀˆiÃÊ
$AILY 6ALUE
-œ`ˆÕ“ÊMG
/œÌ>Ê >ÀLœ Þ`À>ÌiÊG
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?
/œÌ>Ê>ÌÊG
3UGARS G
*ÀœÌiˆ˜ÊG
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.
66. Between which two percents is _?
7
40
65. Mr. Lee asked his students how many
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
75
B _
100
F 15% and 16%
G 16% and 17%
H 17% and 18%
19% and 20%
J
% of
Students
92.5%
5%
2.5%
67. What percent of the square is shaded?
A 20%
IN
B 31.25%
C 44%
5
C _
100
IN
D 62.5%
IN
D 0.075
IN
Write each ratio as a percent.
68. 27 out of 100
(Lesson 5-1)
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
Time (s)
375
750
1,125
1,500
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.
74. {-12, 5, -5, 13, -1}
75. {42, -56, -13, 101, 13}
PREREQUISITE SKILL Solve each proportion.
77.
x
_5 = _
6
24
78.
(Lesson 1-3)
a
2
_
=_
12
15
76. {64, -58, -65, 57, -61}
(Lesson 4-3)
79.
_2 = _5
7
t
80.
10
_3 = _
n
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
94
_
=
w
⎫
89
_
⎬ percent
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
2% of 50
4% of 25
8% of 12.5
16% of 6.25
=
=
=
=
=
1
1
1
1
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.
If the percent remains the same and
the whole is halved, the answer is 1.
J
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
of the right triangle.
24 in.
(Lesson 3-5)
10 in.
PREREQUISITE SKILL Compute each product mentally.
38.
_1 · 422
2
39.
1
639 · _
3
(Lesson 2-3)
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
Grade Level
play basketball. Use mental
math to find the number of
students in the sixth grade
who play basketball.
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
_1
12 % =
_1
_1
37 % =
_2
_2
62 % =
_1
_5
87 % =
3
40% =
_2
33 % =
75% =
_3
60% =
_3
66 % =
100% = 1
80% =
_4
83 % =
4
5
5
7
8
_2
16 % =
_1
5
3
4
1
Percent-Fraction Equivalents
50% =
2
5
8
3
3
3
6
3
3
6
_1
_1
10% =
1
_
_1
_3
30% =
3
_
_1
_5
70% =
7
_
_1
_7
90% =
9
_
2
2
2
2
8
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
_1
Use the fraction form of 20%, which is .
5
10
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
3. 37_% of 72
4. 1% of 52
5. 10% of 350
6. 2% of 630
1
3
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?
Write each fraction as a percent.
39.
9
_
20
40.
(Lesson 5-3)
(Lesson 5-2)
_7
8
41.
3
_
42.
500
_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
3.
5.
2. 8 : 10
13
_
4.
25
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.
3
_
8.
200
4
_
9. 1.72
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)
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«ˆ˜}
Îί
"Ì iÀ
Î
Óx
274
-V œœ
Î
£ä
7œÀŽ
£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`ވ˜}
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)
Example 4
(p. 276)
Examples 5–7
(p. 276)
(/-%7/2+ (%,0
For
Exercises
8–15
16–23
24–25
See
Examples
1–3
5–7
4
Estimate.
2. 66_% of 20
2
3
1. 49% of 160
3. 73% of 65
4. SCHOOL Math is the favorite subject of about 28% of students, according to
a recent study. If there are 30 students in your class, estimate the number of
students who would pick math as their favorite subject.
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
City
%842!02!#4)#%
See pages 690, 712.
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.
40. 0.2a = 7
278
Chapter 5 Percent
41. 20s = 8
(Lesson 2-7)
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
Multiply each side by the whole.
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
Divide each side by 600.
0.7 =
Simplify.
600
600
n
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
_
=_
Divide each side by 0.52.
125 =
Simplify.
0.52
0.52
w
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?
Find the Percent
15 is what percent of 60?
15 = n(60)
Find the Whole
15 is 25% of what number?
15 = 0.25w
p = 0.25(60)
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
See
Examples
1
2
3
4
dealership paid $5,300 for a car, for how much should they sell the car?
Solve each problem using a percent equation.
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?
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?
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.
Compute mentally.
33. 20% of $200
(Lesson 5-5)
(Lesson 5-4)
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.
41. -5fg
43. -10fh
42. 2gh
PREREQUISITE SKILL Evaluate each expression.
44. ⎪17 - 24⎥
45. ⎪340 - 253⎥
(Lesson 1-2)
(Lesson 1-3)
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
new: 6 meters
d. original: 1.25 hours
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?
Estimate.
(Lesson 5-7)
(Lesson 5-6)
34. 21% of 60
35. 25% of 83
Express each rate as a unit rate.
37. 34% of 95
(Lesson 4-1)
38. $36 in 3 hours
39. 1.5 inches of rain in 5 months
PREREQUISITE SKILL Solve each equation.
40. 45 = 300 · a · 3
36. 12% of 31
(Lesson 2-7)
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
_
=_
10,800
10,800
0.105 = r
Divide each side by 10,800.
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
9. $834 at 7.25% for 2 months
8. $725 at 4.5% for 4 years
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.
28. What percent of 70 is 17.5?
(Lesson 5-7)
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 decrease (p. 285)
compound interest (p. 294) percent equation (p. 279)
discount (p. 286)
percent of increase (p. 285)
0ERCENT
Key Concepts
Percent
percent of change (p. 284)
(p. 275)
(Lessons 5-1 and 5-2)
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
part
_
= percent, where
whole
2. (Percents, Compatible numbers) are
Percent–Fraction Equivalents
1
50% = _
2
2
40% = _
5
1
1
_
33 % = _
3
3
3
1
_
_
37 % =
2
8
3
_
30% =
10
3
75% = _
4
3
60% = _
5
2
2
_
66 % = _
3
3
5
1
_
_
62 % =
2
8
7
_
70% =
10
1. A (proportion, percent) is a ratio that
compares a number to 100.
the percent is written as a fraction.
10
Choose the correct term or numbers to
complete each sentence.
(Lesson 5-3)
• A percent proportion is
1
25% = _
4
1
20% = _
5
1
2
_
16 % = _
3
6
1
1
_
_
12 % =
2
8
1
_
10% =
Vocabulary Check
numbers that are easy to divide mentally.
