Saxon Math 87 An Incremental Development by Hake

^^^^™ ii ^^^ I >4n J # Incremental Development SAXOK THIS BOOK IS THE PROPER! Y OF: STATE Book No. PROVINCE Enter information COUNTY spaces PARISH in SCHOOL to the left as DISTRICT instructed OTHER CONP'TION Year *j{ ISSUED TO ISSUED Used ^fLx, RETURNED 1 n<,cA PUPILS to or whom this mark any part of textbook it in is Gvocx* issued must not write on any page any way, consumable textbooks excepted. 1. Teachers should see thai the pupil's every book issued. 2. The following terms should be used Poor; Bad. in name is clearly written in ink in the recording the condition of the book: spaces above New; Good; in Fair; Digitized by the Internet Archive in 2012 http://archive.org/details/saxonmath87incre00hake MATH 87 An Incremental Development MATH 87 An Incremental Development Stephen Hake John Saxon SAXON PUBLISHERS, INC. Math 87: Copyright An Incremental Development © 1997 by Stephen Hake and John Saxon All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America ISBN: 0-939798-54-9 Editor: Nancy Warren Production supervisors: Joan Coleman and David Pond Graphic artists: Scott Kirby, John Chitwood, and David Tenth printing: June 1998 Reaching us WWW: via the Internet http://www.saxonpub.com E-mail: info@saxonpub.com Saxon Publishers, Inc. 2450 John Saxon Blvd. Norman. OK 73071 Pond Contents Preface LESSON A xi Operations of Arithmetic • Parentheses • Arithmetic 1 with Whole Numbers LESSON B Operations of Arithmetic with LESSON 1 Missing Numbers in Addition, Subtraction, and Money 6 11 Multiplication LESSON 2 Number Line • Positive and Negative Numbers • 15 Ordering and Comparing Numbers LESSON 3 Adding and Subtracting on the Number Line 20 LESSON 4 Place Value through Hundred and Writing Whole Numbers 23 LESSON 5 Factors LESSON 6 Lines, Rays, LESSON 7 Fractions LESSON 8 Adding, Subtracting, and Multiplying Fractions 41 LESSON 9 44 LESSON 10 "Some and Some More" Word Problems "Some Went Away" Word Problems LESSON 11 "Larger-Smaller-Difference" • Trillions • Reading 27 Divisibility and Segments 32 and Mixed Numbers Word Problems 36 49 • Time 54 Problems Word Problems LESSON 12 Equal Groups LESSON 13 Part-Part- Whole LESSON 14 Fractions Equal to 1 LESSON 15 Equivalent Fractions LESSON 16 Reducing Fractions, Part LESSON 17 Linear Measure 82 LESSON 18 Pairs of Lines • Angles 87 LESSON 19 Polygons 92 Word Problems • Improper Fractions 59 63 69 73 1 77 VI Contents LESSON 20 Perimeter, Part 1 LESSON 21 Solving Equations LESSON 22 Prime and Composite Numbers 96 100 • Prime 104 Numbers 110 Factorization LESSON 23 Simplifying Fractions and Mixed LESSON 24 Writing Mixed Numbers and Whole Numbers 114 as Fractions LESSON 25 Fraction-of-a-Group Problems 117 LESSON 26 Adding and Subtracting Mixed Numbers 121 LESSON 27 Reciprocals 127 LESSON 28 Reducing Fractions, Part 2 131 LESSON 29 Dividing Fractions 135 LESSON 30 Multiplying and Dividing Mixed Numbers 140 LESSON 31 Multiples LESSON 32 Two-Step Word Problems 147 LESSON 33 Average, Part 1 151 LESSON 34 Rounding Whole Numbers LESSON 35 Common Denominators • Least Common Multiple • • 143 Estimating Answers Adding and Subtracting 155 160 Fractions with Different Denominators LESSON 36 Decimal Fractions LESSON 37 Reading and Writing Decimal Numbers Comparing Decimal Numbers LESSON 38 Rounding Decimal Numbers 176 LESSON 39 Decimal Numbers on the Number Line 180 LESSON 40 Adding and Subtracting Decimal Numbers 184 LESSON 41 Ratio 188 LESSON 42 Perimeter, Part 2 193 LESSON 43 Graphs 197 LESSON 44 Proportions 203 LESSON 45 Multiplying Decimal LESSON 46 Dividing a Decimal LESSON 47 Repeating Digits LESSON 48 Decimals LESSON 49 Division Answers Decimal Place Value • • Numbers Number by to Fractions 166 171 207 a Whole Number 211 215 and Fractions to Decimals 220 225 Contents Number VII LESSON 50 Dividing by a Decimal LESSON 51 Unit Price 233 LESSON 52 Exponents 236 LESSON 53 Powers of 10 240 LESSON 54 Rectangular Area, Part LESSON 55 Square Root 249 LESSON 56 Rates 253 LESSON 57 Percent 256 LESSON 58 Mixed Measures LESSON 59 Multiplying Rates 266 LESSON 60 Rectangular Area, Part 2 271 LESSON 61 Scientific Notation for Large LESSON 62 Order of Operations LESSON 63 Unit Multipliers LESSON 64 Ratio LESSON 65 Average, Part 2 LESSON 66 Subtracting LESSON 67 Liquid Measure LESSON 68 Scientific Notation for LESSON 69 Classifying Quadrilaterals 307 LESSON 70 Area of a Parallelogram 312 LESSON 71 Fraction-Decimal-Percent Equivalents 316 LESSON 72 Sequences 320 LESSON 73 Adding LESSON 74 Fractional Part of a 244 1 Adding Mixed Measures • • 228 Numbers 275 279 Unit Conversion 283 Word Problems 288 292 Mixed Measures • 296 299 Small Numbers 304 Functions, Part 1 Integers a Number, Part 262 on the Number Line Number and Decimal 324 Part of 330 1 LESSON 75 Variables LESSON 76 Classifying Triangles 338 LESSON 77 Symbols of Inclusion 343 LESSON 78 Adding Signed Numbers 347 LESSON 79 Area of a Triangle 352 LESSON 80 Percent of a Number 357 LESSON 81 Ratio Problems Involving Totals 361 and Evaluation 334 VIII Contents LESSON 82 Geometric Solids 366 LESSON 83 Weight 370 LESSON 84 Circles • Investigating Circumference 375 LESSON 85 Circumference and Pi 379 LESSON 86 The Opposite 384 LESSON 87 Operations with Fractions and Decimals 388 LESSON 88 The Addition Rule 393 LESSON 89 More on LESSON 90 Multiplication Rule for Equations 403 LESSON 91 Volume 409 LESSON 92 Finding the Whole Group LESSON 93 Implied Ratios LESSON 94 Fractional Part of a of the Opposite • Algebraic Addition for Equations 399 Scientific Notation When a Fraction Is Known 414 417 Number and Decimal Part of 422 a Number, Part 2 LESSON 95 Multiplying and Dividing Signed Numbers 427 LESSON 96 Area of a Complex Figure 432 LESSON 97 Inverting the Divisor 437 LESSON 98 More on Percent 442 LESSON 99 Graphing Inequalities 447 LESSON 100 Insufficient Information • Quantitative LESSON 101 Measuring Angles with a Protractor 456 LESSON 102 Using Proportions 461 LESSON 103 Area of a Circle LESSON 104 Multiplying Powers of 10 Scientific Notation LESSON 105 Mean, Median, Mode, and Range LESSON 106 Order of Operations with Signed Numbers to • Area of a Trapezoid Comparisons Solve Percent Problems 451 467 • Multiplying Numbers in 472 477 • 482 Functions, Part 2 LESSON 107 Number Families LESSON 108 Memorizing LESSON 109 Multiple Unit Multipliers 488 Common Fraction-Percent Equivalents • Conversion of Units 494 498 of Area LESSON 110 Sum of the Angle Measures of a Triangle Angles • Straight 504 Contents LESSON 111 Equations with Mixed Numbers LESSON 112 Evaluations with Signed IX 510 Numbers • Signed Numbers 514 without Parentheses LESSON 113 Sales Tax 519 LESSON 114 Percents Greater than 100, Part 1 523 LESSON 115 Percents Greater than 100, Part 2 528 LESSON 116 Solving Two-Step Equations 533 LESSON 117 Simple Probability 538 LESSON 118 Volume 544 LESSON 119 Rectangular Coordinates 548 LESSON 120 Estimating Angle Measures 554 LESSON 121 Similar Triangles 561 LESSON 122 Scale and Scale Factor 567 LESSON 123 Pythagorean Theorem 572 LESSON 124 Estimating Square Roots LESSON 125 Multiplying Three or More Signed Numbers Powers of Negative Numbers LESSON 126 Semicircles 588 LESSON 127 Surface Area 592 LESSON 128 Solving Literal Equations LESSON 129 Graphing Functions 600 LESSON 130 Formulas and Substitution 605 LESSON 131 Simple Interest 609 LESSON 132 Compound Probability 613 LESSON 133 Volume 618 LESSON 134 Probability, Chance, and Odds 622 LESSON 135 Volume, Capacity, and Weight in the Metric System 627 Supplemental Practice Problems Selected Lessons 633 APPENDLX of a Right Solid of a Special Angles • • 578 • Transforming Formulas Pyramid and a Cone for 584 596 Glossary 647 Index 661 Preface To the Student We study mathematics because it is an important part of our daily lives. Our school schedule, our trip to the store, the preparation of our meals, and many of the games we play all involve mathematics. Many of the word problems you will see in this book are drawn from our daily experiences. Mathematics is even more important in the adult world. In fact, your personal future in the adult world may depend in part upon the mathematics you have learned. This book was written with the hope that more students will learn mathematics and learn it well. For this to happen, you must use this book properly. As you work through the pages of this book, you will find similar problems presented over and over again. Solving these problems day after day is the secret to success. Work every problem in every practice set and in every problem set. Do not skip problems. With honest effort you will experience success and true learning which will stay with you and serve you well in the future. Acknowledgments We thank Shirley McQuade Davis for her ideas on teaching word problem thinking Stephen Hake Temple City, California patterns. John Saxon Norman, Oklahoma XI REVIEW LESSON Operations of Arithmetic • Parentheses • Arithmetic with A Whole Numbers Operations of arithmetic The operations of arithmetic addition, are subtraction, and division. Each of these operations is a binary operation. The English word binary comes from the Latin word binarius, which means "in two parts." We say that addition is a binary operation because we can add only two numbers in one step. If we wish to add multiplication, 2 we can add two number. get + + 4 3 numbers and then add the other and 3 to get 5 and then add 5 and 4 to of the We can add 2 9. + 2 3 + 4 problem 5 + 4 added 2 and 9 added 5 and 4 3 The two numbers we add are called addends, and the result is called the sum. Subtraction is subtract the subtrahend from the also a binary operation. minuend and We call the result the difference. Subtraction Addition 3 -*— addend addend 5 -<— sum 2 + **— - 9 •*— minuend 7 -*— subtrahend 2 *— difference The other two fundamental operations of arithmetic are multiplication and division. These operations are also binary operations. The numbers that are multiplied are called factors, and the result is called the product. When we divide, we divide the dividend by the divisor, and the result is called the quotient. Multiplication 5 factor x 4 20 factor product Division 4 divisor - 5l20 quotient dividend Math 87 To indicate multiplication, we can use a multiplication symbol shaped like the letter X. We can also use a center dot. We can also use symbols of inclusion such as parentheses. 4x5 multiplication symbol 4 center dot 5 • parentheses 4(5) Many computer programs use an asterisk to indicate multiplication. 4*5 asterisk (computers) There are also several ways to designate division. of the following indicates that 12 is to 3)12 division box 12-3 division symbol 12 be divided by Each 3. division bar or fraction line 3 Computers often use a slanted can be written on one line. slanted fraction line (computers) 12/3 Example 1 Solution When the sum of 3 and the difference? 4, what is fraction line so that everything and 4 is subtracted from the product of 3 We add to find a sum, so the sum of 3 and 4 is to find a product, so the product of 3 and 4 is to find the difference, so the difference of 12 4 + 3 = 7 4 12 Parentheses • 3 - = 12 7 = 5 We multiply 12. We subtract 7. and 7 is 5. sum product difference The numbers inside parentheses are considered to be a single quantity. If an expression contains parentheses, the first step is to simplify within the parentheses. . Arithmetic with Example 2 Solution Simplify: 8 - (4 2) We simplify within the parentheses as the we subtract. 8 Arithmetic with whole + - + (4 simplified 2 subtracted The counting numbers They are first step.' Then problem 2) 8-6 numbers Whole Numbers are the numbers we use i> 2, 3, 4, 5, 6, 7, . to count. . The three dots, called an ellipsis, means the list goes on and on and never ends. Every counting number is also a whole number. The number zero is also a whole number. To make a list of the whole numbers, we write 0, 1,2, 3,4, In the following examples 5, . . . we review the procedures for adding, subtracting, multiplying, and dividing whole numbers. Example 3 Add: 36 + 472 + 3614 Solution To add whole numbers, the ones' digits so we align that we add with the same place value. We add in columns from right to left. We may add the digits in any order. Looking for combinations of digits that total 10 may speed our work. digits Example 4 Solution Subtract: 5207 36 472 + 3614 4122 - 948 To subtract whole numbers, we also align the ones' digits. There an order to subtraction. We write the first number above the second number. Then we subtract. is in 4 11 9 * * i 0207 -94 8 4259 Math 87 Example 5 Solution Multiply: 164 x 23 In a multiplication problem, either number can go on top. We usually put the number with the most digits on top. We begin by multiplying 164 x 3. Then we multiply 164 by 2 and write this result below the result of the first 164 x 23 492 3280 This zero is optional. 3772 multiplication. we either write a zero at the end, or we leave the place empty. Then we add the Since this 2 is really 20, partial products to find the final product. Example 6 Solution Multiply: 468 x 200 We let write the 468 on top. the zeros in 200 "hang out" to the right. below the 468 by 2. Example 7 Solution Example 8 Solution We may Divide: Then we write zeros line. Then we multiply 468 x 200 00 468 x 200 93,600 234 We rewrite the problem using a division box. We put the top number inside the box. This division comes out even. There is no remainder. Divide: 1234 39 3)234 18 54 54 - 56 This division does not come out even. There is a remainder. Note how we write the answer. Other methods for dealing with a remainder will be considered later. 22 r 2 56)1234 112 114 112 Arithmetic with Practice a. When the product of 4 and what Add, Problem set the quotient? is 3014 - 426 e. 42 x 367 365 g- (6 -s- 12 When the If 2) -s- -h 234 sum and 6 what is subtracted from the product the difference? the subtrahend is 9 of 5 2. of 4 42 + 1250 + 568 d. A sum divided by the c. 12 1. is 5 subtract, multiply, or divide, as indicated: b. f. 4, and 4 Whole Numbers and 6, of 5 is and the difference is 8, what is the minuend? 3. If the divisor is 4 and the quotient is 8, what is the sum of 6 dividend? 4. When the and 6, 5. Name 6. If the product of 6 and 6 what is is divided by the the quotient? the four fundamental operations of arithmetic. sum is 12 and one addend is 4, what is the other addend? Add, 7 subtract, multiply, or divide, as indicated: iooo 8 8 - 4374 1659 64 g 10 7 x37 8 4 11. 13. 364 +52+867+9 316 x 18 12. 14. 4000 - 3625 6 9 4360 3 5 7 -s- 20 + 15. 25)767 16. 64 * (8 - 4) 6 Math 87 - - 4 17. (64 19. 76 20. (76 22. 431 23. 3000 - (1200 - 457) 24. 365 x 20 25. 30(40) - 8) (37 - 64 18. 8 21. 3708 * 12 • + 16) 37) + 16 +562+54+29+8 26. 1010 - 234 27. 560 14 28. 4017 - 3952 29. REVIEW LESSON 250 x 80 30. 3634 + 2957 Operations of Arithmetic with Money B Two forms Amounts of writing money dollars of and money cents. measured in To indicate an amount of money, we may use the dollar sign in the United States are ($) or the cent sign Since 1 dollar equals 100 cents, either of these two ways: $1.00 A or ($), but not both. we may write 1 dollar 100c decimal point is used with a dollar sign. Numbers to the left of the decimal point name whole dollars. Numbers to the right of the decimal point name hundredths of a dollar, that Operations of Arithmetic with Money 7 with no cents, it may be written without the two zeros following the decimal point. For example, $1.00 may be written $1. A cent sign means the value is in pennies, not dollars. So a decimal point is used with a cent sign only when naming part of a is, cents. If a dollar value is even, that is, cent. Example 1 Write twenty-five cents with (a) a cent sign and (b) a dollar sign. Solution (a) 25<2 (b) $0.25 In (b) Example 2 we wrote a zero to the left of the decimal point to show that there are no customary use it. we will see the dollar sign to Occasionally used dollars. This zero is not necessary, but it is and the cent sign incorrectly. This sign is incorrect. Soft Drinks 0.500 each Show two ways Solution to correct this sign. We assume that the sign means that soft drinks each. Fifty cents dollars The and 50 sign is may be written with a cost 50 cents dollar sign, $0.50 (no with a cent sign, 50(2 (50 pennies). incorrect because it used a decimal point with the cent sign. cents), or The incorrect sign literally means that soft drinks cost not half a dollar but half a cent! Both of the following signs are correct. Soft Drinks Soft Drinks 50^ each $0.50 each 8 Math 87 Arithmetic with money Example 3 Solution The following examples provide a review of arithmetic with money. Add: $1.45 + $6 + 8tf We begin by rewriting the written with a dollar amounts of money so that each is sign and has two digits to the right of the decimal point. $1.45 + $6.00 + $0.08 we write the numbers in columns $1.45 and we align the decimal points. Then we add. The total is written with a dollar sign and with a decimal point 6.00 Next + 0.08 $7.53 in line with the other decimal points. Example 4 Solution - 25c Subtract: $5 We rewrite the a dollar show amounts so that each amount is written with sign and with two digits after the decimal point to cents. $5.00 - $0.25 We number on top. We are careful to line up the decimal points. The decimal point in the answer is in line Example 5 Solution write the - 0.25 $4.75 with the other two decimal points. Multiply: (a) $5.00 first We (a) set $1.45 x 6 (b) 29c x 5 up the multiplication $1.45 the same way as we do when we multiply whole numbers. We write the product with a dollar sign and have two digits to the right of the X 6 $8.70 decimal point. (b) First we answer multiply is 29(2 x 5. The greater than $1, so we use a dollar sign and a decimal point to write the answer. 290 x 5 145«2 = $1.45 9 Operations of Arithmetic with Money Example 6 Divide: S12.60 - 5 $2.52 Solution Practice We divide the same way we divide whole numbers. We write the quotient with a dollar sign. We place the decimal point in the answer directly above the decimal point in the dividend. a. Write five cents with b. The shown sign Show two ways is (a) a cent sign Problem set and (b) a dollar sign. Lemonade 0.450 per cup subtract, multiply, or divide, as indicated: $1.75 + 60C + $3 e. 65C x 8 1. When the product of 2 and of 4 2. 25 10 10 to correct this c. B 26 incorrect. sign. Add, 5)$12.60 10 and d. f. 5, what Write four cents 47tf $24.00 + 5 subtracted from the sum the difference? is (a) 3 is $2 - with a cent sign and (b) with a dollar sign. 3. The sign shown Show two wavs J Name 5. If incorrect. to correct this Orange Juice 0.75^ per cup sign. 4. is the four fundamental operations of arithmetic. the dividend quotient? is 60 and the divisor is 4, what is the 1 Math 87 6. If the product is 12 and one factor is 4, what is the other factor? Add, 7. subtract, multiply, or divide, as indicated: $42.47 + 63.89 8. $20.00 - 14.79 11. $4.36 + 75C + $12 + 12. $10.00 - ($4.89 + $1.54 9. x $30.00 10. 8 7 6(2 7H) 13. 8 5 14. 3105 - 15 16. 81 -s- (9 * 3) 15. 40)1630 17. (81 + : 9) 4 6 + 5 3 4 3 18. (10)($3.75) 19. 3167 - (450 - 78) 7 2 4 1 20. (3167 - 450) - 78 21. $20.00 22. 70 24. 3714 25. $51.71 + $6.49 + 790 + $15 26. $20 - ($1.47 + $8) 27. $75.00 * 12 • x 16 800 23. $10 - $8.45 +268+47+9 $0.45 28. *- +8 30 29. 45 C x 72 6324 30. 31 1 Missing Numbers in Addition, Subtraction, and Multiplication Missing Numbers in Addition, LESSON and Multiplication Subtraction, 1 Missing numbers 1 in addition problem there are at least two addends. There is only one final sum in an addition problem. Sometimes w e encounter addition problems in which the sum is missing. Sometimes we encounter addition problems in which an addend is missing. Our job is to find the missing number. We can use a letter to represent a missing number. In every addition problem on the the missing sum. In the Missing + Sum left, we use the letter N to represent Missing Addend Missing Addend 2 2 3 + A B + 3 N In the problem in the center, we use the letter A to represent we use the an addend. We can the missing addend. In the problem on the right, letter Example 1 Solution B to represent the missing addend. Find each missing number: In both (a) and (b) N (a) the missing 26 (b) + 53 + A 75 61 number is addend by subtracting the known addend from the sum. Then we check. find each missing Subtract. Try 75 22 -53 + 53 22 75 (a) Subtract. it. (b) check So the missing number in (a) is 22. Try 61 26 -26 + 35 35 61 it. check So the missing number in (b) is 35. 12 Math 87 Missing numbers in Subtraction There are three numbers in a subtraction problem. If any one of the three numbers is missing, our job is to find the missing number. Missing Minuend Missing Subtrahend N - - 3 Missing Difference 5 5 X ^_3 M To find a missing minuend (top number), we add the other two numbers. To find a missing subtrahend or difference, we subtract. Example 2 Find each missing number: (a) P -24 - 17 Solution (a) To find the top number, the bottom two numbers. We find that the missing number in (a) is X 14 Add. we add 32 (bj Try it. 41 17 + 24 -24 41 17 check 41. (b) To find one of the two Subtract. lower numbers, we subtract the smaller given number from the larger. 32 So the missing number in Try it. 32 -14 -18 18 14 check (b) is 18. Missing numbers in multiplication There are three numbers in a multiplication problem. Two of numbers are factors, and the third number is the product. If any one of the three numbers is missing, we can figure out what it is. the Missing Product Missing Factor Missing Factor 3 3 R x2 x F x 2 6 3 Missing Numbers in Addition, Subtraction, and 1 Multiplication To find a missing product, we multiply the factors. To find a missing factor, we divide the product by the known factor. Example 3 Find each missing number: 12 (a) K (b) x N 168 Solution In both (a) and (b) the missing find a missing factor x 7 105 number is a factor. We can by dividing the product by the known factor. 14 Trv it. 12 12)l68 35 12 check So the missing number So the missing number in in (b) (a) is 14. is 15. Find each missing number. Pay close attention you know what type of problem it is. A a. 31 15 14 2. e. + E 420 43 When the If 4, product of 4 and 4 what is to the sign so C c. x 15 180 26 D and 1 B -24 X 1. b. + 12 d. Problem set check 35 168 Practice x 7 105 7 48 48 48 it. 15 7)l05 (b) xl4 12 Try 15 f. 51 - F 20 is divided by the sum of 4 the quotient? the subtrahend is 9 and the difference is 9, what is the minuend? 3. Write the value of 5 quarters with a dollar sign. (a) with a cent sign and (b) 14 Math 87 one addend addend? 4. If 5. A center dot is and the sum 7 may be used is 21, to indicate what is the other what operation of arithmetic? 6. If the product of two identical factors what 36, is is each factor? Find each missing number: X 7. + 83 - R 112 27 127 0. Add, X 7 119 M 11. K 9. Z -137 300 731 + 13. 96 8. 25 12. X N 400 subtract, multiply, or divide, as indicated: 3517 * 14 14. 15. 60(7 00) 8 5 16. 96 -* (16 - 17. 2) (96 *- 16) * 2 6 1 18. 35)2104 8 7 19. $16.47 + $15 + 63C 4 3 20. 21. $50.00 - ($6.48 + $31.75) 47 x 39 $8.79 22. x 24. 1100 - (374 - 87) 25. (1100 - 374) - 87 26. 4736 80 +271+9+88 5 8 474 ° 23. 30 5 + 3 Number 27. 30,145 - 4,299 $40.00 32 40 30. 32(2 X48 Number Line LESSON • and Positive Negative Numbers 2 15 $0.48 28. X 2g Line • Ordering and Comparing Numbers Number line A number order. can be used line help us arrange numbers in to To construct a number line, we draw a straight line with a straightedge. Then we put marks on the line. The distance between any two adjacent marks is the same. 1 1 1 1 1 1 Then we choose one of the marks zero below the mark. We 1 1 1 call this 1 1 1 1 1 on the line and write a mark the origin. 1 1 1 1 o Then below the marks to the right of the origin we write the counting numbers in order beginning with the number 1 as we show here. 1 1 1 1 1 12 1 1 1 1 3 4 The arrowhead reminds us that we have just drawn a part of the number line. The number line goes on and on and does not end. - 16 Math 87 Positive and negative numbers The numbers to the right of the origin are called the positive numbers, and they are all greater than zero. Every positive number has an opposite that is the same distance to the left of the origin. The numbers to the left of the origin are called negative numbers. The negative numbers are all less than zero. We always use a negative sign when we write a negative number, as we show on the number line below. If the number does not have a sign, we know that the number is a positive number. A number is greater than another number if it is farther to the right on the number line. greater less —I— I— -4 -2 -1 Negative numbers Positive numbers We note that each number and its opposite are the same distance from zero on the number line. Note that 5 is 5 to the right of zero and -5 is 5 to the left of zero. We read -5 by saying "negative five." Ordering and Comparing numbers The number line lets us see how numbers can be arranged in order. As we move to the right on the number line, the numbers become greater and greater. As we move to the left on the number line, the numbers become less and less. We compare two numbers by determining whether one number is greater than another number or whether the two numbers are equal. We place a comparison symbol between two numbers to show the comparison. The comparison symbols are the equals sign (=) and the greater than/less than symbol (> or <). The greater than/less than symbol may point in either direction. We write this symbol so that the smaller end (the point) points to the "smaller" number. We may read a comparison from left to right or from right to left. When we read the greater than/less than symbol, we say the end of the symbol we get to first. We read the pointed end by saying "is less than." We read the open end by saying "is greater than." 7 Ordering and Comparing Numbers Below we show three comparisons. -5<4 we First 3 5>-6 + 2 = 5 will read these expressions from left to right. -5 < -5 1 is less 4 3 than 4 5>-6 + 2 = 5 3 plus 2 equals 5 5 is greater than -6 Now we will read the same comparisons by reading the right side first and then the -5<4 4 is left side. 3 greater than -5 5>-6 + 2 = 5 -6 5 equals 3 plus 2 is less than 5 Although comparison expressions may be read in either direction, for the exercises in this book we will assume the left-to-right direction is Example 1 intended. Arrange these numbers in order from least to greatest: 0,1,-2 / Solution Example 2 Solution We arrange the numbers in the order in on the number line. -2, 0, 1 which they appear Rewrite each expression by replacing the circle with the correct comparison symbol. Then use words to write the comparison. O4 (a) -5 (a) Since 1 [b) -5 is less than we 4, + 1 O 1 x 1 write -5 < 4 negative 5 (b) First we than 4 mentally simplify each expression. 1 Since 2 is less is + 1 = greater than 1, 1 1x1 and 2 + 1 we > 1 plus 1 is greater write 1 x 1 than 1 times 1 = 1 18 Math 87 Practice a. Copy number b. Arrange these numbers in order from the line in this lesson. least to greatest: 0,-1, 2,-3 c. Use digits of 2 and comparison symbol to write "The sum less than the product of 2 and 3." and 3 is a Replace each circle with the proper comparison symbol. d. 30-4 e. 312(3321 f. Problem set 1. 2 2 • 2(3 2 + 2 Find the quotient when the product of 10 and 10 by the sum of 10 and 10. divided is • 2. 3. Use digits less than the product to write "The of 3 and 3." and symbols sum of 3 and 3 is Write the value of a nickel and 3 pennies with a dollar sign. 4. Arrange these numbers in order from least to greatest: -1, 1,-3,3 5. If the sum of two identical addends is 16, what are the addends? 6. Replace each circle with the proper comparison symbol, (a) -2 3 7. What 8. How many 2 is the (b) name units for is it numbers from -2 432 3 4 23 that are less than zero? to 3 on the number line? 9 Ordering and Comparing Numbers Find each missing number: 367 9. + P 10. 11. 4 iV -216 8 432 198 6 2 24 12. M 13. X K 576 1 + 329 673 5 3 8 888 14. - 15. X x 147 Y 2 8 7 624 4 + 6 N Add, subtract, multiply, or divide, as indicated: 16. 68 x 706 18. 90(700) 19. $10.00 - ($1.45 + 20. ($10.00 21. $3.75 + $24 + $76.38 22. 348 + 76 + 3859 23. 200 * (10 + 25. 27. - 17. $31.50 - 25 89(2) $1.45) + 89tf +7+15 24. 43)974 3-4-5 26. 10(20)(30) $64.48 - 16 28. 5) $0.75 x 29. ^ 60 30. 36 15(2 X78 1 20 Math 87 Adding and Subtracting on the Number Line LESSON 3 We can use the number line help us add and subtract. In this lesson we will sketch number lines and use arrows to show addition and subtraction. To add, we let the arrow point to the right. To subtract, we let the arrow point to the to left. Example 1 Solution Show First this addition we sketch a problem on a number number line. Next line: 3 + 2 we start at the origin (at and draw an arrow that points to the right that is 3 units long. From this arrowhead we draw a second arrow zero) that points to the right that -7 Solution Show this subtraction units long. -1 -2 The second arrow ends Example 2 is 2 at 5. This shows that 3 + 2 = problem on a number line: 5. 5-3 We sketch a number line. Then, starting at the origin, we draw an arrow 5 units long that points to the right. Now, to subtract, we draw the second arrow that points to the left that is 3 units long. Remember to draw the second arrow from the first arrowhead. +-6 -5 -4 -3 The second arrow ends 7 at 2. This shows that 5-3 = 2. 21 Adding and Subtracting on the Number Line Example 3 Solution Show this subtraction We take numbers problem on a number the in the order given. We -6 The second arrow ends Use arrows number a Problem set 3-5 always begin at the origin. Starting from the origin, we draw an arrow 3 units long that points to the right. From this arrowhead we draw a second arrow 5 units long that points to the left. -7 Practice line: to at show each -2. This shows that 3 - 5 = -2. addition or subtraction problem on line. 4 - 2 - 4 a. 4 + 2 1 What is the difference when the sum of 5 and 4 is subtracted from the product of 3 and 3? . 3 2. If the b. minuend is c. 27 and the difference 2 is 9, what is the subtrahend? 3. What is the name 4. The sign shown is incorrect. Show two ways to correct this sign. for numbers that are greater than zero? Grapefruit 0.150 each and other symbols to write "The product of 5 and 2 is greater than the sum of 5 and 2." 5. Use 6. Arrange these numbers in order from digits -2, 1,0,-1 least to greatest: 22 Math 87 7. Replace each circle with the proper comparison symbol, 402(6) 3 (a) 8. Show 9. Show this this addition (b) -30-2 problem on a number subtraction problem on a line: 2 number line: + 2-3 Find each missing number: X 10. 439 11. x 12 + 372 12. 8 Y 7 512 4 2 13. Z -123 1000 14. = 654 6 101 P 15. 3 M + $1.45 5 8 32 16. X 7 K 5 224 $4.95 2 + 9 N Add, subtract, multiply, or divide, as indicated: 17. $3.67 + 14C + $52.75 18. $100.00 - $36.49 19. 47(31) 20. 5-6-7 21. 9900 * 18 22. 30(20)(40) 23. (130 24. 2014 - 1987 25. $68.60 * 7 26. 46<2 27. 21)6414 28. - 57) + 9 $3.75 X 29. + 30 4640 80 30. 36(2 X48 64(2 3 Reading and Writing Whole Numbers LESSON 23 Place Value through Hundred Trillions 4 Writing Place value In our number system Reading and • Whole Numbers the value of a digit depends upon its The value of each position is its place value. The chart below shows place values from the ones' place to the hundred trillions' place. position within a number. Whole Number Place Values U3 h en C d cd millic billio thous '5 c/3 Cm H asands d 03 — c ions d lions s d ids ex -a 03 mil bill hundre billions ten r~1 tho million hundre hundre ten CS -a thousai ten c z en c 03 | Example 1 Solution Reading and writing whole numbers 'cj C o T3 • (a) Which digit is in the trillions' place in 32,567,890,000,000? (b) In 12,457,697,380,000 (a) The (b) The place value what is the place value of the digit 4? digit in the trillions' place is 2. of the digit 4 is hundred billions. Whole numbers with more than three digits are often written with commas to make the numbers easier to read. Commas help us read large numbers by marking the end of the trillions', billions', millions', and thousands' places. We need only to read the three-digit number in front of each comma and then say "trillion" or "billion" or "million" or "thousand" when we reach the comma. trillion billion million thousand 24 Math 87 We will use the following guidelines when writing out numbers. 1. Put commas million, 2. after the words trillion, billion, and thousand. Hyphenate numbers between 20 and 100 that do not end in zero. For example, 52, 76, and 95 are written fifty-two, seventy-six, and ninetyfive. Example 2 Solution Use words One to write 1,380,000,050,200. hundred eighty three trillion, billion, fifty thousand, two hundred. Note: Since there are no millions, millions' Example 3 Solution Use words First we places as not read the comma. to write start we do 3406521. on the we move right and commas insert every three to the left. 3,406,521 Three million, four hundred six thousand, five hundred twenty-one. Example 4 Use digits to write twenty trillion, five hundred ten million, seventy-eight thousand. Solution It may be helpful to draw a "skeleton" that the number is of the number. more than one trillion, M T so We see we draw this skeleton. T The letters B below the commas stand for trillion, billion, and thousand. We will read to a comma, then pause write what we have read. We read "twenty trillion." We million, to write: 2 , , , T B M • T Reading and Writing Whole Numbers Next we read "five hundred ten million." before the millions' 5 T / M T we write zeros in the three places comma. before the billions' 000 T Then we read 10 B Since there are no billions, i write 510 comma. 2 20 We 25 t 510 B * * M T We "seventy-eight thousand." before the thousands' comma. Since 78 is write 78 just only two digits, we write a zero in front. _20 _ «. —000 _— T * 510— —078 —— ——— B ^ * M T Since there are no hundreds, tens, or ones, we write zeros in the last three places. Now we omit the dashes and write the number. 20,000,510,078,000 Practice a. In 217,534,896,000,000, which digit is in the ten billions' place? b. In 9,876,543,210,000, Use words to write c. 36427580 d. 40302010 Use e. f. digits to write what is the place value of the digit 6? each number. each number. Twenty-five million, two hundred six thousand, forty Fifty billion, four thousand hundred two million, one hundred 26 Math 87 Problem set 1. 4 What is the sum of six hundred seven and two thousand, three hundred ninety-three? 2. Use digits and symbols to write "One hundred one thousand is greater than one thousand, one hundred." 3. Use words 4. Which digit is in the trillions' place in 5. Use digits to write two hundred fifty million, to write 50,574,006. 12,345,678,900,000? five thousand, seventy. 6. Replace the circle with the proper comparison symbol. Then use words to write the comparison. -12Q-15 7. Arrange these numbers in order from -1,4,-7,0, 5, least to greatest: 7 5-4 8. Show this 9. How many units is it from negative 5 to positive 2 on the number subtraction problem on a number line: line? Find each missing number: N x 30 A -1367 960 2500 10. 11. 12. 4 3 1 2 13. B + 571 3142 14. $25.00 5 K 7 $18.70 2 3 8 6400 15. + D 10,000 16. 26 X E 624 5 4 +9 N Factors Add, 17. subtract, multiply, or divide, as indicated: 37,428 31,014 18. -24,767 + 59,775 19. 45 + 362 + 7 + 4319 20. $64.59 + $124 + $6.30 + 21. 144 + (12 * 23. 37tf + 22. (144 40(500) 24. 20)1000 25. 6-5-4 26. $10 - ($4.60 - 27. ($10 28. 27 Divisibility • - 3) $4.60) 29tf - *- 12) 39tf) 39tf 29. ^^5 30. 21 X36 LESSON 3 Factors • $12.47 x 10 Divisibility 5 Factors Recall that a factor is one of the numbers multiplied to form a product. In 3 x 5 = 15, the factors are 3 and and 5, so both 3 5 are factors of 15. In 1 x 15 = 15, the factors are 1 and 15, so both 1 and 15 are factors of 15. Therefore, any of the factor of 15. numbers 1, 3, 5, and 15 can serve as a 28 Math 87 Notice that 15 can be divided by 1, 3, 5, and 15 without a remainder. This leads us to another definition of factor. A factor is a whole number that divides another whole number without a remainder. For example, the numbers 1, 2, 5, and 10 are factors of 10 because each divides 10 without a remainder (that is, with a remainder of zero). Example 1 Solution Example 2 Solution List the 10 5 2 1 1)10 10 2)10 10 5)10 10)10 10 whole numbers 10 that are factors of 12. The factors of 12 are the whole numbers that divide 12 with no remainder. They are 1, 2, 3, 4, 6, and 12. List the factors of 51. As we try to think of whole numbers no and that divide 51 with we may think that 51 has only two factors, 1 51. However, there are actually four factors of 51. Notice that 3 and 17 are factors of 51. remainder, 17 3 is a factor of 51 17 is a factor of 51 3T51 Since 3-17 equals 51, both 3 and 17 are factors of 51. Thus, the four factors of 51 are 1, 3, 17, and 51. Divisibility As we saw Example 2,51 can be divided by 1, 3, 17, and 51, and the remainder is zero. The capability of a whole number to be divided by another whole number with no remainder is called divisibility. Thus, 51 is divisible by 1, 3, 17, and 51. in There are several methods for testing the divisibility of a number without actually performing the division. For example, Factors number Divisibility • 29 or 5. by 5, its last digit is either Therefore, we do not need to divide 12,476,365 by 5 to find out if it is divisible by 5. All we need to do is look at the last digit. The last digit is 5, so the number is divisible by 5. Below are listed some methods for testing whether a number is divisible by 2, 3, 4, 5, 6, 8, 9, and 10. if a is divisible Tests for Divisibility A number can be Example 3 divided by... 2 if the last digit can be divided by 4 if the last two digits can be divided by 8 if the last three digits can be divided by 5 if the last digit is 10 if the last digit is 0. 3 if 6 if 9 if of the digits can be divided by 3. the number can be divided by 2 and by 3. the sum of the digits can be divided by 9. 7 There the or 2. 4. 8. 5. sum is no simple Which whole numbers from test for divisibility 1 by 7. through 10 are divisors of 9060? Solution In the sense used in thing as a factor. this problem, a divisor The number 1 is a same any whole is divisor of the number. As we apply the tests for divisibility, we find that 9060 passes the tests for 2, 4, 5, and 10. The sum of + 6 + 0) is 15, which can be divided by 3 but its digits (9 + not by 9. Since 9060 is divisible by both 2 and 3, it is also divisible by 6. The only whole number from 1 to 10 we have not tried is 7, for which we have no simple test. We have to divide 9060 by 7 to find out if 7 is a divisor. It is not. We find that the numbers from 1 to 10 that are divisors of 9060 are 1, 2, 3, 4, 5, 6, and 10. Practice List the a. 25 whole numbers b. that are factors of each 24 c. 23 number. 30 Math 87 List the whole numbers from 1 to 10 that are factors of each number. d. 1260 e. 73,500 f. Problem set 3600 g. List the single-digit divisors of 1356. h. The number 7000 numbers? 1. If the product of 10 and 5 30, 2. List the 3. Use what is is divisible and 20 single-digit divided by the is sum of 20 the quotient? whole numbers digits by which and symbols that are factors of 30. to write "Negative five is less than positive four." 4. Use digits to write four hundred seven million, six thousand, nine hundred sixty-two. 5. List the whole numbers from 1 to 10 that are divisors of 12,300. 6. Replace the circle with the proper comparison symbol. Then use words to state the same comparison. -70-H 7. The number 3456 numbers? 8. Show this is divisible by which subtraction problem on a number single-digit line: 2-5 Factors 9. Use words • Divisibility to write 604,003,504. Find each missing number: X 10. P 11. + 4.60 $10.00 12. 7 -3850 4 4500 8 6 z 13. 14. 1426 2 K 1 x 8 $50.00 87 6 8 45 15. 16. x P 990 32,800 9 Z 5 + 60,000 4 + 5 N Add, 17. subtract, multiply, or divide, as indicated: 1225 18. 35 20. 800 x 50 46,302 19. $100.00 - 48.37 21. $45.00 - 20 + 49,998 22. 7-11-13 23. 9)43,271 24. 3625 + 59 + 570 + 8 25. 48C + $8.49 + $14 26. 1000 - (430 27. 140(16) 28. 25C x24 -- 58) 29. $43.50 10 30. $0.07 x 50 31 32 Math 87 LESSON and Segments Lines, Rays, 6 Lines A mathematical line has no ends. A mathematical line goes on and on without end in both directions. To represent a mathematical line, we may draw a pencil line like this: Aline The arrowheads ends. To name Below line is indicate that the mathematical line has a line, we identify two points on the no line. AB. A B -•Line AB or line BA may also be named line BA. The symbol -—- above the two letters may be used instead of the word line, as AB or This line BA Rays (read line AB or line BA). A part of a line that has one endpoint is a ray. A ray of sunlight begins at the sun and then continues on and on. To represent a ray, we may draw a is a physical example of a ray. It figure like this: A ray A ray is sometimes called a half line. To name a ray, we name and then name one other point on the ray. The endpoint is called the origin of the ray and must be named first. Below is ray AB. It may also be named by using the symbol - above the letters, as AB. the endpoint — B -•- Ray AB or AB Segments A part of a line that has two endpoints is called a line segment or just a segment. A segment Lines, Rays, and Segments 33 We name a segment by naming its endpoints. Below is segment AB or segment BA. A B Segment AB or segment BA An BA overbar with no arr owh eads means segment. Segment can also be written BA. A segment is part of a line. A segment may also be part of a ray or part of another segment. In this figure we can identify three segments. The A B •- -•- C three segments are AB, BC, and AC. A segment has length. We can describe the length of segment AB by using the letters with no overbar, or we can designate the measure of (the length of) segment AB by writing an m in front, mAB. Thus both AB. On segment AC Example 1 AB = AB and + denote the length of segment above, we note that BC mAC or letters to T Example 2 AC mAB = mAB + mBC Describe each figure as a line, a ray, or a segment. Then use a symbol Solution and name 0b) C ST (a) Figure (a) is a line: (b) Figure (b) is a segment: (c) Figure (c) is a ray: Name it. ( or TS CD or ON three segments in this figure. R DC c) ^ N 34 Math 87 Solution PractiCG The three segments are: RS or SR (2) ST or TS (3) RT or 77? Describe each figure as a line, symbol and name letters to b. Problem set (1) ray, or segment. Then use a it. N x 1. the product of two one-digit whole numbers what is the sum of the same two numbers? 