100% = 1
_4
5
5
1
_
83 % = _
3
6
1
7
_
_
87 % =
2
8
9
_
90% =
80% =
10
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
12. 16.5 out of 100
5
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
Comparing Fractions, Decimals, and Percents
25
1
_1 = _
So, _
= 25%.
100
4
4
× 25
Example 2 Write 35% as a fraction in
simplest form.
35
7
35% = _
or _
20
(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.
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?
Algebra: The Percent Proportion
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?
Chapter 5 Percent
the percent symbol.
0.04 = 0.04
= 4%
Example 5
Multiply by 100 and add the
percent symbol.
_
Write 9 as a percent.
25
9
_
= 0.36
Write as a decimal.
25
Change the decimal to percent.
= 36%
(pp. 263–267)
Write a percent proportion and solve each
problem. Round to the nearest tenth if
necessary.
296
4
× 25
100
15. 120%
Write each decimal as a percent.
5-3
_
Write 1 as a percent.
Example 1
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
32.
2
66_
% of 24
3
31. 10% of 18.3
33. 6% of 200
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
Percent and Estimation
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.
Example 9
38. 41% of 39
Estimate each percent.
39. 33 out of 98
The total duration of the trip is
365 minutes, or 6 hours and 5 minutes.
One half, or 50%, of the trip would be
(pp. 275–278)
Estimate.
37. 67% of 60
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?
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
Percent of Change
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.
(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
The amount of change is 900 - 725 or 175.
47. original: 8
new: 15
new: 10
48. original: 37.5
49. original: 18
new: 30
amount of change
original amount
175
=_
900
percent of change = __
new: 12
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
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.
Express each percent as a decimal.
5. 135%
16. TAXES Sandra estimated that about 35% of
6. 14.6%
7. 0.97%
8. Order the set of numbers 38%, _, and 0.038
3
8
from least to greatest.
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?
Solve using the percent equation.
19. Find 45% of 600.
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
11. 33_% of 90
1
3
13.
10. 1% of 99
12. 62_% of 48
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
G
H
J
The interest earned will be $117.
The interest earned will be $9.75.
The total amount will be $134.75.
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?
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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°.
A
∠ABD and ∠DBC are complementary angles.
The sum of the measures of
supplementary angles is 180°.
Extra Examples at ca.gr7math.com
50˚
40˚ C
F
125˚
∠F and ∠G are supplementary angles.
Angle Measure The
measure of ∠ABD is 50°.
In symbols, m∠ABD = 50°.
B
D
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
12–19
20–23
See
Examples
1
2, 3
4
For Exercises 8-11, refer to the figure at the right.
n
8. Name a line that contains point S.
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⬚
167⬚
64⬚ x⬚
For Exercises 20–23, refer to the figure at the right.
x⬚
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
L
K
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.
314
*/ -!4( Write about a situation in which you use inductive reasoning.
(*/
83 *5*/(
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
8. 14-gon
9. 19-gon
11. ART The sculpture below
7. 11-gon
10. 24-gon
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°
The measure of each interior angle
of an octagon is 135°.
J
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?
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)
(Lesson 5-9)
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
YW
5.
XY
6.
m∠W
Y
W
Z
V
15 in.
N
M
10 yd
P
X
Q
61˚
7 yd
4.
73˚
W
R
Y
Determine whether the polygons shown are congruent. If so, name
the corresponding parts and write a congruence statement.
7.
J
AD
M
3 cm Q
11. m∠G
10.
A
9m
DC
12. m∠H
C
D
A
76˚
B
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
In the figure, quadrilateral ABCD
is congruent to quadrilateral HEFG.
Find each measure.
9.
B
H
5 cm
3 cm
6 cm
5 cm
8.
P
K
H
X
15 in.
In the figure, PQR YWX.
Find each measure.
m∠X
L
12 in.
9 in.
H
3.
K
12 in.
85˚
E
Examples 2, 3
J
45˚
50˚
50˚
2.
M
K
J
L
S
P
N
O
%842!02!#4)#%
Z
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
State whether the following statement is sometimes, always or
never true. Explain your reasoning.
16. CHALLENGE
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.
1.
(Lesson 6-1)
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
11 2 = 121
111 2 = 12,321
Use logical reasoning to find the next
equation. Explain your reasoning. (Lesson 6-2)
In the figure, quadrilateral QRST is congruent
to quadrilateral JKLM. Find each measure.
(Lesson 6-4)
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
Chapter 6 Geometry and Spatial Reasoning
H
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°.
ƒ
ƒ
ƒ
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.
3x˚
(Lesson 6-4)
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)
Example 4
(p. 334)
334
3. ART Copy and complete the pattern
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
congruent.
B 12 ft C
D
(Lesson 6-4)
16 ft
x ft
20 ft
A
PREREQUISITE SKILL Add.
25. -4 + (-1)
336
(Lesson 1-4)
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
right is a traditional Japanese design.
Describe the minimum number of
translations of the original pattern, A,
needed to create the section shown.
Chapter 6 Geometry and Spatial Reasoning
Silvia Otte/Getty Images
4 in.
A
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.
24.
1
1
16%, _
, 1.6, _
6
16
25.
_3 , 0.65, 38%, _5
8
8
(Lesson 5-2)
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)
Polygons
(p. 328)
(p. 314)
supplementary angles
(p. 307)
(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
3. If ABC DEF, then ∠C ∠E.
(Lessons 6-6 and 6-7)
• 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
4
5
16. ∠4 and ∠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
PSI: Use Logical Reasoning
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.
(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
24.
D
E
8 cm
14 in.
A
Y
6 cm
65˚
4 ft
B
C
14 in.
5 ft
P
Q
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
45˚ B
C
P
G
H
Example 5 In the figure below,
ABC RPQ. Find PQ.
A
25.
N
Multiply.
(pp. 320–323 )
Determine whether the polygons shown
are congruent. If so, name the
corresponding parts and write a
congruence statement.
V
Subtract.
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
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.
ƒ
ƒ
ƒ
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.
symmetry? Name its angle(s) of rotation.
13.
3. octagon
4. 15-gon
5.
12. MUSIC Which figure above has rotational
STANDARDS PRACTICE Which of the
following statements is not true
concerning the quadrilaterals in the stained
glass window?