2. What 3. List the 4. Use 6 If is another name whole number digits and symbols is 35, for a half line? divisors of 50. to write "Two minus five equals negative three." 5. Use digits to write ninety million, five hundred thousand, thirty-five. 6. List the single-digit whole numbers that are factors of 924. 7. Arrange these numbers in order from -10, 5,-7, 8. What 9. Use words 10. is the name 0,-2 for part of a line? to write How many units is 8, least to greatest: it 10203045. from 3 to - 4 on the number line? Lines, Rays, 35 and Segments Find each missing number: 36 11. - Z X $100.00 12. 1224 13. 4 K 1 $17.54 5 6 14. R + 8320 X 9 x 20 $36.00 8 15. 10,000 5 W 16. 2 - 98 432 4 6 + 7 Add, 17. N subtract, multiply, or divide, as indicated: 36,475 476 18. + 55,984 9 x 38 20. $80.00 - $72.45 21. 49 + 387 + 1579 + 98 22. 4000 24. 26. 23. (4000 - 200) + 10 (200)(400) 25. $68.00 8-7-5 27. $1.25 -s- (200 - 10) x 28. ^° 35 30. 45S4 ^^ 19. Name 29. 38 99tf x99 three segments in this figure. -s- 40 36 Math 87 LESSON and Mixed Numbers Fractions 7 Fractions A fraction Here the may be used we use 1. see that A part of a whole. a circle to represent -- 1. of 1 is not shaded. may name fraction group. Here - of the group and name A fraction may name Here - of 1 is shaded. We number part of to of the group is part of a is (oooo) shaded not shaded. [ A line. fraction is written with The division line mav - Bar two numbers and a division be a bar or a slash. Slash 1/4 4 The slash is used when entering fractions on computers and by some people when writing fractions by hand. A slash may lead to confusion when fractions become more involved. For this reason we will use a bar instead of a slash and we recommend that students develop a habit of using a bar instead of a slash. The number below the bar is the denominator. This number tells how many equal parts are in the whole. The number above the bar is the numerator. This number of the parts have been selected. Numerator Denominator tells how many 1 4 we say the name of the numerator, and then we say the name of the denominator. If the denominator is 2, we say "half." If the denominator is 3 or more, we say the ordinal name of the number. (The ordinal name of a To read a number etc.) If fraction, is its positional name, such as third, fourth, the numerator with an "s" at the is more than end (two 1, the denominator is thirds, three fourths, etc.). fifth, read . Fractions and Mixed Example 1 Solution Example 2 Solution Draw a circle and shade - of Numbers 37 it. We must be careful to divide the circle into 3 equal parts. Drawing a "spread-out Y" at the center is one way. We shade any one of the parts. Use words We name to name the fraction the numerator — then the denominator: Three first, fourths. Mixed numbers A whole number plus a fraction is a mixed number. To name number of circles shaded below, we use the mixed number 3 2 7 means ^ 2 4 To read a mixed number, We see that ~. 4 4 we first say the whole number, then we say "and," and then the , we — — . . say the fraction. Two and Example 3 Solution Draw and shade To (.1 illustrate 1J « three fourths circles to illustrate one and one sixth. one and one sixth we will draw two circles and divide the second equal parts. circle and into 6 We shade the first 1 part of the second circle. Fractions and mixed numbers on the number line Between the points on a number line that represent whole numbers are many points that represent fractions and mixed numbers. To identify the fraction or mixed number associated with a point on a number line, it is first necessary to discover the number of segments into which each length has been divided. — 38 Math 87 Example 4 Point A represents what number on this number 7 9 H Solution We see that point less than we first A h number represents a greater than 8 but represents 8 plus a fraction. To find the fraction, 9. It notice that the segment from 8 to 9 has been divided From 8 to point A is 2 of the 5 represents the mixed number 8-. into 5 smaller segments. Thus, point Note: PractiCG line? A segments. important to focus on the number of segments and not on the number of vertical tick marks. The four vertical tick marks divide the space between 8 and 9 into 5 segments, just as four cuts divide a candy bar into 5 pieces. It is a. What b. What fraction of this circle is shaded? fraction of this circle is not shaded? Use words to name each d. c. fraction or 3- mixed number. e. 6- 12 3 Draw and shade circles to illustrate each fraction or mixed number. y f. Points 8- i h. 1 24 and represent what mixed numbers on these number j lines? ———— —— 4 3 H — i 12 h I 5 i I I 1 — 14 13 y- H i- 1 H H 1 Fractions and Mixed Problem set Use digits 2. Use digits to write 3. What is nine billion, forty-two. the quotient divided by the 4. 39 and a comparison symbol to write "One and three fourths is greater than one and three fifths." 1. 7 Numbers sum when the product of 20 and 20 is and 10? of 10 (a) List the factors of 39. (b) List the single-digit whole numbers that are divisors of 1680. 5. Point A represents what mixed number on this number line? 4 3 -I 1 1 1 1 1 5 1 1 1 1 h 1 A 6. Replace each circle with the proper comparison symbol. (a) 3 + 2O2 7. Use words 8. (a) + 3 to write What (b) - 3 2O2 - 3 32500000089. fraction of the circle is shaded? (b) What fraction of the circle is not shaded? 9. 10. Draw and shade What is the circles to illustrate 2—. name of the bottom number Find each missing number: 11. A -$4.70 $2.35 12. B +$25.48 $60.00 of a fraction? 40 Math 87 C 13. 14. 10,000 x 8 $60.00 15. D 5,420 5376 + E 7157 Add, 19 16. F 399 X subtract, multiply, or divide, as indicated: 400 17. 18. X500 $50.00 - 48.79 8 19. 9 8 20. 8 3625 + 431 + 687 9 21. 8 6000- 50 2 22. 8 20-10-5 5 6 23. $27.00 + 18 1 +6 24. 1000 - 11 25. 416 - (86 + 119) 26. (416 86) + 119 $0.08 27. x 30. - 28. 3456 ^^R three segments in this figure. Q 79C x30 75 Name 29. R Adding, Subtracting, and Multiplying Fractions 41 Adding, Subtracting, and LESSON Multiplying Fractions 8 Adding fractions Example K 1 Solution To add fractions with the same denominators, we add the numerators and write the sum over the same denominator. Add: (a) — + — 15 15 (a) Subtracting ' fractions - + 7 7 - + - + - = - (b) 15 15 + 7 —+—=— 15 - (b) 7 7 7 7 To subtract fractions that have the same denominators, we write the difference of the numerators over the same denominator. Example 2 K Solution Multiplying fractions Example 3 Subtract: - - - (a) 9 | (a) - J = Practice - - - 9 5 5 (b) f |- | = To multiply fractions, we multiply the numerators to find the numerator of the product. We multiply the denominators to find the denominator of the product. Multiply: (a) 2 4 - x - \ x % 5 3 (a) a. 5 2 - + - 9 d - 1 15 2 4 X 3 4 X 5 1 3 5 40 3 b. c. 5 5 e. 4 2 (b) 9 '88 - 13 1 - x - x - (b) 5 3 Solution (b) —4 5 2 73 X 13 3 x - x — —+—+ 3 f. 4 2 3 3 - h 42 h Math 87 Problem set 1. What is when the quotient divided by the product of 8 2. the 1, 2, sum of 1, 2, and 3 is and 3? Show This sign is incorrect. two ways to correct this sign. Apples 0.450 per pound! 3. 4. Replace each circle with the proper comparison symbol. Then use words to write the same comparison. (b) -20-4 Use digits to write four hundred seventy-five billion, nine hundred forty-two thousand, ten. 5. Use words to write 406000012005. 6. (a) What fraction of the square is shaded? (b) What fraction of the square is not shaded? an imaginary "line" from the earth a ray, or a segment? 7. Is 8. Point to the moon a line, X represents what mixed number on this number line? ———— 4 H 9. 5 i i i -i (a) List the factors of 18. (b) List the factors of 24. (c) Which numbers — are factors of both 18 and 24? 43 Adding, Subtracting, and Multiplying Fractions Find each missing number: 4315 10. A + 2157 + $5.60 $20.00 E 14. 15. x 12 $30.00 F -$98.03 $12.47 subtract, multiply, or divide, as indicated: 16. + - 17. 15 *« 19. D 13. x C 900 Add, B 200,000 60 12. 85,000 11. —3 15 x 4 8 18. 5 8 5 -1 7 4 6 20. +296+8+79 5317 8 1 2 21. $8.97 + $110 + 53tf 3 4 22. (125 -s- 25) -s- 5 5 +6 23. $60.00 - 49.49 607 24. X 78 25. ^^ 17 26. $6.75 + 15 27. $0.09 x 56 28. 50 29. 5 x 2 3 x 1 60 • 70 a + - + 4 9 9 9 30. 3 • 44 Math 87 "Some and Some More" LESSON Word Problems 9 In this lesson we will begin solving one-step story problems. There are hundreds of different story problems but only a few different thought patterns. One common thought pattern in story problems is that someone has some and then gets some more. We will call problems with this thought pattern some and some more problems. The some and some more pattern is an addition pattern. We will sketch the pattern, as we show below, to help us structure our thinking. We encourage readers to sketch the patterns. Patterns occur often in mathematics, and sketching patterns can be a powerful problem-solving tool. Problem Pattern Some + Some more Total + — t Numbers t Words (same word each time) some and some more problems, we write the number for some on the top line, write the number for some more on the second line, and To complete an addition pattern for write the total on the bottom line. Thus, there are three numbers. Any one of the three numbers may be missing in the problem. To solve the problem, missing number. Then we we need to find the can write the answer. We follow these four steps: Step 1 Step 2. . Read and recognize some more pattern. that the problem has a some and Sketch the pattern and record the given information. "Some and Some More" Word Problems Step 3. Find the missing number Step 4. Answer We Example 1 Solution to 45 complete the pattern. the question. will follow these steps as we consider three examples. Robert had 27 dollars. For his birthday he received 18 dollars more. Now how much money does Robert have? Step 1. We recognize that this is a some and some more problem because Robert had some money and got some more money. Step 2. We sketch the pattern and record the given information. Problem Pattern Step 3. Some 27 dollars + Some more + 18 dollars Total dollars We need to find the missing number which is the total. We find the bottom number in an addition pattern by adding. 27 dollars + 18 dollars 45 dollars Step 4. We have completed the pattern. Now we answer the question. Robert has 45 dollars. Example 2 At the end of the first day of camp, Marissa counted 47 mosquito bites. The next morning she counted 114 mosquito bites. Solution Step How many new bites 1. did she get? We recognize this as a some and some more problem. She had some bites and then she got some more bites. Step 2. We and record the given information. She had 47 bites. She got some more sketch the pattern 46 Math 87 bites. Then she had a total of 114 bites. Problem Pattern Some 47 bites + Some more + B bites 114 bites Total Step We 3. find B, the missing number. We find the first or second number of an addition pattern by subtracting. We subtract. Then we complete the pattern. 114 - 47 47 bites ^ + 67 bites 114 bites 67^^ Step Now we 4. answer the question. Marissa got 67 new bites. Example 3 encamped in the ravine. A second troop of 137 scouts joined them, making a total of 312 scouts. How The first many Solution Step scouts were in the 1. Step 2 scout troop . first We recognize this problem has a some and some more thought pattern. There were some scouts. Then some more scouts came. We sketch the pattern and record the given information. Pattern 3. We Problem Some S scouts + Some more + 137 scouts 312 scouts Total Step troop? find S, the missing number. To find the second number in an addition pattern, we We subtract, then complete the pattern. 175 scouts + 137 scouts 312 scouts first or subtract. 47 "Some and Some More" Word Problems Step 4. Now we answer the question. There were 175 scouts in the Practice first troop. a. on the scales. Billy weighed 118 pounds. Then Nathan and Billy stood on the scales. Together they weighed 230 pounds. How much did Nathan weigh? b. Tim cranked Billy stood for a crank 216 turns. how many c. Problem set number If the total turns did Tim Then Dawn gave the number of turns was 400, of turns. give the crank? Before the game began, 87 kids were in the pool. the whistle blew, 49 more kids jumped in. How kids were in the pool then? When many 1. Robin weighed 165 pounds. Little John and Robin together weighed 450 pounds. What was the weight of Little John? 2. In the 9 morning Ricky raked 1057 leaves. That afternoon he raked 2970 leaves. In all, how many leaves did Ricky rake that day? 3. What is the difference when the subtracted from the product of 4. (a) What (b) What sum 2, 3, of 2, 3, and 4 is and 4? fraction of the rectangle is shaded? fraction of the rectangle is not shaded? 5. Use words to write the number 2000450000. 6. Replace each circle with the proper comparison symbol. (a) 2 - 2Q2 - 2 „> (b) 1 - + 2 1^1 -OW2 2 1 x - 2 h 48 h Math 87 7. Point M represents what mixed number on this number line? H ————— I I H I — —I—- 10 11 H I ———— I I I M /\ 8. Draw and shade 9. List the single-digit circles to represent 1 numbers . that are divisors of 420. Find each missing number: 12,500 10. X + 18 11. 12. X Y 396 36,275 8 7 5 4 77,000 13. A 14. Z x 8 $10.00 39,400 < 7 6 15. B -$16.25 C 16. 4_ + $37.50 3 $75.00 2 $8.75 +8 Add, *n 17. subtract, multiply, or divide, as indicated: 5 7 X 5 —3 18. 4 19. 8 20. 2000 - (680 - 59) 21. 436 + 2799 + 68 + 347 22. 89C + 23. 800 57(2 X300 26. 8)38,479 8 N — + — 20 20 + $15.74 24. 768 16 25. 27. $ 96.00 $40.75 - 36.57 * 30 "Some Went Away" Word Problems 28. 30. — + — + — 20 20 20 29. symbol and -• (a) M • 3 Describe each figure as a a - letters to — »- C 3 name each 3 segment. Then use line, ray, or W ^-^ • 49 figure. W m F H "Some Went Away" LESSON Word Problems 10 we considered story problems that had a some and some more thought pattern. The some and some more pattern is an addition pattern. In Lesson 9 we will consider story problems that have a some went away thought pattern. The some went away pattern is a subtraction pattern. In this lesson Problem Pattern Some - Some went away What's left —* — — ** — t Numbers t Words (same word each time) some went away problems, we write the number for some on the top line. We write the number for how many went away on the second line. The number that remains goes on the bottom line. There are three numbers in this pattern. Any one of the three To complete the subtraction pattern for 50 Math 87 numbers may be missing in the problem. To solve the problem, we find the missing number. Then we answer the question in the problem. Thus, we follow the same four steps we followed in the previous lesson: Example 1 Solution Step 1. Read and recognize the type of thought Step 2. Sketch the pattern and record the given information. Step 3. Find the missing number Step 4. Answer to pattern. complete the pattern. the question. Denise spent $63.45 at the grocery store. If she went to the store with $90.00, how much money did she have when she came home from the store? Step 1. We recognize that this problem has a some went away pattern. Denise Step 2. had $90.00, then some went away. We sketch the pattern and record the given information. Problem Pattern Some $90.00 Some went away -$63.45 What's Step 3. IT left Next we find the missing number. To find the bottom number in a subtraction pattern, we subtract. $90.00 -$63.45 $26.55 Step Example 2 4. We answer the question. Denise came home from the store with $26.55. Tim baked 4 dozen cookies. While they were cooling, he went to answer the phone. When he came back, only 32 cookies remained. His dog was nearby, licking her chops. How many cookies did the dog eat while Tim was answering the phone? Solution Step 1. We recognize that this problem has a some went away thought pattern. "Some Went Away" Word Problems Step 2. We sketch the pattern and record the information. We know that Tim had 4 dozen, or 48, cookies. Problem Pattern Some 48 cookies - Some went away What's left Step 3. 51 We - Q cookies 32 cookies find the missing number. To find the second number in a subtraction pattern, subtract and complete the we subtract. We pattern. 48 cookies - 16 cookies 32 cookies Step Example 3 4. Now we answer the question. While Tim was answering the phone, his dog ate 16 cookies. The room was full of boxes when Sharon began. shipped out 56 boxes. Only 88 boxes were left. boxes were in the room when Sharon began? Solution Step 1. We Step 2. We recognize that this problem has a away thought pattern. 3. Some B boxes - Some went away What's left - 56 boxes 88 boxes We find the missing number. To find the top number numbers. Then we we add 144 boxes - 56 boxes 88 boxes Now we answer the question. in the the other two complete the pattern. 144^ 4. some went Problem in a subtraction pattern, Step How many sketch the pattern and record the information. Pattern Step Then she There were 144 boxes room when Sharon began. 52 Math 87 Practice Problem set a. At dawn 254 horses were in the corral. Later that morning, Tex found the gate open and saw that only 126 horses remained. How many horses got away? b. Cynthia had a lot of paper. After using 36 sheets for a report, only 164 sheets remained. How many sheets of paper did she have at first? 1. As 10 the day of the festival people in the 85,000, city. If how many drew near, there were 200,000 the usual population of the city visitors had come was to the city? 2. Syd returned from the store with $12.47. He had spent $98.03 on groceries. How much money did he have when he went to the store? 3. Exactly 10,000 runners began the marathon. If only 5420 runners finished the marathon, how many dropped out along the way? 4. (a) What is (b) group shaded? What fraction of the group is 5. fraction of the not shaded? Arrange these numbers in order from 2'°'" 2 ' Use words to write 407000075. 7. Use two digits is less least to greatest: * 6. 8. oo Oo o o o o "The product of one and than the sum of one and two." and symbols to write Subtract eighty-nine million from one hundred million. Use words to write the difference. "Some Went Away" Word Problems (a) List the factors of 16. (b) List the factors of 24. (c) Which numbers 9. and 24? are factors of both 16 Find each missing number: 8000 10. 1320 11. K 5340 1760 36 12. 13. R + $126 $375 15. T x 40 $220.00 N X 720 S 14. -$8.75 $7.75 Add, -,* 16. subtract, multiply, or divide, as indicated: ._ 4 4 — + — 9 17. 9 24 25 19. 100 - 21. 29,214 + 6037 + 528 22. 36,418 - 989 (5 135 23. M + 30($1.49) 28. 7)64,404 29. 5 9 3 25 (100 ^55. - . I 2 5) 5 - 2 x - 7 3 x 20 $100.00 25. 40 26. I „ 18. 20. 24. x 72 . x 20) 23 - 81.93 27. $140.70 - 35 30. —5 8 + (3 l« 1 1 «J 53 54 Math 87 LESSON "Larger-Smaller-DiflFerence" Word Problems 11 • Time Problems Largersmallerdifference word problems In the previous lesson a we practiced word problems that have some went away thought pattern. is a subtraction pattern. The other type is A some went away pattern of problem that has a subtraction pattern a larger-smaller-difference problem. In larger-smaller- we are asked problems we not only difference problems In these compare two numbers. decide which number is to and which number is less, but also how much greater or how much less. The number that describes how much greater or how much less is called the difference. To set up the pattern, we list the numbers in order: larger, smaller, greater difference. Example 1 Solution During the day, 1320 employees worked at the toy factory. At night, 897 employees worked there. How many more employees worked at the factory during the day than at night? Step 1. Questions such as "How many more?" or many fewer?" indicate that the problem has a "How larger- smaller-difference pattern. Step 2. In the pattern we write the numbers in this order: the larger number, the smaller number, then the difference. Problem Pattern Larger - Smaller Difference Step 3. 1320 employees - 897 employees E employees We find the missing bottom number of a subtraction pattern by subtracting. 1320 employees - 897 employees 423 employees "Larger-Smaller-Difference" Step 4. Word Problems • Time Problems 55 We at answer the question: 423 more employees work the factory during the day than work there at night. Example 2 Solution The number 620,000 Step 1 . how much is less than 1,000,000? The words "how much less" indicate that this problem has a larger-smaller-difference pattern. Step 2. We sketch the pattern and record the numbers. There no words are to write this time. Problem Pattern 1,000,000 Larger - - Smaller 620,000 D Difference Step 3. We subtract to find the missing number. 1,000,000 - 620,000 380,000 Step 4. The difference question. Six less than is "how much less." We hundred twenty thousand answer the is 380,000 1 million. Time Time problems are like larger-smaller-difference problems. problems We arrange the times in this order: later-earlier-difference. At this point we will consider time problems involving only years a.d. Example 3 Solution How many Step 1. We Step 2. We years were there from 1492 to 1776? recognize that this problem has a later-earlierdifference thought pattern. sketch the pattern and record the years. Pattern Later - Earlier Difference Problem 1776 - 1492 Y 56 Math 87 Step 3. We subtract to find the missing number. 1776 - 1492 284 Step 4. Now we answer the question. from 1492 Example 4 Solution to 1776. Abraham Lincoln died was he born? Step 1. This is There were 284 years in 1865 at the age of 56. In a time problem. what year Time problems have a larger- smaller-difference pattern. Step 2. We sketch the pattern. Age is the difference between the birth date (earlier) and the date of death Problem Pattern 1865 Later (death) - - Earlier (birth) 3. To find the middle number in a subtraction pattern, we subtract. 1865 - 56 1865 ^-1809 1809-^ Step 4. Y. 56 Difference (age) Step (later). 56 Now we answer the question. Abraham Lincoln was born in 1809. Practice a. The number 1,000,000,000 is how much greater than 25,000,000? b. How many c. John years were there from 1215 to 1791? Kennedy died year was he born? F. in 1963 at the age of 46. In what "Larger-Smaller-Difference" Problem set 11 Word Problems • Time Problems 57 Problems 1-4 identify the type of problem. Then find the In , answer. 1. Seventy-seven thousand fans filled the stadium. As the fourth quarter began, only thirty-nine thousand, four hundred remained. How many fans left before the fourth quarter began? 2. Mary purchased 18 bananas When she got home, she discovered that she already had some bananas. If she now has 31 bananas, how many did she have before she went to the store? 3AHow many years at the store. were there from 1066 to 1215? The first week 77,000 fans came to the stadium. Only ^49,600 came the second week. How many fewer fans came to the stadium the second week? 4. 5. Draw and shade circles to \ show 2—. 4 6. Use words 7. Twenty-three thousand 8. Replace each circle with the proper comparison symbol, (a) 9. - 2 Name to write 100000000042. is how much less than one million? W |Of 30-1 three segments in this figure in order of length from shortest to longest. Q 10. (a) What R fraction of the triangle is shaded? (b) What fraction of the triangle not shaded? is 58 Math 87 11. The number 100 is by what numbers? divisible Find each missing number: X 12. Y 13. 14. 5 x 15 -2714 8 630 3601 4 7 2900 P 15. 16. + 64 $1.53 6 Q 5 $5.00 7 N 20 17. 4 R 1200 x + 6 58 Add, subtract, multiply, or divide, as indicated: 72,112 18. 19. -64,309 — + —A 5 fa 21. 9 23. I 111 - - - • • 2 25. 7000 40 27. $4.36 + $15.96 + 28. $20.00 - $0.89 29. 30 30. 120($0.15) 60 • 900 76(2 5 3 9 9 + —2 9 24. $37.20 * 15 26. 9)42,847 + $35 74 x68 4 3 • 20. 22. V 9 453,978 + 386,864 Equal Groups Word Problems Equal Groups LESSON 12 We 59 Word Problems have used both the addition pattern and the subtraction word problems. In this lesson we will use a multiplication pattern to solve word problems. Consider this pattern to solve problem: Willie packed 25 marbles in each box. boxes, how many If he marbles did he pack in filled 32 all? This problem has a thought pattern that is different from the addition pattern or subtraction pattern. This problem has an equal groups thought pattern. An outline for an equal groups problem is shown below. Equal Groups x Number in group Number of groups eanh X Total 1 t t NunLbers Same Same words words Since an equal groups pattern is a multiplication pattern, we multiply the top two numbers to find the bottom number. To find one of the top two numbers, we divide. We will consider three examples. Example 1 Solution Willie packed 25 marbles in each box. how many marbles did he pack in all? Step 1. The words If he filled 32 boxes, "in each" are a clue to help us recognize an equal groups pattern. Step 2. We x sketch the pattern and record the information. Number in group Number of groups Total 25 marbles each box x 32 boxes M marbles 60 Math 87 Step 3. To find the missing bottom number, we multiply. 25 x 32 50 75 25 marbles each box x 32 boxes ^800 marbles s 800 Step 4. We answer the question: in Example 2 Movie $820. Solution Step all. The tickets sold for $5 each. How many 1. Willie packed 800 marbles tickets total ticket sales were were sold? The word "each" is a clue that this is an equal groups problem. Step 2. We x sketch the pattern and record the information. Number in group Number of groups 5 dollars 3. tickets 820 dollars The second number is first or second number we T x Total Step each ticket missing. To find a missing in a multiplication pattern, divide. 164^ 5)820 5 dollars each ticket x 164 tickets 820 dollars Step Example 3 Solution 4. We answer the question: 164 tickets were sold. Every truck held the same number of cars. Six hundred new cars were delivered to the dealer by 40 identical trucks. How many cars were delivered by each truck? Step 1. An equal number of cars were grouped on each truck. This problem has an equal groups thought pattern. 61 Equal Groups Word Problems Step 2. We sketch the pattern and record the information. x Number in group Number of groups C x Total Step 3. 40 trucks 600 cars number is missing. To find a missing first second number in a multiplication pattern, we The or cars each truck first divide. 15 40)600 — 15 x^0 cars each truck trucks 600 cars Step 4. We answer the question: 15 cars were delivered by each truck. Practice Beverly bought 2 dozen juice bars for 18c each. much did she pay for all the juice bars? b. Johnny planted a total of 375 trees with 25 trees in each row. How many rows of trees did he plant? c. Problem set 1 2 How a. Every day Arnold did the same number of push-ups. If he did 1225 push-ups in one week, then how many push-ups did he do each day? In Problems 1- 4 identify the type of problem. , Then find the answer. 1. Ashton was 64,309. By the 1990 census, the population had increased to 72,112. The population of Ashton in 1990 was how much greater In 1980, the population of than the population in 1980? 2. Huck had 5 dozen night crawlers in his pockets. He was unhappy when all but 17 escaped through holes in his pockets. How many night crawlers had escaped? 62 Math 87 3. President Franklin D. Roosevelt died in office in 1945 at the age of 63. In what year was he born? 4. The beach were packed 12 in each case. If 75 cases were delivered, how many beach balls were there in all? 5. The product of 5 and sum of 5 and 8? 6. Use digits to write three hundred ninety hundred seven million, forty-two. balls 8 how much is greater than the billion, five 7. How many units is it from - 5 8. Use words 9. Describe each figure as a line, a ray, or a segment. Then use a symbol and letters to name each figure. w _jB 10. 11. to write to +5 on the number line? 104000032. ^ R—»• (a) List the factors of 24. (b) List the factors of 36. (c) What whole numbers What A o 1 1 1 1 1 1 1 —— | A 1 1 —— 2 h I 1 1 1 B Find each missing number: 3674 12. - A 2159 and 36? mixed numbers are represented by and B on this number line? fractions or points H are factors of both 24 13. 4610 B ± 5179 36 14. x Q 1800 Part-Part-Whole D 15. E 16. + $56.45 4 17. 30 7 4500 6 x $80.00 63 Word Problems 8 18. F -$1.64 4 5 $3.77 Add, 5 7 subtract, multiply, or divide, as indicated: 9 + 19. v f 1^ ^2 5 n\ a 5 6 + 20. N y + 8 75 21. +86+7 22. 363 + 4579 23. $12.00 - $11.37 24. 600 25. 25 LESSON 13 x 25 26. $63.75 27. 1000 - (100 - 10) 29. 3 • 600 30 4- • 5 300 28. (1000 30. (5 • 4) 4- -5- 100) + 10 (3 + 2) Part-Part-Whole Word Problems We remember that a some and some more thought pattern is an addition pattern. Another type of pattern that has an addition pattern is the part-part-whole pattern. Here are two 64 Math 87 sample part-part-whole problems: • Seventeen of the 31 students in the class were boys. • One test. How many were girls in the class? A third of the students earned an What earn an A on the fraction of the students did not on the test? In a part-part-whole problem, a whole group described as being made up of two or more parts. From the information we are given, we can figure out the whole or the other part. The part-part-whole pattern is some and some more pattern. This a little is a is different from the some and some more pattern. Some and Some More Some + Some more + Total Nuir ibers Same words This is a part-part-whole pattern. Part-Part- Whole 4 Part + Part + 6 Whole 10 red green marbles marbles total marbles ii i Nuir Lbers Diffe ;rent Sa] me wo rds wo rds In the part-part-whole pattern there are two words to write each number. The first word is different for each number. The second word is the same for each number. We follow the after Word Problems 65 Seventeen of the 31 students in the class were boys. How Part-Part-Whole same four Example 1 many Solution Step steps girls 1. were we have been practicing. in the class? We recognize that this problem has a part-partwhole whole pattern because the problem separates the two class of students into Step 2 . parts: boys and girls. We sketch the pattern and record the given information. Notice how the words "students," "boys," and "girls" are recorded. Part of the students are boys. Part of the students are girls. Part + Part 3. students girl students + Whole Step 17 boy 31 total students We find the missing we complete the pattern. number by 17 boy 31 - 17 +14 girl subtracting. Then students students 31 total students Step 4. Then we answer the question. There were 14 girls in the class. Example 2 One fraction Solution We B on the test. What of the students did not earn a B on the test? third of the students earned a are not given the number of students. We are given only the fraction of students in the whole class who earned a B on the test. A drawing may help us to visualize the problem. All Step 1. We students @ recognize this as a part-part-whole problem. 66 Math 87 Step 2. We sketch the pattern and record the may seem -. as we though are given only The drawing should remind us class of students is information. It one number, that the whole We will use NB to stand for "not -. B" students. Part + Part i NB + > students not B students 3 - Whole students total 3 Step 3. We find the missing number, NB we complete 3 I 3 3 - 3 Then we answer four-step Problems a and Problem set 13 B students students total 3 the question. students did not earn a Use the Then 3 ' 2 Practice subtracting. + - not B students 3 4. by the pattern 1 Step , Two B on the method described of the thirds v3, test. in this lesson to solve b. How many a. Only 396 of the 1000 lights were off? b. Two fifths of the pioneers did not survive the journey. What fraction of the pioneers did survive the journey? In lights were on. of the Problems 1-4, identify the thought pattern used in the problem. Then find the answer. 1. Beth fed the baby 65 grams of cereal. The baby wanted to eat 142 grams of cereal. How many additional grams of cereal did Beth need to feed the baby? Part-Part-Whole 2. Seven tenths of the new haircut. first 3. 4. What Word Problems 67 recruits did not like their first fraction of the new recruits did like their haircut? How many years were there from 1776 to 1789? One hundred twenty poles were needed to construct the new pier. If each truckload contained 8 poles, how many truckloads were needed? 5. The sign shown is incorrect. Milk Show two ways to correct this 0.60^ per quart sign. 6. Draw and shade circles to show 1 3-. 3 7. Use digits to write four hundred seven million, two thousand, six hundred three. 8. Use words 9. (a) List the factors of 40. (b) List the factors of 72. (c) What to write is forty- 37060p43. the greatest number that is a factor of both 40 and 72? 10. Name three segments in this figure in order of length from shortest to longest. • W 11. X m Y (a) What fraction of the group is shaded? (b) What fraction of the group is not shaded? oooooo ooooo 68 Math 87 Find each missing number: A 12. 13. + 295 1000 B 407 623 14. 5 8 7 6 4764 15. 16. D ± 9159 $20.00 E $3.47 5 9 4 3 35 6 F 4 7070 7 17. x 8 Add, subtract, multiply, or divide, as indicated: 5 18. V _ 2 iV 7J7 + 6 89 3) 7 2" '3 - + - 19. 7 20. 5 7 2.2.2 3 3 3 21. $3.63 + $0.87 + 96(2 22. 13,456 - 9714 23. 900 20 24. 145 x 74 25. 7)56,153 26. 1000 - (100 - 10) 27. (1000 - 100) - 10 28. 30(65(2) 29. 2 30. (5)(5 • 3 • 4 • 5 + 5) Fractions Equal to LESSON 14 1 • Improper Fractions 69 Fractions Equal to 1 • Improper Fractions Fractions equal to 1 A fraction is equal to 1 if the numerator and denominator are equal (and are not zero). Here we show four fractions equal to 1. 2 2 Example 1 Which r i (a) Solution The 4 4 3 3 of these fractions —5 6 6 D equal to 1? D fraction equal to 1 fraction - is 5 5 is (b), or -. The fraction - is less than 1, and the Improper A fraction that fractions improper fraction. Improper fractions can be rewritten either as whole numbers or as mixed numbers. To convert an improper fraction to a whole number or to a mixed number, we divide. Example 2 Convert each improper fraction to either a whole number or a mixed number. W (a) Solution is is greater than equal to 1 or is 1. greater than 1 is called an f We perform the division indicated by each fraction. A remainder written as the numerator of a fraction with the is denominator. 2 1 r i (a) 5 3 3J5 3 2 (b) ^ 3 3J6 same . 70 . Math 87 Example 3 Draw and shade circles to show that 1— = — 4 Solution On the we shade a whole circle and three fourths of circle. On the right we shade four fourths of one left another circle 4 and three fourths of another circle. 7 if From 4 we can and three fourths equals seven fourths. If the answer to an arithmetic problem is an improper fraction, the improper fraction is usually changed to a whole number or to a mixed number. Example 4 these circles Simplify: (a) -+3 4 (a) + 5 (b) 5 — x 2 Practice a. see that one - (b) 3 _ 4 8 —8 = „3 1— 5 5 5 —3 = — „7 — = 1— 4 15 x t: 5 15 8 8 8 Write a fraction equal to 1 that has 10 as the denominator. Convert each improper fraction to either a whole number or a mixed number. b. « 12 d. c. e. 1? 7 5 Draw and shade circles to illustrate that 1 9 2— = — 4 Simplify: f. 5+i 6 6 g. 5 —7 X -2 3 3 4 Fractions Equal to Problem set 1 4 1 • Improper Fractions In Problems 1- 4 identify the type of problem. , 71 Then find the answer. 1. Three hundred seventy answered "yes." The rest answered "no." If there were seven hundred thirty answers in all, how many answered "no"? 2. The h y- helicopter could carry only 6 passengers on each trips 150 people needed to be evacuated, would it take to evacuate them all? The great trip. If how many V: 3. was nearly destroyed in the flash flood of 1970. At that time, the tree was estimated to be 550 years old. If the estimate was correct, in what year oak tree did the oak begin its life? 4. Sam spent $4.75 for the ticket. He spent $1.50 for popcorn and 85c for a drink. How much did he spend in all? 5. Use digits to write one hundred forty-two million, seventy- five thousand, three 6. Replace each circle with the proper comparison symbol. (a) 50(2 $0-50 How many units 8. List the single-digit 9. Draw and shade (a) What fOl (b) 7. 10. hundred two. is it from - 4 numbers circles to to 2 on the number line? that are divisors of 27,300. show that fraction of the circle 2 5 1— equals -. is shaded? (b) What fraction of the circle is not shaded? 11. Write a fraction that of 12. is equal to 1 and has a denominator 72 Math 87 12. Convert each improper fraction to either a whole number or a mixed number. 9 (b) (a) (c) Find each missing number: M 13. N 14. 15. 5 - 76 124 + 35 118 8 6 4 15 5 Q 7 210 3 16. x 2 Add, subtract, multiply, or divide, as indicated: 9 17. + i + 4 2 9 19. 9 *<. 18. 9 5 5 + — + 3 3617 + 98 + 249 + 77 21. $100 - $97.74 22. 900 15 360 24. $45.00 * 20 26. (10)(10 + 10) 28. 1 + 2 3 3 x 50 25. 15 27. 12($0.20) 29. • 15 111 - + - + 2 2 5 72 20. 23. 6 N 5.4 3 5 9 9 2 30. 5 3 2 ' 2 73 Equivalent Fractions LESSON Equivalent Fractions 15 name the same number are called Here we show four equivalent fractions. Different fractions that equivalent fractions. 1 2 4 2 4 3 6 8 We see in each case that half the circle is shaded. We also see we have named the shaded portions by using four different fractions. Since each fraction has the same value, we say that that the fractions are equal fractions or are equivalent fractions. Equivalent fractions have the same value. 1 2 3 4 2 4 6 8 We can form equivalent fractions by multiplying a number by a fraction equal to 1. We will multiply - by -, - and 2 show this. 1 — x 2 2 — = 4 2 3 1 — x 2 = x Example 1 4 Find three equivalent fractions and 10 10 6 3 1 2 2 — 4 8 for - by multiplying by 4 4 74 Math 87 Solution We multiply as indicated. - x — = — 3 multiplied by ^ J 9 3 3 14 — —=— — x — = 4 3 1 x 3 The Example 2 fractions 4 12 10 10 10 30 ^, |, multiplied by ^ J u multiplied by F J ,. . ~ ^, and Find an equivalent fraction , . 4 — 4 10 — 10 are equivalent fractions. has a denominator of for - that 12. Solution We want a denominator of 12. 1 = J12 2 We have multiplied 2 by 6 tiply 1 by 6. to get 12. So we must also mul- 16 - x — = 6 2 The Example 3 fractions - and — 6 12 are equivalent fractions. Find an equivalent fraction for | that has a denominator of 12. Solution We can get the fraction we want 2 4 x 3 The Practice a. fractions | Form 2, 5, and ^ 4 if we multiply by 8 12 are equivalent fractions. three equivalent fractions for | and -. - by multiplying by 75 Equivalent Fractions b. Form 5 7 c. md i 5»7' by multiplying by three equivalent fractions for | 3 3' Find an equivalent fraction for - that has a denominator of 16. Find the number that makes these fractions equivalent fractions. 4 — = , d. 5 Problem set 1 5 In — 20 ? e. 9 —3 = — 8 ? Problems 1- 4 identify the type of problem. Then find the , answer. 1. The bikers rode for several hours before stopping for lunch. Then they rode fifty-three miles after lunch. they rode a total of one hundred twenty-one miles, many miles did they ride before lunch? 2. The pickers filled 3. Twenty-three million million, eight how 63 large buckets with strawberries. each bucket held 7 pounds of strawberries, pounds of strawberries did they pick? is how much more If If how many than seven hundred thousand? Use words to write the answer. 4. Three hundred twenty-four girls and boys crowded into the auditorium. If 186 of the students were girls, how many boys were there? 5. Arrange these numbers in order from 1 -2, -, 2 6. Draw and shade 7. Make 2 2 3 , 3 circles to 0, 1, least to greatest: 4 3 show that 2^ equals -. three equivalent fractions for - by multiplying j 4 ,and-. by h 76 Math 87 8. Write a fraction equal to has a denominator of 1 that 100. 9. (a) What fraction of the square is shaded? (b) What fraction of the square is not shaded? 10. Complete each equivalent 1 n, ? fraction. 1 ? ? (b) (a) (c) 12 12 whole numbers 12 6 11. List the 12. What mixed number names point A on this number line? that are factors of 100. ——— ———— ——— —— ——— H 13. | I I I I ( I 1 I I I 1 I I | 1 Convert each improper fraction to either a whole number or a mixed number. 1 (a)1 9 \° (b) 11 (c) 5 5 Find each missing number: A 14. 15. x 18 4500 B + 3619 4087 16. 5 6 24 3 $20.00 17. 18. C X 2 D 7 1760 $18.49 Add, 40 14 8 subtract, multiply, or divide, as indicated: 6 to 19. 5 1 — + — 6 21. 6 20. 4 $0.79 + $6.48 + $15.95 • 2 N + 4 99 . Reducing Fractions, Part 22. 27,400 - 989 23. 1 77 1450 ^^30 875 24. 25. 8)14,404 27. 1591 x 16 26. 25 25 • 11 28. (6 + 29. 7){7) - + 2 30. 5 5 2 2 31. 2 - - 2 LESSON 2 Reducing Fractions, Part 1 16 In the preceding lesson, we formed multiplying by a fraction equal to equivalent fractions by 1. 14 4 4 2 8 The fraction - equals the fraction - because we multiplied by |, which equals 1. When we multiply by 1, the value of a number is not changed. It is also true that when we divide by 1 the value of a number is not changed. Here we divide \ by | i 8 By dividing, ^ i 4 = (4*4 1 2 we have changed (8 - to -. = 1) + 4 = 2) When we divide, both 78 Math 87 the numerator is and the denominator of -o become smaller. This called reducing. The numbers we use when we write a fraction are called the terms of the fraction. The terms of - are 4 and 8. If both o terms of a fraction are divisible by a number greater than 1, then the fraction can be reduced. To reduce a fraction, we divide both terms of the fraction by a number that is a divisor of both terms. In some cases the terms of a fraction are divisible by more than one number. For example, 4 and 8 are both divisible by 2 and by 4. 4 8 ' 2 2 2 ~ 4 ill 8 4 ' ~ 2 Dividing - by - instead of by - results in a fraction with lower terms, since the terms of - are lower than the terms of customary Example 1 Solution Reduce to reduce fractions to lowest terms. - to lowest terms. Both 4 and 6 are divisible by 4 6 Example 2 Solution Reduce || to ' 2 2 2 ~ 3 divisible by divide by 2 over 2. lowest terms. 18 24 is still we so 2, Both 18 and 24 are divisible by This -. It is " 2, so 2 _9^ 2 ~ 12 we divide by 2 over 2. not in lowest terms because both 9 and 12 are 3. JL . 12 " ! 3 • ! ~ 4 We could have used just one step had 18 and 24 are divisible by 6. 18 24 Both methods are correct. 6 3 6 4 we noticed that both One method took two the other took just one step. steps, and Reducing Fractions, Part 1 79 Q Example 3 Solution Reduce 3^ to lowest terms. To reduce a mixed number, we reduce the fraction and leave the whole number unchanged. 12 Thus Example 4 3 First we A 12 3 = 32 3 —3 + —3 Simplify: 8 Solution ~ 4 add. 8 Then we reduce. Add Example 5 Solution Simplify: First we Reduce 3 3 6 6 2 3 8 8 ~ 8 8 2 4 —7 - —1 subtract. Then we reduce. Reduce Subtract Example 6 K Solution Reduce 4 7 1 6 6 3 2 9 9 ~ 9 9 3 3 — 10 to lowest terms. We cannot reduce ^ because divisors. So the answer is 4 Practice Reduce each a. —3 6 3 and 10 have no common a 10 fraction to lowest terms. 8 8 d. c. b. 10 12 12 16 —— — 80 Math 87 e. „4 4- 9 c 6 f f. g. ° 12 8 8 16 u h. 12 24 15 Perform each indicated operation and reduce the . l. Problem set 1 6 5 5 l. 3 ' 1 10 result. ,52 — .-7.1 — + — 12 12 8 „ x - k. 10 6 In Problems 1- 4 identify the type of problem. , 3 Then find the answer. 1. Great Grandpa celebrated his seventy-fifth birthday in 1989. In what year was he born? 2. Austin watched the geese fly south. He counted 27 in the first flock, 38 in the second flock, and 56 in the third flock. How many geese did Austin see in all three flocks? 3. If five twelfths of the eggs were cracked, what fraction of the eggs were not cracked? 4. The farmer harvested 9000 bushels of grain from 60 acres. The crop produced an average of how many bushels of grain for each acre? 5. Use words 6. Use and digits five is to write 29705060. and symbols "The product of three greater than the sum of three and five." to write 7. List the single-digit divisors of 2100. 8. Reduce each 9. Make 3 5 3' 5« mixed number, 6 ?8 (a) fraction or (b) 2 10 three equivalent fractions for a™j1 6 6 - by multiplying by Reducing Fractions, Part 10. For each fraction find an equivalent fraction that has a denominator of 20. M (a) 5 11. 12. 81 1 1 2 Refer to this figure to (a) name the line. (b) name three rays. a Convert each fraction to either a whole number or a mixed number. — (a) r ^ 3 3 I3 3 Find each missing number: A 13. x 14. 6 3300 3977 B + 5000 15. 5 21 30 6 16. $3.24 8 C 4 $0.27 Add, -„ 17. subtract, multiply, or divide, as indicated: 2 3 4 + — + — 5 5 19. 7 * I 3 4 18. 5 5 _ 1 8 8 20. 35 98 + 76 + 56 + 38 + 119 22. $40.00 - $19.80 $26.00 8 5 21 + N 134 21. 23. q • 35 $6.50 24. x 70 82 Math 87 25. 12)72,049 26. 28. I + « 29. 5 5 1001 11 - — — 12 12 (11)(6 30. 5.2 1 LESSON 17 27. 6 + 7) 3 Linear Measure One of the characteristics of any civilization is the use of an agreed-upon system of measurement. The fair exchange of goods and services requires consistent units of weight, volume, and length. In a technological society the necessity for a standard system of measurement is even greater. There are two systems of measurement currently used in the United States. The traditional system of measurement, with units such as feet, gallons, and pounds, was adopted from England. This system used to be known as the English system, but is now referred to as the U.S. Customary System. The second system States of measurement used in the United the system used in the rest of the world. is System It is known Systeme International) or the metric system. The metric system has units such as as the International meters, We liters, (or SI, for and kilograms. will consider both systems of many lessons. measurement over we will consider units of length. The following table shows equivalent measures of length in the U.S. system. We should remember the equivalent measures, have a "feel" for the units so that we can estimate length, and be able to use rulers and read scales when measuring lengths. In this lesson Units of Length (U.S. System) = 3 feet = 1760 yards = 5280 feet = 12 inches (in.) 1 foot (ft) 1 1 1 yard (yd) mile (mi) mile 83 Linear Measure Example 1 Solution how many One yard is One yard is 3 feet yard is Solution A foot is 12 inches long. Thus, 1 yard = 3 x 12 inches = 36 inches 10-speed bicycle We One long. 36 inches long. 1 Example 2 inches? is should develop a about how many various units of measure. Most feel for 10-speed bicycles are about 5^ would be about 5 or 6 feet. Example 3 How long is this line 1 good estimate segment? " I ' TTT TTT TTT ' I I inches Solution feet long, so a i ' I feet long? I I 1 Each inch on this scale has been divided Each part is one eighth inch long. The into 8 equal parts. line segment is 2 inches plus four eighths inch. = 2— inches lone Segment b & 8 But we can reduce 4 — 8 1 to — , so 2 \ = 2— inches Segment 5 2 The following table shows equivalent measures commonly used units of length in the metric system. Units of Length (Metric System) 10 millimeters (mm) = 100 centimeters = 1000 meters = 1 1 1 centimeter (cm) meter (m) kilometer (km) of 84 Math 87 Example 4 Solution Example 5 Solution Example 6 Two meters m) (2 is how many One meter equals 100 A door The height of centimeters (100 cm), so 2 how many about is most doors This line segment centimeters? is m = 200 cm. meters high? about 2 meters. mm is 1° 20 30 40 llllllllllllllllllllllllllllllllllllllll Solution Practice (a) how many centimeters long? (b) how many millimeters long? (a) 3 (b) 30 cm mm a. Name two b. Two c. Five yards d. A e. Your shoe is about How is this piece of f. cm systems of measure used in the United States and identify some units of measure in each system. meters car is is is how many about long how many millimeters? feet? how many meters long? how many inches long? gum? . Linear Measure Problem set 1 7 In Problems 1- 4 identify the type of problem. , 85 Then find the answer. 1. Thirty-five of the one hundred eighteen students who took the test earned an A. How not earn an A on the test? many of the students did 2. At Henry's egg ranch 18 eggs are packaged in each carton. How many cartons would be needed to package 4500 eggs? 3. Three hundred twenty-four ducks floated peacefully on the lake. As the first shot rang out, all but twenty-seven of the ducks flew away. How many ducks flew away? 4. The number 516,824 5. Replace each circle with the proper comparison symbol. how much is Ô^ w 10 (a) 6. I 7. 5 Find the length of the segment an inch. inches I ^ (aj than 804,216? 5 to the nearest eighth of 1 I I I I Reduce each r W (b) 5 less I I I J L fraction or — 12 8 (b) I I I I I I I I I I mixed number. _9^ (c) 12 show 6 10 12 — 1 10 3equals 8. Draw and shade 9. For each fraction, find an equivalent fraction that has a denominator of 24. ( ^ (a) 5 — 6 circles to (b) that (c) 8 86 Math 87 10. (a) What is (b) 11. shaded? What is fraction of the group fraction of the group not shaded? The number 630 is by which divisible single-digit numbers? 12. Convert each fraction mixed number. whole number or to either a 16 a6 r (b) (a) TT 9 15. 43 8 7 16 ci Find each missing number: M 13. -1776 + 2937 3000 87 $16.25 B $10.15 16. K 14. 17. 7 86 24 42 7 D 6 X 1764 + N 175 Add, .,„ 18. subtract, multiply, or divide, as mdicated: : 3 1 - - - 4 o 20. 19. 4 — + 10 10 8 3 ! - x - 4 3 21. 60,310 - 49,157 22. $21.56 + $15 + 790 23. 10,000 16 24. 176 x 84 25. 9)70,000 26. 45 • 45 a Pairs of Lines 27. 1001 + 28. (5 -7 - -1 30. 1.2 9 9 3 LESSON Angles 87 6)(7) 13 29. • 2 Pairs of Lines • Angles 18 A desktop is a flat surface with boundaries. A desktop occupies nes a part of a plane. A plane is a flat surface that has no boundaries. Pairs Of I i Two same plane either cross once or they do When two lines cross, we say that they lines in the not cross at all. They intersect. intersect at one point. Two lines that do not remain the same distance apart. Lines in the same plane that are always the same distance apart are parallel lines. intersect Line AB intersects In this figure, line line CD at point M. QR is parallel to line ST. This statement can be written with symbols, as we show QR II here. ST Lines that intersect and form "square corners" are perpendicular. The small square in the figure below indicates a "square corner." 1 1 >M Line p ~1 MN is MN ± Q The symbol to. 1 'A/ perpendicular to line PQ. _!_ means PQ is perpendicular . 88 Math 87 Angles An angle is formed by two rays that have a common endpoint. Angle DMB is formed by the two rays, MD and MB. The common endpoint is M. Ray and ray MB are the sides of the angle. Point 1 M MD is the vertex of the angle. Angles may be named Angles may be named by using in several ways: three letters in this order: a point on one ray, the vertex, then a point on the other ray. Angle DMB BMD or angle 2. When there is no chance of confusion, an angle may be named with only one letter: the letter at the vertex. 3. An Angle A may be named by angle placing a small letter or number near the vertex and between the rays. The symbol Angle Z is often used instead of the Thus, the three angles just named could be 1 word "angle." referred to as: ZDMB read as "angle DMB" ZA read as "angle A" Z\ read as "angle 1" Angles are classified by their size. An angle that is formed by perpendicular rays is a right angle. An angle smaller than a right angle is an acute angle. An angle that forms a straight line is a straight angle. An angle that is smaller than a straight angle but larger than a right angle is an obtuse angle. Right Acute Straight Obtuse Pairs of Lines Example 1 (a) (b) Which AB? Angles 89 line is parallel to line — Which to line • A B <i »- line is perpendicular AS? * • D Solution Example 2 Solution (a) Line CD (or DC) (b) Line BD (or D5) is parallel to line this figure (a) Name the straight angle. (b) Name the obtuse angle. (c) Name two right angles. (d) Name two acute angles. (a) (c) (d) ZAMD (or ZDMA) ZAMC (or ZCMA) ZAMB {ot ZBMA) 2. ZBMD (or ZDMB) 1. 1. ZBMC{otZCMB) 2. ZCMD (or ZDMC) a. Draw two b. Draw two perpendicular c. Draw two d. Draw e. Draw an f. AB. perpendicular to line AB. There are several angles in (b) Practice is parallel lines. lines. lines that intersect but are not perpendicular. a right angle. acute angle. Draw an obtuse angle. 90 Math 87 Problem set 1 8 In Problems 1- 4 identify the type of problem. , Then find the answer. 1. Prince Caspian assembled his soldiers at the bank of the river. Two thousand, four hundred twenty had gathered by noon. An additional five thousand, ninety arrived after noon. How many soldiers arrived in all? 2. Three twentieths of the test answers were incorrect. What fraction of the answers were correct? 3. There are 210 students in the first-year physical education class. If they are equally divided into 15 squads, how many students will be in each squad? 4. How many 5. Which years were there from 1492 to 1620? of the following does not equal 1-? (a) 6. 03) 1— (d) (c) 1 12 Refer to this figure to answer (a) i and [b). (a) Which (b) Which line is perpendicular to ST? Q< line is parallel to < Reduce each (a) 8. 9. fraction or 12 16 (b) Draw and shade s ST? n 7. • n R T • 1 mixed number 12 3— 18 circles to Complete each equivalent (c) 5 20 show that 3 2— 4 equals —4 11 fraction. ? (a) 18 03) (c) 18 18 Pairs of Lines • 91 Angles 10. Draw a triangle that has one right angle. 11. What factors of 12. Draw a line and identify two points on the line as R and S. Then draw a ray ST that is perpendicular to the line. 20 are also factors of 50? Find each missing number: W 13. X 14. 15. - 2316 1415 x 8 $30.00 58 4 2 62 2715 N Y Z $25.00 1716 + 6 143 $6.30 16. + Add, 18. 17. subtract, multiply, or divide, as indicated: * - I 6 6 19. i ? = 2 3 20. 36+67+59+86+8 21. $20.25 - $15.17 22. $100 00 40 23. 300 x 800 24. 9)31,805 25. 55 26. ^°52 27. 20(20 + 20) - • 55 11 000 28. - + - + 5 30. 5 5 14 9 3 2 2 29. 14 4 15 15 92 Math 87 LESSON Polygons 19 When more segments are connected to enclose a portion of a plane, a polygon is formed. The name of a polygon tells how many sides the polygon has. three or line Names of Polygons Example of Polygon Number of Sides Name of Polygon 3 Triangle 4 Quadrilateral 5 Pentagon r-i 6 Hexagon v 7 Heptagon n 8 Octagon <5 9 Nonagon 10 Decagon vru 11 Undecagon <£> 12 Dodecagon /\ r\ [ i < Cn/l A polygon with more than 12 sides may be referred to as an n-gon, with n being the number of sides. Thus, a polygon with 15 sides is a 15-gon. Polygons Two sides of a polygon of the polygon. meet at a point that is 93 called a vertex The plural of vertex is vertices. A polygon always has the same number of vertices as the number of sides. Letters may be used to identify a particular polygon. The letters S, T, V, and U are used in the drawing below to indicate the points that are vertices of the polygon. To refer to this polygon, we give the letters of the vertices in order around the polygon. Any letter may be first. The rest of the letters can be named clockwise or counterclockwise. This polygon has eight names, which are listed here. USTV UVTS STVU SUVT If all the sides of a polygon have the angles have the TSUV TVUS VTSU VUST same length and same measure, then the polygon is all the a regular polygon. Regular and Irregular Polygons Regular Type A Triangle / Quadrilateral O o Pentagon Hexagon Practice a. What b. What do we c. What kind is Irregular /A £3 <r> Cn the shape of a stop sign? usually call a regular quadrilateral? of angle is each angle of a regular triangle? 94 Math 87 Problem set 1 9 In Problems 1- 4 identify the type of problem. , Then find the answer. 1. The Collins family completed the 3300-mile coast-tocoast drive in 6 days. They traveled an average of how many 2. miles each day? On their return trip, the Collins family drove four hundred miles the first day and five hundred seventeen miles the second day. How far did they travel on the first two days of their return trip? fifty-six 3. Albert ran 3977 meters of the 5000-meter race, but walked How many the rest of the way. meters of the race did Albert walk? 4. One billion is words 5. how much greater than ten million? to write the answer. Arrange these numbers in order from 5 3 3 4 -,-1, -, 6. Use In a rectangle the opposite least to greatest: 0, 1 D sides are parallel. In rectangle ABCD, which to side side is parallel BC? B 7. (a) What icm is 8. WX in millimeters? [234567891 W (b) the length of segment What X is Reduce each Y the length of segment fraction or XY in centimeters? mixed number. 12 (c) (b) (a) 20 20 6 15 20 95 Polygons 9. For each fraction find an equivalent fraction that has a denominator of 30. (b) (a) - (c) 3 10. An octagon has how many more sides than a pentagon? 11. (a) Draw (b) What kind a triangle that has one obtuse angle. of angles are the other two angles of the triangle? 12. (a) What fraction of the circle is shaded? (b) What fraction of the circle is not shaded? each missing number: X 13. Y 14. -5814 15. 4 + 1537 7351 3286 7 8 15 $50.00 16. 300 4 X N 12,000 6 17. M $40.70 5 7 Add, 18 subtract, multiply, or divide, as indicated: 7 3 10 10 19. 8 21 3.2 + 4 2 N 93 20. $3.67 + $14.39 + $0.78 21. 10,000 - 576 22. 2025 45 23. 750 x 80 24. 50)95,714 96 Math 87 25. 21 21 26. 2002 14 27. 30(40 + 50) 28. - + 7 7 29. 15 30. 16 16 LESSON 20 Perimeter, Part 1 Perimeter The distance around a polygon is the perimeter of the polygon. To find the perimeter of a polygon, we add the lengths of its sides. Example 1 What is 3 the perimeter of this cm rectangle? Solution The opposite sides of a rectangle are equal in length. Tracing around the rectangle, our pencil travels 3 cm, then 2 cm, then 3 cm, then 2 cm. Thus, the perimeter is 3 Example 2 What is cm + 2 cm + 3 cm 3 2 + 2 cm cm 2 cm cm cm 3 2 cm = 10 cm the perimeter of this regular hexagon? Solution All sides of a regular polygon Thus the hexagon is are equal in length. perimeter of this 8 or mm + 8 mm + 8 mm + 8 mm + 8 mm 6x8 mm = 48 mm 8 mm + 8 mm = 48 mm Perimeter, Part Example 3 The perimeter of a square is 48 How ft. long is 97 1 each side of the square? Solution A square has four sides whose lengths are equal. The sum of the four lengths this 1. is 48 ft. to think about problem: What number added 4 times equals 48? + + 2. Here are two ways What number multiplied by 4 each use the second way and ft equals 48? = 48 4 x We will = 48 + ft divide to find the length of side. 12 4J48 The length Perimeter of each side of the square is To work some story problems, we need 12 ft. to find a perimeter. word problems Example 4 250 Ray wants to fence some grazing land for his sheep. He made this sketch of his pasture. How many feet of wire fence does he need? 202 A ft Pasture 150 Solution If we add the lengths of the sides to find 175 ft ft how many feet of fence Ray needs, 250 we see that ft + 175 ft + 150 Ray needs 777 ft ft of + 202 ft = 777 ft wire fence. v^ Practice a. What is the perimeter of this 2 quadrilateral? 5 in. in. 98 Math 87 b. What is the perimeter of this regular pentagon? -^ c. If each 12 inches, perimeter? is d. Problem set 20 5 side of a regular octagon what MacGregor has 100 is cm its wire fence that he plans to use to surround a square garden. Each side of his garden will be how many feet long? feet of Problems 1- 3 identify the type of problem. Then find the In , answer. 1. One eighth of the students in the class were left-handed. What fraction of the students were not left-handed? 2. The theater was full when the horror film began. Seventysix people left before the movie ended. One hundred twenty-four people remained. 3. How many people were in the theater when The Pie King restaurant cuts each pie into 6 slices. it was full? restaurant served 84 pies one week. The How many slices of pie were served? 4. Lincoln began his speech, "Four score and seven years ago...." How many years is four score and seven? [Hint: A score is 20 years.) 5. Use words 6. Use digits to write 18720501. and symbols to write "Three minus seven equals negative four." 7. The ceiling in a floor? house is about how many feet above the . Perimeter, Part 8. Find the perimeter of this rectangle. 9. 6 8 Simplify: (a) 99 1 cm cm 15 3±| (b) 24 24 \t> 0. Find a and r l (a) 3 = - b. a 4 _ b_ [b] 36 36 9 11. Draw a sketch of a regular pentagon. 12. What is the name of a polygon that has twice as many sides as a quadrilateral? 13 What kind of angle is every angle of a rectangle? Find each missing number: 14. A + 1547 8998 B 15. 16. 3 x 30 $41.10 8 7 29 18. 32 17. x c 736 2657 D + 3010 4 6 8 N Add, subtract, multiply, or divide, as indicated: + 5 78 n 19. , 21. 2 2 2 - 3 3 3 - + - + 2.3 3 23. 7 4363 + 2791 + 5814 20. 8 22. 8 $15.00 - $9.65 1 00 Math 87 24. 3600 26. 50 • $0.79 25. 18 x 50 27. 48 100,100 11 28. 11(12 + 13) 29. 6 + 5 7 30. 16 16 20 20 LESSON 7 Solving Equations 21 Since Lesson 1 we have practiced finding the missing numbers in arithmetic problems such as these: X M Y + 25 32 24 x 6 84 77 We may arrange the numbers horizontally instead of vertically. Thus, the same three problems x + m 25 = 77 may be written this way: - 32 = 24 6y = 84 Each of these three problems is an equation. We solve an equation by finding the number that correctly completes the equation. Note that we used lowercase letters in writing our equations. The letters in an equation can be capital letters or lowercase lettters. Also, in the equation on the right, we omitted the times sign, because 6y means 6 times y we will learn some special rules For now we will solve equations in In later lessons solving equations. for the same way we have solved missing number problems. Example 1 Solution Solve: x + 25 = 77 To find what number added to 25 equals 77, we subtract 25 101 Solving Equations from 77. Then, on the -25 ^^(52) The solution Solution m check. + 25 = 77 77 = 77 52-""^'' Solve: we x + 25 = 77 77 Example 2 right, x = is equation replaced x with 52 check 52. - 32 = 24 This equation is like this missing number problem: m -32 24 To find m, we can add 24 and 32. Then m (56) - 32 = 24 - 32 = 24 56^ The Example 3 Solution solution Solve: 32 24 = 24 is m we check the result. equation replaced m with 56 check = 56. - p = 24 This equation is like this missing number problem: 32 ~ P 24 To find p, check the we can subtract 24 from 32. Then, -24 8-— Example 4 Solution solution Solve: right, we result. 32 - 32 The on the p = 24 = 24 24 = 24 ^J32_^ (8) " is p = equation replaced p with 8 check 8. 6y = 84 The equation says that 6 times y equals 84. We can missing factor by dividing the product by the known find a factor. 102 Math 87 Then, on the right, we check our answer. 6y = 84 14 6j84 6(14) = 84 84 = 84 The Practice Problem set 21 solution y = is equation replaced y with 14 check 14. Solve each equation: a. 49 = 17 + c. y - 59 = In Problems w 18 b. 59 - d. 84 = 4d x = 18 1- 4 identify the type of problem. Then find the , answer. 1. There were 628 students in 4 dormitories. Each dormitory housed the same number of students. were housed in each dormitory? 2. Thirty-six bright green parrots flew remained in the tree. How many students away while 46 parrots How many parrots were in the tree before the 36 parrots flew away? 3. Two hundred 4. Twenty-one thousand, twenty-five of the six hundred fish in the lake were trout. How many of the fish were not trout? door. Forty-eight swarmed in through the front thousand, nine hundred seventy-two fifty swarmed in through the back door. Altogether, how many swarmed in through both doors? 5. This sign is written incorrectly Show two ways to correct this sign. Pickled Peppers 0.20# each 6. Arrange these numbers in order from least to greatest: 1 1 -, -2, 1, --, 3 2 Solving Equations 7. Which is ^_- the best estimate of how much of this rectangle 103 is shaded? 8. w (b) (a) W) i I Each angle of a rectangle is a right angle. Which two sides are perpendicular to side 9. BC? Simplify each fraction or mixed number. (a) 2 _8_ (b) 16 10. 3 For each fraction find an equivalent fraction that has a denominator of 36. (b) (a) 11. List the factors of (a) 12. - each number, 10 (b) The perimeter inches long is 7 (c) of a certain square is 2 feet. How many each side of the square? Solve each equation: w 13. 36 + a = 54 14. 46 - 15. 5x = 60 16. 100 = 17. y- 18. 60 = 4y Add, 19. K + 64 subtract, multiply, or divide, as indicated: 9 10 22. 14 = 30 = 20 m 20. 10 + 1 | 9 9 $36.45 + $15 + $0.59 21. 1 1 04 Math 87 23. 25. 10,350 - 9764 24. 6345 86 49 360 26. 9 41 23 x 25 51 27. 87 70)16,161 93 28. 4386 4- 1 29. u + ( 2 30. l 4J 1 + 4 Prime and Composite Numbers • Prime Factorization LESSON 22 Prime and Composite f + 46 21 We remember that we the counting numbers are the numbers use to count. They are numbers 1 ^ 3 ^ 5 6 7> 8 9 10 Some counting numbers are called prime numbers and some counting numbers are called composite numbers. In the following list we circle the prime numbers and leave the composite numbers uncircled. We place a box around the number 1 because it falls into neither category. The number 1 is not a composite number and is not a prime number. CD©© 4 © 6 16(17)18(19)20 21 We define a prime © 8 9 22@)24 number A prime number 10(11)12(13)14 15 25 26 27 28 (29) as follows. a counting number greater than 1 whose only divisors are 1 and the number itself. is 30 Prime and Composite Numbers • Prime Factorization 1 05 A prime number has exactly two different divisors. The number 1 does not have two different divisors, so the number 1 is not a prime number. Prime Numbers Divisors 2 1 3 1 5 1 7 1 11 1 13 17 1 19 23 29 1 numbers 1 1 1 and and and and and and and and and and 2 3 5 7 11 13 17 19 23 29 prime numbers because they have exactly two different divisors. Counting numbers that are greater than 1 that are not prime numbers have more than two divisors. These numbers are called composite numbers. All of these are Composite Numbers Divisors 4 6 1,2, 8 1, 2, 9 1, 3, 10 12 14 15 16 18 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, and 4 3, and 6 4, and 8 and 9 5, and 10 3, 4, 6, and 12 7, and 14 5, and 15 4, 8, and 16 3, 6, 9, and 18 numbers are composite numbers because they have more than two divisors. All of these Example 1 Solution Make First a list we of the prime list numbers the counting that are less than 16. numbers from 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 through 10, 11, 12, 13, 14, 15 15. 1 06 Math 87 A prime number must be greater than 1, so we cross out 1. The next number, 2, has only two divisors, so 2 is a prime number. All the even numbers greater than 2 have more than two divisors, so they are not prime. We cross these out. X, 2, 3, /, 5, 0, 7, 0, 9, Xf, 11, ¥Z, 13, The numbers The numbers 9 and 15 Now we 15 we cross that are left are 2, 3, 5, 7, 9, 11, 13, out. H, are divisible by have 2, 3, 5, 15 7,0, 11, 13, 3 so them ^ The only divisors of 2, 3, 5, 7, 11, and 13 are 1 and the numbers themselves. So these are the prime numbers less than 16. Example 2 Solution List the First composite numbers between 40 and we 50. numbers between 40 and write the counting 50. 41, 42, 43, 44, 45, 46, 47, 48, 49 We circle the even numbers because these numbers are divisible by and 2 are composite numbers. 41,(42), 43, (44), 45,(46), 47, Forty-five is divisible by 5, and 49 is (48), 49 divisible by 7. Now we have 41,(42), 43,(44), The numbers not (45), @, 47,@,(49) prime numbers. These numbers are divisible by only 1 and the number itself. The numbers 42, 44, 45, 46, 48, and 49 are composite numbers. Prime factorization circled are Every composite number can be formed or composed by multiplying two or more prime numbers. Here we show each of the first eight composite numbers written as a product of prime numbers. 4 = 9 = 2-2 3-3 14 = 2 • 7 6 = 2-3 10 = 2 • 5 15 = 3 • 5 8 = 2 12 = 2 • 2 2 2 3 Prime and Composite Numbers Example 3 When we write a composite numbers, number. we have 07 Prime Factorization 1 as a product of prime written the prime factorization of the Write the prime factorization of each number. (a) Solution number • We 30 (b) 81 number will write each 420 (c) as the product of two or more prime numbers. (a) 30 = 2 3 • We do not use 5 • 5 • 6 or 3 • because neither 6 nor 10 10 is prime. (b) 81 = 3 3 • We do 3 • 9 (c) 420 = 2 • 2 • : 7 is not use 9 not prime. • 9 because Two methods for doing one are shown below. this Factoring composite numbers can be difficult without a method. There are two commonly used methods for factoring composite numbers. One method uses a factor tree. The other method uses repeated division. We will factor 420 using both methods. Factor tree method To factor a number using a factor tree, we write the number, and below the number we write any two whole numbers greater than 1 that multiply to equal the number. If these numbers are not prime, then we continue the process until there is a prime number at the end of each "branch" of the factor tree. These numbers are the prime factors of the original number. Factor Tree 108 Math 87 division To factor a number using repeated division, we method write the Repeated number Repeated Division in a ©)420 box and begin dividing by the smallest prime number that is a division (D)210 ©)105 Then we divide factor. ©135 answer by the smallest prime number the that is a factor. © We repeat this process until the 420 == 2 • 2 • 3 • 5 • 7 answer is itself a prime number. When we use this method, we custom- each division answer below the division box rather than above it. The prime factors are all the divisors arily write and the Practice final answer. 10 prime numbers. a. List the first b. whole number greater than kind of number is it? If a 1 is not prime, then what Write the prime factorization of each number. c. Problem set 22 27 d. 360 In Problems 1- 4 identify the type of problem. , Then find the answer. 1. Two thirds of the students Day. What fraction St. Patrick's wore green on St. Patrick's of the students did not wear green on Day? 2. There were 343 quills carefully placed into 7 compartments. If each compartment held the same number of quills, how many quills were in each compartment? 3. Twenty-one million is how much Use words to write the answer. less than two billion? L . Prime and Composite Numbers 4. Last year the price Prime Factorization was $14,289. This year the been increased $824. What 5. • 1 09 price has the price this year? is Simplify each fraction or mixed number. 12 (a) 3 (b) 21 prime numbers between 50 and 6. List the 7. Write the prime factorization of each number. (a) 8. 50 60 (b) 60. (c) Which point could represent 1610 on this number line? 1000 2000 + B 9. Complete each equivalent 2 — 10. Draw and shade 11. Draw 12. This rectangle perimeter? circles to (c) 15 5 show - = — 3 12 ? ? - = (b) 15 that —8 8 9 equals 2-, a sketch of a regular quadrilateral. 'cm . fraction. q ? (a) 13 300 is twice as long as 12 3 4 it is 5 wide. What is its 6 Find the width in millimeters of the rectangle in Problem 1 2 110 Math 87 Solve: 14. p + 58 = 17. 8y = 96 19. 51 Add, o21. m - 85 15. x - 46 = 20 18. 45 = 22 + 20. 51 = 3c 16. 5 w -A = 17 subtract, multiply, or divide, as indicated: 2 2 2 + + 3 3 22. 3 -T 2.2.2 -4~ 3 9 3 3 -^ 23. 36 + 47 + 52 + 116 + 45 + 8 24. $370.47 - $296.65 25. 75)$36.00 26. 960 * 40 27. 25(30)(40) 28. 6 29. - - 30. 3 v 3 71 1, LESSON AT 5 • • 4 3 • 3 1 3 3 3 1 • 2 • 1 • Simplifying Fractions and Mixed Numbers 23 We can simplify some proper fractions by reducing the fractions to lowest terms. _9_ 3 12 4 We can simplify improper fractions by writing them as mixed numbers. 9 = 2! 4 4 If an improper fraction can be reduced to lowest terms, use both ways to simplify the fraction. It we does not matter . Simplifying Fractions and Mixed whether we reduce first or convert first. Reduce first 12 Reduce we simplify — Here Convert 4 Convert: first —= 1— 9 9 4 1 — = 1— Convert: 1 Solution Simplify: this doesn't matter improper way we show both ways. Convert: 24 = 16 16 simplifies to 8_ 16 1 2 24 — 16 1 8 Reduce: find that It 2 1 1— 2 -7 + -7 8 we will 2 2 First We - = i± Convert: Solution first. 3 16 Simplify: we reduce and convert. fraction, which we do 24 Reduce: Example 2 1- 24 16 To simplify Either Reduce: 1- = 3 3 Example 111 Numbers 8 add. converting to Then we simplify a mixed number. i+i 8 = the result by reducing and — added 8 8 7 reduced 4 converted 4 O Example 3 Simplify: 26 Solution The mixed number 2| means 2 + | . The fraction tol|. « = 6 Thus 2 1 equals 2 + l|, ll = ll 6 3 which equals 3| | simplifies 112 Math 87 Practice Simplify: 20 b. a. 3 —8 c. 8 Problem set 23 3 x 5 — 4 Problems 1- 4 identify the type of problem. Then find the In , answer. 1. Eight hundred thirty of the one thousand, twenty villagers tsunami struck. How before the tsunami struck? fled before the not flee 2. many villagers did Tony discovered that 87 of the 402 watermelons field were ripe. How many of the watermelons field were not ripe? in the in the 112 students were equally grouped into 4 different classrooms, how many students would there be in each classroom? 3. If 4. Eight million, five hundred thousand is how much less than seventeen million, five hundred sixty thousand? Use words to write the answer. 5. Write the prime factorization of each number, (a) 6. 16 (b) 525 Simplify each fraction or mixed number. 16 9 (b) (a) 7. 1 12 Replace each circle with the proper comparison symbol. (a) 5 - 3(33 - 5 w |Oi| ; 8. (a) What is (b) shaded? What is fraction of the rectangle fraction of the rectangle not shaded? Simplifying Fractions and Mixed 9. For each fraction, write an equivalent fraction that has a denominator of 24. — 12 (a) 10. 113 Numbers - (b) 8 The perimeter of the rectangle is 20 cm. What is the width of the rectangle? 6 11. Draw AB to be 1 cm inch long. Then draw BC, also long, perpendicular to AB. Complete triangle 1 ABC drawing AC. prime numbers between 20 and 12. List the 13. Draw and shade circles to show that 30. —9 1 equals 2—. 4 i 4 Solve: 14. x + 47 = 129 15. y- 54 = 49 16. 4 8 17. 108 = 18 + a 18. 32 = 50 - d 3 9 19. 5 84 = 12q 7 Add, 9n 2 °- 22. subtract, multiply, or divide, as indicated: N 6 5 8 + 5 ' 8 4 5 ?1 21 3 * 4 5 3 7 " ^ £ 12 12 5 + 8 23. $5.47 + $16.79 + 85C + $28 + $0.08 24. 30,175 - 9.757 25. .678 X 94 26. 83 1410 ^~ inch by 114 Math 87 27. 400(5000) 28. 60,000 * 100 2». 5 — + 6 1) 2 J (1 V 3 2 1 3 2 30. Numbers and Whole Numbers as Fractions Writing Mixed LESSON 24 Mixed numbers as fractions We have used pictures to illustrate that mixed numbers can be rewritten as improper fractions. The illustration below shows 3j equals 13 4 3i 4 13 4 We can use a "circular" process to convert a mixed number to an improper fraction. To convert 3^ to a mixed number, we multiply 4 by 3 and then add 1, as we show here. + 4x3 1 + 1 13 (I x 4 Example 1 Write each mixed (a) Solution 3- number (b) In each case 4 as 1 3 fraction. (c) 2- 12 we multiply the whole number by the denominator and then add the numerator improper fraction. (a) an improper = 10 3 3 (b) 4 to find the = 11 4 numerator of the .0 U| = f 115 Numbers and Whole Numbers as Fractions Writing Mixed Whole The number 3 means 3 "wholes." A fraction that means 3 numbers as wholes is -. To write a whole number as a fraction, we fractions Example 2 simply write the whole number over a denominator of Write each whole number as a fraction, (a) Solution We (a) Practice 4 (b) write each whole 4 = number over (b) | 100 a denominator of 1. ^° 100 = Write each number as a fraction. a. 3- b. 7 c. 4- d. 10 4 6 Problem set 24 1. In Problems 1- 4 identify the type of problem. , Then find the answer. 1. 600 If manv fish fish would be 2. How many 3. If 4. Two each were separated into 8 equal schools, in each school? years were there from 789 to 1215? ticket cost $2.75, fifths of what was the cost of 12 tickets? the students in the class were boys. fraction of the students in the class 5. Write the prime factorization of 720. 6. What number 10 is halfway between 20 and 50? 20 40 30 Simplify: f1 (a) o 10 3- n, (b) 24 - What was made up girls? 7. how 50 60 of 116 Math 87 8. What (a) fraction of the group is shaded? What (b) fraction of the group is not shaded? 9. Complete each equivalent fraction. ? , \ (c) (a) 24 24 8 10. Sketch a regular pentagon. 11. Refer to triangle ABC to —? = 4 18 — 24 answer these questions. (a) What kind of angle is ZACB? (b) What kind of angle is ZABC? (c) Which side of the triangle appears to be the longest side? 12. a Write each number 2- (a) as an improper (b) fraction. 5 (c) 8- 2 13. 3 Draw and shade show circles to that 5 2— equals 17 — 6 6 Solve: 14. m + 412 = 1000 15. w - 59 = 63 8 16. 7 17. 300 = 30 + a 18. 59 = 63 - x 5 2 19. 6 300 = 15£ 4 Add, ^777 — 20. 7 subtract, multiply, or divide, as indicated: + — —+ 9 9 9 4 21. 5 5 3 4 N + 5 81 Fraction-of-a-Group Problems 22. 8 10 24. 10 1 4 2 - + 1 1 8 4 2> 8 476 + 380 + 767 + 1289 $100.00 25. - 27. 3 23. +79+8 26. 806 x 790 28. 5 97.49 2718 117 18 ( 29. 100,000 - 100 LESSON 25 30. \ Fraction-of-a-Group Problems One way to describe part of a group is by using a fraction. Consider this statement: Two thirds of the students wore green on St. Patrick's Day. We can draw a diagram of rectangle to represent all this statement. We will use a the students in the class. Next will divide the rectangle into 3 equal parts. we Then we describe the parts. Students ^ wore green 1 did not { d wear green we know how many students figure out how many students are If are in the class, we in each part. Two thirds of the 27 students in the class wore green on St. Patrick's Day. can 118 Math 87 There are 27 students in all. If we divide 27 students into 3 equal parts, there will be 9 students in each part. We write these numbers on our diagram. 27 students 9 students wore green < 9 students 1 did not _». J wear green 9 students wore green, we add 2 of the parts and find that 18 students wore green. Since ^ of the students did not wear green, we find that 9 students did not wear green. Example 1 Since | Draw a diagram of this statement. of the students Then answer the questions that follow. Two (a) (b) Solution fifths of the 30 students in the class are boys. How many boys are in the class? How many girls are in the class? We draw a rectangle to represent all 30 students. Since the statement uses fifths to describe a part of the class, we divide the 30 students into 5 equal parts with 6 in each part. Then we describe the parts. 30 students 6 students are boys < 6 students 6 students ^ are girls 6 students << 6 students Now we (a) Two answer the questions. of the five parts are boys. Since there are 6 students in each part, 12 students are boys. (b) Since 2 of the 5 parts are boys, 3 of the 5 parts must be girls. Thus, 18 students are girls. Another way to find the answer to (b) after finding the Fraction-of-a-Group Problems answer Practice to (a) is to subtract. Since 12 of the 119 30 students are boys, the rest of the students (30 - 12 = Draw a diagram Then answer the questions of this statement. 18) are girls. that follow. Three fourths of the 60 pumpkins were Problem set 25 a. How many pumpkins were ripe? b. How many pumpkins were not ripe? In Problems ripe. 1- 4 identify the type of problem. Then find the , answer. 1. In Room 7 there are 28 students. In 2. 9 there are 30 Room 11 there are 23 students. Altogether, students are in all three rooms? students. In how many Room the students in Problem 1 were equally divided If all among 3 rooms, how many students would be in each room? 3. One hundred 4. Two thousand, seven hundred is how much less than ten twenty-six thousand scurried through the colony before the edentate attacked. Afterward only seventy-nine thousand remained. How many were lost when the edentate attacked? thousand, three hundred thirteen? 5. Draw a diagram of this statement. Then answer the questions that follow. Five ninths of the 36 spectators were happy with the outcome. (a) How many spectators were happy with the outcome? (b) How many outcome? spectators were not happy with the 120 Math 87 6. Use less 7. (a) and symbols to write "Two and three than two and three fourths." digits fifths is What fraction of the rectangle shaded? is I (b) What fraction of the rectangle not shaded? is 8. The top many 9. about meters above the floor? —— (b) 2 15| (b) 6; Draw and shade circles to 12 show that 3 2— equals 8 12. (a) Draw rectangle ABCD so 1 (b) 13. that 8 AB is 2 cm and BC is cm. What is the perimeter of rectangle ABCD? 32 (b) 840 For each fraction, write an equivalent fraction that has a denominator of 60. (a) — (0 (b) 6 15. 19 — Write the prime factorization of each number, (a) 14. how Write each number as a fraction. (a) 11. is Simplify: (a) 10. of the chalkboard in the classroom Arrange these numbers in order from -' 3' 1 - -2 '2' i 12 least to greatest: Adding and Subtracting Mixed Numbers 1 21 Solve: 16. 475 + a = 754 17. 900 - c = 90 18. 4 7 19. 131 = x + 13 20. 131 = y- 13 8 21 21. 130 = lOn 4 6 Add, 22. subtract, multiply, or divide, as indicated: 5 6 + 5 + 6 5 23. 6 3 10 15 8 • 2 3 N 24. H - A 12 12 25. $47.58 + $63.75 + $114 + $9.47 + 26. 37,104 - 9,865 29. 9-9-9 7 + 6 81 27. $12.50 x 48 30. 8(2 $27 00 - 28. 15 1,000,000 - 100 Adding and Subtracting Mixed Numbers LESSON 26 we have practiced adding and subtracting fractions. In this lesson we will practice adding and subtracting Since Lesson 8 mixed numbers. Example 1 There are 2| pies on the top shelf and l| pies on the bottom shelf. Altogether, how many pies are there on both shelves? 4 4 ' 122 Math 87 Solution There are some pies on the top shelf and some more pies on the bottom shelf. To find how many pies there are in all, we add 2 1 pies and l| pies. Before doing the arithmetic we will study the picture. 7 We > 4i see that there are 3 whole pies and 7 pieces, which 3| pies. However, we can put 5 of the pieces together to fill another pie pan, making 4 whole pies and 2 pieces. Thus, is we see that there are 4| pies on the shelves. When adding mixed numbers, we add the fractions and add the whole numbers. Then we simplify the result. We will arrange the numbers vertically to add them. ~3 pies pies 5 pies We notice that | and add is an improper fraction. We convert to , n this to 3. 7 35 = 5 2 = A5 There are 4| pies on the shelves. Example 2 9 1 There are 3- pies on the shelf. If the baker takes away 1- Adding and Subtracting Mixed Numbers pies, Solution how many To answer before we pies will be this question, subtract we we 1 23 on the shelf? subtract if from 3|. However, will look at the picture again. 2% / 1 1 5 5 7iJ In order for the baker to to slice one of the whole pies into into fifths, there are 2 can remove 1- pies, we remove 1- pies, fifths. By 3* 2 + first rename 3| 1 2 2 X- 5 5 Simplify: 3- + We add the Then we left on the shelf. 1-| 8 Solution as 2-, as 5 5 Example 3 cutting one pie whole pies and 6 fifths. Then the baker which leaves l| pies still on the shelf. To perform the subtraction, we show. Then we can subtract. There are 1~ pies will be necessary it 8 fractions. Then we add simplify. 8 + 18 4* = 4 8 the whole numbers. 1 24 Math 87 Example 4 Solution 6-1- Simplify: First we arrange the problem vertically. 6 4 We need a fraction on top. Thus we rewrite 6 as 4 5—. 4 4 4 Practice Add or subtract as indicated. Then simplify the answer if possible. a. 3 i + l| b. 4 4 6 d. e. 6 Problem set 26 1 1 5+ 2- 6 7- c. 6 2i 3 2 2 3 3 3- + 6- f. 4 6-1 - 15 5 In Problems 1- 4 identify the type of problem. , Then find the answer. 1. Willie shot eighteen rolls of film for the school annual. were thirty-six exposures exposures were there in all? If 2. there Fifty million is fifty 3. how much in each roll, greater than how many two hundred thousand? Use words to write your answer. There were 259 people who attended on opening night. On the second night 269 attended, and 307 attended on the third night. How many people attended on the first three nights? — — Adding and Subtracting Mixed Numbers 4. The 16-pound turkey for each pound? 5. Draw cost $14.24. What was a diagram of this statement. 1 25 the price Then answer the questions that follow. Three eighths of the 56 restaurants in town were closed on Monday. (a) How many of the restaurants in town were closed on Monday? (b) How many of the restaurants in town were open on Monday? 6. What one number can be put in both boxes to make both sides of the equation equal? Qx = 50 + 50 was made, the spheroid sailed four thousand, one hundred forty inches. How many yards did the spheroid sail after contact was made? 7. After contact 8. What 9. What number is halfway between 2000 and 3000? 