F
A
E
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
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6-7
5-8
4-3
3-7
1-5
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6-4
1-1
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5-8
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MG3.2
NS1.7
AF4.2
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NS1.2
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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.
Example 1
(Lessons 2-3 and 2-9)
1
1. _ · 8 · 12
3
2.
_
_1 · 4 · 9 2
Multiply 1 · 5 · 6 2.
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
that are the same distance from a
given point in the plane, called the
center. The segment from the center
to any point on the circle is called
the radius. A chord is any segment
with both endpoints on the circle.
A diameter is a chord that passes
through the center. It is the longest chord.
center
radius
(r)
circumference
(C)
diameter
(d)
The diameter of a circle is
twice its radius or d ⫽ 2r.
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
3.14
5.1 ≈ d
Divide each side by 3.14.
Divide.
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)
(/-%7/2+ (%,0
For
Exercises
8–11
12–15
16–19
See
Examples
1, 2
3, 4
5
7.9 inches long. What was the diameter of the bracelet?
Find the circumference of each circle. Round to the nearest tenth.
8.
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
*/ -!4( Explain how to find the diameter of a circle if you
(*/
83 *5*/(
38.
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
Z
circle x
3π
F _
A 16π in.
4
4π
G _
3
B 64 in.
C 32 in.
12 cm
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
e. ∠ECB
f. BA
g. DC
A 20˚
B
d. ∠b
130˚
h. CB
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
For
Exercises
5–10
11, 12
13–16
See
Examples
1
2
3
Find the area of each figure. Round to the nearest tenth if necessary.
CM
5.
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
11. CARPENTRY Scott is constructing
CM
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.
˜ÌiÀÃiVÌʈ˜Ê>ʈ˜i
˜ÌiÀÃiVÌÊ>ÌÊ>Ê*œˆ˜Ì
!
A
B
D
C
J
œÊ˜ÌiÀÃiV̈œ˜
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
̜«ÊۈiÜ
vÀœ˜ÌÊۈiÜ
SIDE
È`iÊۈiÜ
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.
*œÀV Ê-Ìi«Ã
-VՏ«ÌÕÀiÊ*i`iÃÌ>
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
is the length of the label?
IN
IN
(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
For
Exercises
6–9, 12, 13
10, 11,
14, 15
16, 17
See
Examples
1, 2
3
4
Find the volume of each solid. Round to the nearest tenth if necessary.
6.
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.
42. 0.12%
43. 225%
44. 135%
(Lesson 5-1)
45.
_3 %
49.
_1 · 6 2 · 20
8
PREREQUISITE SKILL Multiply.
46.
378
_1 · 6 · 10
3
47.
_1 · 7 · 15
3
Chapter 7 Measurement: Area and Volume
48.
_1 · 4 2 · 9
3
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
11 m
_
B = 1 · 8.1 · 6.4, h = 11
2
1 1
about _ _ · 8 · 6 (11)
3 2
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.
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
For
Exercises
10–13
18
14–17
See
Examples
1
2
3
Find the volume of each pyramid. Round to the nearest tenth.
10.
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.
904.3 ft 3
FT
(Lesson 7-5)
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.
27.
_1 · 2.8
2
28.
(Lesson 2-3)
_1 · 10 · 23
2
29.
_1 · 2.5 · 16
2
30.
_1 3_1 (20)
2 2
( )
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.
œ`iÊœvÊ,i}Տ>ÀÊ*ÞÀ>“ˆ`
iÌʜvÊ,i}Տ>ÀÊ*ÞÀ>“ˆ`
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
10.2 m
4 ft
3. HISTORY Refer to the lesson opener. Each face of the parachute has a base of
(p. 394)
(/-%7/2+ (%,0
For
Exercises
4–9
10, 11
See
Examples
1
2
12 m
12 m
4 ft
Example 2
15 m
12 m
about 12 yards and a height of about 17 yards. Find the amount of material
needed to make the parachute.
Find the lateral and total surface area of each regular pyramid. Round to the
nearest tenth if necessary.
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.
23.
396
16
12
_
=_
n
40
24.
_3 = _x
5
8
Chapter 7 Measurement: Area and Volume
(Lesson 4-2)
25.
a
7
_
=_
13
39
26.
10
30
_
=_
26
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 %
&
'
(EIGHT 3URFACE 6OLUME
!REA
3HEET The spreadsheet evaluates the
formula 2*C3*D3⫹2*C3*E3⫹2*D3*E3.
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
ޏˆ˜`iÀÊ
ޏˆ˜`iÀÊ
radius cylinder A
height cylinder A
__
is proportional to __
Words
radius cylinder B
Variable
_5
8
h
_5 = _
12
Find the cross products.
5 · 12
8·h
_
=_
Divide each side by 8.
7.5 = h
8
=
h
_
12
Write the proportion.
5 · 12 = 8 · h
8
height cylinder B
Let h represent the height of cylinder A.
Equation
8
FT
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>“ˆ`Ê
3 IN
*ÞÀ>“ˆ`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
___
= _a
( )2
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 ,
3
replace a with 1 and b with 2 in _a .
(b)
volume of smaller cube
__
= _a
( )3
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 )
27 · 8 = V · 1
216 = V
3
=_·_·_
1
2
1
2
1
2
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
The volume of the smaller cube is
125 cubic feet. Find the volume of
the larger cube.
F 375 ft 3
C 3,376 in 3
G 3,375 ft 3
D 6,752 in 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)
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
Circles
Volume
(Lesson 7-1)
(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
Surface Area
volume (p. 373)
3
(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
PSI: Solve a Simpler Problem
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?
Area of Complex Figures
CM
MM
13.
CM
CM
MM
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.
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
A = πr 2
A ≈ 3.14 · 5 2
A ≈ 78.5 yd 2
(pp. 363–367)
Find the area of each figure. Round to the
nearest tenth if necessary.
12.
The radius is 5 yards.
C = 2πr
C ≈ 2 · 3.14 · 5
C ≈ 31.4 yd
YD
(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
YD
MM
MM
YD
YD
MM
V = 1,170 ft 3
(pp. 380–384)
Example 6 Find the
volume of the pyramid.
21.
The base B of the
pyramid is a rectangle.
22.
CM
FT
IN
IN
IN
1
V=_
Bh
CM
FT
FT
)
(2
Find the volume of each solid. Round to
the nearest tenth if necessary.
FT
FT
1
· 13 · 10 18
V= _
diameter of 10.5 centimeters and a
height of 13 centimeters. Find its
volume.
Volume of Pyramids and Cones
FT
The base of this
prism is a triangle.
V = Bh
20. FOOD A can of green beans has a
7-6
Example 5 Find the
volume of the solid.
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 Pyramids
Surface area of a
cylinder
S = 2πr 2 + 2πrh
2
Simplify.
(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
FT
similar. If the total
surface area of cylinder
A is 84 square feet,
what is the total
FT
surface area of
cylinder B?
ޏˆ˜`iÀÊ
ޏˆ˜`iÀÊ
Example 9 Two
similar cones are
shown at the right.
Find the volume of
the smaller cone.
9 cm
3
V ⫽ 184 cm
volume of smaller cone
__
= _a
volume of larger cone
FT
V
1
_
=_
184
27
( b )3
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.
1.
2.
FT
Find the volume of each solid. Round to the
nearest tenth.
8.
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.
STANDARDS PRACTICE Find the volume
of the solid.
FT
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
F
G
H
J
2,160 ft 3
2,520 ft 3
3,600 ft 3
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.
w = -8
_
4
= -5
Solve 44 = k - 7.
Write the equation.
Add 7 to each side.
Simplify.
17. 12 = _
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.
74. $5 for 4 loaves of bread
(Lesson 4-1)
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.
_a = -15
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.
Use the Subtraction Property
of Equality.
3x + 1 = 7
- 1 =-1
____________
3x
= 6
x
x
3x
1
1
1
1
1
1
3x = 6
3
422
Aaron Haupt
Chapter 8 Algebra: More Equations and Inequalities
3
x=2
6
There are 2 1-tiles in each
group, so the solution is 2.
Write the equation.
Subtract 1
from each side.
Use the Division Property
of Equality.
3x
6
_
=_
x
Use symbols.
METHOD 2
Divide each side by 3.
Simplify.
BrainPOP® ca.gr7math.com
_
2 Solve 25 = 1 n - 3.
4
METHOD 1
METHOD 2
Vertical method
1
25 = _
n-3
4
+3=
+3
1
28 = _
n
Write the equation.
Add 3 to each side.
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.
Write the equation.
6 + (-3x) = 21
Rewrite the left side as addition.
6 - 6 + (-3x) = 21 - 6
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.
Check
6 - (-15) 21
Multiply.
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
_
=_
-1
-y = -1y; divide each side by -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)
i. -3 = 6 - 5w + _w
5
2
h. 10 = 2a + 13 - a
Solve each equation. Check your solution.
1. 6x + 5 = 29
4.
_2 x - 5 = 7
3
2. -2 = 9m - 11
3. 10 = _ + 3
5. 3 - 5y = -37
6.
a
4
c
_
-4=3
-2
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
37.
a-4
_
= 12
5
36. -4_ = _(t + 1)
6
2
5
5
6+z
_
39.
= -2
10
35. 6(w - 2) = 54
38.
n+3
_
= -4
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.
48. 6(a + 6)
49. -3(x + 5)
(Lesson 8-1)
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
2f + 3 - 3 = 19 - 3
+
19
Subtract 3 from each side.
Simplify.
2f
16
_
=_
Divide each side by 2.
2
=
Combine like terms.
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)
Example 5
(p. 428)
(/-%7/2+ (%,0
For
Exercises
6–9
10–13
14, 15
See
Examples
1–3
4
5
4. BOOK FINES You return a book that is 5 days overdue. Including a previous
unpaid overdue balance of $1.30, your new balance is $2.05. How much is
the daily fine for an overdue book?
5. SHOPPING Marty paid $121 for shoes and clothes. He paid $45 more for
clothes than he did for shoes. How much did Marty pay for the shoes?
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
Solve each equation. Check your solution.
30. 5x + 2 = 17
31. -7b + 13 = 27
Simplify each expression.
34. 5x + 6 - x
4
(Lesson 8-2)
32. -6 = _ + 1
33. -15 = -4p + 9
36. 7a - 7a - 9
37. 3 - 4y + 9y
n
8
(Lesson 8-1)
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.
40. 2x - 8 + 2x
41. -5n + 7 + 5n
(Lesson 8-1)
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
2x 1 1
x
1
1
1
1
Remove the same number of x-tiles
from each side of the mat until there
are x-tiles on the only one side.
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
51
1
1
1
1
x
2x
1
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
x4
1
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
Check for Reasonableness
Write the equation.
Subtract 0.10m
from each side.
Subtract 19.95 from
each side.
Divide each side
by 0.10.
$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.
34. 9 + 5y = 19
(Lesson 8-2)
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.
• Guess
11. How many do not prefer apples?
and check.
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)
1. 3(x + 2)
2. -2(a - 3)
3. 5(3c - 7)
4. -4(2n + 3)
Translate each sentence into an equation. Then
find each number. (Lesson 8-3)
16. Nine more than the quotient of a number
and 3 is 14.
17. The quotient of a number and -7, less 4,
Simplify each expression.
is -11.
(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
19. 32 ≤ 2x, x = 16
20.
n
_
≥ 5, n = 12
18. -3y < 21; y = 8
21.
4
-18
_
> 9, x = -2
x
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.
41. y + 15 = 31
444
(Lesson 1-8)
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?
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.
Write the resulting inequality and decide
whether it is true or false.
Family
Member
Dad
Age
43
Mom
41
Greg
16
Victoria
13
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
22 - 8 15
Write the inequality.
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.
a. t + 3 > 12
b. n +
_1 ≥ 4
c. y - 1.5 < 2
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
Write the inequality.
968 - 968 + w ≤ 1,300 - 968
w ≤ 332
Check for Reasonableness
Subtract 968 from each side.
Simplify.
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
9. 3 < y + 8
7. 10 + n ≥ -2
8. -4 < k + 6
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 -
_2 ≤ _1
3
2
23. 5 > f + 1 _
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.
41. 18 - n > 4, n = 11
42. 13 + x < 21, x = 8
(Lesson 8-5)
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 $7 per person.
Write and solve an equation to determine how many trips to the zoo a
family of four could make for the cost of a membership to equal regular
admission. (Lesson 8-4)
PREREQUISITE SKILL Solve each equation.