10. name for numbers that are less than zero? | • f Of W §Oi Simplify each fraction or mixed number. r , (a) 12. the Replace each circle with the proper comparison symbol, (a) 11. is 3 18 10 n , lb J 220 — 12 Write each number as an improper fraction. (a) l\ 4 (b) 3| 5 126 Math 87 13. Complete each equivalent 3 - = (a) 4 14. 15. — 40 ? fraction. 2 ? , (b) —? = , (c) 40 8 — 40 15 Write the prime factorization of each number. (a) 42 (b) 600 Refer to this figure to answer the following questions. (a) What type of angle is ZADB? (b) What type of angle is ZBDC? (c) What type of angle is ZADC? (d) Which ray is perpendicular toDB? Solve: 16. 7w = 105 17. m - 34 = 25 18. 4 7 19. 36 + x = 115 8 2 Add, 20. subtract, multiply, or divide, as indicated: si + i^ 3 21. 4 4— + 1— 8 3 6 9 8 5 22. 23. 4 4 i- 2 5-3^ 5 8 3 7 4 24. 7 3 3 1 3 N 25. 3524 + 4617 + 2819 + 568 + 70 26. $50.00 - 41.74 $0.89 27. x 76 + 3 77 28. 26,880 42 . . . Reciprocals 7 29. h 12 7 fl 1 4 I V 30. ) 3 1 4 I 2 LESSON 27 1 f 8 1 J Reciprocals 27 If we exchange the position of the terms in a fraction, we will write the reciprocal of the fraction. The The The The reciprocal of F — reciprocal of F — reciprocal of r —1 — is 4 3 — is 4 3 4 — which is , 4 reciprocal of 4 — is To find the reciprocal of a mixed number, mixed number as an improper fraction. 3 1 = 16 5 we first write the improper fraction 5 Now we can write the reciprocal of 3-5 of is 4. 1 , which is the reciprocal 16 . 5 The 16 — — 16 5 reciprocal of F is 5 We note a very important property of reciprocals. The product of any fraction and its reciprocal 16 16 5 3 4 4 3 is the 80 80 1 12 12 1 1-4 4 4 4 1 number 1. .. 1 28 . Math 87 Example 1 Find the reciprocal of each number. | 5 (a) Solution (a) The 3 (b) l\ (c) 3 —3 reciprocal of 5 — is O 3 — (b) The reciprocal of 3 (c) First we write the mixed number as an improper fraction. is 1 7 2- = - 3 The Example 2 Solution reciprocal of A fraction times is its —3 N= • which 3 4 12 4 3 ~ 12 —7 which , is 3 — 1 reciprocal equals the reciprocal of |, —3 , b. —8 5 n 1. Thus, the missing is -. 5 c. d. 7 2± e. 34 3 ind each missing number. f. 8 27 the reciprocal of Write the reciprocal of each number. a. Problem set is Find the missing number: number Practice 1 2- 3 A = 1 1 g. 5 fl • = 1 h. 3- 6 • C = 1 3 In Problems 1- 4 identify the type of problem. , Then find the answer. 1. All the lines for concert tickets were long. There were 112 people in the the third line, people were in first line, and 105 all 117 in the second in the fourth line. four lines? line, 98 in How many Reciprocals 2. 1 29 the people in Problem 1 arranged themselves in 4 If all how many would be in each line? equal lines, 3. The Viking Bjarni Herjolfsson is believed to have seen North America in about 986. This was how many years before Columbus sighted North America in 1492? 4. Nathan left a total of $21.33. when he 5. Draw and three tapes that cost much money did Nathan have the store with $5.42 How entered the store? a diagram of this statement. Then answer the questions that follow. Five eighths of the 160 seats were empty during the matinee. (a) How many seats were empty during the matinee? (b) How many seats were occupied during the matinee? 6. How many thousands equal one million? 7. A 20. score is Three score and 10 years is how many years? 8. (a) What fraction of the square is shaded? (b) What fraction of the square is not shadepl? 9. Simplify each fraction or mixed number. 75 25 (b) (a) 4 (c) 12 6 10. Write each nunfber as a fraction. (a) 1 ; 34 2 (b) 1 (c) 6 1 30 Math 87 11. For each fraction, write an equivalent fraction that has a denominator of 45. (b) (a) 12. Write the prime factorization of 540. 13. Write the reciprocal of each number. 5 4 (b) (a) (c) 2- 8 14. Find each missing number. (a) 15. ^ n = - l (b) k • 1- = 1 2 This rectangle is twice as long as it is wide. What is the perimeter of the rectangle in millimeters? 15 mm Solve: 16. 370 + d -- = 530 17. 500 - x = 125 52 18. 12 19. 8m = 144 4 5 Add, 20. 22. 24. subtract, multiply; or divide, as indicated: 3- + 38 8 7 1 8 8 6- - 1- 21. 28 5* + li 6 6 23. 14 5 7 + N 126 - 16 6- - 45 5 25. $96.74 + $59.87 + $115 + $4.68 26. 15,470 14 27. 43,050 - 8,313 $6.59 28. x 78 131 Reducing Fractions, Part 2 I 29. ^3 2 - + 5 1 LESSON 28 2^ 2 5 1 + 30. v 5 ; Reducing Fractions, Part 2 Reducing fractions We have been practicing reducing fractions by dividing the numerator and the denominator by a common factor. In this we will practice another method of reducing fractions. This method uses prime factorization. If we write the prime factorization of the numerator and of the denominator, we can see how to reduce a fraction easily. lesson Example 1 Solution Use prime We 420 1050 factorization to reduce rewrite the numerator and the denominator as products of prime numbers. 420 1050 ~ Next we equals 2 • 2 • 3 • 5 • 7 2 • 3 • 5 • 5 • 7 look for pairs of factors that equal 1 if the numerator and denominator 1. -, |, -, and Thus the -. Below we have indicated each fraction equals 1 • 1 • 1 • 1 fraction are equal. In this fraction there are four pairs of factors that equal 7 A 1. 5 are of these pairs. —2 which , They is 2 — 5 may be reduced before they are multibef ore plied. Reducing before multiplying is also known as canceling, Reducing multiplying The terms of fractions Consider this multiplication: 3 2 _6_ _6_ 8 3 24 24 reduces to , 1 32 Math 87 can be reduced. The product, ~ can be reduced. We can avoid reducing after we multiply by reducing before we multiply. To reduce, any numerator may be paired with any denominator. Below we have paired the 3 with 3 and the 2 with 8. We nor - see that neither - Then we reduce these pairs: - reduces to -, and - reduces to - as we show below. Then we multiply the reduced terms. , 111 l l 4 ' t 4 Example 2 Simplify: — • 16 Solution Solution 3 Simplify: We - pairs. • — • we and 2 with 16 and Then we multiply the reduced terms. Before multiplying, reduce these Example 3 1 - pair 9 with 3 3 1 y6 $ 8 ' mentally pair 8 with 4, 3 with 9, and 5 with 10 and reduce. 1? 2 3 We can still 1 1 2 1 reduce by pairing 2 with l I l l A 3 1 ys £ t 1 1 3 2. Then we multiply. Reducing Fractions, Part 2 Practice Use prime a. 1 33 factorization to reduce each fraction. 48 144 , b. 90 324 Reduce before multiplying. c. —5 8 Problem set 28 6 _5_ 3 7 16 identify the type of problem. Then 10 In Problems 1 and 2, A 10 8 d. _8^ 5_ 15 12 find the answer. 1. From Hartford to Los Angeles hundred ninety-five miles. is From from Hartford 2. Use to is eight Hartford to Portland three thousand, twenty-six miles. Hartford to Portland two thousand, is The distance from how much greater than the distance Los Angeles? Hal ordered 15 boxes of microprocessors. contained two dozen microprocessors, microprocessors did Hal order? this information to answer questions 3 and If each box how many 4. Jason had completed reading 36 pages of a 310-page book. He read 47 more pages that morning and 139 more pages that afternoon. how many 3. Altogether, 4. How many more pages had Jason read? pages did Jason have to read to finish the book? 5. Nancy descended the 30 steps that led to the floor of the cellar. One third of the way down she paused. How many more steps were there to the cellar floor? 6. What number 100 is halfway between 100 and 400? 200 300 400 . 134 Math 87 7. Write the] reciprocal of each number 3 (a) 2- (b) 3 8. Simplify: 15 11 fal 540 600 fhl 12 9. Write each number as a fraction. 9- (a) 8 (b) 2 10. Complete each equivalent 3 fraction i = 30 ? (b) (a) 30 10 as a product of prime factors. 11 Write 8 1 12. Draw two parallel lines that are intersected by a third line that is perpendicular to the parallel lines. 13. 14. The perimeter of a square is one yard. How many inches each side of the square? long is (a) Estimate the length of this segment in centimeters. (b) Use a centimeter scale to find the length segment to the nearest centimeter. Solve: 15. 514 = x + 50 17. 9t Add, 18. w - 75 = 57 = 144 subtract, multiply, or divide, as indicated: 5 3 5- + 3- 8 20. 16. 8 3- + 24 19. 4 2 2 7- + 63 21. 7 4— 12 3 - 1 1— 12 of this Dividing Fractions 22. 6- - 4 8 5.2 6 3 13,513 + 15 28. (13 30. Solve mentally: • 110) + 7590 8 3 18 4 2 25. 26. 9 27. 73,604 - 64,416 29. 13(110 + 7590) 4 + 7 + 6 + 3 + 5 + 8 + 1^ + 7 + id + LESSON 29 35 7 5-1 - 1- 23. 3 24. 1 N= 78 Dividing Fractions We remember when we multiply fractions we multiply the numerators to form the new numerator. We multiply the denominators to form the new denominator. that 4 2 8 3 5 15 Sometimes when we divide fractions, we can divide the numerator by the second numerator and divide the denominator by the second denominator. 8 ^ 2 = 4 9 3 3 first first (8-5-2 = 4) (9+3 = 3) Often the numerator and denominator of a fraction are not divisible by which by the numerator and denominator of the it is fraction divided. 8 ^ 3 = 9*4 ? ? Thus we need to learn another method for dividing fractions. The method we will learn uses reciprocals to find the answer. -< 1 36 Math 87 To divide two fractions, we may multiply the first fraction by the reciprocal of the second fraction. In a later lesson we will explain the reason that this procedure works. 8 3 9 4 8 4 9 3 division multiply by reciprocal 32 27 result 5 simplified 1 27 Example 1 (a) Simplify minators. (b) Simplify + g 3 + g the divisor, (a) o n o Solution by dividing the numerators and deno- 3 by multiplying 933 - - - = - = 13 8 2 9 3 division 8 3 9 2 multiply by reciprocal 24 18 = 1 Example 2 Solution by the reciprocal ~. (b) We g see that both Simplify: - To divide by -*- | result 1 simplified methods produce the same - , we will multiply "J r~J by ' 2 4 8 3 3 9 3 result. of — ? Dividing Fractions Example 3 To divide before 10 f we by ^ we will multiply f , 3 _^ _9_ ™ We will simplify . b >d 3 i 2 How many -'s are in 5 " 6 ' 10 4 Solution by multiply. 1 Example 4 37 - + Simplify: 4 Solution 1 3 -? This question can be restated as, "What is the answer if | is 2 divided by f?" 3 - 3 2 4 3 1 3 4 2 division multiply by reciprocal 9 result 8 1 = 1— Practice Simplify: a. —3 5 d. -h 2 — —7 u b. 1 -; How manyy —2 . 's are in c. 4 8 3 5 Problem set simplified 5 — 6 — 2 h 3 —3 4 1. Three hundred twenty-four students were treated to ice cream for receiving A's in citizenship. If each box of ice cream contained a half dozen ice cream bars, how many boxes of ice cream were needed? 2. Martin's pockets bulged with the coins he had been paid for redeeming bottles and cans. If he had 11 quarters, 14 29 dimes, 15 nickels, and 17 pennies, did he have in all? how much money ? 138 . Math 87 Use this information to answer questions 3- The family picnic was 5 a success, as 56 relatives who attended played in the big game. However, the two teams were not equal since one team had only 7 players. attended. Half of those 3. How many 4. If 5. If relatives played the game? one team had 7 players, other team have? how many players did the the teams were rearranged so that the number of on each team was equal, how many players would be on each team? players 6. Draw a diagram of this statement. Then answer questions that follow. Jason has read 7. How many pages has Jason read? (b) How many pages has Jason not read? How manyy —3 are in 's Which is 2 the best estimate of how „ , is much shaded? 2 , (b) (a) , 2 (c) Simplify each number. (a) 10. —7 8 of this rectangle 9. of the 310 pages in the book. (a) 4 8. — 6 20 12 (c) 84 210 (0 2§ 54 (b) 8 Write the reciprocal of each number. 9 (b) (a) 10 8 the — Dividing Fractions 11. 20. t 4 (a) t 5 (b) (c) 10 Write the prime factorization of each number, 35 (a) 13. 640 (b) Write each number as an improper fraction. 5^ (a) 6 (b) (c) Points A number 3| o Z 14. and B represent what mixed numbers on this line? 4 1 5 I 1 1 I 6 I 1 I I I | I | A 15. Draw (a) AB. Then draw ray AB. What kind (b) B line to line of angle is BC perpendicular ZABC? Solve: 16. 126 + y = 310 18. 175 = 7m Add, 17. 4§ + 5f 21. 3 - 1-7- 20. 6- + 1- 22. 3 4 2 - 5 „_ —3 7 0/1 24. 5 — 6 -8 5 9 -s- -6 5 1 -^ . 4 12 25. x - 53 35 = subtract, multiply, or divide, as indicated: 19. 23. 39 For each fraction, write an equivalent fraction that has a denominator of 12. 1 9R 2b. 4386 — 9 5 . A 15 140 Math 87 $6.98 27. x 30. $120.00 - 108.89 28. 74 29. $110.16 27 Solve mentally: 4 + 13 + 2 + 8 + 12 + 15 + 4 + l + iV=65 Multiplying and Dividing LESSON 30 Mixed Numbers To multiply or divide mixed numbers, we first rewrite the mixed numbers as improper fractions. Then we multiply or divide as indicated. Example 1 Solution Simplify: 3- x We first 1- rewrite 3| as y and l| as |. Then we multiply and simplify. 11 I X 11 $ 2 " 2 l 2 Example 2 Solution Simplify: 3-^-2 We first rewrite 3| as y and rewrite 2 as y by | and |. Then we multiply simplify. 11 2 3 1 —x— divide 11 1 11 3 2 6 multiply by reciprocal 6 . Practice Simplify: a. 6^ x 3 b. 2- x 3- 5 1- d. -r 3 e. 3- 30 In c. 3- x -s- 2- 3 3 4 2 3 3 Problem set 141 and Dividing Mixed Numbers Multiplying f. 5 + - 2 3 Problems 1- 4 identify the type of problem. Then find the , answer. 1. hour of the monsoon, 23 millimeters of precipitation had fallen. After the second hour, a total of After the first 61 millimeters of precipitation had fallen. How many millimeters of precipitation fell during the second hour? 2. Each enlargement cost 85c and Willie needed 26 enlargements. What was the total cost of the enlargements Willie needed? 3. The Byzantine Empire many 4. lasted from 395 to 1453. How years did the Byzantine Empire last? Dolores went to the theater with $20 and came home with $11.25. How much money did Dolores spend at the theater? 5. 6. A gross is a dozen dozen. A gross of pencils many pencils? Draw a diagram of this statement. is Then answer how the questions that follow. Two 7. fifths of the 60 marbles in the bag were blue. (a) How many (b) How many of the marbles in the bag were not blue? Draw and shade of the marbles in the bag circles to show that were blue? — 2 12 2— equals 5 5 1 42 Math 87 8. (a) What is (b) 9. fraction of this square not shaded? Simplify: (a) 10. shaded? What is fraction of this square 9^ (b) Write the reciprocal of each number. 5 (b) (a) 11. 210 252 Complete each equivalent 7 (a) - = 8 (c) (c) 7 fraction. — 48 = — — 16 48 r ? (b) ? 28 48 J_ 12 prime numbers between 50 and 12. List the 13. Draw segment AB 60. Then draw BC 1 cm long perpendicular to AB. Complete triangle ABC by drawing 2 cm long. AC. 14. Arrange these numbers in order from i, *> 6 least to greatest: >"> 3 Solve: 15. 250 = 700 - x 17. 12w = 276 Add, 18. 16. 53 - y = 35 subtract, multiply, or divide, as indicated: 8-7 19. 9 5 6— 12 + 11 8— 12 Multiples 20. 9± 9 Least • Common 143 Multiple 5 4i 9 21. 10 8 6 V^ 22. 2 24. 3i 3 3 * 4 2 5 3 5 • 14 - 26. 25. 3 1- - 2- 23. 3 27# 16,524 36 29. 80(64)(25) 2 28. $100 - 30. Add (78tf x 48) mentally: 14+4+11+12+5+6+8+42 LESSON Multiples • Least Common Multiple 31 Multiples The multiples of a number are the numbers that are produced by multiplying the number by 1, by 2, by 3, by 4, and so on. Thus the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, The multiples . of 6 are 6, 12, 18, 24, 30, 36, If . . 42, 48, 54, . . . we inspect these two lists, we see that some of the numbers are the same. A number that appears in both of in both lists these common multiple of 4 and 6. Here we some of the common multiples of 4 and 6. lists is circled have a @, 28, 32, @, 18, @, 30, @, 42, 48, 54, Multiples of 4: 4, 8,(12), 16, 20, Multiples of 6: 6,(12), . . . . . . 144 Math 87 We see that If and 36 12, 24, are common multiples of 4 and 6. we continued both lists, we would find many more common multiples. Least common multiple Example 1 Solution Of particular interest is the least (smallest) of the common The least common multiple of 4 and 6 is 12. It is the smallest number that is a multiple of both 4 and 6. The term "least common multiple" is often abbreviated LCM. multiples. Find the least common multiple of 6 and We will list some multiples of 6 8. and of 8 and circle common multiples. The Example 2 Solution Multiples of 6: 6, 12, 18,(24), 30, 36, 42,(48), ... Multiples of 8: 8, 16, common least Find the LCM of 4 @, 32,40,(48), 56, 64, multiple of 6 and 8 and . is 24. 8. 4: 4,®, 12,(16), 20, @), 28, Multiples of 8:®, (16), The Solution . The initials LCM stand for least common multiple. We list some multiples of 4 and 8 and circle the common multiples. Multiples of Example 3 . LCM Find the We The of 4 LCM circle and of 8 . . @, 32,40,48, 56, . is 8. 3, 4, and 6. only the numbers that appear in all three Multiples of 3 3,6,9,(12), 15, 18, 21,(24), . . Multiples of 4 4,8,(12), 16, 20,(24), 28, 32, . Multiples of 6 6,(12), 18,(2?), 30, LCM of 3, 4, and 6 is lists. 36,42,48, 12. not necessary to list the multiples each time. Often the search for the least common multiple can be conducted It is mentally. Multiples Practice Find the numbers. a. Problem set 31 8 least common and 10 b. • Least Common Multiple 1 45 multiple of each pair or group of 9 and 12 4, 6, c. In Problems 1- 4 identify the type of problem. , and 10 Then find the answer. 1. There were three towns in the valley The population of Brenton was 11,460. The population of Elton was 9420. The population of Jennings was 8916. What was the total population of the three towns in the valley? 2. Norman 3. If is 6 feet tall. How many inches the cost of one dozen eggs was tall is $0.96, Norman? what was the cost per egg? 4. One billion is how much greater than ten million, nine hundred thousand? Use words 5. Draw to write your answer. a diagram of this statement. Then answer questions that follow. Three eighths of the 712 students bought their lunch. \ 6. (a) How many students bought their lunch? (b) How many students did not buy their lunch? The perimeter is 30 inches. of this rectangle What is the length 6 in. of the rectangle? 7. (a) List the first six multiples of 6. (b) List the first six multiples of 10. (c) What is the least common multiple of 6 and 10? the 1 46 Math 87 8. What number is halfway between 3000 and 4000? H 1 2000 9. h 1 4000 3000 5000 S 4 2 [b] 36 180 Write the reciprocal of each number. (a) 11. 1 Simplify: (a) 10. 1 1 5| (b) «! 12 For each fraction, write an equivalent fraction that has a denominator of (a) 36. _5_ (0 (b) 12 12. Write the prime factorization of 384. 13. Write each number as an improper fraction. (a) 5- (b) 12 (c) 7 ~ 9 12 6 14. Figures ABCF and FCDE are squares. (a) What kind of angle is ZACD? (b) Name two segments parallel to FC. 15. Refer to the figure in Problem 14. If AB the perimeter of rectangle ABDFP. is 3 cm, what Solve: 16. lOy = 360 18. 56 - n = 14 17. p + 16 = 144 is Two-Step Word Problems Add, 19. 4- + 6- si- 20. 3 3 23. 4 6 4 9 25. S16.25 9 $150.00 28. - 8 Use mental addition 5 1 15 8 x 30. 8 26. 6 2 5i + 3 24. 5 8 si'. 22. 5 3 li 8 10 - 1- 27. v9 29. V S12.00 J.U 97.75 to find N: 4 + 12 + 8 + ll + 13 + 5 + 4 + 31 + 6 + iV=105 Two-Step Word Problems LESSON 32 47 subtract, multiply, or divide, as indicated: 3 21. 1 We have considered five types of one-step word problems thus far. Their thought patterns are: 1. Some and some more 2. Some went away 3. Larger-smaller- difference 4. Equal groups 5. Part-partwhole Word problems often require more than one step to solve. In this lesson we will begin practicing problems that require two steps to solve. Two-step problems involve a combination of the one-step problems we have practiced. 148 Math 87 Example Julie went to the store with $20. food for 67 cents per can, how she bought 8 cans of dog much money did she have If left? Solution This is a two-step problem. First spent. This is we find out how much Julie an equal groups problem. Number Number in group $0.67 each can of groups x Total 8 cans $5.36 Now we can find out how much money Julie has left. This is some went away problem. a $20.00 - 5.36 $14.64 After spending $5.36 of her $20 on dog food, Julie has $14.64 left. Practice Work each problem as a two-step problem. a. Jody went to the store with $20 and returned home with $5.36. If all she bought was 3 bags of dog food, how much did she pay for each bag? b. Sam bought a pair of headphones for $15.89 and for $7.89 each. Altogether, how much money 3 tapes did he spend? c. Problem set Three eighths of the 32 students were boys were in the class? girls. How many 1. Quentin purchased 5 arrowheads for $1.75 each. He paid for them with a $10 bill. How much did he receive in change? 2. 379 students attended the assembly. If 198 girls were in attendance, then how many fewer boys than girls were in attendance? 32 In all, Two-Step Word Problems 3. When tall. on 4. How many his toes, inches tall was he was 49 5 feet 11 inches Gilbert when he stood his toes? Cynthia picked 176 apricots, but there were apricots left tree before 5. on Gilbert stood 1 Draw on the tree. still 394 How many apricots were on the Cynthia picked them? a diagram of this statement. Then answer the questions that follow. Five ninths of the 270 carrot seeds sprouted. 6. (a) How many carrot seeds sprouted? (b) How many carrot seeds did not sprout? the perimeter of a regular hexagon If many 7. 8. inches long List the first six multiples of 4. (b) List the first six multiples of 10. (c) What On this is number LCM of 4 3380 line, closest to is which multiple of 100? (b) which multiple of 1000? 1 1 1 how and 10? (a) H 1 1 1 1 h 1 4000 3380 3000 Simplify: (a) 10. the yard, 1 each side? (a) -I 9. is is ^ (b) 5^- Complete each equivalent W 1 ' 1 = 5 ± 40 (b) 2^- (c) 2 = 12 10 40 fraction. 5=28 (c) 40 1 50 Math 87 11. Write the prime factorization of each number, 62 (a) 12. (b) 312 Write each number as an improper fraction. 1 (b) 7- (a) 15 13. Draw rectangle ABCD so that AB is What is the perimeter of ABCD? 14. In triangle RST, 2 cm and BC is 3 i 15. (a) ZS is what kind of angle? (b) ZT is what kind of angle? (c) Which segment is perpendicular to RS? What mixed number is halfway between 7 and 10? Solve: 16. 84 = 7x 17. 210 = 21 + p 18. t Add, 19. - 56 = 14 subtract, multiply, or divide, as indicated: 6« - 1° 7 21. 8 20. - 22. 8 3 • 14 $375 25. x 64 1- 6 *- 14 2 1 23. 7- + 16 7 ft • 24. 9 26. 3-5 1,000,000 400 cm. Average, Part 27. 30. - + 1 6 2 A 14,507 28. 29. - 9,618 Use mental addition 151 1 36,444 12 to find N: 4 + 13 + 3 + 12 + 22 + 5 + 7 + 14 + 12 + iV = 99 LESSON 33 Average, Part 1 Here ^ we show z=^3 5 stacks of coins: =z mm 16 m mttttlM fflBffl 11IIIIIIIIIIIIII &.".' ^ There are 15 coins in all. If we made all the stacks the same size, there would be 3 coins in each stack. - ~ ~3p ^^ . We say that the average number of coins in each stack Consider the following problem. is 3. There are 4 squads in the physical education class. Squad A has 7 players, squad B has 9 players, squad C has 6 players, and squad D has 10 players. What is the average number of players in a squad? The average number of players in a squad is the number of players that would be on each squad if each squad had the same number of players. To find the average of a group of numbers, we begin by finding the sum of the numbers. 7 players 9 players 6 players + 10 players 32 players Then we divide the sum of the numbers by the number of 1 52 Math 87 numbers. There are 4 squads, so sum we numbers number of numbers A — Avprspp of divide by = 4. 32 players 4 squads = 8 players per squad Finding an average numbers is a two-step process. to find the total. Then we divide We first add the the total to make equal groups. Example 1 When the people were seated, there were 3 in the first row, second row, and 20 in the third row. What was the average number of people in each of the first 3 rows? 7 in the Solution The average number of people in the first 3 rows is the number of people that would be in each row if the number in each row were equal. First we add to find the total number of P e °P le ' 3 7 people people + 20 people 30 people Then we divide by 3 to separate the total into 3 equal groups. —— rows 30 people 3 „„ = 10 people f f per f row The average was 10 people , in each of the first 3 rows. To find the average of several numbers, we add the numbers, and then divide the total by the number of numbers. Example 2 Solution What is the average of 26, 42, 57, 49, and 16? There are 5 numbers. To find the average of these numbers, we first find the total. Then we divide the total bv 5. 26 42 57 49 + 16 190 The average of the 5 numbers 38 5)l90 15 40 40 is 38. Average, Part Practice Room there were 28 students, in Room 1 1 53 were 29 students, in Room 3 there were 30 students, and in Room 4 there were 25 students. What was the average number of students in the 4 rooms? a. In b. What c. What is is 1 2 there the average of 46, 37, 34, 31, 29, and 24? the average of 40 and 70? What number is halfway between 40 and 70? Problem set 1. 33 The on the weighed 242 pounds, 236 pounds, 248 pounds, 268 pounds, and 226 pounds. What was the average weight of the players on the front 5 players front line line? How many 2. Matt ran a mile in 5 minutes 14 seconds. seconds did it take Matt to run a mile? 3. Ginger bought a pair of pants for $24.95 and 3 blouses for $15.99 each. Altogether, how much did she spend? 4. The 5. Draw Christopher Columbus was 41 years old when he reached the Americas in 1492. In what year was he born? Italian navigator a diagram of this statement. Then answer the questions that follow. Bill led for three fourths of the 6. (a) Bill led the race for (b) Bill as it is how many did not lead the race for This rectangle is wide. 5000-meter race. meters? how many twice as long What is the perimeter of this rectangle? 8 7. (a) List the first six multiples of 3. (b) List the first six multiples of 4. (c) What is the LCM of 3 and 4? cm meters? 1 54 Math 87 8. On this number 283 line, is (a) which multiple of 10? (b) which multiple of 100? -\ 1 closest to 1 1 h+- 1 200 9. 283 Simplify each fraction or mixed number. (a) -^- 1™ (b) 240 10. — 10 4- (b) 5 For each fraction, write an equivalent fraction that has a denominator of 24. (a) 12. 8 Write the reciprocal of each number. (a) 11. 300 — — (b) Write the prime factorization of each number. (a) 27 13. What 14. (a) is 2800 (b) the average of 45, 36, 42, 29, 16, and 24? Draw square ABCD so that each side is about 1 inch long. (b) Draw segments AC and BD. Label the point at which they intersect point E. (c) 15. Shade triangle CDE. Arrange these numbers in order from -1, — — 10 10 least to greatest: 11 1 , 1, , Solve: 16. 12y = 360 17. 123 = m + 64 18. 45 = 54 - w Rounding Whole Numbers Add, 19. 4 A_ a JL 20. 12 12 - 8-1 22. 7 s8 1 2- -f- 4 25. 28. 55 + 3 3 8 2 6- 8 23. 1 subtract, multiply, or divide, as indicated: 12 21. Estimating Answers • 5 3 7-1 24. 8 * 2 8000 x 600 26. $47.63 78.49 10,000 80 27. 1 fi V 2 1 2 $4.56 30. -36,295 + 35.24 3 4 37,484 29. 2 23 X 9 ,1 Rounding Whole Numbers • Estimating Answers LESSON 34 Rounding The first sentence below uses an exact number to state the whole size of a crowd. The second sentence uses a round number. numbers There were 3947 fans at the game. There were about 4000 fans Round numbers are often at the game. used instead of exact numbers work with. One way to round because they are easier to numbers is to consider where the number number Example 1 line. Use a number line to hundred. (a) round 283 to the nearest (b) round 283 to the nearest ten. is located on the 1 56 Math 87 Solution (a) We draw a number line and mark the location of hundreds as well as the estimated location of 283. 283 200 100 We 300 400 between 200 and 300. Since 283 is closer to 300 than it is to 200, we say that 283 rounded to the nearest hundred is 300. see that 283 We draw (b) a is number and mark the location of the line tens from 200 to 300 as well as the location of 283. 283 H 200 We h» 1 1 1 1 1 1 1 210 220 230 240 250 260 270 280 290 300 between 280 and 290. Since 283 is 280 than it is to 290, we say that 283 rounded see that 283 closer to h 1 is to the nearest ten is 280. Sometimes we are asked round a number to a certain place value. We can use a circle and an arrow to help us do this. We will circle the digit in the place to which we are rounding, and we will draw an arrow above the next place. Then we 1. to will follow these rules. If the arrow-marked digit is 5 crease the circled digit by marked digit 2. digit is less Solution 5, 1. If we in- the arrow- we leave the circled unchanged. We change the arrow-marked digit and all digits to the right of the Example 2 than or more, arrow-marked (a) Round 283 to the nearest (b) Round 283 to the nearest ten. (a) We circle the 2 since we draw an arrow it is digit to zero. hundred, in the hundreds' place. over the digit to Then its right. I I ©8 3 Rounding Whole Numbers Since the arrow-marked digit the circled 2 to digit and 3. Then we • Estimating Answers 157 more, we increase change the arrow-marked 5 or is zero and get all digits to its right to 300 (b) we Since are rounding to the nearest ten, and mark the tens' digit we circle the with an arrow. digit to its right i 2(8)3 Since the arrow-marked digit 8 unchanged. is less Then we change than 5, we leave the the 3 to zero and get 280 Example 3 Round 5280 (a) to the nearest thousand, and (b) to the nearest hundred. \ Solution (a) To the nearest thousand, ©280 rounds to 5000. I (b) Estimating answers To the nearest hundred, 5(2)80 rounds to 5300. Rounding can help us estimate the answer to arithmetic problems. Estimating is a quick and easy way to get close to an exact answer. Sometimes a close answer is "good enough," but even when an exact answer is necessary, estimating can help us decide if our exact answer is reasonable. To estimate, we round Example 4 Solution the numbers first. Mentally estimate: 396 x 312 (a) 6879 + 3145 (a) We round both numbers to the (b) same place before we (c) 6879 + 3145 4160 - 19 — 7000 + 3000 ~~* 10,000 add. (b) We round each number so there is one nonzero digit before we multiply. 396 x 312 —+ — ** x 400 300 120,000 1 58 Math 87 (c) We round each number so there is one nonzero digit before Practice we 4160 ~"•" I9 ~~* divide. 4000 _ 200 20 a. Sketch a number line to round 231 to the nearest hundred. b. Round 1760 c. Round 186,282 to the nearest hundred. to the nearest thousand. Estimate each answer. d. Problem set 7986 - 3074 e. 297 x 31 f. 1. Larry jumped 16 feet 8 inches on his inches did he jump on his first try? 2. If 3. On her first six tests 34 pounds pound? of bananas cost $3.68, 8 85, and 90. What 5860 first try. what is -* 19 How many the cost per Sandra's scores were 75, 70, 80, 80, was her average score on her first six tests? 4. Two hundred nineteen billion, eight hundred million is how much less than one trillion? Use words to write your answer. 5. Draw a diagram of this statement. Then answer questions that follow. Two (a) (b) 6. fifths of How many How many What is the the 80 chips were blue. of the chips were blue? of the chips were not blue? LCM of 4, 6, and 8? the Rounding Whole Numbers 7. What • Estimating Answers 1 59 the perimeter of is this square? 8. Round 366 (a) to the nearest hundred. Round 366 (b) to the nearest ten. 9. 10. Mentally estimate the sum of 6143 and 4952 by rounding each number to the nearest thousand before adding. Simplify each number. 4 (a) 11. (b) 125 500 Write each number as an improper fraction. 10 (a) 12. 60 36 (b) Complete each equivalent 2 3 fraction. n ? 15 , ? (b) (a) 30 13. 25 30 Write the prime factorization of each number, 102 (a) (b) 14. What 15. Draw rectangle ABCD so is 1020 the average of 374, 286, 397, that and 423? AB is 1 cm and BC is 2 cm. Next draw segment AC. Then shade triangle ABC. Solve: 16. 96 = Add, 19. 4m 17. x + 37 = 105 18. 45 = n - 54 subtract, multiply, or divide, as indicated: 15 - 44 20. 5 7 3- + 4- 9 9 160 Math 87 2 1 6 21. I" 5 3 22. I 4 6* + 5 23. 3± 24. 3 25. 28. 3 749 x 86 26. $350.00 - 127.48 29. 8 3 400,000 2 27. 25 5 $7.25 x 2 + 30. 6 u I 5 * 3741 2639 4814 + 68 Common Denominators LESSON V • Adding and Subtracting Fractions 35 with Different Denominators Common When two fractions have the same denominator, we say they denominators have common denominators. 3 6 3 3 8 8 8 4 These two fractions have These two fractions do not common have common denominators. denominators. two fractions do not have common denominators, then one or both fractions can be renamed so both fractions do If denominators. We remember that we can rename a fraction by multiplying the fraction by a fraction equal to 1. Thus we can rename - so that it has a denominator of 8 by multiplying by -. have common 3 2 6 4 2 8 Common Example 1 Solution 2 1 Rename - and — so that they have 161 Denominators common denominators. The denominators are 3 and 4. A common denominator for these two fractions would be any common multiple of 3 and 4. The lowest common denominator would be the least common multiple of 3 and 4, which is 12. We want to rename each denominator fraction so that the is 12. 2 1 12 3 We multiply - by 2 3 Thus 3 and - - 4 and multiply - ~ 12 by 4^ 13 4 4 12 can be written with 3 -. _3_ ~ 12 common denominators as 4 8 3 j and 12 12 Fractions written with common denominators can be compared by simply comparing the numerators. Example 2 common Write these fractions with compare them. 6 Solution The common denominator 6 and 9, which is 18. In place of - W9 for these fractions is the 5 3 15 7 6 3 ~ 18 9 15 we will write — Then we compare denominators and then 14 ~ 18 2 l the fractions. — O w— 18 18 14 — > — 18 18 15 of 2 r 7 In place of t . LCM renamed i compared F we .ii .. will write 14 18 162 Math 87 Adding and SUbtracting fractions Example 3 To add or subtract fractions that do not have common denominators, we first rename one or both fractions so that they do have common denominators. Then we can add or subtract. Add: - + 4 Solution First we 8 common denominators. The least common multiple write the fractions with The denominators are 4 and 8. of 4 and 8 is 8. We rename - so multiplying by -. We that the do not need to denominator rename -. by Then we add is 8 the fractions and simplify. + 3 2 6 renamed — 4 2 8 4 3 =3 8 8 9 added 8 mplifying |. II 8 Example 4 Solution Subtract: First we 5^ 3 6 4 write the fractions with common The LCMof6 and 4 is 12. We multiply 3 3 so that both denominators are by b | y 3 4 the renamed fractions. i ~ denominators. by § and multiply 12. Then we subtract f 5 2 10 renamed — 6 3 2 3 12 6 _9_ renamed — 4 3 12 4 subtracted 12 Example 5 Subtract: 8- - 5^ 3 Solution 6 We first write the fractions with common denominators. The LCM of 3 and 6 is 6. We multiply \ by | so that the denominator Adding and Subtracting Fractions with is 6. Then we subtract and Denominators Different 1 63 simplify. renamed 8— 3 6 -5-1 1 = 5- 6 6 3 = 3 Example K 6 Solution Add: - + - + - 2 4 3 The denominators are 2, 3, and 4. The LCM of 2, 3, and 4 is 12. We rename each fraction so that the denominator is 12. Then we add and simplify. + 1 6 6 2 2 6 12 4 8 renamed - 3 3 4 12 — 3 9 3 A 3 renamed 4 3 12 23 12 Practice subtracted and simplified 2 Write the fractions with compare them. ^— Ô w 10 7 3 a. renamed 2 — 4 = 1 — 12 addec and simplified added common L b. 5 Add c. —3 + —5 1. + 6 —3 The d. 8 7- - 2- 8 „3 c 5 4+ 54 Problem set 35 r~\ 7 5 -O w 6 or subtract. 4 e. denominators and then 8 5 starters f. 6 2 9 6 4 on the basketball team were tall. Their heights were 76 inches, 77 inches, 77 inches, 78 inches, and 82 inches. What was the average height of the 5 starters? — 1 64 Math 87 2. 3. Marie bought 6 pounds of apples for $0.87 per pound and paid for them with a $10 bill. How much should she get back in change? On the first day of their 2479-mile trip, the Curtis family drove 497 miles. How many more miles do they have drive until they complete their trip? 4. to One hundred thirty-six of the two hundred sixty students in the auditorium were boys. How many girls were in the auditorium? 5. Draw a diagram of this statement. Then answer the questions that follow. The Daltons completed 2140-mile three tenths of their trip the first day. (a) How many miles did they travel the (b) How many miles of their trip do they first day? still have to travel? 6. If the perimeter of a square long 7. is Rewrite 9. 2 3 and — 4 to the nearest thousand, (b) Round 36,467 to the nearest Mentally estimate the quotient hundred. when 49. Simplify each number. ^ (a); inches common denominators. Round 36,467 r 1 11. so that they J have (a) by 10. how many each side of the square? 3 8. is 5 feet, 24 ^ 15 no (b) 12 —w O— 15 Compare: ^ 12 90 — 16 29,376 is divided Adding and Subtracting Fractions with 12. 1 65 1 11 (b) 12- Write the prime factorization of each number, 51 (a) 14. Denominators Write the reciprocal of each number. (a) 13. Different What (b) is the average of 2592 5, 7, 9, 11, 12, 13, 24, 25, 26, and 28? 15. List the single-digit divisors of 5670. Solve: 16. 6w Add, 19. = 8 • 9 1 + 1 2 3 20. 2 24. 23. 3 4 3 2 + l5 3 6 3 1 4 3 26. V3 43,164 36 3 3. X 63 - d = 35 8 -7 5 i-ii 4 8 ] " 5 8 3 -1 + 2 29. 18. 22. 6 27. 356 + a = 527 subtract, multiply, or divide, as indicated: 21. 25. 17. $3.87 28. 4J x 30. 96 $3.15 4.80 7.67 8.98 7.42 + 8.65 166 Math 87 LESSON Decimal Fractions • Decimal Place Value 36 Decimal fractions We have used fractions to name parts of a whole. We remember that a fraction has a numerator and a denominator. The denominator indicates the number of equal parts in the whole. The numerator indicates the number of parts that are being considered. Number of parts considered Number of equal parts in the whole In a common fraction the numerator 3 see that the numerator is 3. I I i I I I and denominator both written. The fraction diagramed above We I I 10 I E are is three tenths. We see that the denominator is 10. Parts of a whole can also be fractions. In a decimal fraction named by we using decimal can see the numerator, we cannot see the denominator. The denominator of a decimal fraction is indicated by place value. Here is the decimal but fraction three tenths. 0.3 If we write one digit after a decimal point, the denominator is 10. understood to be 10 because the to the right of the Yo are place ways . indicate that We see the numerator, which is we do not see the denominator, which is is we 3 10. The decimal is It is first customary ^, and both to write a zero before the decimal point. 0.3 = three tenths equals three tenths 10 place the tenths' place 0.3 equals the fraction to write three tenths. but The denominator occupies the decimal point, which 3, Decimal Fractions • 167 Decimal Place Value A decimal written with two digits after the decimal point is understood to have a denominator of 100, as we show here. 0.03 = three hundredths 100 21 0.21 = twenty-one hundredths 100 Example 1 Solution Write seven tenths Solution Decimal place value as a fraction and (b) as a decimal. 7 (b) (a) 10 Example 2 (a) Name the shaded part of this square (a) as a fraction. [b] as a decimal. (a) 23 100 In our 0.7 (b) 0.23 number system called place value. the place a digit occupies has a value, We remember that places to the left of the decimal point have values of 1, 10, 100, 1000, and so on, becoming greater and greater. Places to the right of the decimal point have values of — j^, and so on, becoming less and less. This chart shows decimal place values from the 11— , 1 , millions' place through the millionths' place. o re o Q. re o Q. CD "to oc re to o re Q. "to c o 3 O D i_ T3 c 3 re O re ocw re co 3 O CD o re Q. O o o re Q. re o. "co c <z re CD I— w 3 O d 1,000,000 100,000 10.000 1000 100 10 "O c 10 3 o O re Q. re T3 re to c 3 oc re to Q. c T3 Q. re "to "co T> o IS) 3 o 1— -a c o c 3 3 o c 1 1 1 100 1000 10.000 100,000 1,000.000 c 3 1 68 Math 87 Example 3 Solution What the value of the sixth decimal place to the right of the decimal point? The is sixth decimal place is six places to the right of the decimal point. The sixth decimal place has a value of one millionth Example 4 In the • 1,900,000 ' number 12.34579, which digit is in the thousandths' place? Solution Example 5 Solution We can name the places to the right of the decimal point by saying "tenths, hundredths, thousandths." The thousandths' place is the third place to the right of the decimal point and is occupied by the 5. Name The This Practice the place occupied by the 7 in 4.63471. 7 is in the fourth place to the right of the decimal point. is the ten-thousandths' place. a. Write three hundredths as a fraction. Then write three hundredths as a decimal. b. Name the shaded part of this circle both as a fraction and as a decimal. c. In the number 16.57349, which digit is in the thousandths' place? d. Name e. The number 36.4375 has how many decimal f. What the place occupied by the 8 in 4.634718. is places? the value of the third decimal place? Decimal Fractions Problem set • Decimal Place Value 1 69 1. James and his brother are putting their money together to buy a radio that costs $89.89. James has $26.47. His brother has $32.54. How much more money do they need to buy the radio? 2. Norton read 4 books during his vacation. The first book was 326 pages, the second was 288 pages, the third was 349 pages, and the fourth was 401 pages. What was the average number of pages of the 4 books he read? 36 3. A one-year subscription to the monthly magazine costs $15.96. At this price, 4. what is the cost for each issue? Jamestown began in 1607. This was how many years after Columbus reached the Americas The settlement at " in 1492? 5. A square and a regular hexagon share a common side. The perimeter of the square is 24 cm. What is the perimeter of the hexagon? 6. Draw a diagram of this statement. Then answer the questions that follow. Nelson correctly answered four questions on the the 20 test. (a) How many questions (b) How many questions did Nelson answer incorrectly? 7. What 8. Round 481,462 9. fifths of is the least common did Nelson answer correctly? multiple of (a) to the nearest hundred thousand. (b) to the nearest thousand. 3, 4, and 6? Mentally estimate the difference between 49,623 and 20,162. 1 70 Math 87 10. Name the shaded part of this square 11. (a) as a fraction. (b) as a decimal. In the number 9.87654, which digit is in the hundredths' place? 12. Replace each circle with the proper comparison symbol. — O w 10 (a) 13. (b) 100 3 45 30 (b) Complete each equivalent 15 5 n 360 744 fraction. ? 7 , (b) (a) 24 15. O0.3 Simplify each fraction or mixed number. (a) 14. 0.3 Draw two (c 12 parallel lines. 