45. 3y = -15
448
(Lesson 1-9)
46. -18 = -2a
Chapter 8 Algebra: More Equations and Inequalities
47.
w
_
= 12
4
8-8
Solving Inequalities by
Multiplying or Dividing
Main IDEA
Solve inequalities by using
the Multiplication or
Division Properties of
Inequality.
Standard
7AF1.1 Use
variables and
appropriate operations to
write an expression, an
equation, an inequality, or a
system of equations or
inequalities that represents a
verbal description (e.g. three
less than a number, half as
large as area A.
Standard 7AF4.1
Solve two-step linear
equations and inequalities in
one variable over the
rational numbers, interpret
the solution or solutions in
the context from which they
arose, and verify the
reasonableness of the
results.
SHOPPING The table shows the prices of the
same brand name of shoes at a sports store.
Notice that walking shoes cost less than
cross-training shoes since 80 < 150. Will this
inequality be true if the store sells both pairs
of shoes at half price?
Shoe
Style
Regular
Price ($)
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.
7
7
y > -6
Simplify.
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.
a. 3a ≥ 45
b.
n
_
< -16
c. 81 ≤ 9p
4
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.
c
_
< -14
e. -5d ≥ 30
-7
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.
8
8
c ≥ 2.25
Simplify.
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
5. -4y > 32
2.
_3 < _7 y
4
3. 8x ≤ -72
9
6. -56 ≤ -7p
7.
(p. 451)
Example 5
g
_
< -7
-2
4.
_h ≥ -6
8.
d
_
≥ -3
4
-3
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.
22.
_x ≤ -3
9
t
_
25.
≤ -2
-5
v
_
>4
-4
n
_
23.
< -14
7
y
26. -8 ≤ _
0.2
21.
a
_
≥5
24.
m
_
< -7
27.
-1
_
k > -10
-3
-2
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
33. 7 + _ < 4
34.
n
3
w
_
- 4 ≤ -5
8
32. -3g + 8 ≥ -4
35. 10 - 3x ≥ 25 + 2x
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
_
≥_
8
8a
-56
_
≤_
8
8
8
a ≥ -7
a ≤ -7
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
48. j - 8 ≥ -12
47. a - 5 ≤ 2
Write an inequality for each sentence.
49. -14 > 8 + n
(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
coefficient (p. 417)
Be sure the following
Key Concepts are noted
in your Foldable.
%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
• When used to compare a variable and a number,
inequalities can describe a range of values.
(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
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
(Lesson 8-6)
Inequality Properties
Vocabulary Check
Chapter 8 Algebra: More Equations and Inequalities
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.
9. 4(a + 3)
10. (n - 5)(-7)
Simplify each expression.
11. p + 6p
12. 6b - 3 + 7b + 5
13. SOCCER Pam scored n goals. Leo scored
Example 1 Use the Distributive
Property to rewrite -8(x - 9).
-8(x - 9)
Write the expression.
= -8[x + (-9)] Rewrite x - 9 as x + (-9)
= -8(x) + (-8)(-9) Distributive Property
= -8x + 72
Simplify.
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
16.
_c + 2 = 9
5
15. 4 = -3y - 2
17. 39 = a + 6a + 11
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
Writing Two-Step Equations
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.
(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
16
4n
_
=_
4
4
n=4
is
10.
=
10
Write the equation.
Add 6 to each side.
Simplify.
Divide each side by 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.
Example 4
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
PSI: Guess and Check
-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.
(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
Solve -7x + 5 = x - 19.
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
Solve each inequality.
44.
n
_
<6
4
45.
k
_
≤3
1.7
46. 0.5x > 3.2
47. -56 ≥ 8y
48. 9 > _
49. -_a ≤ 2
x
-4
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
(pp. 449-453)
Example 9
solution.
-2n ≥ 26
26
-2n
_
≤_
-2
-2
n ≤ -13
Solve -2n ≥ 26. Check your
Write the inequality.
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.
_k - 11 = 5
5. 3n + 18 = 6
6.
7. -23 = 3p + 5 + p
8. 4x - 6 = 5x
9. -3a - 2 = 2a + 3
2
10. -2y + 5 = y - 1
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.
STANDARDS PRACTICE The perimeter of
the rectangle is 44 inches.
21.
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
G
H
J
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
B
C
D
Less than $25,000
More than $25,000
At least $25,000
At most $25,000
22 in 2
120 in 2
392 in 2
440 in 2
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?
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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
!MBER
4IM
+AELA
BLUE
RED
GREEN
)NPUT
2ELATION /UTPUT
!LONSO
-ALINA
#HEN
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)
Input
Rule
Output
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
-2
-1
Substitute each value of x,
or input, into the function
rule. Then simplify to find
the output.
0
1
The domain is {-2, -1, 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
10. f(5) if f(x) = 2x + 5
11. f(-5) if f(x) = 4x - 1
9. f(4) if f(x) = 3x - 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
(_56 ) if f(x) = 2x + _13
22. f
(_58 ) if f(x) = 4x - _14
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
See pages 699, 716.
Self-Check Quiz at
ca.gr7math.com
H.O.T. Problems
Depth
(ft)
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.
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,
the total cost of t tickets for a movie.
Which table contains values that
satisfy this equation?
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
29. Stephanie received a $25 gift certificate
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?
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.
31. b + 15 > 32
(Lesson 8-6)
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
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)
y
-1
y
(x, y)
(3, -1)
(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
7
8
y
-1
Extra Examples at ca.gr7math.com
5
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
13. y = 2x + 3
10. y = x - 3
11. y = x + 1
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)?
-4
y
-2 -1
y
A
0
x
⫺2
4
8
0
1
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
O
4x
2
G
H
J
Find each function value.
29. f(6) if f(x) = 7x – 3
x
O
C
O
y
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
(Lesson 9-1)
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.
34.
O
476
35.
y
x
Chapter 9 Algebra: Linear Functions
36.
y
O
x
(Lesson 4-9)
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
• However, the
coordinates of both
points must be
used in the same
order.
To check Example 5,
let (x 1, y 1) = (-4, 3)
and (x 2, y 2) = (1, 2).