24 - = 6 Then draw two more — 24 4 parallel lines that are perpendicular to the first pair of lines. Label the points of intersection A, B, C, and D. of quadrilateral is What kind ABCLP. Solve: 16. 9n = 6 Add, ..« 19. 1 2 + - 17. a + 57 = 220 20. 3 3- - 6 23 22. 18. 1 -s- ^ 2 - - 4 3 -5 8 3 2 23. 12 60 - subtract, multiply, or divide, as indicated: 2 21. • 14 24. 25 7 18 4 2 • • i- 9 m = 48 Reading and Writing Decimal Numbers 25. 1 5 3- + 1- x 29. 87 8 2 28,308 14 30. 1 (3 27. 4 $4.67 28. 1 26. 6 2 Comparing Decimal Numbers • [4 71 2 '3 $420.15 - 398.75 Reading and Writing Decimal LESSON Numbers Comparing Decimal Numbers 37 Reading and writing decimals We remember that in a decimal of the decimal point identify a • number the digits to the whole number. The digits to the right of the decimal point identify a fraction. Here show two and three as a mixed number. tenths written as a decimal 2.3 = left we number and 2- 10 To read a decimal number, we read the whole number part, then we read the fraction part. To read the fraction part of a decimal number, we read the digits to the right of the decimal point as though we were reading a whole number. This number is the numerator of the decimal fraction. Then we say the name of the last decimal place. This number is the denominator of the decimal fraction. Example 1 Solution Read this first decimal number: 123.123 read the whole number part. When we come to the decimal point, we say "and." Then we read the fraction part, ending with the name of the last decimal place. First we 123.123 One hundred thousandths. We say "and" for the decimal point. twenty-three and one hundred twenty-three 1 72 Math 87 Example 2 Solution Use digits to write these (a) Seventy-five thousandths (b) One hundred and (a) The last word decimal numbers. eleven hundredths tells us the number. "Thousandths" means the right of the decimal point. We place in the decimal there are three places to last 75 into these places so that the 5 is in the place. We write zero in the remaining place. fit • last 7 5 Decimal numbers without a whole number part are usually written with a zero in the ones' place. Either .075 (b) or 0.075 To write one hundred and eleven hundredths, we remember that the word "and" separates the whole number part of the number from the fraction part. First we write the whole number part followed by a decimal point for "and." 100. Then we write the fraction part. We shift our attention to the last word to find out how many decimal places there are. "Hundredths" means there are two decimal places. 100. Now we fit __ 11 into the two decimal places. 100.11 Comparing decimals With decimal numbers it is important to consider the values of the places occupied by the digits. Each place to the left of a particular digit has a value 10 times greater than the value of the place of the particular digit. Terminal zeros to the right of the decimal point have no value. 1.3 equals 1.30 equals 1.300 equals 1.3000 When we compare decimal numbers, it is convenient to Reading and Writing Decimal Numbers • Comparing Decimal Numbers 173 both numbers will have the to the right of the decimal point. insert terminal zeros so that same number of Example 3 Solution digits O 0.012 Compare: 0.12 So that each number has the same number of decimal places, we insert a terminal zero in the number on the left and get 0.120 O 0.012 One hundred twenty thousandths is greater than twelve thousandths, so 0.120 > 0.012 Example 4 Solution Compare: We may right 0.4 O 0.400 delete and two terminal zeros from the number on the get 0.4 We = 0.4 could have added terminal zeros to the number on the left to get 0.400 = 0.400 Example 5 Solution Compare: 1.232 We insert O 1-23185 two terminal zeros in the number on the get 1.23200 and since 23200 is O 1-23185 greater than 23185, we 1.23200 > 1.23185 Practice Use words a. 25.134 b. 100.01 Use c. to write digits to write each decimal number. each decimal number. One hundred two and three tenths write left and 1 74 Math 87 d. One hundred e. Three hundred and seventy-five thousandths twenty-five ten-thousandths Write terminal zeros as necessary so that each number has the same number of digits to the right of the decimal point. Then replace the circle with the proper comparison symbol. f. g. h. Problem set 37 O 5.06 O 5.60 O 1-099 10.30 10.3 1.1 1. There were 3 towns on the mountain. The population of Hazelhurst was 4248. The population of Baxley was 3584. The population of Jesup was 9418. What was the average population of the 3 towns on the mountain? 2. The film was How many minutes long was the film? 1760 yards. Claudia ran 440 yards. How many more yards does she need to run in order to run 1 mile? 3. A mile is 4. One seats 5. a long one, lasting 3 hours 26 minutes. sixth of the seats were not empty? A square a and a regular pentagon share common side. The perimeter of the 20 cm. What of the pentagon? 6. 7. were empty. What fraction of the square is Round 3,197,270 is the perimeter (a) to the nearest million. (b) to the nearest Rewrite -,-, and inator. hundred thousand. - so that they have a common denom- . Reading and Writing Decimal Numbers 8. 9. • Mentally estimate the product of 313 and 489. Draw 175 Comparing Decimal Numbers a diagram of this statement. (p Then answer J the questions that follow. Five eighths of the troubadour's 200 songs were about love and chivalry. (a) How many of the songs were about love and chivalry? (b) How many of the songs were not about love and chivalry? 10. What (a) is fraction of the rectangle not shaded? What decimal (b) tangle is part of the rec- not shaded? 11. Use words 12. Use 13. Insert terminal zeros as necessary. digits to write seventy-six circle Then replace each with the proper comparison symbol. O 12.60 3.14159 O 3.1416 (b) Simplify each fraction or mixed number. 3 (a) 15. and five hundredths 12.6 (a) 14. to write 3.025. 30 18 (b) 81 24 (c) 480 1000 Write the prime factorization of 3008. Solve: 6 = Aw 16. 10 17. 340 = 18. x - 28 = 82 • m + 74 <} 1 76 Math 87 Add, 19. 21. subtract, multiply, or divide, as indicated: ± + * 4 8 + ± 2 5 1 4- - 1- 8 25. 5 — 10 3 7 5- + q 2 + 3 29. 24. --6 4 8 ii 9 5 26. ? 6 1 ' V 2 8600 28. 1 a l 2 1 6- - 2-1 8 f x $52.48 78.69 47.78 86.94 42.03 + $§.16 30. 10 ^ 1 4- + 1- 8 7- + 14 2'A 22. 2 2 23. 20. 90 $368.46 18 Rounding Decimal Numbers LESSON 38 To round decimal numbers, we can use the same circle and arrow procedure that we use to round whole numbers. Example 1 Solution Round 3.14159 to the nearest The hundredths' place decimal point. is two places We circle the digit to its right hundredth. to the right of the digit in that place with an arrow. 3.1(4)159 and mark the Rounding Decimal Numbers Since the arrow-marked digit circled digit unchanged. than change is less Then we 5, we 177 leave the all digits to the right of the circled digit to zero. 3.14000 Terminal zeros to the right of the decimal point do not serve as placeholders as they do in whole numbers. After rounding decimal numbers, we should remove terminal zeros to the right of the decimal point. — 3.14006- Example 2 Solution Round 4396.4315 We to the nearest 3.14 hundred. are rounding to the nearest hundred, not to the nearest hundredth. 4@96.4315 Since the arrow-marked digit is 5 or more, we increase the circled digit by 1. All of the following digits become zeros. 4400.0000 Zeros at the end of a whole number are needed as placeholders. Terminal zeros to the right of the decimal point are not needed as placeholders. We remove these 4400.0000 Example 3 Solution Round - zeros. 4400 38.62 to the nearest whole number. To round a number to the nearest whole number, we round to the ones' place. 3®.62 Practice a. Round 3.14159 b. Round 365.2418 c. Round — 39.00 —• 39 to the nearest ten-thousandth. to the nearest hundred. 57.432 to the nearest whole number. 1 78 Math 87 Problem set 1. 38 2. The high jumper new school record cleared 5 feet 8 inches. How many inches inches? During the set a week first of November when is she 5 feet 8 the highest daily temperatures in degrees Fahrenheit were 42°, 43°, 38°, 47°, 51°, 52°, and 49°. What was the average high daily temperature during the first week of November? 3. In 10 years the population increased 120,310. from 87,196 to By how many people did the population increase in 10 years? 4. The than the largest 5. A odd number is how much four-digit even number? largest three-digit regular hexagon less and a regular octagon share a common side. If the perimeter of the hexagon is 24 cm, what is the perimeter of the octagon? 6. Draw a diagram of this statement. Then answer the questions that follow. Three twentieths of the 100 questions on the test were true-false. (a) How many of the questions on the test were true- false? (b) How many of the questions true-false? LCM 7. Find the 8. Round 15.73591 of 5, 6, and 10. (a) to the nearest hundredth. (b) to the nearest whole number. on the test were not — Rounding Decimal Numbers 9. 10. 11. Use words 150.035 (b) 0.0015 Use digits to write each of these decimal numbers. (a) One hundred twenty-five thousandths (b) One hundred and twenty-five thousandths Insert terminal zeros as necessary. (a) 12. 13. 0.128 O O-l 4 (a) in centimeters. (b) in millimeters. (b) 0.03 O 0.0015 .cm I Simplify each fraction or mixed number. 1 (a) 810 630 CU\ [bj A 4 72 48 Write the reciprocal of each number, (a) 15. Then replace each with the proper comparison symbol. Find the length of this segment f 14. each of these decimal numbers. to write (a) circle 7 (b) Complete each equivalent ( , (a) 20 = 9 36 ? Solve: 16. 8m 17. 125 + 18. 54 = 54 - = 4 x 18 • = 500 y 179 3| fraction. „> ^ 7 = 14 ? 36 1 80 Math 87 Add, 19. subtract, multiply, or divide, as indicated: 5 1 —3 + — + - 4 8 3 1 4 6- 20. 2 "> 21. 4 I_3 23. 7 2— 4- 2 5- 10 25. 4 5 26. 4 + 22 5 fl 2 K 2^ 28. 3j • . 3i 3 + 4i 6 l-2^ 6 2 740 X800 $500.00 29. 24. • 6 8- + 53 3 8 5 8 27. 22. 8 2 30. 47.74 48,080 16 865.62 478.95 + 3 0.72 Decimal Numbers on the LESSON Number Line 39 each centimeter segment on a centimeter scale is divided into 10 equal segments, then each segment is 1 millimeter long. Each segment is also one tenth of a centimeter long. If Example 1 Find the length of this segment cm MllllllllllllllllllHlilll Solution (a) in millimeters. (b) in centimeters. (a) Each centimeter the scale is 1 is mm. mm. Thus, each small segment on The length of the segment is 23 mm. 10 1 1 1 Decimal Numbers on the Number Line 1 81 Each centimeter on the scale has been divided into 10 equal parts. The length of the segment is 2 centimeters (b) plus three tenths of a centimeter. In the metric system we use decimals rather than common fractions to indicate part of a unit. The length of the segment is 2.3 cm. whole numbers on a number line is divided into 100 equal units, then numbers corresponding to the marks on the number line can be named using two decimal places. the distance between consecutive If ABC Find the number on the number Example 2 line indicated by each arrow. i 1 1 1 1 ifi 4.0 Solution \ I i n I mUn 1 1 1 u U 1 1 4.2 4.1 1 1 1 n 1 1 1 i»i 1 4.3 1 1 1 1 1 1 1 4.4 1 1 1 i 1 H n 4.5 Hi ii i U ii Mlu »ili n 4.7 4.6 i I 4.8 We are considering a portion of the number line from 4 to 5. The distance from 4 to 5 has been divided into 100 equal segments. Tenths have been identified. The point 4.1 is one tenth of the distance from 4 to 5. However, it is also ten hundredths of the distance from 4 to 5, so 4.1 equals 4.10. Arrow A indicates 4.05 Arrow B indicates 4.38 Arrow C indicates 4.73 Practice a. 2 Find the length of this segment MllllllllllllllllllMMllllllllll in centimeters. b. What and c. 3 point on a number line is halfway between 2.6 2.7? What decimal number names this number line? marked by the point A -I 10.0 1 1 1 1 1 1 1 1 h 10.1 A on 182 Math 87 Problem set 1. 39 2. In 3 boxes of cereal Jeff counted 188 raisins, 212 raisins, What was the and 203 raisins. in each box of cereal? average number of raisins The pollen count had increased from 497 parts per million to 1032 parts per million. By how much had the pollen count increased? 3. 4. Sylvia spent $3.95 for lunch but still had $12.55. How much money did she have before she bought lunch? In 1903 the Wright brothers airplane flight. Just 66 years made the first 5. first moon? The perimeter of the square equals the perimeter of the regular hexagon. of the If each side hexagon 6 inches long, 6. landed land on later astronauts first on the moon. In what year did astronauts the powered Draw how is long is each side of the square? a diagram of this statement. Then answer the questions that follow. Each week Jessica saves two fifths of her $4.00 allowance. (a) How much allowance money does she save each week? (b) How much allowance money does she not save each week? 7. Estimate the product of 396 and 71. 8. Round 7.49362 9. Use words to the nearest thousandth. to write (a) 200.02 (b) 0.001625 each of these decimal numbers. — Decimal Numbers on the Number Line 10. 11. Use digits to write each of these decimal numbers. (a) One hundred seventy-five millionths (b) Three thousand, 83 and three hundredths Replace each circle with the proper comparison symbol, 6.174 (a) 12. thirty 1 O 6.17401 (b) Q 14.276 1.4276 2 1 Find the length of this segment 3 llllllllllllllllllllllitlllll 13. (a) in centimeters. (b) in millimeters. What decimal number names the number line? —— marked X on this point X -| 1 1 1 $ 1 i 1 1 8.3 8.2 14. Simplify each fraction or mixed number. 75 \ ( n i (bl (a) 1 } 15. 1- 1 300 o 3 30 -. ( (cj 12 What decimal number is halfway between 7 and 8? Solve: 20 = 12y 16. 15 17. 300 = 74 + c 18. 36 = x - 24 Add, 19. • subtract, multiply, or divide, as indicated: 5 + ? + I 6 21. 3— - 1- 23. 1 ? 5- - 1- 4 I6 22. -l-.2f.34 10 4 12 3 - 20. 2 3 1 24. \ 4 3 3^-5-4 5 288 720 4 184 Math 87 25. 6^ + 4i 8 4 —1 27. + 26. 6 $375.64 124.36 468.95 876.43 + 984.86 3 28. 90,900 18 30. 370 (5 8 29. 5-1 - 1-2 X800 Adding and Subtracting Decimal Numbers LESSON 40 When we add or subtract whole numbers, we align the ones' digits so that we add digits that have the same place value. When we add or subtract decimal numbers, we align the decimal points vertically. Aligning the decimal points ensures that we will be adding or subtracting digits that have the same place value. We will consider several examples. Example 1 Solution Add: We 3.6 + 0.36 + 36 align the decimal points vertically. A 3.6 number is is Example 2 Solution written without a decimal point a whole number, so the decimal point to the right of 36. Add: We 0.1 0.36 +36. 39.96 + 0.2 + 0.3 + 0.4 and 0.1 record only one digit in each column. The sum is 1.0, not 0.10. Since 1.0 equals 1, we can simplify the answer 0.2 align the decimal points vertically add. to 1. We 0.3 + 0.4 1.0 = 1 Adding and Subtracting Decimal Numbers Example 3 Subtract: 12.3 Example 4 number above the second 12.3 0O number, aligning the decimal points. We write zeros in the empty places and subtract. - 4.567 first 7.733 - 4.32 4 Solution Practice Problem set 129 1 We write the Subtract: 5 We write the whole number 5 with a decimal point and write zeros in the two empty decimal places. Then we subtract. Add 9 3.00 -4.32 0.68 or subtract as indicated: + 3.45 + 23.6 a. 1.2 b. 4.5 + 0.51 + 6 + 12.4 c. 0.2 d. 36.274 - 5.39 e. 16.7 f. 12 - 0.875 1. In the first 6 + 0.4 + 0.6 + 0.8 - 1.936 monthly 40 85 - 4.567 1 Solution 1 months of the year the electricity bills were $128.45, $131.50, $112.30, $96.25, $81.70, and $71.70. monthly electricity bill Montgomerys' What was during the first 6 their average months of the year? 2. The price was reduced from two thousand, four hundred ninety-eight dollars to one thousand, nine nine dollars. By how 3. hundred ninety- much was the price reduced? A 1-year subscription to a monthly magazine costs $15.60. The regular newsstand price is $1.75 per issue. How much price? is saved per issue by paying the subscription 186 Math 87 4. Carlos ran one lap in one lap did 5. it 1 minute seconds faster than Carlos. take Orlando to run one lap? 5 The perimeter Orlando ran many seconds 3 seconds. How of the square equals the perimeter of the regular pentagon. Each side of the pentagon is 16 cm long. How long is each side of the square? 6. Draw Then answer a diagram of this statement. the questions that follow. Two ninths of the 54 fish in the tank were guppies. (a) How many of the fish were guppies? (b) How many of the fish LCM and were not guppies? 7. Find the 8. (a) What (b) What decimal part of this of 6, 8, 12. fraction of this square is not shaded? square is not shaded? 9. Round 2375.4174 (a) to the nearest hundredth. (b) to the nearest hundred. 10. Use words 11. Use 12. Find the length of segment to write 100.075. digits to write twenty-five this (a) in centimeters. (b) in millimeters. hundred-thousandths. 2 cm lllllllJJ imlmi 4? 3 iimIiiii iimIiiii imiImm Adding and Subtracting Decimal Numbers 13. What decimal number names arrow on this number line? 187 marked with an the point I i | i i i i i i i i 3.2 14. i i i i i i ' i i | 3.3 3.4 Simplify each of these numbers. 4 (a) 15. i 1 44 (b) 444 44 What decimal number is halfway between 1.2 and 1.3? Solve: 16. 15x = 9 17. / + 46 = 200 18. 512 Add, - y = 10 215 subtract, multiply, or divide, as indicated: + + 15 3.65 36.45 - 4.912 22. 15 - 4.29 1- + 2- + 3- 24. 1- 3.4 21. 2S „ 5.63 - + -<\ 27. 46,731 - 30 29. $36.24 + $1.79 + A 1 3 2 4v 3- + 1- 28. ( 38<£ 4-38 A 2- + 8 + 15.23 3 • 34 3. 3— - 1- 12 V \* • -2 1 26. 0.9 2n 1 (<\ 1 1 6 30. + 20. 19. 25. • 3 4 30(40)(25) + $15.60 + $12 1 88 Math 87 LESSON 41 Ratio A ratio is a comparison of two numbers. Ratios can be written several ways. The ratio 7 to 10 can be written as follows. With the word "to" As As Usually, we A 10 — 10 a fraction a decimal number 0.7 With a colon 7:10 will write ratios as fractions. Ratios should be written in reduced form just as form. 7 to we write reduced ratio should not be written as a mixed number. fractions in Almost every ratio problem has three ratio numbers. Usually only two of the ratio numbers are given. We must calculate the hidden ratio number. If we are told that the we can ratio of frogs to fish is 2 to 5, write 2 frogs 5 fish The third ratio number equals a total of is the total. Two frogs plus five fish 7. 2 frogs 5 fish 7 total Now we can write the ratio of frogs to fish as 2 5 and the ratio of froes to the total is 2 7 and the ratio of fish to the total is 5 7 Ratio 1 89 Sometimes the total is given and the hidden ratio number is one of the other numbers. If we are told that in every group of 30 dogs and cats there are 12 dogs, 12 dogs cats ? 30 total we see that the ratio number for cats must be 18. 12 dogs 18 cats 30 total Whenever we work a ratio problem, it is a good idea to begin by writing all three ratio numbers. Example 1 Solution In a class of 28 students there are 12 boys. (a) What is the boy-girl ratio? (b) What is the girl-boy ratio? We will begin by writing all three ratio numbers. 12 boys 12 boys ? 16 -> girls 28 total 28 total Now we (a) The girls can write the answers. boy-girl ratio is — , which reduces to -, a ratio of 3 to 4. (b) girl-boy ratio is ^7 which reduces to -, a ratio of 4 to 3. Note that we do not change the ratio to a mixed The « number. Example 2 The team won | of its games and lost the rest. What was the team's won-lost ratio? Solution We are not told how many games the team played. we are told that the team lost - of its team won However, - of its games. Therefore the games. Thus, on the average, the team won 1 90 Math 87 4 out of every 7 games. So the three ratio numbers are 4 won 3 lost 7 total The won-lost Example 3 ratio was 4 to 3, which we write -. were red marbles and green marbles. If the ratio of red marbles to green marbles was 4 to 5, what fraction of the In the bag marbles was red? Solution First we write all three ratio numbers. 4 red 5 green 4 red 5 green —> 9 total ? total The question was what fraction of the marbles There were 4 red marbles and a fraction that Practice was red was In the pond were 240 was the b. c. marbles so the all three ratio numbers. little fish fish. What girls. What and 90 big ratio of big fish to little fish? Fourteen of the 30 students in the class were was the red. -. Begin each problem by writing a. total of 9 was boy-girl ratio in the class? The team won - of its games and lost the rest. What was the team's won-lost ratio? Problem set d. The bag contained red marbles and blue marbles. If the ratio of red marbles to blue rnarbles was 5 to 3, what fraction of the marbles was blue? 1. Fourteen of the 32 students in the class were was the ratio of boys to girls in the class? 2. During the 41 last 3 girls. What years the annual rainfall has been 23 inches, 21 inches, and 16 inches. What has been average annual rainfall during the last 3 years? the 191 Ratio 3. Sean reads 35 pages each he read in a week? 4. Shannon swam 100 meters swam many night. How many pages does in 56.24 seconds. Donna 100 meters in 59.48 seconds. Donna took how seconds longer to swim 100 meters than Shannon? 5. Draw a diagram of this statement. Then answer the questions that follow. Two fifths of the 30 players in the game had never played rugby before. How many (a) of the players had never played rugby before? What was (b) who had to those 6. 7. AB is 40 the ratio of those mm. AC is 95 not played rugby? mm. Find BC. • • A B C The length of the rectangle is width. What perimeter of the rectangle? 9. played rugby -«-• greater than 8. who had its Estimate the sum each number to 5 cm is the » 8 cm and 896 by rounding the nearest hundred before adding. of 3624, 2889, Round 6.857142 (a) to the nearest (b) to three whole number. decimal places. 10. Use words 11. Use digits to write (a) Twelve millionths (b) One hundred to write 120.0305. each number. thirty thousand and four hundredths 1 92 Math 87 12. 13. Find the length of segment i (a) in centimeters. (b) in millimeters. i i i i i i i i 1 1 II 1 1 ll 1 1 ll 1 1 II I ItlllH lllltl i i 50 llllllllll M on this i 13.5 13.4 13.6 Simplify each of these numbers. (a) 15. II 40 30 What decimal number names the point marked number line? M -i 14. 20 10 mm this 27 18 144 600 15 (b) 8 (c) which angle In this figure, (a) a right angle? (b) an acute angle? (c) an obtuse angle? is Solve: 16. 8y = 144 17. 63 = 91 - 18. 213 = Add, w p + 57 subtract, multiply, or divide, as indicated: 19. 4.27 + 16.3 + 10 21. 3- + 1- + 2- 23. 25. 111 2 3 b (2 6 3 2* + 4± 4 2 0.9 + 0.8 22. 3- l) 2 + 0.7 + 0.5 111 4 3 I 20. • 1- 2 24. 8 26. 5 J A 12 - • 24 3 (2 ,3 3 . § l) 2, I Perimeter, Part 2 LESSON 42 27. (437)(86) 28. 38,015 29. $6.47 + $5.24 + $11 + 30. $20.00 - 1 93 19 4- (3 53(2 x $4.98) Perimeter, Part 2 In this lesson we will practice finding the perimeter of shapes such as the shape shown here. All the angles in the figure are right angles. A E 3 cm B F 6 4 cm cm D 7 cm We can find the perimeter of this shape by adding the lengths Although we are given the lengths of only four of the sides, we can figure out the lengths of the other two sides. of all six sides. To find EF, we observe that the length of side EF plus the length of side AB equals the length of side DC. Thus EF is 4 cm. We the length is 2 AF in a similar way. The length of side AF plus of side ED equals the length of side BC. Thus AF find cm. The perimeter 3 cm + 6 cm of the hexagon + 7 cm + 4 cm is + 4 cm + 2 cm = 26 cm 194 Math 87 Example Find the lengths of the two unmarked sides of this polygon. Then find the perimeter of the polygon. Dimensions are in meters. All angles are right 12 • angles. 15 Solution We will use the to the letters unmarked a and b to refer sides. The length of side a is equal to the combined lengths of the two shorter vertical sides. Thus the length of side a 12 b is m 12 The lengths + 4 m of the = 16 m 15 two shorter horizontal sides together equal 15 m. Thus the length of side b 15m-8m The perimeter 8m Practice + 12 of the polygon m + 7m + = is 7m is 4m + 15 m + 16 m = 62 m Find the length of each unmarked side and find the perimeter of each polygon. Dimensions are in feet. All angles are right angles. 12 a. b. 4 I Problem set 1. 42 2. 3 The team won - of its games and was the team's won-lost ratio? 5 lost the rest. What During the first 6 months of the year the car dealership sold 47 cars, 53 cars, 62 cars, 56 cars, 46 cars, and 48 Perimeter, Part 2 What was cars. the 3. first 6 number the average 1 95 of cars sold during months of the year? The relay team carried the baton around the track. Darren ran his part in eleven and six tenths seconds, Robert ran his part in eleven and three tenths seconds, Orlando ran his part in eleven and two tenths seconds, and Claude ran his part in ten and nine tenths seconds. What was the team's total time? 4. Jenny went to the store with $10 and returned home with 5 gallons of milk and $1.30 in change. What was the cost of each gallon of milk? 5. Draw a diagram of this statement. Then answer questions that follow. Kevin shot par on two thirds of the 18 holes. (a) (b) 6. On how many On how many holes did Kevin shoot par? holes did Kevin not shoot par? Sketch this figure on your Then find the length of each unmarked side. Then paper. 6 find the perimeter of the polygon. Dimensions are in inches. All angles are right angles. 7. Complete each equivalent 5 (a) ? 8. Find the 9. (a) (b) 8 LCM of 9, 6, What decimal square is is and A 24 12. part of this shaded?. What decimal square 1 = (b) 6=18 fraction. part of this not shaded? , , 3 {c) 4 = 15 T the 196 Math 87 10. 11. 12. 13. Round 3184.5641 two decimal (a) to (b) to the nearest Name hundred. each decimal number. (a) 0.001875 (b) 600.007 Use places. digits to write each number. (a) Three hundred seventy-five ten-thousandths (b) Sixty and seven hundredths Simplify each of these numbers. 15 n 5 (a) , (b) ( ) 14. 144 30 (c) 720 1080 Find the length of segment AB. 2 inches _l_Ll_ ±1 m 9 • m B 15. Draw a pair of parallel lines. Next draw another pair of parallel lines that intersect the first pair of lines but are not perpendicular to them. Then shade the region enclosed by the intersecting pairs of lines. Solve: 8 = 4x 16. 7 17. 4.2 18. 134 = d - 27 Add, 19. • = 1.7 + y subtract, multiply, or divide, as indicated: 4.375 + 12.125 + 1.3 20. 0.1 + 0.2 + 0.3 + 0.4 i 21. * . 12 4± • 23. 22. 6 5 5 — + 3^ (l 24. 8 I 2 *) 4 3± + 5 26. 3 4 - 3 40(60)(80) 28. 29. $12 + $18.57 + $7.98 + 7(2 30. $10 - (4 x 2" 2 12 27. V* 1 2 - 13- f 25. 197 2i 2 + ii + 6 ' ' * 6 3raphs ' 3, \ 3000 - 24 78(2) Graphs LESSON 43 We use graphs to help us understand quantitative information. A graph may use pictures, bars, lines, or parts of circles to help the reader visualize comparisons or changes. In this lesson we will practice gathering information from graphs. Example 1 Find the information in this pictograph to answer the following questions. Donut Sales Jan. Feb. Mar. ©©©© ©©©©©© ©©©©©$ Co) Represents 10,000 donuts (a) About how many donuts were sold (b) About how many donuts were sold of the year? in March? in the first 3 months 198 Math 87 Solution The key at the bottom of the graph shows us that each picture of a donut represents 10,000 donuts. (a) For March we whole donuts, which represents 50,000 donuts, and half a donut, which represents 5000 see 5 mean donuts. Thus, the 5- donuts pictured that about 55,000 donuts were sold in March. (b) We see a total of 15 - donuts pictured for the first 3 months of the year. Fifteen times 10,000 is 150,000. Half of 10,000 5000. Thus, about 155,000 donuts were sold in the first Example 2 is 3 months of the year. Find the information in this bar graph to answer the following questions. Number of Aluminum Cans Collected by Each Homeroom 10,000 Rm12 Rm14 Rm (a) Solution Rm 18 About how many cans were collected by the students Room (b) 16 in 14? The students in Room 16 collected about as many cans as what other two homerooms combined? We look at the scale on the left side of the graph. We see that the distance between two horizontal lines on the scale represents 2000 cans. Thus, halfway from one 1000 cans. (a) (b) The students in Room line to the next represents 14 collected about 8000 cans. The students in Room 16 collected about 9000 cans. This was about as many cans as Room 12 and Room 18 combined. ' Graphs Example 3 This line graph shows during the year. how Paul's test scores have 1 99 changed Paul's Test Scores uu 90 80 70 bU Test Solution 1 Test 2 Test 3 Test 4 Test 5 Test 6 (a) What was (b) In general, are Paul's scores improving or getting worse? (a) We Paul's score find Test 3 on the on Test 3? scale across the bottom of the graph and go up to the point that represents Paul's score. We see that the point is halfway between the lines that represent 70 and 80. Thus, on Test 3, Paul's score was about 75. (b) With only one exception, Paul scored higher on each succeeding test. So, in general, Paul's scores are improving. Example 4 Find the information in this circle graph to answer the Where Dina Spends Her Day following questions. (a) how Altogether, many hours are included in this graph? (b) What fraction of Dina's day is Solution A spent at school? graph (sometimes called a pie graph) shows the relationship between parts of a whole. This graph shows circle parts of a whole day. (a) This graph includes 24 hours, one whole day. (b) Dina spends 8 of the 24 hours O at school. We reduce — A to -. 200 Math 87 Practice Find the information from the graphs in each question. a. How many more this lesson to answer donuts were sold in February than in January? b. How many aluminum cans were collected by four all homerooms? c. On which test was Paul's score lower than his score on the previous test? d. What than Problem set 43 1. The 7. fraction of Dina's at home day was spent somewhere other or at school? was 3 to the outpost were ratio of soldiers to civilians at the outpost What fraction of the people at soldiers? 2. Denise read a 345-page book in 3 days. What was the average number of pages she read each day? 3. Christine ran a mile in 5 minutes 52 seconds. many seconds did it How take Christine to run a mile? Refer to the graphs in this lesson to answer questions 4 and 4. 5. How many fewer cans were collected by the students in Room 18 than by the students in Paul scores 85 on Test average for all 7 tests? 5. If 6. Draw 7, what Room 16? will be his test score a diagram for this statement. Then answer the questions that follow. Mira read three eighths of the 384-page book before she could put it down. (a) (b) How many pages did she read? How many more pages does she need to halfway through the book? read to be 201 Graphs 7. Sketch this figure on your 18 Then find the length each unmarked side and paper. of 20 find the perimeter of the 12 polygon. Dimensions are in centimeters. All angles are right angles. 8. Complete each equivalent - = 9 9. 10. 11. 12. — 18 ? 7 (a) fraction. 20 36 1 = (b) 9 4 24 (c) 5 Round 2986.34157 (a) to the nearest thousand. (b) to three Use words decimal places. each number. to write (a) 0.00325 (b) 3,280,004,000 Use digits to write (a) One and 03) Twenty and each number. seventy-five thousandths five twelfths Find the length of this segment cm ', , hiiIiiiiIiiiiIiiiiIhiiIiiiiIiiiiIhii 13. (a) in centimeters. (b) in millimeters. What decimal number names the point marked A on this number line? A i | 12.8 14. i i i i i i i i i i | i i i 9lf (b) i i i | 13.0 12.9 Simplify each of these numbers. (a) i 288 90 202 Math 87 15. In the figure shown: Which segmen t is perpen- (a) dicular to AD? Which segment appears (b) be parallel to to AD? Solve: 16. 4.3 + a 17. m 18. 5m Add, - = 6.7 3.6 = 10 = 4.7 • 7 subtract, multiply, or divide, as indicated: 19. 5.37 + 27.7 + 4 20. 345.6 + 14 + 1.58 21. 5-6 •2 1 22. 10 9 — + 10 25. 5^ 1 3 + - + - 8 3 23. 5 — 4 2 3 (l l) 10 [2 5) 24. I 2 5j 31 3 26. 10 - '3 -4-2 v 27. 470(6C>0)' 28. 10,000 - 16 29. $0.89 + $15 + $5.47 + 89tf + $1.42 30. ($20 - $5.24) h- 6 4 "1 J 203 Proportions LESSON 44 Proportions We remember comparison of two numbers. Ratios may be written as fractions. The ratio 16 to 20 and the that a ratio is a ratio 4 to 5 are equal ratios. Whenever we 16 4 20 5 write two ratios connected by an equals sign, we are writing a proportion. we multiply the upper term of one ratio by the lower term of the other ratio, we form a cross If product. The cross products of equal ratios are equal. We illustrate by finding the cross products of the proportion above. • 4 = 80 16 • 5 X ^ 16, 20' If 20 = 80 the cross products are equal, the ratios are equal. We will use cross products to help us find the missing terms in equal ratios. We Example 1 Solution will follow a two-step process. Step 1. Find the cross products. Step 2. Divide the known product by the known factor. Solve the proportion: — = — 20 30 We by finding the missing term. 12 solve a proportion Step 1. we First find the cross products. Since we are completing a proportion, the cross products must be equal. 30 20 • n = 12 20n = 360 • 30 cross products simplified 204 Math 87 Step 2. Divide the (20). The known product n = 18 Solution Practice Step 1. Step 2. simplified —x = — 32 20x = 480 cross products x = 24 divided by 20 Solve each proportion. a 6 b. a. 12 8 14 , d. c. 21 Problem set 44 factor divide by 20 20 Solve: by the known result is the missing term. 360 n = Example M 2 (360) Dan made 15 a line graph to show 30 b 1 ~~ 20 16 24 30 — = — 25 d his height Refer to this graph to answer questions 1 on each and 2. birthday. Dan's Height 13 12 Age in years 1. How many 2. Between which two birthdays did Dan grow the most? inches did Dan grow between his twelfth and thirteenth birthdays? Proportions 3. 205 There were 12 princes and 16 princesses in the palace. What was the ratio of princes to princesses in the palace? On 4. 4 days of their trip, the Curtis family drove 497 miles, 513 miles, 436 miles, and 410 miles. What was the average number of miles they drove on each of the first 4 days of their trip? 5. Don the first receives a weekly allowance of $4.50. How much allowance does he receive in a year (52 weeks)? 6. Draw a diagram for this statement. Then answer the questions that follow. Three sevenths of the 105 adults in the Khoikhoi clan were less than 5 feet tall. 7. (a) How many of the adults were less (b) How many of the adults were 5 feet tall or taller? than 5 120 Sketch this figure on your Then find the length of each unmarked side and paper. 30 80 find the perimeter of the polygon. All angles are right angles. Dimensions are in 55 millimeters. 8. Find the 9. Name shaded 10. LCM the of 8, 12, number and 9. of circles (a) as a decimal (b) as a number. mixed number. Round 0.9166666 (a) to the nearest hundredth. (b) to the nearest hundred-thousandth. feet tall? 206 Math 87 11. 12. 13. Use words. to write each number. (a) 3.0175 (b) 23,310,050,000 Use digits to write (a) One hundred and seventy-five thousandths (b) Ten and eleven twelfths each number. Simplify each of these numbers. 26- (a) (b) 3 14. —O w 15 Compare: F 24 126 24 10 16 Solve: 15. 18. A = 12 277 Add, x 1" = * y 3 19. 21 77 14 4 17. 1.4 w - = 0.8 subtract, multiply, or divide, as indicated: + 12 + 8.59 9.6 22. — + 10 7 26. 16. + 0.36 = 0.75 20. 24. 6 fl ^ 2^ + — 8-1 4 1- 3 5 28. 970(480) 30. ($20 - $5.24) - 12 3.15 - 7 23. 25. (2.1 3 f 12 2 5 4a 5 + c 6— 12 8 - 21. 2 ) ^4 3J 4-L-i! 4 5 27. 6 29. 300,000 + 2^ 0.06) Multiplying Decimal LESSON 45 Multiplying Decimal Numbers 207 Numbers To multiply decimal numbers, we set up the problem as though we were multiplying whole numbers. (We do not need to align the decimal points.) Then we multiply. After we have multiplied, we place the decimal point in the answer so that there are the same number of decimal places in the product as there are decimal places in 4.2 x 0.36 all the factors combined. one decimal place two decimal places 4.2 x 0.36 252 126 three decimal places 1.512 in the product Example 1 Solution Multiply: (0.23)(0.4) We we 0.23 were multiplying whole numbers, and then we multiply. After multiplying, we go back and count the x 0.4 92 set up number factors. the problem as though of decimal places in both There are a total of three decimal places, so we write the product with three decimal places. We count from right to left, writing one 0.23 x 0.4 O.Ogg 2 places 1 place 3 places more zeros in front as is necessary. The product of 0.24 and 0.4 is 0.092. or Example 2 Solution Multiply: 35 x 0.4 we 35 were multiplying whole numbers. x 0.4 1 place we count the total 14.0 1 place We set up the problem as though After multiplying, number of decimal places in the factors. Then we place a decimal point in the product so that the product has the same number of decimal places as there are in the factors combined. After placing the decimal point, we simplify the result. places 14 .0 = 14 208 Math 87 Example 3 Solution Multiply: (0.2)(0 3)(0.04) S Sometimes we can perform the multiplication mentally. First we multiply as though we were multiplying whole numbers: 4 = 24. Then we count decimal places. There is a total of four decimal places in the three factors. Starting from 2 • 3 • the right side of 24, zeros in the empty we count to the left four places. We write places. 0.0024 .2.4 Practice Problem set Multiply. Try to do Problems d and e mentally. a. 4.2 x 0.24 b. (0.12)("0.06) c. 5.4 x 7 d. 0.3 e. (0.04X25) 1. The bag contained only red marbles and white marbles. If the ratio of red marbles to white marbles was 3 to 2, what fraction of the marbles was white? 2. John ran 4 laps of the track in 6 minutes 20 seconds. 45 0.045 x 0.6 f. (a) How many (b) John's average time for running each lap many 3. x 0.2 x 0.1 The seconds did it take John to run 4 laps? was how seconds? Curtis 's car traveled an average of 24 miles per gallon of gas. At that rate, how far could the car travel on a full tank of 18 gallons? 4. Normal body temperature is 98.6°F. Allan's temperature was 103. 4°F. His temperature was how many degrees above normal? 70 5. The length twice its of the rectangle width. What is perimeter of the rectangle? is the mm Multiplying Decimal Draw 6. 209 Numbers Then answer a diagram for this statement. the questions that follow. Five eighths of the 200 sheep in the flock grazed in the meadow. The rest drank from the brook. (a) How many of the sheep grazed in the (b) How many of the sheep drank from the brook? meadow? AB is 30 mm. CD is 45 mm. AD is 100 mm. Find BC 7. -«-• • • A B C 8. The length is 9. 10. 11. 12. CD D . CD in Problem of segment the length of —•- • 4 7 is 45 in centimeters? Round 0.083333 thousandth. (a) to the nearest (b) to the nearest tenth. Use words to write (a) 12.0545 (b) 10 each number, 11 100 each number. Use digits to write (a) Twenty-five millionths (b) Five billion, two hundred fifty thousand What decimal number names the point marked B on this number line? B i i i i i | » i i i t i i i i i i I 0.8 0.7 13. mm. What Write seventy and seven hundredths (a) as a decimal. (b) as a mixed number. i i i 1 0.9 21 Math 87 14. In this figure, which angle (a) a right angle? (b) an obtuse angle? is D Solve: w 8 15. 10 25 12 m 3.7 = a + 17. 18. 16. Add, 1.8 2.7 x 0.18 22. 15 x 1.5 23. 0.4 x 0.3 x 0.2 x 0.1 24. 5.6 26. 6- - 9 = t - 3.1 4 (4 - 1.25) • 1- (0.15)(0.05) 25. 5 27. 7-5- - 8 ($100 - $32.50) * 15 (3.14 + 1.2) 6 5 4- + 26 21. 3 1 30. 3.9 6 subtract, multiply, or divide, as indicated: 20. 28. 19. n 15 29. 4- - 15 4 1 Dividing a Decimal Number by a Whole Number Dividing a Decimal by a Whole LESSON 46 21 Number Number Dividing a decimal number by a whole number is like dividing money. The decimal point in the answer is straight up from the decimal point in the division box. Example 1 Divide: 3.425 5 -s- 0.685 Solution We rewrite the problem with a division box. We place a decimal point in the 5J3.425 30 answer directly above the decimal point 42 40 25 25 Then we divide as though we were dividing whole numbers. The answer is 0.685. in the division box. Example 2 - Divide: 0.0144 8 0.0018 Solution We place the decimal point in the answer directly above the decimal point inside the division box. We 10.0144 _8 64 64 write a every place following the decimal point until the division is completed. If we cannot perform a digit in division, we write a zero in that place. The answer Example 3 Solution is 0.0018. Divide: 1.2 + 5 0.2 We do not write a decimal division answer with a remainder. Since a decimal point fixes place values, we may write a zero in the next decimal place. This zero does not change the value of the number, but it does let us continue dividing. The answer is 0.24. 5Jl^2 1 0.24 1.20 1 20 20 212 Math 87 Example 4 Divide 126 by 8 and write the answer with a remainder. Then continue the and write the answer division 15 1126 as a 8 decimal number. Solution 46 40 we divide 126 into 8 equal groups, there are 15 in each group. The If 6 remainder is 6. We may divide the remainder by placing the decimal point after 126. Then we 15.75 1126.00 write zeros in the 8 following decimal places and continue to divide. answer The decimal point 46 40 60 56 40 40 in the placed directly above the decimal point we placed in the division box. Practice Problem set 46 is The answer is 15.75. Find each decimal answer. a. 13.464 * 6 c. 3.4 e. 27.4 * 8 1. * 5 b. 0.0288 * 8 d. 145 - 4 f. 371 -s- 10 Two hundred wildebeests and 150 gazelles grazed on the savannah. What was the ratio of gazelles to wildebeests grazing on the savannah? 2. In their first 5 games the Celtics scored 105 points, 112 points, 98 points, 113 points, and 107 points. What was the average number their first 5 3. of points the Celtics scored in games? The crowd watched with bar was set to 19 feet 6 19 feet 6 inches? anticipation as the pole vault inches. How many inches is 3 Dividing a Decimal Number by a 21 Whole Number 4. Estimate the sum of 4387, 2914, and 796 by rounding each number to the nearest hundred before adding. 5. Draw a sketch to help with this problem. From Tad's From John's house to John's house is 2.3 kilometers. house to school is 0.8 kilometer. Tad rode from his house to John's house and then to school. Later he rode from school to John's house to his house. Altogether, how far did Tad ride? 6. About seven tenths of the earth's surface is water. (a) About what fraction of the (b) On the earth's surface, earth's surface is land? what the ratio of water to is land? 7. The tally lift indicated by 8. III number 8. What number is indicates the JHf |Hl jjfl JHl III ? Sketch this figure on your paper. Find the lengths of the unmarked sides and find 6 the perimeter of the polygon. Dimensions are in feet. All 3 4 I angles are right angles. 9. Name the point marked M M | on 10. 11. this number line i i i i i i i | 3.0 (a) as a decimal number. (b) as a 4.0 mixed number. Round 5.142857142857 (a) to the nearest millionth. (b) to the nearest What is the sum hundredth. of the first four prime numbers? — 214 Math 87 12. 13. 14. Use words to write (a) 1000.02 (b) 0.102 each number. Use digits to write (a) Sixty-seven hundred-thousandths (b) One hundred and each number. twenty-three thousandths Simplify each of these numbers. r o 3 i (a) 21 no [bj 936 12 15. and in order from Write 36 ° -, -, - with a common denominator and arrange least to greatest. Solve: *- = ™ 24 16 16. 18. 3.45 + a = 7.6 Add, (3.4)(5.6) 22. 4.315 24. 7.4 5 15 3- + 1- + 6 152- 35 30. -s- * 8 3 28. 19. 18 m 8 20 2.7 - b = 1.49 subtract, multiply,. or divide, as indicated: 20. 26. 17. • 8 — 12 21. (0.4)(0.6)(0.02) 23. 0.0144 + 9 25. 4— 12 7 27. 17 6.4 + 0.27 + 12 + 9.36 1 4- - 6 ( 4-14y 6 3 • - 3- 1 29. 42 + 0.144 -5- 6 5 21 Repeating Digits LESSON 47 Repeating Digits When a decimal number is divided, the answer sometimes a decimal number that will not end with a remainder of zero. Instead the answer will have one or more digits in a pattern that repeats indefinitely. Here we show two examples. 7.1666. 6)43.0000. 42 . . . . is 0.31818. ll)3.50000. .. . . 3 3 10 _6 40 36 40 36 40 36 20 11 90 88 20 11 90 88 4 2 decimal number are called the the repetend is 6. In 0.31818. the repetend. In 7.1666. repetend is 18 (not 81). One way to indicate that a decimal number is a repeating decimal number is to write the number with a bar over the repetend where the repetend first appears to the right of the decimal point. We will write each answer above as a decimal with a bar over the repetend. The repeating digits of a . 7.1666. Example 1 . = 7.16 . . ., 0.31818. . = 0.318 Rewrite each of these repeating decimals with a bar over the repetend. Solution . ., (a) 0.0833333. (b) 5.14285714285714. (c) 454.5454545. (a) The repeating . . . . . . digit is 3. 0.083 (b) This is a six-digit repeating pattern. 5.142857 21 6 Math 87 (c) The repetend We always decimal point. do not write a bar over a whole number. is to the right of the 454.54 Example 2 Solution Round each number (a) 5.316 (a) We decimal places. to five 25.405 (b) remove the bar and write the repeating right of the desired decimal place. 5.316 = 5.316666. Then we round digits to the . . to five places. I 5.3166(6)6. (b) . . remove the bar and continue the repeating pattern beyond the fifth decimal place. Then we round — 25.405405 to five places. I 25.4054©5. Solution 5.31667 We 25.405 Example 3 —* . — . 25.40541 Divide 1.5 by 11 and write the quotient (a) with a bar over the repetend. (b) rounded (a) Since place value is fixed by the decimal point, we can write to the nearest hundredth. zeros in the places to the right of the decimal point. The repetend is apparent. 36 (not 63). We write the quotient with a bar over 36 where it first is appears. 0.13636. . . = 0.136 ll)l.50000. 1 1 We continue dividing until the repeating pattern 0.13636 £jj 70 66 — 70 66 4 217 Repeating Digits (b) The hundredths' place is the second place to the right of the decimal point. i 0.1(3)636. Practice . —* . 0.14 Write each repeating decimal with a bar over the repetend. a. 2.72727. . Round each number c. 0.816666. b. . . . to the nearest thousandth. 0.6 d. 5.381 Divide 1.7 by 12 and write the quotient Problem set 47 e. with a bar over the repetend. f. rounded 1. Two decimal places. children in the nursery were boys. was the ratio of boys to girls in the nursery? fifths of the What 2. to four Four hundred thirty-two students were assigned classrooms. What per classroom? 3. The migrating birds flew for 7 hours at miles per hour. 4. Draw was the average number of students an average How far did the birds travel in a diagram for this statement. to 16 rate of 7 hours? Then answer the questions that follow. Seven ninths of the 450 students in the assembly were enthralled by the speaker. (a) (b) 5. How many How many of the students were enthralled? of the students were not enthralled? Round each number (a) 5.16 to four decimal places, (b) 5.27 23 218 Math 87 Refer to this pie graph to answer questions 6 and 7. Class Test Grades 6. How many more students a C or earned an A or B than earned D? 7. What 8. Sketch this figure on your fraction of the students in the class earned an 16 paper. Find the length of each unmarked side and find the perimeter of the polygon. Dimensions are A? 18 10 30 12 in inches. All angles are 48 right angles. 9. 10. 11. 12. Divide 1.7 by 11 and write the quotient (a) with a bar over the repetend. (b) rounded Use and to three digits to write the fifty-eight Use words decimal places. sum of twenty-seven thousandths hundredths. to write each number. (a) 760.005 (b) 3,524,000,000,000 Write the prime factorization of each number. (a) 71,000 (b) 1296 9 Repeating Digits 13. Find the length of segment BC to the nearest eighth of an inch. inches 1 I I I I I I I ' i I I I I I - • B 14. What is the least common multiple of 12 and 15? Solve: 21 w 24 40 15 20 9 y 15. 17. 18. m 19. n - Add, + 9.6 4.2 16. 12 9 x 6 = 14 = 1.63 subtract, multiply, or divide, as indicated: 20. 12 x 8.6 21. (0.4X0.5X0.6) 22. 6.165 - 9 23. 0.288 4- 24. 7.1 25. 7- - 1— * 4 6- + 4 65 30. 8.3 • 2^ 8 + \ 3 6 5 2 28. 5~ 12 • 2^ 27. 4 - ' 4-1 - 1- V 29. 6-4 7 + 0.72 + 15 + 3.96 + 0.108 8 12 8 26. 21 6 4y 220 Math 87 LESSON 48 Decimals and Fractions to Decimals Decimals to fractions To write a decimal number as a fraction, we write the digits after the decimal point as the numerator of the fraction. For the denominator of the fraction we write the place value of the last digit. Example to Fractions 1 Solution Then we reduce. Write 0.125 as a fraction. The digits 125 form the numerator of the fraction. The denominator of the fraction is 1000 because 5 is in the thousandths' place. 0.125 = 125 -=^- 1000 Now we reduce. 125 1000 Example 2 Solution 1 8 Write 11.42 as a mixed number. The number 11 the fraction is is 42, the whole number part. and the denominator is The numerator 100 because 2 is of in the hundredths' place. 11.42 = 11 Now we 42 100 reduce the fraction. — ^42 11 = „„21 11 50 100 Fractions to decimals To change a fraction to a decimal number, we perform the division indicated by the fraction. The fraction - indicates that 1 is divided by 4. 4JI 221 Decimals to Fractions and Fractions to Decimals may appear that we cannot perform this division. However, if we fix place values with a decimal point and write zeros in the decimal places to the right of the decimal point, we can It perform the division. The result equal to the fraction is a decimal number that is -. 0.25 Thus, 1.00 - = 0.25 4 _8 20 20 Example 3 Write each of these numbers as a decimal number. 23 7 100 Solution (a) 2 (b) (a) (c) 3- (d) 4 3 Fractions with denominators of 10, 100, 1000, etc., can be written directly as decimal numbers, without performing the division. 23 = 0.23 100 (b) An improper fraction is equal to or greater than 1. When we change an improper fraction to a decimal number, the decimal number will be greater than or equal to 1. 1.75 7 4 4)7.00 4 = 1.75 3 2 8 20 20 (c) To change a mixed number to a decimal number, we may change the mixed number to an improper fraction and then divide. Another way is to separate the fraction from the whole number and change the fraction to a 222 Math 87 decimal number. Then we write the whole number and the decimal number as one number. Here we show both ways. 3 = 19 4 5 5 5 0.8 5)4.0 5J19.0 40 15 "0 40 40 3Z 5 3.8 5 (d) To change 2 3 5 3.8 3- = 4 + - 4 3- = or to a = 3.8 decimal number, 0.666 _^ 3 we divide. = 0.6 3)2.000 18 20 18 20 18 2 We will write repeating decimal numbers with a bar over the repetend unless directed otherwise. Practice Change each decimal number a mixed number. a. 0.24 b. Change each fraction or 23 d. e. 4 Problem set 48 1. The 4 to a 45.6 c. to 2.375 mixed number to a decimal number. f. 5 ratio of Celtic soldiers to What reduced fraction or 5 g- 8 Roman fraction of the soldiers ° 6 soldiers was were Celts? 2 to 5. Decimals to Fractions and Fractions to Decimals 2. 223 Eric ran 8 laps in 11 minutes 44 seconds. (a) How many (b) What seconds did the average is it take Eric to run 8 laps? amount of time it took Eric to run each lap? 3. Some gas was gas, which still in the tank. Jan added 13.3 gallons of tank held a total of 21.0 gas was in the tank before Jan filled the tank. If the gallons of gas, how much added the gas? 4. From 1750 to 1850, the estimated population of the world increased from seven hundred twenty-five million to one billion, two hundred thousand. How many more people were living in the world in 1850 than in 1750? 5. Draw a diagram for this statement. Then answer the questions that follow. The 6. Jets won two (a) How many (b) What was The tally games did the the jiff thirds of their 15 games. Jets' Jets win? won-lost ratio? indicates the number 12. What is the tally for 16? 7. A square and a regular hexagon share a common side, as shown. The perimeter of the hexagon is 120 mm. What is the perimeter of the square? 8. Write each of these numbers as a fraction or as a mixed number. (a) 9. 0.375 (b) 5.55 Write each of these numbers as a decimal number. (a) 2§ 5 M 224 Math 87 10. 11. Round each number (a) 0.45 (b) 3.142857 to the nearest thousandth, Divide 1.9 by 12 and write the quotient (a) with a bar over the repetend. (b) rounded to three decimal places. 12. Four and five hundredths is how much greater than one hundred sixty-seven thousandths? 13. Simplify each of these numbers. (a) Q^- 90— 500 IO 14. Draw segment AB to be 1 Draw segment AC inch long. perpendicular to AB. Let segment AC be inch long. Complete triangle ABCby drawing segment BC Measure segment BC with a ruler. How long is segment BCl Solve: 15. £ = 21 . 12 8 17. — = — 45 18 19. 7.8 Add, = y- 16. " j b 18. 1.9 = A 10 = w + 0.42 6.9 subtract, multiply, or divide, as indicated: 20. 4.8 x 32 22. 24.156 24. 6.5 4- -s- 4 6 21. (0.12)(0.5)(0.02) 23. 0.072 25. 3-^7 - 10 -* 3 It 4 Division 26. 5- + 6— 10 2 30. LESSON 49 - 27. 7- 3- 2 3 5 - si 1* 3 6 5 14- 1 28. + 29. • 10- * 5 Answers 7 2 5.267 + 3.4 + 15 + 12.35 + 0.015 Division Answers We have considered three ways of writing the answer division problem in which there is a remainder. Example 1 1. Write the answer with a remainder. 2. Write the answer as a mixed number. 3. Write the answer as a decimal number. Divide 54 by 4 and write the answer (a) with a remainder. (b) as a mixed number. (c) as a decimal number. 13 Solution (a) We divide and find the result is 13 r 2. r 2 1|54 4 (b) The remainder is the numerator of a fraction and the divisor is the 14 12 denominator, so this answer can be written as 13t = 13-. 2 13.5 (c) 225 We place values by placing the decimal point to the right of 54. Then we can write zeros in the the following places and continue dividing to comfix pletion. The result is 13.5. 4)54.0 4 14 12 2 20 to a 226 Math 87 Sometimes answer written as a decimal number will be a repeating decimal number or will have more decimal places than the problem requires. In this book we will show the complete division of the number unless the problem states that the answer is to be rounded. Example 2 Solution a.division Divide 37.4 by 9 and round the quotient thousandth. We continue dividing until the answer has four decimal places. Then we round to the nearest thousandth. 4.1555 9)37.4000 36 1 4 ' * 9 4.156 4.15(5)5 Practice to the nearest Divide 55 by 4 and write the answer 50 45 50 45 50 45 a. with a remainder. b. as a mixed number. c. as a decimal number. d. Divide 5.5 by 3 and round the answer to three decimal places. Problem set 49 1. The rectangle was 24 inches long and 18 inches wide. What was the ratio of its length to its width? 2. Amber's test scores were 90, 95, 90, 85, 80, 85, 90, 95, and 100. What was her average test score? 3. The report stated two out of every five young people find a job. What fraction of the young were unable to people were able 4. 80, that to find a job? Rachel bought a sheet of fifty 29-cent stamps from the post office. How much did she have to pay? Division 227 Answers 5. Ninety-seven thousandths is how much less than two and ninety-eight hundredths? Write the answer in words. 6. Draw a diagram for this statement. Then answer the questions that follow. Five sixths of the 30 students passed the (a) How many (b) What was 7. students did not pass the test? who the ratio of students to students test. who passed the test did not pass the test? Sketch this figure on your paper. Find the length of each unmarked side and find the perimeter of the polygon. Dimensions are in meters. All angles are right 4 8 1 ia 15 angles. 16 8. Write 0.75 as a fraction. 9. Write — as a decimal number. 8 10. Round 11. Divide 54 by 5 and write the answer 12. 123.6 to the nearest thousandth. (a) with a remainder. (b) as a mixed number. (c) as a decimal number. Divide 5.4 by 11 and write the answer with a bar over the repetend. 13. What composite number is equal to the product of the four smallest prime numbers? 14. Arrange these numbers in order from 1.2, -12, 0.12, 0, i least to greatest: 228 Math 87 15. What is the sum number line? numbers marked of the M 1 i i i i ' 1 i 1 * V i 1 M and N on this N 1 i i i 1 1 1 | i i 1 1 V I i 1 1 1 1 i 1 1 2.4 2.5 2.6 Solve: 16. 12 — 8 = 25 18. 12 x 20. 0.1 = P 4 90 18 - z 1 subtract, multiply, or divide, as indicated: 21. (2.5)(2.5) 23. 16.42 25. _1 5 c 3 5+ - + 24 34 -5- 8 6 3 27. — = —n 15 4 = 3.14 + Add, n 17. m 9 19. + 1.2 24. 0.153 - 9 26. 3 2 3- • i- 5-1-5^ f 3 6J l 1 1 • x 0.4 x 0.05 22. 8 28. 7 + 10- 3 2 29. 5.46 + 2.791 + 21.4 + 10 + 0.199 30. Compare: 45.0 /c\ 4.5 30 LESSON 50 Dividing by a Decimal Number We remember that we do not change the value of a fraction we if multiply the fraction by another fraction that has the same numerator and denominator. If we multiply - by 10 over 10, 1 2 10 10 ~ 20 10 Dividing by a Decimal we get 10 over 20, which is another name for Number 229 one We half. use this fact to change division by a decimal number into a division by a whole number. If we want to divide 1.36 by 0.4, we have 1.36 0.4 We can change the divisor to the whole number 4 by multiplying bv J 10 over The value 10. 1.36 10 0.4 10 = 13.6 of 13.6 divided by 4 by 0.4. This means have the same answer. 1.36 divided 4 is the as the value of that both of these divisions 4)13.6 equals 0.4)1.36 same To divide by a decimal number, we move the decimal point in the divisor to the right to make the divisor a whole number. Then we move the decimal point in the dividend the Example 1 Solution same number of places Divide: 3.36 We to the right. - 0.06 use a division box and write 0.06)3.36 First we move right to make the decimal point in 0.06 two places to the it 6. 0Q6J3.36 Then we move the decimal point in 3.36 the same number of places to the right. 0Q6.)33,6. The decimal point in the answer location of the decimal point. 6)336. is just above the new 230 Math 87 Now we divide. 56. 6J336. 30 36 36 Thus, 3.36 * 0.06 = 56. Example 2 * Divide: 0.144 0.8 0.18 Solution We 0.8 need to move the decimal point in one place to the right to get the whole number 8. Then we move 08.) 1.44 8 the decimal 64 64 point one place to the right in 0.144 to get 1.44. Example 3 Solution Then we Divide: 15.4 -s- divide. 0.07 We move both decimal 220. points two places. This makes an empty place in the division box, which we fill with a zero. We keep dividing until we reach the decimal point. We find 220 as the answer. Example 4 Solution Example 5 007.)l54p. 14 14 14 Divide: 21 + 0.5 42. We move the decimal point in 0.5 one place to the right. Then we move the other decimal point one place to the right. We find 42 as the answer. Divide: 1.54 *- 05.)210. 20 10 IP 0.8 1.925 Solution We do not write a remainder. We write zeros in the places to the right of 4 and continue dividing until the remainder is zero. Og.) 15^.400 _8 74 7_2 20 16 40 40 Dividing by a Decimal Practice Problem set 231 Number Divide: a. 5.16 * 0.6 b. 0.144 c. 23.8 + 0.07 d. 24 1. Raisins and nuts were 50 -s- -h 0.09 0.08 mixed in a bowl. If five eighths the mixture was made up of nuts, what was the ratio of of raisins to nuts? 2. The new coupe traveled 702 kilometers down the autobahn in 6 hours. The new coupe averaged how many kilometers per hour? 3. Fifty-four than fifty and five hundredths is how much and forty thousandths? greater Refer to this election tally sheet to answer questions 4 and 5. Vote Totals 4. Judy Jflf Carlos %m Jiff JKM mi Yolanda Jiff Khanh jjtfuri wtf The winner IWf II iiu of the election received how many more votes than the runner-up? 5. What fraction of the votes did Carlos receive? 6. Draw a diagram for this statement. Then answer the questions that follow. Four sevenths of those who rode the Giant Gyro at the fair were euphoric. All the rest were vertiginous. (a) What fraction of those who rode the ride were vertiginous? (b) 7. What was the ratio of euphoric to vertiginous riders? What is the least common multiple of 10 and 16? 232 Math 87 8. The perimeter of this rectangle is 56 cm. What is the length of „ 10 the rectangle? 9. 10. 9 Write as a decimal number. 6 Round each number decimal places. to five 23.54 (a) 12. / Write 62.5 as a mixed number. 100 11. cm 0.916 (b) Divide 51 by 6 and write the answer (a) with a remainder. (b) as a (c) as a decimal mixed number. number. 13. Divide 5.1 by 9 and write the quotient rounded to the nearest thousandth. 14. Use and 15. Draw segment product of twelve hundredths eight tenths., Then write the answer in words. digits to write the XY to be 2 cm long. Draw segment YZ perpendicular to XY and 1.5 cm long. Complete triangle XYZ by drawing segment XZ. How long is segment XZl Solve: 16. 50 25 19. 777 Add, 21. t = 17. 10 + 0.23 = 1.2 3 7 w 28 12 3 44 a 18. 20. r - 1.97 = 0.65 subtract, multiply, or divide, as indicated: (0.15)(0.15) 22. 1.2 x 2.5 x 4 Unit Price 23. 25. 14.14 * 5 5 s + 6 8 — 7 27. • 15 29. A 12 + 10 • 24. 0.096 - 0.12 26. 4 2 1 2- f 1 ll 2- - 1- 3 l 4J 1 28. I 7 Add i- 233 10 5 mentally: 4 + 6 + 9 + 8 + 7 + 5 + 3 + 4 + 1 + 7 + 4 + 3 30. Solve mentally: 4 + 14 + 8 + 9 + 12 + 14 + 5 + 3 + 7 + LESSON N=84 Unit Price 51 As an aid to grocery store customers, the unit price for various products is often posted. The unit price is the cost for a single unit measurement of the product. The unit price can be found by dividing. Example 1 What is the unit price of a 24-ounce box of cereal that costs $3.60? Solution The cereal of 1 ounce. is measured We in ounces. 24 ounces Example 2 unit price What is $4.50? unit price is the cost divide the price by 24 ounces. $0.15 $3.60 The The is 1 ounce $0.15 per ounce, which is 15c per ounce. the unit price of a 36-ounce box of cereal that costs 234 Math 87 Solution The unit price- for the cereal the price per ounce. is We divide the price by 36 ounces. $4.50 $0,125 36 ounces The unit price is ounce 1 12.5c per ounce. Unit pricing helps customers determine which brand or which size package provides the better buy. From the two examples in was the Practice a. this lesson, better What is we buy because see that the larger box of cereal it cost less per ounce. the unit price of a 28-ounce box of cereal that costs $1.12? b. What is the unit price of an 11 -ounce can of soup that costs 55C? c Which the better buy: an 18-ounce jar of jelly that is costs $1.98, or a 24-ounce jar of jelly that costs $2.28? Problem set 1. Brand X costs $2.40 for 16 ounces. Find the unit price the better buy? for 12 ounces. 51 brand 2. The is taxi ride cost 1 dollar plus mile traveled. 3. for Y costs $1.92 each brand. Which 40c more for each quarter What was the total cost for a 2-mile trip? on the farm. Thirty-six cows on the farm. What was the ratio of sheep to Forty-eight sheep were were also cows? 4. Brand At 4 different stores the price of 1 gallon of $1.86, $1.83, $1.98, milk was and $2.09. Find the average price per gallon rounded to the nearest cent. 5. Two and three hundredths is how much and two tenths? Write the answer less in words. than three 6. Draw a diagram for this statement. Unit Price 235 Then answer the questions that follow. Three eighths of the 48 roses were red. 7. (a) How many roses (b) How many roses were not red? (c) What 9. 10. were not red? fraction of the roses Replace each circle with the proper comparison symbol, 3.0303 (a) 8. were red? O 3.303 © 0.6 (b) 0.600 From goal line to goal line, a football field long. How many feet long is is 100 yards a football field? Write 0.080 as a fraction. Divide 48 by 5 and write the answer (a) with a remainder. (b) as a mixed number. 11. Round 14.285714 12. Estimate the sum to three decimal places. and 28,519 and 43,456 by the nearest thousand before of 37,142 rounding each number to adding. 13. Write — 1 as a decimal number. 11 14. What is the average of the first five prime numbers? Solve: 15. ^= 12 17. 4.7 25 a _ w = 1.2 18. 6 = _b_ 8 100 43 = 821 - m 236 Math 87 19. Sketch this figure on your paper. Find the length of each unmarked side. Then find the perimeter of the polygon. Dimensions are in angles All inches. are 4 4 5 right angles. Add, subtract, multiply, or divide, as indicated: - 20. (5 21. 10 22. 3- + 4- + 1- 5 • 5 10 • 3 5) • 10 • 3 • (5 10 • 5) 10 • < 1 4 8 24. • 23. 1 3 8 - x 4 x 1- 26. 3.45 + 6 + (5.2 28. 0.35 x 2.4 30. Add 3-1 6 2 2 5- - 0.57) 25. 6- + 27. 2.4 29. 4.26 * 40 4 * 0.016 mentally: 4+6+5+8+12+14+3+6+8+9+4 LESSON 52 Exponents We remember that we can show repeated addition by using multiplication. 5 There has the same value as + 5 + 5 + 5 is also a show repeated 4x5 way to show repeated multiplication. We can multiplication by using an exponent. 5 • 5 In the expression 5 4 the 4 , • 5 is • 5 = 54 the exponent and the 5 is the 237 Exponents The exponent shows how many times the base base. used as a is to be factor. — base **~ 54 exponent The following examples show how we read expressions with exponents, which we call exponential expressions. 42 "four squared" or "four to the second power" 23 "two cubed" or "two 54 "five to the fourth 10 5 to the third power" power" "ten to the fifth power" To find the value of an expression with an exponent, we write the base the number of times shown by the exponent. Then we multiply. 54 Example 1 Solution Example 2 Solution Simplify: 42 (a) 42 = 4 (b) 23 (c) 10 5 = 10 • 2 • .Simplify: 4 2 5 • 5 • • 5 23 (b) = 625 (c) 10 5 4 = 16 (a) = 2 = 5 - • • = 8 2 10 • 10 • 10 • 10 = 100,000 23 We first find the value of each expression. Then we subtract. 42 k _ 23 16 - 8 = 8 Practice Use words to show how each exponential expression is a. 43 b. 52 c. 10 6 d. 3 In the expression 10 number is , read. what number is the base and what the exponent? . 238 Math 87 Simplify: e. Problem set 1 5 10 4 : f. g. h. 3; &1 32 In 1 803 the United States purchased the Louisiana territory , from France for $15 million. In 1867, the United States purchased Alaska from Russia for $7.2 million. The purchase of Alaska occurred how many years after the purchase of the Louisiana territory? 52 2. Red and blue marbles were in the bag. Five twelfths of the marbles were red. 3. (a) What (b) What was the ratio of red marbles to blue marbles? Which 5. 6. were blue? A 6-ounce can of peaches sells for 90c\ A 9-ounce can of peaches 4. fraction of the marbles Find the unit price the better buy? sells for $1.26. size is for each size. two numbers is the number halfway between the two numbers. What number is halfway between two thousand, five hundred fifty and two thousand, nine hundred? The average of Five hundred thirty-three thousandths is how much more than forty-five hundredths? Use words to write the answer. Draw a diagram for this statement. Then answer the questions that follow. Five sixths of the 30 students smiled their teacher told the joke. (a) (b) 7. Use is How many How many digits less when students smiled? students did not smile? and symbols to write "Twenty-five thousandths than three hundredths." Exponents 8. (a) Estimate the length of segment (b) Use a centimeter AB to AB in e -•- Write each of these numbers as a decimal number. 5 3- (a) (b) 8 3 Divide 2.5 by 22 and write the answer (a) with a bar over the repetend. (b) rounded to the nearest thousandth. 11. Estimate the product of 596 and 306. 12. In the expression 5 3 what number 13. 14. is , what number is the perimeter of a regular hexagon is how many (a) inches long? What fraction of this rectangle is shaded? What fraction of this rectangle is not shaded? Refer to quadrilateral ABCD to answer the following questions. (a) Which angle is a right angle? (b) Which angle appears to be obtuse? (c) What kind ZC? the exponent and the base? If (b) 15. segment the nearest centimeter. A 10. centimeters. scale to find the length of -•- 9. 239 of angle is is 1 foot, each side 240 Math 87 Solve: 16. 18. 28 m 49 72 + m Add, 50 100 17. = 340 19. 12 = 7.2 - 0.27 77 subtract, multiply, or divide, as indicated: 20. 32 + 23 21. 10 4 - 10 3 22. - + -+- 23. 3- - 4 5 24. 3 3 3- x 1- x 4 6 ( l\ 1- -- 2 V V 4-6- 25. 9 3 26. 24 - 15.8 + (12 - 3.64) 27. 0.12 x 0.15 28. 100 x 0.0125 29. 0.1 - 4 30. 10 + 0.25 LESSON Powers of 10 53 The powers of 10 are easy positive to write. The exponent matches the number of zeros in the product. 10 2 = 10 10 3 = 10 10 4 = 10 Place value • 10 = 100 • 10 • 10 = 1000 • 10 • 10 • (two zeros) (three zeros) 10 = 10,000 (four zeros) We can use powers of 10 to show place value, as the chart below. Notice that 10° equals 1. Etc. Trillions Millions Billions we see in Units (ones) Thousands -c I "o CO CO D •o o "D C 3 CO co TD T5 0) CO W CD CD C= c t5 C 3 X H O X co c o h- oC C 3 O X 52 a> co c <D I- DC ^ o x «S CD =J CO to CD CD O c i- Hundreds ^ 1014 1013 1012 1011 1010 10 9 10 8 10 7 10 6 10 5 10 4 10 3 Tens o o Ones <=> O Powers Powers of 10 are notation. In expanded notation we write a number sum Example 1 Solution sometimes used to write numbers in expanded of each nonzero digit times Of 1 Example 2 Solution place value. The number 5206 means 5000 + 200 + 6. each number as a digit times its place value. (5 by powers its as the Write 5206 in expanded notation using powers of 10. 5000 Multiplying 241 of 10 + x 10 3 ) + 200 (2 + We will write 6 x 10 2 ) + (6 x 10°) When we multiply a decimal number by a power of 10, the answer has the same digits in the same order. Only their place values are changed. Multiply: 46.235 x 10 2 This time we will write 10 2 as 100 and multiply. 46.235 x 100 4623.500 We see that the same digits occur in the same order. Only the place values have changed as the decimal point has been x shifted two places to the right. Thus, to multiply by a power of 10, we need only to shift the decimal point to the right the number of places indicated by the exponent. Example 3 Solution Multiply: 3.14 x 10 4 The power of 10 shows us the number decimal point to the right. We move of places to move the the decimal point four places to the right. 3.14 x 10 4 = 31,400 Practice Write each a. 456 number in expanded notation by using powers of 10. 242 Math 87 b. 1760 c. 186,000 Multiply: Problem set d. 24.25 x 10 3 e. 25 x 10 6 Refer to the graph to answer questions 1-3. 53 How Walk 1. Answer (a) Ride a bike to School Ride in a bus Ride in a car true or false. Twice as many school in a (b) Come Students students walk to school as ride to car. The majority of the students ride to school in a bus or car. 2. What who is the ratio of those who walk to school to those ride the bus? 3. What fraction of the students ride the bus? 4. What is the average of these numbers? 1.2, 1.4, 1.5, 1.7, 2 Powers 5. What is 6. Draw and one and the product of twelve thousandths two tenths? Write the answer 243 of 10 in words. a diagram of this statement. Then answer the questions that follow. Only one eighth of the 40 students answered question 5. correctly (a) How many students correctly answered question 5? (b) How many students did not correctly answer question 5? 7. Replace each circle with the proper comparison symbol, 4.0102 (a) 8. A cubit (a) is (b) 5.014 O 50.140 the distance from the elbow to the fingertips. Estimate the your (b) O 4.0120 number elbow of inches from your to fingertips. Measure the distance from your elbow your to fingertips to the nearest inch. 9. Write 0.375 as a fraction. 10. Divide 59 by 4 and write the answer as a decimal number. 11. Round 53714.54;ô the nearest (a) thousandth. (b) thousand. 12. Write 5280 in expanded notation using powers of 10. 13. The point marked by the 1 arrow represents what decil mal number? i i i i 3 q Solve: .„ 14. — = 10 6 w 100 - K 15. 36 16 = — — x 24 i i i i i l 31 244 Math 87 x +8.9 16. 9.8 = 18. In figure ABCDEF, are right angles 400 - y = 263 17. all and AF = AB b = BC. Segment BC is twice the length of segment CD. If CD is 3 cm, what is the perimeter of the figure? 19. -> C D subtract, multiply, or divide, as indicated: - 92 20. 53 22. 4- + 5- + 5 3 3 5 3.6 23. el3i 8 25. \6 26.7 + 3.45 + 0.036 + 12 + 8.7 27. 5 29. 3.625 (0.4 4- - ( - 5 8 26. - x 10 3 21. 2 8- x 3- x 3 LESSON 54 B V Sketch a circle. Within the circle sketch a regular hexagon so that each vertex of the hexagon "touches" the circle. Add, 24. A <o angles 0.032) 100 A + 4 28. 0.06 x $12.50 30. 3.8 + 0.16 Rectangular Area, Part The diagram below represents the been covered with square side. How many floor of the 1-ft floor of a hallway that has floor tiles that are 1 foot square tiles does it on each take to cover the hallway? floor 1ft 4 1 tile 1ft ft 8ft We see that there are 8 there are 32 1-ft floor tiles in square tiles. each row and 4 rows. So Rectangular Area, Part 245 1 The floor tiles cover the area of the hallway. Area is an amount of surface. Floors, ceilings, walls, sheets of paper, and polygons all have an area. If a square is 1 foot on each side, it is a square foot. Thus the area of the hallway is 32 square feet. Other standard square units in the U.S. system include square inches, square yards, and square miles. Units of area in the metric system include square centimeters, square meters, and square kilometers. It is important to distinguish between a unit of length and a unit of area. Units of length, such as an inch or a centimeter, are used for measuring distances, not for measuring areas. To measure area, we use units that take up area. Square centimeters and square inches take up area and are used to measure area. We include the word "square" or the exponent 2 when we designate units of area. Units of Area Units of Length 1 cm 1 in. cm 2 = 1 square centimeter 1 1 1 square ft How many square floor tiles 1 foot on each side would be needed to cover the floor of a rectangular room 12 feet long and 10 Solution = in. inch 12 Example 2 1 feet use parallel lines to draw squares. Twelve tiles will fit in each row. There are 10 rows. Ten rows with 12 tiles in each row equals 120 tiles. Since each tile square ft wide? We is 1 10 foot, the area of the room is 120 square feet (120 2 ft ). 12 fl 10 ft 246 Math 87 Notice that, the area of the rectangular room equals the length of the room times the width. Area of rectangle = length x width Example 2 What is the area of 5 this cm rectangle? 3 Solution The the area of the rectangle number meters it cm is of square centi- 5 We cm 3 cm). (3 3 Area of rectangle = 5 cm = 15 cm 2 3 • Example 3 cm takes to cover the can find this number by multiplying the length (5 cm) times the width rectangle. cm 3 The perimeter cm cm of a certain square is 12 inches. What is the area of the square? Solution length of the sides. A square we know the has 4 equal sides, so we divide each side is 3 inches. Then we To find the area of the square, need first 12 inches by 4 and find that multiply the length (3 in.) by the width to (3 in.) to find the area. 3 in. 3 Practice Area = 3 = 9 in. x 3 in. in. in. 2 Find the area of each rectangle. 15m a. 2 b. C in. 10m 5 d. If 4 4 cm cm in. the perimeter of a square is 20 cm, what is its area? Rectangular Area, Part Problem set 54 247 1 1. During January the precipitation was 4.5 inches. During February the precipitation was 5.7 inches and during March, 4.2 inches. What was the average amount of precipitation per month for the 3-month period? 2. If 6 ounces of sushi costs $1.86, what is the price per ounce? 3. The parking 4. Sixty girls and 75 boys were seated charges $2 for the first hour plus 50$ for each additional half hour. What is the total charge for parking a car in the lot for 4 hours? What was lot the ratio of boys to on the bleachers. girls seated on the bleachers? 5. Five billion, three hundred ten million is more than two billion ninety-seven million? 6. The Smiths are covering their kitchen floor 1 foot square. If with tiles 8 ~12T 12 feet long and 8 feet wide, how many tiles will they need? Draw ft the kitchen is 7. how much a diagram of this statement. Then answer the questions that follow. Three fifths of the 120 people in attendance agreed with the scholar. (a) How many of those in attendance agreed with the scholar? (b) How many of those in attendance did not agree with the scholar? 8. Find the length of the seemen (a) in millimeters. .mm in centimeters. 20 30 40 iiliiiilii nliiiiliiiiliiiiliiiil i I (b) 10 cm 248 Math 87 9. Write 6— as a decimal number. 6 10. Divide 6.3 by 11 and write the answer rounded to the nearest ten-thousandth. 11. Estimate the difference between 39,875 and 21,158 by rounding to the nearest thousand before subtracting. 12. Write 50,704 in expanded notation using powers of 10. 13. (a) What is the perimeter of this square? (b) What is the area of this 10 square? 14. We know that 10 2 = is 15. 100 and 12 2 = 144. If cm AT 2 = 225, what AT? (a) What fraction of this square (b) is shaded? What decimal this square part of is not shaded? Solve: 12 3^ 100 d 16. 18. 453 = 76 + a Add, 15 y 27 19. c - 3.5 = 1.47 subtract, multiply, or divide, as indicated: 20. 82 + 62 22. 4 4 + 5 10 17. 4 + 4 3 21. 4.7 x 10 4 23. 3- 2 249 Square Root 3- x 3- x 4. 3 6- * 3- 25. 8 5 8 3 26. 0.52 + 0.76 + 0.8 + 0.29 + 0.016 27. 3.6 29. 0.05 * 25 - (7 - 5.437) 28. (12)(1.2)(0.12) 30. 300 + 0.015 LESSON Square Root 55 We remember that the exponent of an exponential expression tells how many times the base squared means 5 is to say, "Five squared to be used as a factor. Five be used as a factor twice. 52 We is is = 25 twenty-five." The inverse operation of squaring a number is the operation "undoes" squaring. This operation is called finding the square root of a number. We indicate square root with a that radical sign. To show the square root of 25, we write V25 We say, "The square root of twenty-five." The square itself 25 root of 25 produces 25. Since is the number which multiplied by 5x5 equals 25, the square root of is 5. V25 = Example 1 Find the square root: (a) Vl6 5 (b) VT00 (c) a|625 250 Math 87 Solution Example 2 Solution (a) Since 4 (b) Since (c) Since 25 10-10 Subtract: a/25 We 4 equals 16, a/16 = • • - equals 100, a/Too 10. 25 equals 625, a/625 25. a/16 radicals before we can - radicals must simplify a/25 a/16 5-4 The term "square The length 4. = 1 root" subtract. simplified comes from of the side of a square a geometric idea. the square root of the is area of the square. Example 3 Solution The area of this square (a) What is the length of each side? (b) What is the perimeter? (a) Since a square is a rectangle with sides of equal length, we can find the length of each side if we know the area is 36 in. 2 . of the square. Side x side = 36 Since each side 6 is in. in. equal, each side x 6 = 36 in. 2 must be in. 2 Each side equals the square root of 36 36 (b) The perimeter = 6 in. 2 of the square is Practice in. 2 . in. the Perimeter = 4 x 6 = 24 6 in. sum of its 4 sides. in. in. Simplify: a. a/64 b. a/121 c. a/9 + a/16 Square Root Problem set 251 d. What is the perimeter of a square whose area is 100 cm 2 ? 1. Alaska was purchased by the United States in 1867. Alaska became the forty-ninth state 92 years later. In what year did Alaska become a state? 2. Brand 55 X for 16 ounces. 3. The 4. 4. Brand Y costs $1.58 the better buy? costs $1.26 for 14 ounces. Which brand is green beans to peas in the garden was 11 to was the ratio of peas to green beans? ratio of What During the month of February, Christy's weekly grocery bills were $110.47, $115.68, $96.40, and $120.10. Find her average weekly grocery bill in February to the nearest dollar. 5. Six and seven hundredths Write the answer in words. 6. Draw is how much less a diagram of this statement. than eight? Then answer the questions that follow. Seven twelfths of the 60 buttons in the box had 4 holes. 7. (a) What fraction of the buttons did not have 4 holes? (b) How many buttons Replace the circle with the proper comparison symbol. 4.06 8. O4 -0 60 Write each decimal as a fraction or as a mixed number, (a) 9. did not have 4 holes? 0.12 (b) 0.012 Divide 5.9 by 12 and write the answer with a bar over the repetend. 252 Math 87 10. Find the length of the segment inches . 1 . 2 1 1 1 1 1 1 1 1 3 | I 1 1 1 1 1 cm 1 1 2 3 4 6 5 (a) to the nearest centimeter. (b) to the nearest eighth of 7 an inch. two million is the dividend and two hundred divisor, what is the quotient? 11. If 12. Simplify each of these numbers. 8 (a) 20 (b) is the 560 640 13. Write 250,000 in expanded notation using exponents. 14. If 15. Draw segment AB the perimeter of a square is 36 inches, what 18. area? be 4 cm long. Next draw segment BC perpendicular to segment AB and 3 cm long. Then form a triangle by drawing segment AC. Measure to find the length of segment AC. to Solve: 16. is its 30 25 18 = _9_ 100 75 / 3w Add, 17. 19. 7.8 4 - m = 1.24 subtract, multiply, or divide, as indicated: - 34 20. 92 22. 26- + 15- + 83 8 2 - V9 21. a/25 23. 7 15— 10 - 3 8- 4 ; 253 Rates 24. 1111— 7- x 5- x 2 10 3 26. 3.7 27. 10 29. 1.3 3 1 9- - 5- 25. 5 3 + 18.9 + 0.65 + (0.125 x 10 2 ) - (0.1 - 0.099) x 0.7 x 1.1 28. 0.1001 * 13 30. 7 *- 0.035 LESSON 56 Rates A rate is two measurements. Either measurement Leo can walk 6 miles in 2 hours, we can a ratio of can be on top. If write two rates. When writing rates, we simplify the fraction. —6 miles 2 — hours = n 6 miles miles cereal costs 27 cents for 2 ounces, or 2 we can write ounces 27 cents 2 ounces 27 cent Jim can drive 100 miles on 3 gallons of gas, 100 miles = __1 miles oo Some rates gallons — 100 miles 3 or 3 gallon 3 gallons hour 3 cents 27 cents = „_1 13 2 ounce 2 ounces If , ., n— mile 3 .-, 3 miles per " ,"1, read - hour per mile r hour 1 „„ , read -j hour hours 2 If _ = 3 we can 3 gallons 100 mile = have special names. — distance rp, The rate The rate — — gallon The rate -3—pj dollar The rate -=- : . , is speed. is mileage, is a rate of exchange. is also a rate of exchange. time r= — dollar francs write 254 Math 87 Example Solution Edmund (a) Write two rates for this statement. (b) What was (a) hours 24 miles his speed? 3 (b) a. 1 hour or 8 mile Speed has time 8 Practice rode 24 miles in 3 hours. 24 miles 3 hours 8 miles hour as the denominator, so his speed is miles ' hour When Monica landed in Belgium, she exchanged S40 for 1640 francs. What was the rate of exchange in francs per dollar? What was the rate of exchange in dollars per franc? Problem set 56 b. Their car traveled 322 miles on 14 gallons of gas. Write two rates for this statement. What was the mileage? c. The Smiths drove 416 miles in 8 hours. Write two for this statement. What was their average speed? 1. The train traveled 384 kilometers in 4 hours. Write the two rates for this statement. What was the train's average rates speed? 2. Twenty-four ounces of cereal cost $3.12. What is the unit price? 3. During 1 hour of television programming, there were 12 minutes of commercials. (a) What fraction (b) What was the ratio of commercial to noncommercial of the hour was commercial time? time? 4. What number is halfway between 6.7 and the average.) 11.9? [Hint: Find 255 Rates 5. Jason bought 2 pounds of meat for $1.85 per pound, 3 cans of tomato sauce for $0.39 per can, and a package of spaghetti for $1.49. What was the total cost of the items? 6. If 7. the perimeter of a square is 1 foot, its many square inches? Draw a diagram of this statement. area is how Then answer the questions that follow. It (a) (b) rained two How many How many fifths of November? days did it rain in days did it not rain in November? 8. Find the reciprocal of 3—. 9. Write — the days in November. as a decimal number. 8 10. 11. Divide 123 by 4 and write the answer (a) with a remainder. (b) as a mixed number. Round 5.45^ (a) to the nearest thousandth. (b) to the nearest hundredth. 12. Write three billion, two hundred million in expanded form using exponents. 13. (a) What fraction of this square (b) is shaded? What decimal part of this square shaded? is not 256 Math 87 14. Copy on your this figure paper. Find the length of the unmarked sides 6 12 and 20 find the perimeter of the 14 polygon. Dimensions are in centimeters. All angles are right angles. 15. The moped moped 24 traveled 78 miles on 1.2 gallons of gas. averaged how many The miles per gallon? Solve: 35 16. 17. 60 18. - 24 + x = 2 19. 40 100 15 w 58 = 4 Add, subtract, multiply, or divide, as indicated: - 20. 15 2 53 21. V36 + V64 22. 7 „1 c 5 5+ 4- + c6— 23. 5 6 24. ( { 4-4-3- 25. 4 5 f* 4-1 - 3-2^ 4 V 7i 2 + 0.375 + 14.84 + 15 + 0.09 26. 7.6 27. 3 29. 0.1001 * 110 - - 9 2 6 (0.2 - 0.001) LESSON 57 - 467 28. 2.4 x 1.2 x 10 30. $14.52 * 0.06 : Percent Percent percent is is a Latin a means by the hundred. Thus a with a denominator of 100. The not written. The denominator of 100 word fraction denominator of 100 is that 257 Percent is indicated by the 1 word "percent" or by the symbol %. means percent 100 13% means v 100% 13 100 means 130 100 means 100 = 100 1 A percent describes a whole as though there were 100 parts, even though the whole may not actually contain 100 parts. For instance, we may say that 50 percent of this square is shaded because if this square were divided into 100 equal parts, 50 of the parts would be shaded. -i Thus 50 percent equivalent to - equivalent to -. is a way to describe -. because 50 percent means Fifty percent is — , and 50 100 IS We note that 100 percent equals 1, so 100 percent of a number is the number. One hundred percent of 42 is 42. One hundred percent of 130.655 is 130.655. We remember that we can multiply or divide a given number by 1 without changing the number. 