Then find the slope.
y2 - y1
m=_
x2 - x1
C(⫺1, ⫺4)
(x 1, y 1) = (1, 2)
(x 2, y 2) = (-4, 3)
-4 - 1
1
1
m= _
or -_
-5
S(⫺4, 3)
Slope formula
3-2
m=_
Check
x
O
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)
Using the Slope
Formula
• It does not matter
which point you
define as (x 1, y 1)
and (x 2, y 2).
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
y
9
4
4
16.
6
-1 -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
Graph each function.
(Lesson 9-2)
38. y = 5x
39. y = x - 2
3
_
G -3
4
x
40. y = 2x - 1
3
H _
4
4
J _
3
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.
43. 42 = -14x
482
44. 144 = 18a
Chapter 9 Algebra: Linear Functions
(l)First Light, (r)Yellow Dog Productions/Getty Images
(Lesson 1-10)
45.
n
_
=7
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
2
_y
Symbols
k = x or y = kx,
where k ≠ 0
Example
y = 3x
4
2
O
y
y 3x
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
_
=_
3
6
21 · 6 = 3 · x
126 = 3x
126
3x
_
=_
3
3
42 = x
human equivalent age
actual age
Find the cross products.
Multiply.
Divide each side by 3.
Simplify.
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.
5
10
15
20
12.5
25
37.5
50
Days, x
Height, y
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.
Proportional Linear Function
Graph
x
-2
-1
1
2
y
-4
-2
2
4
_yx
2
2
2
2
Equation
y = 2x
y
O
x
Nonproportional Linear Function
Table
Graph
x
-2
-1
1
2
y
-5
-3
1
3
_yx _5
3
1
_3
2
Extra Examples at ca.gr7math.com
Equation
y = 2x - 1
y
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
10
15
20
25
0.70
1.05
1.40
1.75
Price, x
Tax, y
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
15.24 22.86 30.48
15
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
Find the slope of each line.
29.
O
J
8
32
A _
=_
48
8
x
C _
=_
12
x
8
32
_
_
B
=
12
x
12
32
8
x
D _
=_
32
12
(Lesson 9-3)
30.
y
G 36
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?
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
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.
34. 7 + a = 15
488
35. 23 = d + 44
Chapter 9 Algebra: Linear Functions
(Lesson 1-9)
36. 28 = n - 14
37. t - 22 = -31
CH
APTER
9
Mid-Chapter Quiz
Lessons 9-1 through 9-4
Find each function value.
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)
(Lesson 9-1)
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?
2
(Lesson 9-4)
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)
8. y = x + 6
(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)
y
12
Gallons
Graph each function.
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
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.
Graph each equation.
e. y = _x - 1
1
2
Personal Tutor at ca.gr7math.com
d. y = x + 3
-4 = -3 - 1
-4 = -4
2
Step 2 Graph the y-intercept -1.
y = -3x - 1
2
_
slope = - 3 , y-intercept = -1
✓
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
of posters that can be printed for $45.
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.
y
60
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
4
3
2
1
D
y
1 2 3 x
3000
2000
1000
x
4
3
2
1
1 2 3 x
Total Earnings
A
graph below?
y
y
4321 O
1
2
10
0
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.
46. M(4, 3), N(-2, 1)
47. S(-5, 4), T(-7, 1)
(Lesson 9-3)
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.
51. 3a - 12 = -3
52. -2 = -n + 4
53. -_p - 7 = -3
1
3
(Lesson 8-2)
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?
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.
(Lesson 9-4)
a
_
≥ 3, a = -12
-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.
Water Depth
Temperature (°F)
Source: The World Book
0
10
20
25
30
35
40
50
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)
EDUCATION For Exercises
3–5, use the table.
300 x
200
Distance Traveled (mi)
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
1930
25.7
1970
45.6
4. Write an equation for
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
data. Then draw a line of fit.
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
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
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
to show the relationship
between home runs and runs
batted in.
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
CITIES For Exercises 31–33, use the table.
As Franco bought more pounds of
fruit, the cost of the fruit increased.
Largest U.S. Cities
(Lesson 9-7)
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)
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
950,000
1,850,000
Detroit, MI
Solve each equation. Check your solution.
(Lesson 8-5)
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
Study Guide
and Review
9
Download Vocabulary
Review from ca.gr7math.com
Key Vocabulary
constant of variation
Be sure the following
Key Concepts are noted
in your Foldable.
-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.
Direct Variation
(Lesson 9-4)
• A direct variation is a relationship in which the
ratio of y to x is a constant, k.
Slope-Intercept Form
(Lesson 9-5)
Vocabulary Check
Choose the correct term or number to
complete each sentence.
1. The (domain, range) is the set of input
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
(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.
x2 - x1
)
_
6. The slope formula is _
x2 - x1 , y2 - y1 .
2
Scatter Plots
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
13. FITNESS Wilson’s Fitness Club charges
$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?
x
2x - 1
f(x)
-2
2(-2) - 1
-5
0
2(0) - 1
-1
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
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.
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.
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
Gas (gal)
2
100 150 200 250
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
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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
Write each expression using
exponents. (Lesson 2-9)
9. 6 · 6 · 6 · 6
10. 3 · 7 · 7 · 3 · 7
11. FUND-RAISER The students at
y
yx1
O
x
Example 2
Write n -3 using a positive exponent.
1
n -3 = _
3
n
definition of negative exponent
Example 3
Write 5 · 4 · 5 · 4 · 5 using exponents.
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.
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
x
Chapter 10 Algebra: Nonlinear Functions and Polynomials
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
Ticket Sales
$5
2
3
$10 $15
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
y
-2
x
4
8
12
16
y
3
0
-3
-6
3
5
7
-18 -50 -98
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
y
2
1
35. FOOTBALL The graphic shows
36. MEASUREMENT Make a graph
8
-1 -4
: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.
4
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.