1 = 50 5° =50 50 x 1 Since 100 percent is equal to 1, we can multiply or divide a number by 100 percent without changing the number. To write a given number as a percent, we multiply by 100 percent. 0.75 x 100% = 75% 258 Math 87 To write a given percent as a number, we divide by 100 percent. 75% 100% Example 1 Solution Write — 7 75 3 100 4 = 0.75 as a percent. To change a number to a percent, we multiply the number by 100 percent. 100% x 10 700 10 multiplied by % multiplied 70% Example 2 Solution 100% simplified 2 Write - as a percent. We multiply by 100 percent. - x 100% multiplied by 100% 3 200 % multiplied 66-% simplified 3 1 Example 3 Solution Write First 1— as a percent. we write 1— either as a fraction or as a decimal. 4 15 1- - 4 Then we can change 4 1 1- = 1.25 4 either of these numbers to a percent by multiplying by 100 percent. 4 x 100% = —% 4 = 125% 1.25 x 100% = 125% Percent Example 4 Solution Change 70% to a fraction. To remove the % symbol, we divide by 100%, 70% 100% 7 10 Thus 70 percent means the same thing Example 5 259 as (a) What (b) What percent of the square is shaded? (c) What fraction of the square is — shaded? percent of the square is not shaded? Solution (a) One (b) To change fourth of the square shaded. is —\ to a percent, i x we 100% = ^5°% = 25% 4 4 (c) The whole square equals 100%. Since 25% 75% Practice multiply by 100 percent. is is shaded, not shaded. Write each percent as a fraction or as a mixed number. a. 30% b. 120% Write each number as a percent. c. - d. 1.5 5 Problem set 57 1. pedaled hard. He traveled 80 kilometers in 2.5 hours. What was his average speed in kilometers per hour? Sam 2. Write the prime factorization of 2016. 3. Write each percent as a fraction or as a mixed number, (a) 8% (b) 150% 260 Math 87 The graph shows how one How Income is Spent family spends their annual income. Use this graph answer questions 4-6. 4. What percent family's income of is to the spent on "other"? 5. What of fraction family's income is the spent on food? 6. If $3200 is spent on insurance, how much is spent on taxes? 7. Draw a diagram of this statement. Then answer the questions that follow. Van has read 8. five eighths of the 336-page novel. (a) How many pages has Van read? (b) How many more pages are left to read? Write each of these numbers as a percent. (a) 0.25 (b) 1^ 5 9. Divide 2016 by 20 and write the answer as a decimal number. 10. Write 0.54 as a decimal number rounded to three decimal places. A 11. package contained 17 pieces and cost 85 cents. Write the two rates implied by this statement. 12. Write 623 in expanded notation using exponents. 261 Percent 13. What the perimeter of this is shape? Dimensions are in inches. 5 All angles are right angles. 12 7~ Refer to the rectangles in figure ABCDEFto answer questions 14 and 15. cm 6 cm 8 C E 14. 15. 5 cm B 7-cm D (a) What is the area of rectangle ABCG? (b) What is the area of rectangle DEFG? What the perimeter of hexagon is ABCDEF? Solve: 15 16. w 40 18. 25 22. 3- - + l4 ( 6 11 16 26. 19. 20 15 x 12 1=1+/ 2 3 J subtract, multiply, or divide, as indicated: 20. 24. 6m 1.44 = Add, 17. + 33 ^ 1 1+ 1- 8.7 + 9.64 VlOO - V36 23. 8l + 25. 9 + 25 + 1.456 ( 4 V V 24 25 21. V V iU 3 2 4 - ) r 3 1 y 4 J 262 Math 87 27. 10 - (0.9 - 0.876) 29. 0.1 * 25 28. 4 x 3.16 x 10 4 30. $13.93 * 0.07 Mixed Measures • Adding Mixed Measures LESSON 58 Mixed The measurement measures 18 inches can be changed into feet and inches. Since 12 inches equals 1 foot, we can divide 18 by 12 to find the number of feet in 18 inches. The remainder is the remaining number of inches. 1 ft 12)l8 12 6 18 Example 1 Solution Change 17 Since 3 ft of yards. ft to yards and = 1 ft 6 in. feet. we divide 17 by 3 to find the number remainder is the remaining number of feet. equal 1 yd, The in. in. 5yd 3)l7 15 2ft 17 Example 2 Solution Simplify: 1 The units is greater ft 18 ft = 5 yd 2 ft in. and inches. However, the number of inches than the 12 in. that are in 1 ft. So we change 18 in, are feet Mixed Measures to feet 263 Adding Mixed Measures • and inches. Then we find the number total of feet and inches. 18 1 ft 1 ft So 1 1 18 ft in. 6 in. ft = 2 in. 6 ft in. Adding mixed To add mixed measures, we align the numbers so that we measures add the same units. Then we simplify when possible. Example 3 Solution Add and We add yd simplify: 1 like units, 2 yd 2 yd 2 yd 4 + 2 3 15 in. to 1 ft 3 in. 3 Then we change 5 ft to 1 yd yd Add and We add. Then we 7 in. ft 8 in. ft 15 5 2 ft ft right to left. in. to 4 ft. Now we have 3 in. and add ft 8 in. to 3 yd. Now we have 3 in. 40 min 35 sec hr 45 min 50 sec 2 hr simplify: + Solution ft and add 4 yd 2 Example 4 ft and then we simplify from 1 We change + 2 yd 2 7 in. ft 1 simplify from right to left. 40 min 35 sec hr 45 min 50 sec 2 hr + We we 1 3 hr change 85 sec to have 1 85 min 25 3 hr 86 Then we simplify 86 min 85 min sec and add min 25 to 1 hr sec to 85 min. Now sec 26 min and combine hours. 4 hr 26 min 25 sec 264 Math 87 Practice Problem set 58 a. Change 70 inches b. Simplify: 5 c. Add: 2 d. Add: 5 hr 42 1. What yd to feet ft 20 in. 1 ft 8 in. + min 53 1 and inches. yd 2 ft 9 in. sec + 6 hr 17 min 27 sec the quotient when the sum of 0.2 and 0.05 divided by the product of 0.2 and 0.05? is is 2. Darren carried the football 20 times and gained a total of 184 yards. What was the average number of yards he gained on each carry? Write the answer as a decimal number. 3. Robin bought two dozen arrows for six the two rates implied by this statement. 4. Jeffrey dollars. Write t counted the sides on three octagons, two hexagons, and two quadrilaterals. sides did he count? a pentagon, 5. What is Altogether, how many the average of these numbers? 6.21, 4.38, 7.5, 6.3, 5.91, 8.04 6. Draw Then answer a diagram of this statement. the questions that follow. Only two ninths of the 72 gruff. The rest were cordial. (a) How many (b) What was of the billy goats billy goats were were cordial? the ratio of gruff billy goats to cordial billy goats? 7. Arrange these numbers in order from least to greatest: 0.5, 0.5, 0.54 8. (a) Estimate the length of segment AB in inches. B Mixed Measures (b) 9. 10. 265 Adding Mixed Measures Measure the length of segment AB to the nearest eighth of an inch. Divide 365 by 12 and write the answer (a) with a remainder. (b) as a mixed number. Write each percent as a fraction or as a mixed number, (a) 11. • 3% 175% (b) Write each of these numbers as a percent. 3 (a) . 0.1 1 (b) Use exponents to write sixteen million in expanded notation. 13. Estimate the sum of 2,198,475 and 3,315,497 by rounding each number to the nearest hundred thousand before adding. Refer to figure ABDEFG to answer questions 14 and 15. 14. (a) What is the area of rectangle (b) 15. E 7 in. J G 8 6 is U 5 ABCGl e the perimeter of hexagon ABDEFG? Solve: 16. 18. X 45 18 15 2 Add, 20. 35 14 40 m 17. 1 - n = i 19. 9d = 2.61 6 subtract, multiply, or divide, as indicated: a/100 + 4 3 A in. What is the area of rectangle DEFC? What in. 21. 3.14 x 10 4 in. 266 Math 87 1^ 3- + 4-1 - 2( 22. 4 24. 3 + 2 6 2y 3 min days 15 hr 45 min days 8 hr 15 25. 25.875 + 4.36 + 19.9 + 15 27. 0.8 29. 0.06 x $18.00 - (7 6.543) LESSON 1 + 2 26. - f«a 34 I 6- 23. yd yd 2 -- ,11 1- V ft 1 ft 6 in. 9 in. 28. 3.45 x 0.2 x 0.05 30. $18.00 * 0.06 Multiplying Rates 59 There are two forms of every rate. To solve rate problems we simply multiply by the correct form of the rate. Consider the following statement. There were We can use this statement to write ( 5 chairs — 1 row x (a) If we multiply rate will find the (a) number each row. 5 chairs in two n, rates. 1 row lb) 5 chairs by 6 rows, the rows will cancel and we of chairs in 6 rows. 5 chairs __ _ x 6 rows" , . = 30 chairs lxow" we multiply rate (b) by 20 chairs, the chairs will cancel and we will find the total number of rows that contain 20 If chairs. ^ 1 „„,„ row ^ i_ x 20jdftairs~ = 4 • rows 5jchaiTS~ Example 1 Eight ounces of the solution cost 40 cents. (a) Write the two rates given by this statement. (b) Find the cost of 32 ounces of the solution. (c) How many ounces can be purchased for $1.20? Multiplying Rates Solution (a) The two 267 rates are 8 oz hi j- v 40 cents 40 cents 8 oz Rates have an original form and a reduced form. In Lesson 56 we learned to reduce rates to lowest terms. If we reduce these rates to lowest terms we get i_25_ (!) (2) s^nti oz 5 cents This step is not necessary as the rates can be used without reducing them first as we will do in this problem. This saves a step. (b) To find the cost, 40 cents we use the rate that x 32j02T has money on top. canceled ounces 8j02T multiplied cents 8 160 cents simplified We usually write answers equal to a dollar or more by using a dollar sign. Thus the cost is $1.60. (c) Again we will not bother to reduce the rate before we use it. Using the rate in unreduced form is more convenient because it saves a step. To find the number of ounces, we use the rate that has ounces on top. x 120jc©ftts" canceled cents 40£eftts" Example 2 Jennifer's 960 oz 40 multiplied r 24 oz simplified speed was 60 miles per hour. (a) Write the two rates given by this statement. (b) How far (c) How long would did she drive in 5 hours? it take her to drive 300 miles? j 268 Math 87 Solution (a) The two , rates are 60 miles , how To find To find hour , far, 60 miles we -— (c) ? hour 1 (b) , use the rate with miles on top. x 5 JiQttrs" = how much time, we 300 miles use the rate with time on top. x 300jftiles" = 5 hours 60jniies" Practice In the lecture hall there were 18 rows. Fifteen chairs were in each row. a. Write the two rates given by this statement. b. Find the total A car could travel Problem set number of chairs in the lecture hall. 24 miles on one gallon of gas. c. Write the two rates given by this statement. d. How many 1. When 59 2. would the product of 3.5 sum of 3.5 (a) What is (b) gallons 0.4, what take to go 160 miles? and is 0.4 is subtracted from the the difference? fraction of this circle marked with a What is and it 1? percent of this circle marked with a number greater than 1? 3. The 13-ounce box of cooked cereal costs $1.17, while the 18-ounce box costs $1.44. Find the unit cost for both sizes. Which size is the better buy? Multiplying Rates 269 4. Nelson covered the first 20 miles in 2- hours. What was his average speed in miles per hour? 5. The parking lot charges $2 for the first hour plus 50c for each additional half hour or part thereof. What is the total charge for parking in the lot for 3 hours 20 minutes? 6. The an average speed of 60 miles per train traveled at hour. 7. (a) Write the two rates given by this statement. (b) How Draw long did it take the train to go 420 miles? a diagram of this statement. Then answer the questions that follow. Two of the 30 football players fifths were endomorphic. (a) How many of the football players were endomorphic? (b) What percent of the football players were not endomorphic? 8. Which percent best identifies the shaded part of this 9. circle? (a) 25% (b) 40% (c) 50% (d) 60% Write 3— as a decimal number rounded to four decimal places. 10. Write 250% 11. Write 5 — 12. Use exponents to write seventy-five thousand in expanded as a mixed number. as a percent. notation. 270 Math 87 13. List the prime numbers between 90 and 100. Refer to figure H ABCDEFGH to answer questions 14 and A 9 15. All angles are right angles. Dimensions are in centimeters. 14. (a) What is the area of rectangle (b) What is the area of rectangle 15. What is ABGW CDEF? the perimeter of octagon ABCDEFGH? Solve: 16. w 1.5 1° 17. 42 18. 3.56 = 5.6 Add, 20. - y -n 19. —3 = w + -1 4 8 subtract, multiply, or divide, as indicated: 12 2 - V81 21. 4 $48.85 tires 1 tire 22. 24. 48 min 45 sec + 6 hr 20 min 20 sec 5 hr 5- - ( 1- * 2-1\ 4 6 23. 25. (3.26 x 10 3 ) + (8.36 x 10 2 ) 27. 2 29. 1.44 -r 160 A 3- + 37 ^ 7 in. 5 in. ( 3 (0.86 + 0.9) ft + 3 yd 26. - 4 yd 2 • 8 25) 28. 0.625 x 80 x 0.02 30. 72 4- 0.018 271 Rectangular Area, Part 2 LESSON Rectangular Area, Part 2 60 We have practiced finding the areas of rectangles. we can find the area of more complex shapes by Sometimes dividing the shape into rectangular parts. We find the area of each part and then add the areas of the parts to find the total area. Example 1 Find the area of this figure. Dimensions are in centimeters. All angles are right angles. 10 12 23 Solution We show two ways to solve this Solution 1 Solution 2 10 10 c (5) (13) 12 problem. A B i D 7 i 23 23 Total area = area Area A = 10 + Area B = 13 cm cm • A 12 • 7 + area B cm = cm = Find the area of 91 = 211 Total area Example 2 120 Total area = area cm 2 cm 2 cm Area + Area D 10 = 23 Total area 2 this figure. C = cm cm • • C + area cm = 7 cm = 5 = D cm 161 cm 2 211 cm 2 50 2 Dimensions are in inches. All angles are right angles. 7 10 6 15 272 Math 87 Solution many ways to We show just one way to There are divide this figure into rectangles. find the answer. 5 (4) 7 10 A 6 c B (3)| 15 Total area = area Area A = 10 Area B = 6 + Area C = 7 A + area B + area in. 2 3 in. = 50 = 18 4 = 28 in. 2 = 96 in. 2 in. 5 in. in. in. in. Total area Example 3 C in. 2 14 Find the area of this figure. Dimensions are in meters. All 10 angles are right angles. 20 Solution This time we will imagine this figure as a large rectangle with a small rectangular piece removed. If we find the area of the large rectangle and then subtract the area of the small rectangle, the answer, will be the area of the figure shown above. 14 i5 10 20 Area of figure = area of large rectangle - area of small rectangle Area of large rectangle = 20 - Area of small rectangle = 6 Area of figure m m • m 5 m 10 m 2 = 200 = 30 m- = 170 m 2 did not need to use subtraction to find this area. We could have added the areas of two smaller rectangles as we did in Example 1. However, sometimes subtraction is easier. We 273 Rectangular Area, Part 2 Practice Find the area of each figure. Try finding the area of the figure in Problem c by subtracting. All dimensions are centimeters. All angles are right angles. b. a. J 5 7 C. 7 5 10 15 24 12 20 18 20 Problem set Refer to the graph to answer questions 1 and 2. 60 Test Scores 100 90 80 70 8 CO 60 50 2 1 3 Test number Bob's scores Class average 1. On how many tests was Bob's score better than the class average? 2. What was 3. Jim's car could go 75 miles Bob's average score on these five tests? on Write the could his car 3 gallons of gas. two rates given by this statement. go on 12 gallons of gas? How far and one hundred eight thousandths is how much less than one hundred one and one hundredth? 4. Fifty-two 5. When is 9 squared the quotient? is divided by the square root of 9, what . 274 Math 87 6. Draw a diagram of this statement. Then answer the questions that follow. Five ninths of the 3960 voters supported Mayor Cobb. (a) How many voters (b) What was mayor 7. (a) What is (b) to those who who supported the did not support the mayor? the ratio of voters percent of the circle shaded? What is did not support Mayor Cobb? percent of the circle not shaded? 160% mixed number. 8. Write 9. Write - as a percent. as a 2 10. Write 0.06 as a decimal number rounded to the nearest thousandth. 11 Write twelve billion in expanded notation using exponents. 12. Jose bunted the ball many 13. and ran 90 feet to first base. yards did he run? Divide 365 by 7 and write the answer (a) with a remainder. (b) as a mixed number. Solve: 10 16. 50 2 x + 4.35 = 7 17. m k - | = -^- How Scientific Notation for hexagon to answer questions 18 and 19. Dimensions are in feet. All Refer to Large Numbers 275 12 this 11 angles are right angles. 18. What is the perimeter of 21 the hexagon? 19. What Add, 20. is the area of the hexagon? subtract, multiply, or divide, as indicated 1J + 2J + 3^ V36 3 V miles • 4 5 hours hour 1- + 2- 5 48 A f 22. 21. 23. 3- * 8^ 3 f.,z 2- Jl 1- 3 2) { i 24. 3 days 15 hr 17 min 16 hr 50 + 25. min 4 yd 2 + 2 ft 9 in. ft 5 in. A' 26. 4.296 + 15.47 + 0.0416 + 1 27. 90 - 29. 0.06 x $24.00 (8.7 - 6.54) 28. 0.04 x 7.5 x 10 6 30. $24.00 -T- 0.06 LESSON Scientific Notation 61 for Large Numbers We use scientific notation as an easy way to write large numbers. We use a decimal number followed by a power of 10 that indicates the true location of the decimal point. The power of 10 tells us where the decimal point really should be. Consider this notation. 4.62 x 10 6 The power of 10 tells us that the decimal point really should 276 Math 87 be six places to the right of where use zeros as placeholders. We get 4&20QQO. To write a number in -* it is written. We have to 4,620,000 customary to place the decimal point to the right of the first nonzero digit. Then we use a power of 10 to tell us where the decimal point really should be. To write scientific notation, it is 405,700,000 in scientific notation, to the right of 4 we begin by placing the decimal point and counting the places to where the decimal point really should be. 4 -05Z99999 8 places We see that the decimal point really should be eight places to the right of where we put it. We omit the terminal zeros and write 4.057 x 10 8 The 10 8 tells us the decimal point really should be eight places to the right of where it is written. Example 1 Solution Write 2.46 x 10 8 in standard form. The 10 8 us that the decimal point really should be eight places to the right of where it is written. We use zeros as placeholders and get tells 24£QQQQQ0. Example 2 Solution — 246,000,000 Write 40720000 in scientific notation. We begin by placing the decimal point after the 4. 4.g7^2gggg 7 places Now we discard the terminal zeros and write 10 7 to show that the decimal point really should be seven places to the right of where it is written. We get 4.072 x 10 7 Scientific Notation for Practice 277 Write each number in scientific notation. a. 15,000,000 Write each d. Problem set Large Numbers 1. 61 2. 3.4 b. number 400,000,000,000 c. 5,090,000 f. 1 in standard form. x 10 6 e. 5 x 10 8 x 10 5 Twenty-three billion, nine hundred fifty million is how much less than two hundred seven billion? Use words to write the answer. on a soccer ball, a regular hexagon and a regular In the pattern pentagon share a common side. If the perimeter of the hexagon is 9 in., what is the perimeter of the pentagon? 3. Five dozen apples cost $1.25. Write the two rates given by this statement. What would 7 dozen apples cost? 4. The 40$ per can or 6 cans for $1.98. can be saved per can by buying 6 cans at the store sold juice for How much 6-can price? 5. Five sevenths of those people were convinced. (a) What fraction of those who saw the phenomenon who saw the phenomenon were unconvinced? (b) What was the ratio of the convinced to the unconvinced? 6. Write twelve million in scientific notation. 7. Write 1.2 x 10 4 in standard form. 278 Math 87 8. Write — as a decimal number. 8 9. Round (a) to the nearest thousand. 29,647 95% 5280.08 (b) 10. Write 11. Divide 96 by 5 and write the answer as a decimal number. 12. Consider the quadrilateral WXYZ. For each statement write true or as a fraction. w P" false. (a) (b) (c) WX WZ WX YZ ZWXY is a right angle. _L II Refer to this figure to answer questions 13 and 14. Dimensions are in meters. All angles are right angles. 13. What 14 6 4 the perimeter of the is figure? 2 18 14. What the is area of the figure? 15. What percent of this regular hexagon is shaded? Solve: 16. 18. 24 x 5.37 + 60 25 777 17. = 8.4 19. _6_ 77 42 7 6.5 - y = 5.06 Order of Operations Add, subtract, multiply, or divide, as indicated: 5Z 20. + 3J + V64 21. 10 cm 16 1 22. 279 days 18 hr 50 min days 8 hr 25 min 5 + 2 23. 3 + 1 yd yd 2 ft mm cm 5 in. 9 in. ft ( a 7 6- + 5- - 3- 24. 4 3 ; A 3 4.367 + 38.54 + 8.59 + 15 27. 4.5 29. 3.75 (3 -s- - \\ 2- + 1- 3- 8 26. - f 1 25. 2.875) 75 3 28. $40.00 30. 3 • 2 0.065 * 0.08 Order of Operations LESSON 62 The four basic operations of arithmetic are addition, subtraction, multiplication, and division. When more than one operation occurs in the same expression, we perform the operations in the order listed below. Order of Operations Example 1 Solution 1. Multiply and divide in order from 2. Then add and Simplify: 2 + We add left to right. subtract in order from left to right. 4x3-2-4 multiply and divide in order from or subtract. left to right before we 280 Math 87 2 + 2 + 2 + 4x3-2-4 problem 12-2-4 6-4 multiplied divided 12 by 2 added and subtracted 4 Example 2 Solution Simplify: 10 We • - 3 5 - 2 + 6(3) perform the multiplications and division 30 - 5 -s- + 2 multiplied 3 6(3) 6 - 2 + 6(3) 3 + 3 + 18 Problem set first. • 10 divided 30 by 5 divided 6 by 2 6(3) multiplied added 21 Practice 4x3 Simplify: + 5-5-5-5 a. 5 b. 50-8-5 c. 24-8-6-2-4 d. 24 - 8 - 4 1. If + • 6-3 2 + (3)4 the product of the by the sum of the 62 numbers is divided three prime numbers, what is the first first three prime quotient? 2. Sean counted a 100 sides on the heptagons and nonagons. If there were 4 heptagons, how many nonagons total of were there? 3. Twenty-five and two hundred seventeen thousandths is how much less than two hundred two and two hundredths? 281 Order of Operations 4. was the 5. cost per tape to the nearest cent? Ginger is starting a 330-page book. Suppose she reads for 4 hours and averages 35 pages per hour. (a) How many pages will she read in 4 hours? (b) After four hours, have 6. What Albert bought a pack of 3 blank tapes for $5.95. Draw how many to read to finish the pages will she still book? Then answer a diagram of this statement. the questions that follow. Three fourths of the 60 passengers disembarked at the terminal. (a) How many passengers disembarked at the terminal? (b) What percent of the passengers did not disembark at the terminal? 7. Write 3,750,000 in scientific notation. 8. Write 2.05 x 10 6 in standard form. 9. Write 7.6 as a mixed number. 10. Write 3.27 as a decimal number rounded to the nearest thousandth. 11. Write 12. Divide 70 by 9 and write the answer 13. 3 2— as a percent. number with a bar over the repetend. (a) as a decimal (b) decimal number rounded thousandth. as a What decimal number names the point marked by 1 | the arrow? the nearest to n.9 i i i m i i i i | 1.0 282 Math 87 Draw a rectangle that is answer questions 14 and 3 cm cm long and 2 wide. Then 15. 14. What is the perimeter of the rectangle in millimeters? 15. What is the area of the rectangle in square centimeters? Solve: 16. 18. 56 105 | p + Add, 17. = 20 6.8 19. w 15 2.5 q - 3.6 = 6.4 subtract, multiply, or divide, as indicated: - 10 2 - V25 20. 53 22. 24 mi lgal — _ _ • 21. 4 + 4 23. 5 hr 4 • ! 5.5 gal - 4 * 4 45 min 30 sec + 2 hr 53 min 55 sec > ( 24. 12 4 • 2- 25. 26. 8.575 + 12.625 + 8.4 + 70.4 27. 4.26 29. 0.1001 * 77 - (9 - 5.74) 3 5- - 3+ 2 4 2 2 3 A ' \ 6- + 5- 28. 0.8 x 1.25 x 10 6 30. $2.60 -s- 0.065 Unit Multipliers LESSON Unit Multipliers • Unit Conversion • 283 Unit Conversion fraction. moment to review the procedure for reducing a When we reduce a fraction, we remove pairs of numbers that appear as factors in both the Let's take a numerator and denominator. 24 [ _ 36 j 2-2-2-3 2 _ ^2-2-3-3"3 We may reduce before we multiply. This is sometimes called canceling. l 2 ' 3 3 2 " 5 5 l We may apply cancel units before procedure to units as well. this we 5 We may multiply. K„ 12 in. -T-F7- K 1 = 60 in. We remember that we change the name of a number by multiplying by a fraction whose value equals 1. Here we have changed the name of 3 to — by multiplying by -. 4 The fraction — Whenever is 4 name for another the numerator 3 because 12 + 4 equals and denominator of a 3. fraction equal to 1. There of fractions that are equal to 1. A are equal (and are not zero), the fraction an unlimited number fraction equal to 1 may have is units, is such as 12 inches 12 inches Since 12 inches equals 1 that equal 1. foot, we can write two more fractions 12 inches 1 foot 1 foot 12 inches 284 Math 87 Because these fractions have units and are equal them unit to 1, we call multipliers. Unit multipliers are very useful for converting from one unit of measure to another. For instance, if we want to convert 5 feet to inches, we can multiply 5 feet by a multiplier we get that has inches on top. The units cancel and 60 inches. 12 r^ 5 K • . 1 = 60 Krv in. 96 inches to feet, we can multiply 96 inches by a multiplier that has feet on top. The units cancel If we want in. and we get 8 feet. Notice that unit to convert we selected unit multipliers that canceled the we wanted to remove and kept the unit we wanted in the answer. When we write the set up unit conversion problems, numbers involved Given measure Example 1 will in this order. Unit multiplier Converted measure Write two unit multipliers for these equivalent measures. 3 Solution we ft = 1 yd We write one measure as the numerator and its equivalent as the denominator. 3ft =- lyd Example 2 Solution A and i yd ^— 3ft Use one of the unit multipliers from Example (a) 240 yards to (b) 240 (a) 1 to convert feet r feet to yards. We are given a measure in yards. We want the answer in feet. We write this down. 240 yd Unit ft multiplier Unit Multipliers We want Unit Conversion • 285 "yd" and keep the unit "ft," so we select the unit multiplier that has ft on the top and yd below. Then we multiply and cancel units. to cancel the unit —~r Q yd 240 The answer • ft = 720 ft reasonable because feet are smaller units than yards, so it takes more feet than yards to measure the same distance. (b) We is measure in feet, and we want the answer in yards. We choose the unit multiplier that has yd on the top. are given the yd 1 240if The answer is than it same Example 3 Solution so feet, • -~r = 80yd reasonable because yards are longer units takes fewer yards than feet to measure the distance. Convert 350 millimeters to centimeters We are given millimeters centimeters. that has and cm = 10 mm). are asked to convert to We form a unit multiplier from the equivalence cm on the top. — cm 1 350jb«T • = 35 lOjaffl Practice (1 cm Write two unit multipliers for each pair of equivalent measures. yd = 36 a. 1 b. 100 c. 16 oz = cm = in. 1 m 1 lb Use unit multipliers to d. Convert 10 yards e. Twenty-four f. In old perform the following conversions. to inches. feet is how many yards? England 12 pence equaled (1 yd = how ft) Merlin had many pence? 1 shilling. 24 shillings. This was the same as 3 286 Math 87 Problem set Refer to this bar graph to answer questions 1-3. 63 Distribution of Birthdays in Class Months Jan-Mar Oct.-Dec 2 1 3 Number 1. 2. 4 of 6 5 students (a) How many boys are in the class? (b) How many girls are in What percent the class? of the students have birthdays in January through June? 3. What fraction of the boys have birthdays in April through June? 4. At the book fair. Bill bought 4 books. One book $3.95. Another book cost $4.47. The other 2 books cost cost $4.95 each. 5. (a) Altogether, (b) What was Draw a how much did Bill spend? the average price of the books? diagram of this statement. Then answer the questions that follow. Seven twelfths of the 840 gerbils were hiding in their burrows. (a) What fraction of the gerbils were not hiding in their burrows? (b) How many gerbils were not hiding in their burrows? Unit Multipliers • 287 Unit Conversion 6. Write one 7. Write 7 x 10 2 in standard form. 8. Use unit multipliers to perform the following conversions. 9. trillion in scientific notation. (a) 35 yards to feet (b) 2000 cm to m (3 ft (100 = cm 1 = yd) 1 m) Write the prime factorization of 10,000. 10. Estimate the difference of 19,827 and 12,092 by rounding to the nearest thousand before subtracting. 11. Write 12. Divide 430 by 20 and write the answer as a decimal J number. 13. Big Bill 140% m mixed number. Stephanie is 165 cm tall. Big centimeters taller than Stephanie? is 2 how many as a tall. Bill is Refer to this figure to answer questions 14 and 15. Dimensions are in feet. All angles are right angles. 14. What 15. What is the area of the figure? is the perimeter of the 12 figure? Solve: 16. li = 90 14 18. 8 = 7.25 + Add, 20. 17. : D 1.5 m 19. 1.5 = 10-11 subtract, multiply, or divide, as indicated: \<81 + 92 - 2 5 21. 16 h- 4 -s- 2 + 3 x 4 288 Math 87 22. lft 84 23. in. 12 yd 3 + ( 24. LESSON 64 1^ V 6 25. 5 3, 2 ft 6^ ; 27. 4.06 - 3.975 29. 5.4 4.5 in. 35 y 28. 0.065 x $12.00 30. 2.6 Ratio we • 2 10.6 + 4.2 + 16.4 + (3.875 x 10 1 ) In this lesson l\ in 8- - 1-1 26. * ft ' 5 12- + 5- * 23 1 in. 4- 0.052 Word Problems will use proportions to solve ratio word problems. Consider the following ratio word problems. The ratio were 750 macaws was 5 to 7. If there how many macaws were there? of parrots to parrots r problem there are two kinds of numbers, ratio numbers and actual count numbers. The ratio numbers are 5 and 7. The number 750 is an actual count of parrots. We will arrange these numbers into two columns to form a ratio box. In this Ratio Actual Count Parrots 5 750 Macaws 7 M We were not given the actual count of macaws, so we have used Mto stand for the number of macaws. The numbers in this ratio box can be used proportion. By solving the proportion, we find count of macaws. to write a the actual Ratio Ratio Actual Count Parrots 5 750 5 750 Macaws 7 M 7 M 5M = 5250 M= We Example The We macaws was find that the actual count of ratio of boys to girls was how many the auditorium, Solution 289 Word Problems 5 to 4. If there 1050 1050. were 200 girls in boys were there? begin by making a ratio box. Ratio Actual Count Boys 5 B 5 B Girls 4 200 4 200 4B = 1000 B = 250 We use the numbers in the ratio box to write a proportion. Then we solve the proportion and answer the question. There were 250 boys. Practice Solve each of these ratio word problems. Begin by making a ratio box. a. The girl-boy ratio many boys Problem set 64 The was 9 to 7. If 63 girls attended, how attended? b. sparrows to bluejays in the yard was 5 to 3. If there were 15 bluejays in the yard, how many sparrows were in the yard? c. untagged fish was 2 to 9. Ninety fish were tagged. How many fish were untagged? 1. Thomas ratio of The ratio of tagged fish to Jefferson died on the fiftieth anniversary of the signing of the Declaration of Independence. He was born in 1743. The Declaration of Independence was signed in 1776. live? How many years did Thomas Jefferson 290 Math 87 2. The heights of the five basketball players are 190 cm, 195 cm, 197 cm, 201 cm, and 203 cm. What is the average height of the players to the nearest centimeter? 3. Use a ratio box to solve this problem. The ratio of winners to losers was 5 to 4. If there were 1200 winners, how many losers were there? 4. What the cost of 2.6 is pounds of cheese at $1.75 per pound? 5. Maria shut the front door, but not before two hundred eighty-five thousand, six hundred slipped in. Meanwhile, another two million, fifteen thousand slipped in through the back door. How many slipped in altogether? 6. Draw a diagram of this statement. Then answer the questions that follow. Four fifths of the 80 trees were infested. (a) How many trees were infested? (b) How many trees were not infested? 7. Write 405,000 in scientific notation. 8. Write 0.04 x 10 5 in standard form. 9. Use unit multipliers to perform the following conversions. (a) 5280 (b) 300 ft cm to yards (3 = 1 yd) to millimeters (1 cm ft = 10 mm) 10. Write 3.1415926 as a decimal number rounded decimal places. 11. Write 2- as a percent. 1 to four Ratio 12. 291 Word Problems Divide 100 by 6 and write the answer (a) as a decimal number with a bar over the repetend. (b) as a decimal number rounded to the nearest hundredth. 13. The positive square root of 100 how much is greater than 2 cubed? Refer to this figure to answer questions 14 and 15. Dimensions are in centimeters. All angles are right angles. 14. What the perimeter of the is <r figure? 15. What is the area of the figure? Solve: 16. 18. 6.2 = x + 4.1 24 II 2 + l 3 21. 1000 25. 0.21 subtract, multiply, or divide, as indicated: 19. 23. y- = 1.2 36 27 r Add, 17. 3— 10 - Vl21 1 cm ( 100 m cm 1 2^ 2 3 9- - 6- + V 3.47 + 6.3 + 12 20. - 4 x 24 22. l + 2 week week ( 24. 5 7- 23.6 + 3 - (10 5 days 14 hr 6 days 10 hr 5 6^3-2" 3 26. 2 -s- - 8.91) 292 Math 87 27. $4.50 x 0.06 29. 7.35 * 70 28. 6.25 x 0.16 30. 24 + 0.016 LESSON Average, Part 2 65 • If we know the average numbers of a group of numbers are in the group, we can and how many figure out the sum of the numbers. Example 1 Solution The average of three numbers is 17. What is their sum? We are not told what the numbers are. We are only told their average. All of these sets of three numbers have an average of 17. 16 + 17 + 18 51 3 3 10 + 11 + 30 51 3 3 1+1+49 51 3 3 = 17 = 17 = 17 Notice that for each set the sum of the three numbers is 51. Since average means what the numbers would be if they were "equalized," the sum is the same as if each of the three numbers is 17. 17 + 17 + 17 = 51 Thus the number of numbers times sum Example 2 Solution of the numbers. The average of are 16, 26, and If numbers is 25. If three of the numbers what is the fourth number? four 30, numbers four numbers were 25. the average of four if all their average equals the is 25, the sum 25 + 25 + 25 + 25 = 100 is the same as 293 Average, Part 2 Thus the sum of the four numbers is 100. We are given three of the numbers. The sum of these three numbers plus the fourth number must equal 100. N 16 + 26 + 30 + = 100 The sum of the first three numbers is 72. For the sum of the four numbers to total 100, the fourth number must be 28. 16 + 26 + 30 + (28) = 100 100 - 4 = 25 Example 3 After 4 tests, Annette's average score does Annette need on her Solution check fifth test to was 89. What bring her average score up to 90? Although we do not know the specific scores on the first 4 tests, the total is the same as if each of the scores was 89. Thus the total after 4 tests is 4x89= The 356 her first 4 scores is 356. However, to have an average of 90 after 5 tests, she needs a 5-test total of 450. total of 5x90= 450 Therefore she needs to raise her total from 356 to 450 on the fifth test. To do this, she needs to score 94. 356 + 94 450 Practice — — 4-test total -«— 5-test total fifth test a. Ralph scored an average of 18 points in each of his first 5 games. Altogether, how many points did Ralph score in the first 5 games? b. The average of four numbers is 45. If three of the numbers are 24, 36, and 52, what is the fourth number? c. After 5 tests, Mike's average score his average score sixth test? was 89. was 91. What was his After 6 tests, score on the 294 Math 87 Problem set 1. Use a ratio box to solve this problem. The ratio of sailboats to rowboats in the bay was 7 to 4. If there were 56 sailboats in the bay, how many rowboats were there? 2. The average of four numbers is 85 If three of the numbers are 76, 78, and 81, what is the fourth number? 65 3. . A one-quart container of oil costs 89c. A case of 12 onequart containers costs $8.64. How much is saved per container by buying the 4. oil by the case? Segment BC is how much longer than segment AB? inches J I I I J I I J I 1 I I I I I • » B 5. Draw a diagram of this statement. Then answer the questions that follow. Three tenths of the 30 students earned an A. (a) How many (b) What percent students earned an A? of the students earned an A? 6. Write 675,000,000 in scientific notation. 7. Write 1.86 x 10 5 in standard form. 8. Use unit multipliers to perform the following conversions. 9. (a) 24 feet to inches (b) 500 mm to centimeters and symbols to write "The product of two hundredths and twenty-five thousandths is five tenUse digits thousandths." 10. Write 48% as a fraction. 295 Average, Part 2 11. 12. Divide 75 by 8 and write the answer (a) with a remainder. (b) as a mixed number. Refer to quadrilateral to answer the ABCD D following questions. (a) (b) 13. Which side to side BC? Which side is is parallel cular to side BC? (c) Which is Don is angle B an obtuse angle? 6 feet 2 inches how many c perpendi- tall. Bob is 68 inches tall. Don inches taller than Bob? Refer to this figure to answer questions 14 and 15. Dimensions are in inches. All angles are right angles. 14. What is the area of the figure? 15. What is the perimeter of the figure? Solve: 16. 18. 4.56 + 4.7 Add, w = 10 - n = 17. a 35 6 10 4.7 subtract, multiply, or divide, as indicated: - 43 - 24 36 in. 19. 12 2 21. 10 yd lyd - Vl44 20. 50 + 22. 8 + yd 30-5-2-6 2 ft 7 in. 5 in. is 296 Math 87 15 7 2 6 8 3 4 10 2- + 6- + 4- 23. LESSON 66 26. 3.47 + (6 29. 2.5 - 3 l 25. 27. 7 l- 6 - 24. 3 U 3 + 3 1.359) * 1000 28. (0.6)(0.28)(0.01) 30. 6.3 Subtracting -s- 0.018 Mixed Measures We have practiced adding mixed measures. In this lesson we will practice subtracting mixed measures. When subtracting mixed measures, we may need to borrow in order to subtract. When we borrow, we need to keep in mind the units in the problem. If we borrow 1 hour, it becomes 60 minutes. If we borrow 1 day, it becomes 24 hours. Example 1 Subtract: 5 Solution 1 days 10 hr 15 min day 15 hr 40 min Before we can subtract minutes, we borrow 1 hour, which is 60 minutes. We combine 60 minutes and 15 minutes, making 75 minutes. Then we can subtract. 9^^ (60 min) 5 1 75 9 days Kf hr 15 min day 15 hr 40 min 5 - 1 days Kf hr y5 min day 15 hr 40 min 35 Next we borrow 1 day, which is min 24 hours, and complete the subtraction. 33 (24 hr) 9 4 75 % days %Q hr y5 min 15 hr 40 min 1 day 35 min • 75 $ days Kf hr V5 min 15 hr 40 min 1 day 3 days 18 hr 35 min 297 Subtracting Mixed Measures Example 2 Solution Subtract: 4 We yd - 3 in. carefully align the which equals 1 yd, 3 2 yd 1 8 in. ft numbers with 1 3 in., making 15 in. yd 1 ft 8 in. which is 12 in. This combines with Then we can subtract. 15 3 2 A yd 3 ft in. 2 yd 1 ft 8 in. 1 yd 1 ft 7 in. Subtract: a. c. Problem set 2 3 in. ft, - Practice borrow ft) A yd we borrow We ft. 3^(3 Next like units. 1. 66 2. 3 hr 1 hr 15 3 sec min 55 2 days 3 hr - sec 30 min - 8 1 3 yd yd 1 ft 5 in. 2 7 in. ft day 8 hr 45 min Three hundred twenty-nine ten-thousandths is how much greater than thirty-two thousandths? Use words to write the answer. Use box to solve this problem. The ratio of the the width of the rectangle is 4 to 3. If the length a ratio length to of the rectangle 3. b. is 12 feet, (a) what is its width? (b) what is its perimeter? The parking lot charges $2 for the first hour and 50c for each additional half hour or part thereof. What is the total charge for parking a car in the lot from 11:30 a.m. until 2:15 p.m.? 4. After four tests Trudy's average score was 85. If her score is 90 on the fifth test, what will be her average for all five tests? 298 Math 87 5. 6. Twelve ounces of Brand X costs $1.50. Sixteen ounces of Brand Y costs $1.92. Find the unit price of each. Which brand is the better buy? Five eighths of the rocks in the box were metamorphic. The rest were igneous. (a) What (b) What was the ratio of igneous to metamorphic rocks? fraction of the rocks were igneous? 7. Write six hundred ten thousand in scientific notation. 8. Write 1.5 x 10 4 in standard form. 9. Use unit multipliers to perform the following conversions. 10. (a) 216 hours to days (b) 5 minutes to seconds Write 5-b as a decimal number rounded to the nearest hundredth. 11. How many pennies equal one million dollars? Write the answer in —1 scientific notation. 12. Write 13. Which even and 5 as a percent. two-digit 7? Solve: 14. 3 2.5 = 48 — c 15. x + 56 = 500 16. k - 0.75 = 0.75 number is a common multiple of 299 Liquid Measure 20 Refer to this figure to answer questions 17 and 18. Dimensions are in millimeters. All angles are right angles. 17. What 7 f 30 the perimeter of is 15 the figure? 15 18. What the area of the is figure? 10 Add, 19. subtract, multiply, or divide, as indicated: 15 2 - 5 3 21. yd yd ,5 + 2 - VTOO 20. 22. £ ft 3 in. 2 ft 9 in. 3-2-3 6 + 12 + 5 - 2 yd yd 2 ft 3 in. 2 ft 9 in. ? 2 23. —km 88 1 25. . : 34 • 24. hr • 4 2- -*- 3- 27. 4.87 + 12 29. 2.5 - + 2±- 4 8 2 7.363 x 4 x 10 4 6 { 33 26. -A 3i 8 2 28. $24.50 x 0.06 30. 3.6 - 10 3 Liquid Measure LESSON 67 2^ + , 4 hr The volume of a container tells us will hold. Thus, the volume how much of a container is the container a measure of the capacity of the container. To measure quantities of liquid, we use units of capacity- Units of capacity in the U.S. Customary System include ounces* (gal). (oz), pints (pt), quarts (qt), and gallons Units of capacity in the metric system include liters (L) *The word "ounce" is used to describe a weight as well as an amount of liquid. An ounce of liquid is often called a fluid ounce. Although ounce has two meanings, a fluid ounce of water does weigh about 1 ounce. 300 Math 87 and Equivalent measures are summarized in milliliters (mL). the tables below. U.S. Customary Metric Equivalents Equivalents = 4qt = 2 pt = 2 cups lgal lqt ipt ipt = 16 oz 1 cup = 8 oz From = 1000 1 liter Note: these equivalents we can form mL 1 liter is a little more than 1 quart. unit multipliers to help us convert one measurement to another. Example 1 Solution Convert 3.5 From liters to milliliters. the equivalent 1 liter = 1000 mL we can form two unit multipliers: 1 liter 1000 To cancel liters 1000 or mL mL 1 liter and give us multiplier that has milliliters 1000 we milliliters, on use the unit top. mL 3.5 liters = 3500 mL 1 liter Example 2 Solution (a) Convert 5 gallons to quarts, (b) Convert 5 gallons to pints. (a) To convert gallons we to quarts, multiply by a unit multiplier that has quarts on top. 5gd.i^=20qt (b) To convert gallons to pints, we take two steps. For the convert gallons to quarts. We did this in part (a). The second step is to convert quarts to pints. Since we showed the first step in (a), we will show only first step we the second step here. 2 0#.f! = 40pt Liquid Measure Example 3 Add: + Solution First 1 qt 1 pt 7 oz 1 qt 1 pt 12 oz we add + Then we simplify from like units. 1 qt 1 pt 7 oz 1 qt 1 pt 12 oz added 2 qt 2 pt 19 oz simplified ounces oz 2 qt 3 pt 3 right to (19 oz = 1 pt 3 oz) 3 qt 1 pt 3 oz simplified (3 Practice a. Complete the chart. b. pt = 1 qt 1 pt) Compare O 2L c. One liter equals how many quantities. 2 qt milliliters? Complete each unit conversion. d. f. 3 gal 2.5 = L = pt e. 2 qt = mL g. 600 mL oz = (decimal answer) h. i. Add: 1 gal 3 qt 1 pt 9 oz + 2 gal 1 qt 1 pt 8 oz Subtract: 3 qt - Problem set 67 1. 