56. y = 2x
57. y = x + 3
(Lesson 9-2)
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)
2
1
(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
2
-1 -2(-1) = -2
528
0
-2(0) 2 = 0
1
-2(1) 2 = -2
2
2
-2(2) = -8
y
(x, y)
-8
(-2, -8)
-2
(-1, -2)
4
O
⫺8
⫺4
4
(0, 0)
⫺4
-2
(1, -2)
⫺8
-8
(2, -8)
0
Chapter 10 Algebra: Nonlinear Functions and Polynomials
y
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.
x
x2 + 2
-2 (-2) 2 + 2 = 6
-1
0
1
2
y
(x, y)
6
(-2, 6)
2
3
(-1, 3)
2
2
(0, 2)
2
3
(1, 3)
2
6
(2, 6)
(-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
0
3
(-1, 3)
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.
d
1,000
(10, 66)
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(0) = 0
0.66(10) = 66
0.66(40) = 1,056
Height (ft)
(0, 0)
2
10
Source: structurae.de
h
(d, h)
2
0
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 = 0.66d 2
800
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
x
O
x
x
Determine whether each equation represents a linear or nonlinear function.
3
37. y = 3x + 2
x
O
O
36. y = x - 5
y
D
38. x + y = -6
(Lesson 10-1)
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
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õÕ>Ài
CHECK
ÓÊ}ÀœÕ«ÃʜvÊ{Ê>˜`
ÓÊ}ÀœÕ«ÃʜvÊÓ
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õÕ>Ài
ÓÊ}ÀœÕ«ÃʜvÊxÊ>˜`Ê
ÓÊ}ÀœÕ«ÃʜvÊÎ
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
a diagram.
eck.
• Guess and Ch
l.
• Make a mode
• Draw
if one or more switches are closed. How
many combinations of open and closed
switches will result in the light being on?
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
s2
(s, A)
2
0 =0
(0, 0)
2
s
0
Graph
1
1 =1
(1, 1)
2
22 = 4
(2, 4)
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.
x
y = x3
(x, y)
-1.5
(-1.5) 3 ≈ -3.4
(-1.5, -3.4)
-1
Graphing
It is often helpful to
substitute decimal
values of x in order
to graph points that
are closer together.
3
(-1, -1)
3
(-1) = -1
0
(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)#%
See pages 703, 717.
Self-Check Quiz at
ca.gr7math.com
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.
19. MEASUREMENT The formula for the volume V of a tennis ball is given by
4 3
the equation V = _
πr where r represents the radius of the ball. Graph
3
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.
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
(Lesson 3-2)
39. √
8.67
PREREQUISITE SKILL Write each expression using exponents.
41. 3 · 3 · 3 · 3 · 3
42. 5 · 4 · 5 · 5 · 4
43. 7 · (7 · 7)
36. y = 4x 2 + 3
40. - √
19.85
(Lesson 2-9)
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
base is x.
=x
2
Add the
exponents.
x 4 · x -2
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
-5
33.
(_78 ) (_78 )
36.
(_25 ) (_25 )
3
31. 4a 2b 3(7ab 2)
13
-2
4
-7
(_25 ) (_25 ) (_25 )
2
37. (_)
(_72 )
7
34.
-2
6
-3
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
Graph each function.
2
5x ft
8
6x ft
30x 16
(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)
Express each number in scientific notation.
Time
(min)
46
53
57
60
Number
of cells
100
200
300
400
(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.
53. 1_ + p = -6_
3
4
52. k - 4.1 = -9.38
1
2
54.
Find each sum or difference. Write in simplest form.
55.
3
_7 - _
8
10
PREREQUISITE SKILL
59. 3 · 3 · 3 · 3
542
(Lesson 2-7)
56. -_ + _
1
5
5
12
1
6
Write each expression using exponents.
60. 5 · 4 · 4 · 5 · 4
61. 8 · (8 · 8)
Chapter 10 Algebra: Nonlinear Functions and Polynomials
10
(Lesson 2-6)
57. 9_ + _
2
3
c
_
= 0.845
58. -2_ - 1_
3
4
1
8
(Lesson 2-9)
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.
13. y = -2x
4IME MIN
Graph each function.
(Lesson 10-2)
6. y = 2x 2
2
7. y = -x + 3
2
(Lesson 10-4)
3
14. y = 3x 3
15. y = 2x 3
16. y = 0.1x 3
Simplify. Express using exponents.
4
17. 10 · 10
(Lesson 10-4)
7
8. y = 4x - 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
5·5·5·5
54
4 factors
+%9 #/.#%04
Words
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.
n9
2 _
4
48
1 _
2
4
Common Error
When dividing
powers, do not divide
n
48
_
= 48 – 2
42
=4
6
The common base is 4.
Simplify.
n9
_
= n9 – 4
n4
=n
The common base is n.
5
Simplify.
48
the bases. _2 = 4 6,
4
6
not 1 .
Simplify. Express using exponents.
a.
544
57
_
54
b.
x 10
_
Chapter 10 Algebra: Nonlinear Functions and Polynomials
x3
c.
12w 5
_
2w
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.
d.
11 -8
_
e.
11 2
b -4
_
f.
b -7
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
F _
(6)
5
1
G _
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
Simplify. Express using positive exponents.
(pp. 544-545)
Example 5
(p. 545)
1.
76
_
5.
9c 7
_
7
3c 2
(p. 546)
546
29
_
6.
24k 9
_
2
13
6k 6
y
_
7.
15 -6
_
y
5
15 2
4.
z
_
8.
35p
_
z2
1
5p -4
22 · 33 · 44
2·3 ·4
B 2
1
C _
2
1
D _
(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
8
3.
9. Simplify _
.
3
5
A 22
Example 6
2.
(/-%7/2+ (%,0
For
Exercises
11–26
27–30
31–34
See
Examples
1–4
5
6
Simplify. Express using positive exponents.
11.
8 15
_
12.
29
_
15.
h7
_
16.
g
_
19.
36d 10
_
20.
23.
22 -9
_
24.
27.
x 6y 14
_
84
h6
6d 5
22 4
x 4y 9
13.
43
_
14.
13 2
_
17.
x8
_
18.
n
_
16t 4
_
21.
20m 7
_
22.
75r 6
_
3 -1
_
25.
42w -6
_
26.
12y
_
30.
3x 4
_
2
18
g
6
8t
3 -5
47
x 11
5m 5
29.
n8
25r 5
-6
7w -2
2y -10
-6
2
63 · 66 · 64
28. _
=
62 · 63 · 64
13 5
_1 × _1
(__
(5)
5)
(_15 )
2
3 4x -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.
35.
17 _
= 17 8
17 4
36.
k6
_
= k2
k
37.
5
_
= 53
5 -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.
-1
38.
Earthquake
p
_
= p 10
p
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
D 10
2 9 feet
Simplify. Express using positive exponents.
4
48. 6 · 6
7
Graph each function.
3
52. y = x + 2
C 10 3
3
49. 18 · 18
-5
(Lesson 10-5)
50. (-3x 11)(-6x 3)
51. (-9a 4)(2a -7)
54. y = -2x 3
55. y = -0.1x 3
(Lesson 10-4)
53. y = _ x 3
1
3
State the slope and the y-intercept for the graph of each equation.
56. y = x – 3
57.
2
y=_
x+7
3
58. 3x + 4y = 12
(Lesson 9-5)
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)
548
Simplify. Express using positive exponents.
(Lesson 10-5)
61. 5n · 3n 4
63. (-5b 7)(-2b 4)
62. (-x)(-8x 3)
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
35.
3 -6 9 2
(_
a b )
5
1
3
34. (_w 5z ) 2
33. (0.3p 7) 3
36. (3x
-2 4
4
6 2
) (5x )
37. (-2v 7) 3(-4v -2) 4
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
(3h 2k) 6
Simplify. Express using positive exponents.
15
_
15 4
(Lesson 10-6)
10
7
50.
D the volume in cubic units of a cube
whose side length is 4m 3
51.
y
_
y2
52.
18m 9
_
6m 4
3
53.
24g
_
3g 8
54. MEASUREMENT Find the area of a rectangle with a length of 9xy 2 and a width of 4x 2y.
(Lesson 10-5)
Find each square root.
55. √
49
552
(Lesson 3-1)
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
4y 2 .
√
4y 2 = √4 · √
y2
√
36q 6 .
√
36q 6 = √
36 · √
q6
√
= √
6 · 6 · √
q3 · q3
1 Simplify
Absolute Value
Use absolute value to
indicate the positive
value of y and q 3.
2 Simplify
2 · 2 · √
y·y
= √
= 2⎪y⎥
= 6 ⎪q 3 ⎥
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
64g 6 .
√
64g 6 = √
64 · √
g6
√
4 Simplify
3
3
3
3
= √
4·4·4·
Product Property of Cube Roots
3
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.
d.
3
√
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
17.
9p 8q 4
√
18.
225x 4y 6
√
22.
√
64k 3
26.
√
216x 12w 15
15.
√
36z 12
16.
√
144k 4m 6
19.
3
√
h3
20.
√
v3
21.
√
27b 3
23.
√
125d 9e 3
24.
8q 9r 18
√
25.
√
343m 3n 21
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
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
value as √27
m 3n 6 ?
2
√144g
?
A 12g
C 12g 2
B 12⎪g⎥
D 12⎪g
2
A the length of the side of a square
whose area is 27m 3n 6
⎥
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
Simplify.
49. (6 3) 5
(Lesson 10-7)
50. (n 7) 2
5
9
_
93
D the length of the side of a cube
whose volume is 3mn 2
200 ⎪h⎥ k 2
Simplify. Express using positive exponents.
53.
√
81a 4b x = 9a 2 ⎪b 5⎥
*/ -!4( Explain why absolute value is necessary when
(*/
83 *5*/(
45.
to
44.
54.
k 15
_
k6
51. (2a 3b 2) 4
52. (-4p 11q) 3
(Lesson 10-6)
4
55.
24y
_
4y 2
3
56.
45g
_
3g 7
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 functions do not have constant rates of
change.
• Quadratic functions are functions in which the
greatest power of the variable is 2.
• Cubic functions are functions in which the
greatest power of the variable is 3.
Monomials
nonlinear function (p. 522)
quadratic function (p. 528)
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 expression y = x 2 - 3x is an example
of a monomial.
2. A nonlinear function has a constant rate of
change.
(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.
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
Number of Pages
98
3
4
5
147 199 248
Graphing Quadratic Functions
15. y = -2x + 1
2
16. y = 3x - 1
17. SCIENCE A ball is dropped from the
PSI: Make a Model
-2
-3
-1
-1
0
1
1
3
Graph y = -x 2 - 1.
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
y
As x increases by 1,
y increases by 2. The rate
of change is constant, so this
function is linear.
Example 2
14. y = x 2 + 4
2
x
(pp. 528–531)
Graph each function.
13. -4x 2
Example 1 Determine
whether the table
represents a linear or
nonlinear function.
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
(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
Multiplying Monomials
25. 4 · 4 5
26. x 6 · x 2
27. -9y 2(-4y 9)
28.
_3
-4
3
· _
(7) (7)
2
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?
Dividing Monomials
n5
31. _
n
21c
_
-7c 8
x
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
33.
Example 7
68
6
Simplify_
. Express using exponents.
3
-1
3
11
32.
y x 3
(pp. 544-548)
Simplify. Express using exponents.
59
30. _
52
y
(pp. 539-542)
Simplify. Express using exponents.
10-6
Graph y = -x 3.
Example 4
Graph each function.
(_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
Simplify.
45. √
a2
47.
49.
36x 2y 6
√
3
p6
√
3
51. √
64c 6d 21
(pp. 553-556)
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
2
y
10
1
3
18
26
13. 15 3 15 5
15.
3 15
_
14. -5m 6(-9m 8)
16.
37
-40w 8
_
8w
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
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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)
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)
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)
Solve each problem.
(Lesson 5-7)
12. Find 52% of 360.
13. What is 36% of 360?
Example 2
3. -4 + (-8)
Order each set of rational numbers
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.
Example 4
What is 72% of 360?
72% of 360 = 0.72 × 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
8
7
9
7
5
2
3
7
4
6
8
5
9
9
8
6
pH number
Tally
Frequency
Less than 7
9
Equal to 7
4
Greater than 7
7
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
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
a table.
Tally
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
Diameter (in.)
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
• Make
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
1. WEATHER The list gives the
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–
12
5–
74
50
–
recorded at the Indianapolis 500?
14
9
0
4. What is the fastest winning speed
–1
24
(p. 571)
112 100 128 120 134 118 106 110 109 112
10
0
Examples 2, 3
State Record High Temperatures (°F)
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
(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
25
35 200 280
40
45
40
Average Speed (mph), Selected Animals
80
80
90
70
61
50
50
50
45
8
50
50
60
90
70
43
42
40
40
40
35
0.17
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=`e[Dfjk
8eefp`e^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