1 qt 5 1 pt oz 7 oz Find the average of these numbers: 6.38, 8.9, 7.14, 8, 9.32, 10.3 301 left. 302 Math 87 2. Use a ratio box The ratio of tall the bay was 2 to 7. If to solve this problem. ships to small ships floating in there were 28 tall ships, how many small ships were there? 3. Al can rent a chain saw for $35 per day or for $8.75 per hour. If he can finish the job and get the saw back in 3 hours, how much will he save by renting by the hour instead of by the day? 4. If lemonade costs 1.5c per ounce, what is the cost per pint? 5. 6. Five and six hundredths is how much less than six and five thousandths? Use words to write the answer. Draw a diagram of this statement. Then answer the questions that follow. Five ninths of the 720 students in the assembly were girls. (a) How many boys (b) What was were in the assembly? the boy-girl ratio in the assembly? 7. Write 300,000,000 in scientific notation. 8. Write 9. Use unit multipliers 1 x 10 5 in standard form. to change 5 liters to milliliters. 10. The velocity of light is 299,792,458 meters per second. Round this number to the nearest million. 11. Write 12. Divide 16.3 by 12 and write the answer as a decimal with a bar over the repetend. 13. Estimate the product of 586 and 716. 110% as a mixed number. 303 Liquid Measure Refer to this figure to answer questions 14 and 15. Dimensions are in feet. All angles are right angles. 3 7 14 14. What is the area of the figure? 15. What is the perimeter of the figure? Solve: 16. = 0.01 + a 1 100 8 Add, subtract, multiply, or divide, as indicated: 23 21. + 23. + 42 + l3 + V49 3 gal 2 qt 1 pt 1 gal 3 qt 1 pt 1000 2.5 liters 20. - 2 3 35 • 2- 12 12 3 2— 3 5 12 27. 12 + 0.8 + 1.46 29. 0.6 x 3.5 x 10 3 - 7 2^ V 2^ 1- -- V V 28. 4.37 30. 4.2 oz 4-1 + 6- ( 26. 2 • V 1 • 1 pt 1 qt ( 24. 6 -s- 4 qt 1 pt 3 oz 22. mL - 12 24 1 liter 25. - c 0.01 = 1 x 18. 19. 17. - -s- (2 (6 * 0.416) 0.24) 2(5)2 304 Math 87 LESSON Scientific Notation 68 for We have used may Small Numbers numbers. We notation to write small numbers. scientific notation to write large also use scientific When we write a number in scientific notation, the power of us where the decimal point really should be. If the exponent is a positive number, the decimal point really should be that many places to the right of where it is written. 10 tells 6.32 x 10 7 The exponent is positive seven, so we know that the decimal point really should be seven places to the right of where written. This means the number it is is 63,200,000 6^2,99990. 7 places number, the decimal point really should be that many places to the left of where it is written. If the exponent is a negative -7 6.32 x 10 The exponent is negative seven, so we know that the point really should be seven places to the written. .g9QQ9gg32 — left of decimal where it is 0.000000632 7 places We Example 1 Solution had to use zeros as placeholders. Write 4.63 x 10~ 8 in standard notation. The negative exponent tells us that the decimal point really should be eight places to the left of where have to insert zeros as placeholders. -*- 0.0000000463 8 places Example 2 it is Write 0.0000033 in scientific notation. written. We Scientific Notation for Small Solution We is place the decimal point to the right of the Numbers 305 first digit that not zero. 0QQQQQ3.3 6 places The decimal point really should be six places where we have placed it. So we write to the left of 3.3 x lO" 6 Practice Write each a. number in scientific notation. 0.00000025 b. 0.000000001 c. 0.000105 Write each number in standard form. Problem set d. 4.5 x 10- 7 1. Make f. 1.25 x 10" 5 box to solve this problem. The ratio of riders was 5 to 3. If 315 were walkers, how many were 2. x 10~ 3 1 a ratio walkers to 68 e. riders? After five tests Allison's average score was 88. After six her average score had increased to 90. tests What was her score on the sixth test? 3. When Richard rented 18(2 per mile. 300 miles, 4. If If how lemonade a car, he paid $34.95 per day plus he rented the car for 2 days and drove much did he pay? costs $0.52 per quart, then what is the cost per pint? 5. Draw a diagram of this statement. Then answer the questions that follow. Jason finished his math homework in two fifths of an hour. (a) How many minutes did it take Jason to finish his math homework? (b) What percent of an hour did finish his math homework? it take for Jason to 306 Math 87 6. (a) 7. number Write each in scientific notation, 186,000 (b) 0.00004 Write each number in standard form. (a) 3.25 x 10 1 (b) 1.5 to convert 2000 8. Use unit multipliers 9. Write - as a decimal x 10~ 6 mL to number rounded liters. to five decimal places. » asa 10. Write 11. Divide 330 by 24 and write the answer as a decimal , percent. number. s 12. (a) What decimal square (b) shaded? What percent of the square not shaded? is 13. is part of the Compare: 2.5 x 10~ 2 O 2 -5 + 10 2 Refer to this figure to answer questions 14 and 15. Dimensions are in yards. All angles are right angles. 2 14. What is the perimeter of the figure? 15. What is *H 1.5 the area of the figure? \/ Z^J 307 Classifying Quadrilaterals Solve: 16. 18. 17.3 = 19. 5.7 17. 3.5 28 w 20 = q - 3.6 8.4 Add, d + subtract, multiply, or divide, as indicated: 20 2 + 10 3 - V36 3 - 1 ( 48 -s- 12 * 2 + 2(3) * i' 21. 20. yd I ft yd 2 ft 2 1 in. 22. 4 gal 3 qt - 3 in. 1 gal l 48 oz <Q f Pt 24. • 5 25. 15 5- + 3- + 6 69 7 2— x 10- 2 ) + 0.46 27. (4.6 29. 0.24 x 0.15 x 0.05 LESSON - 26. '; * i \ ^^ ,A 14J V 12 8 a -- 7 16 oz pt 6 oz 2 qt 1 pt 5 oz \ 23. 1 1 3 2 5 5- - 3- 28. 10 - 30. 10 * (0.14 * 70) (2.3 0.575) Classifying Quadrilaterals We remember from Lesson 19 that a four-sided polygon is called a quadrilateral. Here we show several kinds of quadrilaterals. Trapezium Trapezoid Parallelogram Parallelogram Rectangle Parallelogram Rectangle Square Quadrilaterals are classified by certain characteristics. The 308 Math 87 chart below names the some identifies five illustrated quadrilaterals and characteristics of each one. Type of Quadrilateral Characteristics Trapezium No Trapezoid Exactly parallel sides one pair of parallel sides Parallelogram Two pairs of parallel sides A Rectangle parallelogram with four right angles A rectangle with all sides equal in length Square Notice that a square rectangle Example 1 Solution Answer is is a special kind of rectangle and a special kind of parallelogram. true or false: - (a) A (b) All rectangles are parallelograms. (c) Some (a) True, a square (b) True, (c) False, all squares have square is a rectangle. squares are trapezoids. all is a parallelogram with four right angles. rectangles have two pairs of parallel sides. two pairs of parallel sides. Trapezoids have only one pair of parallel sides. Example 2 Solution Sketch a trapezoid. We that a begin by drawing two parallel segments of different lengths. Classifying Quadrilaterals Then we connect 309 the endpoints of the segments to form a quadrilateral. Practice Answer true or false: a. All rectangles are squares. b. Some c. No parallelograms are rectangles. trapezoid is a parallelogram. Sketch the following figures. d. A e. A trapezoid parallelogram that is In this figure, quadrilateral not a rectangle ABDE is a rectangle and AC FD. Classify the following quadrilaterals. Problem set f. Quadrilateral ACDF g. Quadrilateral ABDF 1. The bag contained red marbles, white marbles, and blue marbles. If half the marbles were red and one third of the marbles were white, what fraction of the marbles 69 were blue? 2. One hundred twenty-five millionths is how much less than two thousandths? Use words to write the answer. 3. What is pound? the cost of 3- pounds of bananas at 38c per . 310 Math 87 4. Use a problem. The ratio of numismatists to philatelists at the auction was 7 to 5. If there were 140 philatelists, how many numismatists ratio box to solve this were there? 5. Jenny's average score for six tests is 89. The teacher said would not be included in the If her lowest score was 79, what was the average that the lowest score average. of the remaining scores? O 4000 m L 6. Compare: 4 7. The survey found taste of (a) liters that 4 out of 5 teenagers prefer the Brand A. According prefer the taste (b) 8. of teenagers According to the survey, what fraction of teenagers do not prefer the taste of Brand A? Write each number in scientific notation, (a) 9. what percent of Brand A? to the survey, 30 trillion (b) 0.0000037 Write each number in standard form, (a) 8 x 10 4 (b) 4x 10- 6 10. Use unit multipliers to convert 16 11 Which of the following is not a parallelogram? (a) square 2% (b) pints to quarts. trapezoid (c) rectangle 12. Write 13. Divide 5.43 by 0.11 and write the answer as a decimal with a bar over the repetend. as a fraction. 1 Classifying Quadrilaterals Refer to this figure to answer questions 14 and 15. Dimensions are 3 in meters. All angles are right angles. - 14. What 6 4 the perimeter of the is figure? 15. What the area of the figure? is 10 Solve: 16. 18. 101 + / = 1000 4 19. subtract, multiply, or divide, as indicated: 10 2 - VlOO - 3 4 21. 3 + 23 1 wk wk 4 days 20. 5 hr 25. 8 5 gal - 6 days 21 hr 4- • - • 24. 1 gal 3- 6 • 30 29. 0.24 - 4.9 (5 4- 3 gal j- 1 gal 2 qt 26. 1 6 3 4- + 3-5-4 2 - 15 0.101) pt 4- + 5-1 - 3-2 5 5 27. 48 - 12 ^ 2 - 2(3) 22. 4qt . 10 - h = 9.7 150 20 y_ Add, 17. 28. (0.37)(5.1)(10 30. $175 H- 0.25 :i ) A 31 31 2 Math 87 Area of a Parallelogram LESSON 70 As we saw in the preceding lesson, a parallelogram quadrilateral in which both pairs Parallelogram is a of opposite sides are parallel. Parallelogram Not a parallelogram We also saw that a rectangle is a special kind of parallelogram. we have practiced finding the areas of rectangles. In this lesson we will practice finding the area of a parallelogram. We may use a paper parallelogram and For several lessons scissors to help us understand the concept. Project Cut a piece of paper to form a parallelogram as shown. Next, cut the paper into two pieces along a line by the dotted line below. Finally, shown the triangular piece as shown to form a We see that the area of the parallelogram move rectangle. equals the area of this rectangle. to form We find the area of a rectangle by multiplying the length times the width. When describing a parallelogram we do not 3 Area of a Parallelogram use the words "length" and "width." Instead words base and height. we 31 use the Height ^J1 Base Notice that the height is not one of the sides of the parallelogram (unless the parallelogram is a rectangle). Instead, the height perpendicular to the base. Multiplying the base and height gives us the area of a rectangle. However, as we saw in the is project, the area of the rectangle equals the area of the parallelogram we are considering. Thus, we find the area of a parallelogram by multiplying its base and height. Area of a parallelogram = base Example Find height and (b) the area parallelogram. Dimensions are (a) of this • the perimeter in inches. Solution (a) We find the perimeter by adding the lengths of the sides. The opposite sides of a parallelogram are equal in length. So the perimeter 5 in. (b) We + 8 is in. + 5 in. + 8 = 26 in. by multiplying the 8 in. and the height is find the area of a parallelogram base and the height. The base 4 in. So the area is (8 in.)(4 in.) Practice in. is = 32 in. 2 Find the perimeter and area of each parallelogram. Dimensions are in centimeters. a. 31 4 Math 87 c. Problem set gallon of milk costs $1.12, what 1. If - 2. Use a is the cost per pint? 70 ratio box to solve this problem. The cookie recipe and brown sugar in the ratio of 2 to 1. If 3 cups of oatmeal were called for, how many cups of brown sugar were needed? called for oatmeal 3. Matt ran the 400-meter race 3 times. His fastest time was 54.3 seconds. His slowest time was 56.1 seconds. If his average time was 55.0 seconds, what was his time for the third race? 4. It is A ^ miles to the end of the end of the trail and back trail. If Paul runs to the what in 60 minutes, is his average speed in miles per hour? 5. Sixty-three million, one hundred thousand is how much greater than seven million, sixty-five thousand? Write the answer in words. 6. Only three tenths of the print area of the newspaper carried news. The rest of the area was filled with advertisements. (a) What percent of the print area was filled with advertisements? (b) What was the ratio of news area to advertisement area? 7. Write 0.00105 in scientific notation. r 8. Write 3.02 x 10 5 in standard form. 5 31 Area of a Parallelogram 9. Use unit multipliers 11. 12. FB. Refer to this C Classify each of the following quadrilaterals. (a) ECBF (b) ECBA In the figure above, if AB BD = is the area of quadrilateral 8 cm. then what BC = 4 cm, - 6 cm, and BCEFt Classify each of the following angles. (a) 13. II to feet. Problems 10-12. D 10. 1760 yards ABDE is a rectangle and EC Quadrilateral figure in to convert ZECB Write — (b) ZEDC (c) ZFBA as a percent. Refer to this parallelogram to answer questions 14 and 15. 12 cm | 14. What the perimeter of is this parallelogram? 15. What is n 1.5 17. p + 18. r - cm the area of this parallelogram? Solve: 16. 16 21 3.6 15 = = 5 1.5 10 cm 316 Math 87 Add, 19. subtract, multiply, or divide, as indicated: - 10 10 + 10 X 10 21. 3 gal 1 pt 3 qt 9 oz 20. 10 4 22. 3 - 7 oz ± 23. 10 -s- 2.75 liters 1000 mL • 24. 4 4- • 8 12 29. 0.2 • * x 10 2 LESSON ) 7 in. 2 -s- 8 5 3 + 0.97) ft 1 6- * 3- 8 (0.8 (4 26. 2 3 f 1- 9 5 27. 17 1- yd yd 23 ' 1 7 5+ 3- - 1- 1 liter 25. 1 - V81 + 28. (2.4)(0.05)(0.005) 30. 0.36 + (4 * 0.25) Fraction-Decimal-Percent Equivalents 71 We remember that if we multiply a number by another number whose value we do not change the value of the number. We just change its name. All numbers have many fraction names and many decimal names, but only one percent name. To find the percent name for a number, we multiply the number by 100 percent. is 1, We can write 4.3 as a percent by multiplying by 100 percent because 100 percent is another way to write 1. 4.3 We can write one x fifth 100% = 430% as a percent by multiplying by 100 percent. i x 100% = 5 ^^ = 20% 5 To write any percent as a fraction, we divide by 100 percent. We can change 53 percent to a fraction by dividing by 100 percent. 7 Fraction-Decimal-Percent Equivalents 53% And 53% 100% = 53 100 of course this fraction equals the decimal 53 31 number 0.53. = 0.53 100 Future problem sets will contain problems that allow us to practice changing from percents to fractions to decimal numbers. The problems will require that we complete a table as we show in the following example. Example Complete the Fraction table. Decimal Percent i 1.5 (c) (e) Solution For (a) and (b) we (b) (a) 3 ffl find the decimal (d) 60% and percent equal to -. 0.3 (a) 3)l.00 (b) - x 100% = ^^ = 33-% 3 3 3 For (c) and (d) we find a fraction percent equal to 1.5. (c) 1.5 = 1-5- 10 (d) For (e) (f) 1.5 (e) (f) 60% = 60% mixed number) and a = \\ 2 100% = 150% x and (or a = we 60 100 find a fraction 3 60 = 0.6 100 and decimal number for 60%. 318 Math 87 Practice Complete the table. Fraction Decimal Percent a. b. 2 3 1.1 C. e. Problem set 1. 71 2. 4. 4% f. Eva bought forty 25-cent stamps, thirty 20-cent stamps, and twenty 15-cent stamps. If Eva paid for the stamps with a $20 bill, how much did she get back in change? At the post office When Jim is resting, his heart beats 70 times per minute. When Jim is jogging, his heart beats 150 times per minute. During a half hour of jogging, Jim's heart beats how many more 3. d. times than it would if The product of the number Nand 12 isN? Use a ratio box to solve this he were resting? is 288. What number problem. The ratio of brachiopods to trilobites in the fossil find was 2 to 9. If 720 trilobites were found, how many brachiopods were found? 5. In her first 5 points. basketball What was games Sherry scored the average number a total of 72 of points Sherry scored per game? 6. Draw a diagram of this statement. Then answer the questions that follow. The survey found that recommend Brand X. (a) According to the survey, 3 out of 4 doctors what percent of the doctors recommend Brand X? (b) According to the survey, what fraction of the doctors do not recommend Brand X? Fraction-Decimal-Percent Equivalents 319 7. Use a unit multiplier to convert 20 centimeters 8. Compare: 30(150 - 9. Light travels at a speed of about three hundred thousand 70) O 30 150 • kilometers per second. Write that to meters. -30-70 number in scientific notation. 10. Write each number in standard form. 6.05 x 10 6 (a) 11. 4 x 10" 5 (b) Divide 1000 by 32 and write the answer as a decimal number. 12. Complete the table. Fraction 1.2 (a) 13. Decimal Percent (b) Write the prime factorization of 5000. Refer to this figure to answer questions 14 and 15. Dimensions are in inches. 14. What is the perimeter of this parallelogram? 15. What is the area of this parallelogram? Solve: 16. 5a = 1.7 Add, 17. 8.1 + x = 12 subtract, multiply, or divide, as indicated: + 7 18. 7 20. 5.8 - • (6 7 - - 7 -s- 3.17) 7 19. 25 2 - 5 3 - a/100 21. (2.4X0.75X0.05) 320 Math 87 22. 1 _ 24. yd 1 ft 1 in. 2 2 1 3l- 2-^3 r 3 I 2 min 45 45 min 15 hr 15 + 5 11 8 6 6- 27. 5- • 4 J + 32 30. 6.4 + 5.88 + 15.7 + 24 2 • 3 33 $15.00 + 0.75 29. Sequences sec 111 1 0.8 sec 3- + 5- + 2- 25. 28. LESSON l in. 24 mi Igal , 1 12gal - 26. ft 23. + 0.09 + 23.86 Functions, Part 1 • 72 Sequences A sequence is an ordered list of numbers that are arranged by following a certain rule. To find the next number in a sequence, we first determine the rule for the sequence. Then we use the rule to find the next number. Example 1 Find the next three numbers in this sequence. 5, 20, 35, 50, Solution . . . As we inspect the numbers in the sequence, we see that each number is 15 greater than the preceding number. This is an addition sequence. 5 5, 5+15+15+15' 5+15+15 V v ' v 35, v 50 We continue this pattern and find that the next three numbers in the sequence are 65, 80, 95 Example 2 Find the next three numbers in 5, 10, 20, this sequence. 40, . . . * Sequences Solution • Functions, Part 321 1 As we inspect the numbers in the sequence, we see that each number is twice the preceding number. This is a multiplication sequence. 5x2x2 5x2 5 y 5, v ' v v 10, 20, v ' 5x2x2x2 ' v 40 We continue this pattern and find that the next three numbers in the sequence are Functions A function certain rule. number a set of is We 80, 160, 320 number pairs that are related will be finding a by a missing number from a The thinking we use pair. for these problems is use for sequence problems. We study pairs of numbers to determine a rule for the function. Then we use the rule to find the missing number. similar to the thinking Example 3 Solution we Find the missing number, We study each "in-out" number pair to determine the rule for the function. We see that for each complete pair, if the number "in" is multiplied by 3, it equals the number Thus the "out." U N 3 Practice — (J 5 1 — — 15 T O N 10 — 30 rule of is We multiply 7 Find the next three numbers in each sequence. 5, 11, 17, a. 23, . . . b. 3, 9, 27, 81, Find the missing number in each diagram. In c. 4 3 7 9 F Out U N — 20 C —- 15 T — T O -* N 9 ** 7 "multiply by 3." We use this rule to find the missing number. by 3 and find that the missing number is 21. the function Out F In In d. 1 3 5 - F U N C T I Out — — ^ O —- N 4 9 322 Math 87 Problem set 1. 72 from Jim's house to school. How far does Jim walk going to and from school every day for 5 It is 1.4 kilometers days? 2. The parking charges 25 C for each half hour or part of a half hour. If Edie parks her car in the lot from 10:45 a.m. lot how much money until 1:05 p.m., 3. 4. If the product of the sum of the The football number N and 17 N and number will she pay? team won is 340, what is the 17? 3 of their 12 games but lost the rest. (a) What was (b) What fraction of the (c) What percent of the games did the team win? the team's won-lost ratio? games did the team lose? bowling average after 5 games was 120. In his next 3 games Will scored 118, 124, and 142. What was Will's bowling average after 8 games? 5. Will's 6. Draw a diagram of this statement. Then answer the questions that follow. Three fifths of the 60 questions on the test were multiple-choice. (a) How many of the 60 questions were multiple-choice? (b) What percent of the 60 questions were not multiplechoice? 7. Write 0.0000001 in scientific notation. 8. Write 1.5 x 10 7 in standard form. 9. Compare: 20 10. qt O 5 gal Divide 3.45 by 0.18 and write the answer rounded to the nearest whole number. Sequences 11. Find the next three numbers in 20, 15, 10, 12. Complete the table. • Functions, Part 1 323 this sequence. . . . Decimal Fraction Percent i 13. (b) (a) 6 Find the missing number. F In Out U N C 3 T 6 12 24 I O 2 N Refer to this figure to answer questions 14 and 15. D 14. 15. In this figure, AE II C CD and AC B II ED. (a) Classify quadrilateral ACDE. (b) Classify quadrilateral ABDE. If AB = 24 cm, AC = 25 cm, and the area of quadrilateral Solve: 1.5 16. 7.5 w - y = 0.17 17. 1.7 18. 3x = 0.45 ACDE? AE = 30 cm, what is 324 Math 87 Add, 19. subract, multiply, or divide, as indicated: 10-1 10 3 - 10 2 + 21. + 1 gal 2 qt 1 pt 1 gal 2 qt 1 pt - 4 - 6 + 3(2) 20. 22. day l 2 min hr 30 min - 8 t> 5280 mi ft \ 5- - 1- ( 4 6 > - \ \ 2- + 5^ - 25. 10 - 24. mi 1 ( 5 - 27. 5 29. 4.6 2 3- 26. • 6 4 h- 2 (4.3 ') - 0.021) 80 -s- 3) 3 hr 15 \ 23. + (5 28. (3.6)(2.5)(10 30. 15 * 0.015 2 ) Adding Integers on the LESSON Number Line 73 IntegGrs The whole numbers are zero and the counting numbers. the numbers in this sequence are whole numbers. 0, 1, 2, 3, Integers include the whole all . is, also the opposites their negatives. All the in this sequence are integers. . The . numbers and of the positives integers, that numbers . All . . dots on this , o, Z, number 1, U, 1, Z, o, line mark the . . . integers from -5 through +5. -5 -4 -— -3 -2 -1 - 4 12 3 4 4 5 *—~ Notice that the numbers between the whole numbers, such as 3 j and All 1.3, are not integers. numbers except zero are signed numbers, or negative. Zero is either positive neither positive nor negative. Positive Adding Integers on the Number Line 325 and negative numbers have a sign and a value, which is called absolute value. Numeral Number Sign Absolute Value +3 Positive three + 3 -3 Negative three 3 The absolute value of both +3 and -3 is 3. Notice on the number line that +3 and -3 are both 3 units from zero. Absolute value can be represented by distance, whereas the sign can be represented by direction. Thus positive and negative numbers are sometimes called directed numbers because the sign of the number (+ or -] can be thought of as indicating direction. When we add, subtract, multiply, or divide signed numbers, we need pay attention to the signs as well as to the absolute values of the numbers. In this lesson we will practice adding positive and negative numbers. Number line model for addition A to number may be used line to illustrate the addition of signed numbers. A positive 3 is indicated by a 3 -unit arrow that points to the right. A negative 3 is indicated by a 3-unit arrow that points to the left. -3 +3 To show the addition of +3 and -3, we begin at zero on the number line and draw the +3 arrow. From this arrowhead we draw the -3 arrow. The sum of +3 and -3 is found at the point on the number line that corresponds to the second arrowhead. -3 -1 +3 h H -3 -I We of see that the sum two opposites is of +3 and -3 always zero. is 0. h We find that the sum H 326 Math 87 Example 1 Sketch a number line to show each addition problem, (-3) + (+5) (a) Solution ^ (a) -5 -4 (-4) + (-2) (b) -3 -2 -1 4 3 + +- 5 +- -3 + H We begin points to —I +5 h and draw an arrow 3 units long that the left. From this arrowhead we draw an at zero arrow 5 units long that points the sum of -3 and +5 is 2. Jl (b) -6 — I- -5 — -3 -4 +— + We used the arrows Example 2 -2 see that 2 3 + -4 -2 is We —h —I- +- I- to the right. to -+- show that the sum of — 4 and - 2 -6. Sketch a number line to show this addition problem. (-2) + (+5) + (-4) Solution This time we draw three arrows. We always begin the first arrow at zero. We begin each remaining arrow at the arrowhead of the previous arrow. —+ -4 — -3 -2 +- -+- -1 2 3 4 5 -2 +5 -»H arrowhead corresponds to - 1 on the number line, the sum of -2 and +5 and -4 is -1. The Practice last Sketch a number line to show each addition problem. a. (-2) + (-3) b. (+4) + (+2) so Adding Integers on the Number Line c. (-5) + (+2) d. (+5) e. (-4) + (+4) f. Problem set 73 1. 327 + (-2) (-3) + (+6) + (-1) an 8 by 10 print. They cost $2.35 for a 5 by 7 print, and 60c for each wallet-size print. What is the total cost of two 5 by 7 prints and six School pictures cost $4.25 for wallet-size prints? Refer to this graph to answer questions 2 and 3. Maximum Temperature Readings for Tri-City Area, August 1-7, 1990 Average maximum for August temperature 2. The highest temperature reading on August how much greater than the average maximum for 3. 1990, was temperature 6, August? average maximum temperature during the seven days of August 1990? What was the first 4. The sum of the number n and 12 of n and 12? 5. Use a ratio box to solve this problem. The ratio of sonorous to discordant voices in the crowd was 7 to 4. If 56 voices were discordant, how many voices were sonorous? is 30. What is the product 328 Math 87 6. Draw a diagram of this statement. Then answer the questions that follow. The Celts won three fourths of their first 20 games. (a) How many of their first 20 games did the Celts win? (b) What percent first fail to of their 20 games did the Celts win? 7. Write 4,000,000,000,000 in scientific notation. 8. from the sun is 3.67 x 10 9 miles. Write that number in standard form. 9. Pluto's average distance A micron is 1 x 10 -6 meter. Write that number in standard form. 10. Use a unit multiplier 11. Complete the table. 300 to convert Fraction 13. Sketch a number line (a) (-5) + (+2) (b) (+5) to Decimal Percent 12% (b) (a) 12. mm to centimeters. show each addition problem, + (-2) Find the missing number. In 2 F Out U N -+ 12 C T 8 T O N -*> 14 24 — — 12 Adding Integers on the Number Line 329 Solve: 4.4 = 16. 2 n + - = x 0.8 8w 14. 15. —3 4 3 Refer to this figure to answer questions 17 and 18. Dimensions are in millimeters. All angles are right angles. 50 20 30 60 15 35 17. What is 18. What is Add, 19. - 4 + 5(6) 3 yd 2 (7 ft 1yd + 2±hr + 8) 7f in - 9 20. 22. f 4-1 26. 6 - 0.1 29. 0.364 + 7 (0.2)(0.3) \/T 6 oz 1 pt 12 oz ,6? -^ 3 J La h- 6+ u0 » 4J l 27. + pt 1 12 9 I \ 23 N 24. • - 1 qt (s 1 hr 3- - \/64 5 1 in. 50mi 2 25. the area of this figure? subtract, multiply, or divide, as indicated: 21. 23. the perimeter of this figure? 6 28. 0.065 X $18.00 30. 3 -s- (30 -s- 0.03) 330 Math 87 Fractional Part of a Number and Decimal Part of a Number, Part 1 LESSON 74 Problems about parts of a number are stated by using decimal fractions or common fractions. The procedure for solving both kinds of problems is the same. We can solve fractionalpart-of-a-number problems by translating the question into an equation. Then we solve the equation. To translate, Example 1 Solution What number is We replace We replace of with x is with = 0.6 of 31? This problem uses a decimal number to ask the question. We will use N to represent what number. We will translate is by writing an equals sign. We will translate of by writing a multiplication symbol. W What number MM question = 0.6 x 31 equation is I WH 0.6 of 31? To find the answer, we multiply. WN = Example 2 Solution Three fifths of 120 is 18.6 multiplied what number? This time the question is phrased by using a common The procedure is the same. We translate directly. Three fifths of J 3 5 is what number? I \ 120 = W-, 120 M x N fraction. question equation Fractional Part of a To find the answer, Number and Decimal we Problem set 74 Number, Part 331 1 multiply. = WN multiplied 72 = WN simplified 5 Practice Part of a Write equations to solve each problem. a. What number b. Three eighths of 3y c. What number d. Seventy-five hundredths of 14.4 1. 2. is is - of 71? o is what number? 0.6 of 145? is what number? Five and seven hundred eighty-four thousandths is how much less than seven and twenty-one ten-thousandths? Cynthia was paid 20c per board for painting the fence. If she was paid $10 for painting half the boards, how many boards were there? 3. When divided by n, the quotient product when 72 is multiplied by n? is 4. Four test. 72 is fifths of the students passed the 12. What is the (a) What percent (b) What was the ratio of students who passed to students who of the students did not pass the test? did not pass? 5. The average height of the five players on the basketball team was 77 inches. One player was 71 inches tall. Another was 74 inches tall, and two were 78 inches tall. How tall was the tallest player on the team? 6. Write each number in scientific notation. (a) 0.00000008 (b) 67,500,000,000 332 Math 87 7. Draw Then answer a diagram of this statement. the questions that follow. Two thirds of the 96 members approved of the plan. (a) How many members approved of the 96 of the plan? (b) What percent of the members did not approve the plan? Write equations to solve problems 8 and 8. What number is of 17? f 4 9. What number is 0.7 of 6.5? 9. O 0.33 10. Compare: - 11. Complete the table. Decimal Fraction Percent i show 12. Sketch a number line to 13. Find the next three numbers in 1, 4, 9, 14. (b) (a) 8 the addition (-3) + (-1). this sequence. 16, 25, . . . Write the prime factorization of 360. Solve: 15. 17. 1 p - 2 48 32 v 20 16. 9m = 0.117 of Fractional Part of a Number and Decimal Part of a Number, Part 333 1 Refer to this figure to answer questions 18 and 19. Dimensions are in feet. 18. What is the area of the parallelogram? 19. What is the perimeter of the trapezoid? Add, 20. subract, multiply, or divide, as indicated: 32 + 23. 2) - 8 4 + -r 3 days 16 hr 48 min 1 day 21. + + 4(3 15 hr 54 m 24. 19- + 27- + 244 • u 5 4 26. 27. 4.6 + 0.375 + 12 28. 6.5 29. (0.4)(0.3)(0.2)(0.1) 30. 0.3 - -r (0.65 (3 -r - 5^ 1 1+ * 24 4 / 0.065) 0.03) 8 — ( \ 25. cm 1 1 sec 3- - (5 - 20 cm = 22. ? min m 344 5 sec a/36 12 J 6 334 Math 87 LESSON Variables 75 we In algebra and Evaluation often use letters in place of numbers. The expression x + y means that two numbers are added. The value of the expression depends upon the numbers we choose for x and y Since the value of a letter may vary, a letter is called a variable. we use the symbols +, - and + to indicate the operations of arithmetic. In algebra we use the + and signs, but sometimes we do not use the x and + signs. To show In arithmetic x, we can write the variables together with no sign between. Thus, xy means that x and y are multiplied. To show division, we can use a division line. The chart below summarizes how the operations of arithmetic multiplication of variables, can be indicated in algebra. Meaning Expression a + b b is added a - b b is subtracted from a b is multiplied by a a is divided by b ab . a to a b The value of an expression depends are substituted for the variables. by finding Example 1 Solution its We upon the numbers that evaluate an expression value. Evaluate: a + ab if a = 3 and b = 4 We will begin by writing parentheses in place of each variable. This step may seem unnecessary, but avoided if this step is the first step. a + many errors can be ab () + ()() parentheses • Variables and Evaluation Then we replace a with We we a + (3) + Solution 4. ab substituted (3)(4) follow the order of operations by multiplying add. (3) + 3 + First we Then multiplied 12 added xy Evaluate: first. problem (3)(4) 15 Example 2 and b with 3 335 if x= 9 and y - - replace each letter with parentheses. - xy Then we - )( ( x^ parentheses ) write 9 in place of y. "2 ^ v . in place of X xy (9) 2 x and — (9) substituted 3; We follow the order of operations by multiplying and dividing before we subtract. (9) ^ v 6 3 (9) ; 4 multiplied and divided 2 subtracted Practice Evaluate: a. ab - be if a = 5, b = 3, and c = 4 336 Math 87 b. — ab + a = if 6, b = and c = 4, 2 c c. Problem set 75 1. x - xv y if 3^4 x = - and y = — At 1:30 p.m. David found a parking meter that still had 10 minutes until it expired. He put 2 dimes into the meter and went to his meeting. If 5 cents buys 15 minutes of parking time, at what time will the meter expire? Use the information 2 and 3. The Barkers in the next paragraph to started their trip a total of 39,872 miles on with a their car. answer questions tank of gas and They stopped and full tank 4 hours later with 8.0 gallons of At that time the car's total mileage was 40,060. filled the gas 2. How 3. The did they travel in 4 hours? far Barkers' car traveled an average of per gallon during the 4. When 24 5. Use a ratio first how many miles 4 hours of the trip? multiplied by w, the product the quotient when 24 is divided by w? is box to solve this Bolsheviks in the crowd. czarists gas. was 9 to 8, If is 288. What is problem. There were 144 the ratio of Bolsheviks to how many czarists were in the crowd? 6. Draw a diagram of this statement. Then answer the questions that follow. Exit polls showed that 7 out of 10 voters cast their ballot for the incumbent. (a) According to the exit polls, what percent of the voters cast their ballot for the incumbent? (b) According to the exit polls, what fraction of the voters did not cast their ballot for the incumbent? - Variables and Evaluation Write equations to solve Problems 7 and 7. What number is 5 — of 337 8. 1 3-? 6 3 8. What number 9. Write 1.6 x 10 7 in standard form. Then use words to is 0.06 of 23.5? write the number. 10. Use a unit multiplier 11. Sketch a number line to show this addition problem. to convert 1.2 liters to milliliters. (-3) + (+4) 12. Complete the table. Decimal Fraction 0.9 (a) 13. 2 —*> 5 10 Evaluate: ab - a + be if a = Refer to this figure to answer questions 15 and 16. — — 5, (b) F In Find the missing number. -*- 14. Percent b = Out U N 1 C 4 T 10 I O N 3, 20 and c = 4 4 \ Dimensions are in inches. All angles are right angles. 15. What is the perimeter of the figure? 16. What is the area of the figure? QV 338 Math 87 Solve: 12 _9 16 m 17. Add, - - 6 * 19. 10 - 20. I4 + 23 + 32 + 41 21. lm (9 - 8) 20mm 1 + x 3 + 1 mm = 2 qt 1 pt 2 oz 1 gal 1 344 ^4+ opt m 14-o-z 1 r 60 sec 1 sec 25. = ^gal^qt/Tpt't - 23. 5.4 subtract, multiply, or divide, as indicated: 22. 1 24. min 1 i i) 10- - (3- 2- 2 LESSON 18. l 3 26. 2i 2 4 4- + 6 l • 7 1- • 8 5 2y 3-2-1 27. $20 - 0.25 X $2 0) 28. 4.36 4 400 29. (0.1)(0.01)(0.001) 30. 0.1 ( 1 1- 9 - + (0.001 + 0.01) Classifying Triangles 76 Certain kinds of triangles have special names. These triangles are classified according to the relative lengths of the sides or the measures of the angles. 339 Classifying Triangles When we we classify triangles by the lengths of the sides, determine whether two or more sides are equal in length. Scalene triangles have three unequal sides and three unequal angles. Isosceles triangles equal sides and at have at least two least two equal an- Marks may be used to show sides of equal length and angles of equal gles. measure. Equilateral triangles have three equal sides and three equal angles. An equilateral triangle is a regular triangle. When we classify triangles by their angle measures, we determine whether the largest angle of the triangle is acute, right, or obtuse. We use the same words to describe the triangles that contain these angles. Acute triangles have three acute angles. Right triangles have a right angle. Obtuse triangles have an obtuse angle. Example 1 Solution The perimeter of an equilateral triangle inches long each side? is is 2 feet. How many All three sides of an equilateral triangle are equal in length. Since 2 feet equals 24 inches, we divide 24 inches by 3 and find that the length of each side is 8 inches. Example 2 Sketch an isosceles right triangle. 340 Math 87 Solution means the triangle has at least two equal sides. "Right" means the triangle contains a right angle. We sketch a right angle, making both segments equal in length. Then "Isosceles" we complete Practice the triangle. Classify each triangle /x. a. angles. by c. its sides. e. - 3 g. If 3 f. we know that two sides of an isosceles cm and 4 cm and that its perimeter is not what Problem set 76 its b. Classify each triangle d by is its triangle are 10 cm, then perimeter? homework during the first 4 nights of the week. What was the average number of minutes he spent on homework each night? 1. Dan spent 2. Eva has just enough money to buy thirty 25-cent stamps. If Eva bought 15-cent stamps instead of 25-cent stamps, how many stamps could she buy? 3. head penny was minted in 1859. latest Indian head penny was minted in 1909. For how many years were the Indian head pennies minted? The The a total of 9 hours doing earliest Indian (Be careful.) Classifying Triangles 4. The product of y and 15 is 600. What is the 341 sum of y and 15? 5. Thirty percent of those gathered agreed that the king should abdicate his throne. All the (a) What (b) What was who 6. Draw rest disagreed. fraction of those gathered disagreed? the ratio of those who agreed to those disagreed? a diagram of this statement. Then answer the questions that follow. Five eighths of the 600 trees in the forest were deciduous. (a) How many of the trees in the forest were deciduous? (b) What percent of the trees in the forest were not deciduous? 7. Write an equation to solve this problem. is I o What number of 510? 8. Write twenty-five thousandths in scientific notation. 9. Write 1.86 x 10 5 in standard form. Then use words to write this number. O 10. Compare: 11. Sketch a number line to show (-3) + (+4) + (-2). 12. Complete the 1 qt table. 1 l iter Fraction Decimal Percent 5 8 (a) (b) 342 Math 87 13. X x + Evaluate: y x = 12 and y = if 3 y 14. Find the next three numbers in this sequence. 113 15 8'4'8'2'8 15. This figure contains an acute triangle, a right triangle, and an obtuse (a) Which triangle. triangle is an acute triangle? (b) Which triangle is (c) Which triangle is a right triangle? an obtuse triangle? Solve: 16. 7q = 1.428 18. 3.4 Add, 19. = 52 15 w 1 5 + 10-5-2 20. 3(8) yd 22. 1 24. 4- + 7- + 3^ 60 mi 1 hr 22 - + 2 5 - V49 + 25. - x subtract, multiply, or divide, as indicated: 21. 23. 5 17. 2 ft 3f in. 2 ft 6^ in. 1 hr 60 min 1 3^ + f 4* 1- 5 4 L - 50 mL 13 20 - 'A 7V 2 = 1 4 5 2H. (5)(2) 2 - ?] V ? mL Symbols 27. 6.7 29. 4.72 + 0.98 + 12 + 9.9 - (6 - 1.375) 28. $30 - 30. 5 * of Inclusion (0.3 (2.5 343 x $30) * 0.05) Symbols of Inclusion LESSON 77 Parentheses, brackets, and braces Parentheses are called symbols of inclusion. We have used parentheses to show which operation to perform first. To simplify the following expression, we add 5 and 7 before subtracting from 15. 15 and braces - (5 + 7) symbols of inclusion. When an expression contains multiple symbols of inclusion, we simplify within the innermost symbols first. Brackets [ ] { } are also To simplify the expression 20 - [15 - we Next we (12)] we - Then we subtract. [20 + (10 - 5)] simplify within the parentheses. 50 - [20 + 50 - 25 simplified within parentheses subtracted Simplify: 50 First first. simplified within brackets [3] 17 Solution 7)] simplify within the brackets. 20 - 1 + simplify within the parentheses 20 - [15 - Example (5 [25] (5)] simplified within parentheses simplified within brackets subtracted 344 Math 87 Division Mne Example 2 Solution A division line also serves as a symbol of inclusion. We simabove and below the division line before we divide. We follow the order of operations within the symbol of inclusion. plify 4 + Simplify: 5x6-7 10 - (9 - 8) We will simplify above Above the line and below the line before we divide. we multiply first. Below the line we simplify within the parentheses first. This gives us 4+30-7 10 - We {1) continue by simplifying above and below the division/ line. 27 Now we Practice divide and get Simplify: a. 30 - [40 - (10 - 2)] 10+9-8 5-4-3 c. Problem set 77 1. b. 100 - 3[2(6 - 1 d. + 2 2)] + 2(3 + 4) 10 - 9(8 - 7) 5 and Jason each earn $5 per hour doing yard work. On one job Jennifer worked 3 hours and Jason worked l\ hours. Altogether, how much money were Jennifer they paid? 2. If p is prime number greater than 90 less than p? a two-digit prime number is 90, what . Symbols 345 of Inclusion 3 Use a ratio box to solve this problem. The ratio of favorable to unfavorable outcomes was 3 to 5. If 45 unfavorable outcomes were possible, how many favorable outcomes were possible? 4. During the first 5 days of the journey, the wagon train averaged 18 miles per day. During the next 2 days the wagon train traveled 16 miles and 21 1017 miles, the wagon train have to travel? If the total journey is miles, respectively. how much farther does What number 5. Write an equation to solve this problem. is 0.35 of 840? 6. The average distance from the Earth to the Sun 1.496 x 10 8 km. Use words to write that number. 7. Draw a diagram of this statement. Then answer is the questions that follow. « Three tenths of the 40 cars pulled by the locomotive were tankers. 8. 9. (a) How many (b) What percent of the cars were tankers? of the cars were not tankers? The top speed of Dan's pet snail is 2 x 10" 3 mile per hour. Use words to write that number. Use a unit multiplier to convert 3000 sec to minutes. 10. Divide 4.36 by 0.012 and write the answer with a bar over the repetend. 11. Sketch a number line and draw arrows to show (-3) + (+5) + (-2). Fraction 12. Complete the Decimal Percent table. (a) (b) 125% 346 Math 87 13. Find the missing number. U N — C — 3 12 14. Evaluate: a = 15. ab b = 6, 4, In this figure, and c = AB (a) Classify (b) What if = AD ABCD by = its *> T 6 2 I O N 15 BD = Out F In CD = — — 4 2 5 5cm sides. the perimeter of is the equilateral triangle? Which triangle appears to (c) be a right triangle? Solve: 16. - = x + 4 17. = 0.4p 2 3 14 n 20 5 18. Add, subtract, multiply, or divide, as indicated: 19. 3[24 20. 33 21. 1 - + (8 3 • 2)] - 6 | 4 - V32 + 4 2 m - 95 cm = ? cm 22. 1 week ? 5 5 3 6 9 - 23. 20 mi 1 gal 1 gal 4 qt 1 25. 122 • 24. 4 45 days 7 hr 5 days 9 hr 2 4- + 3- + 2( 26. 2 6- - 13 3 -s- 3 y — i 347 Adding Signed Numbers 27. 4 * (0.08 29. 10 - 4- - (0.1 16) 28. (0.25)(0.08)(0.05)(10 0.001) 30. $1.56 ) 0.06 h- Adding Signed Numbers LESSON 78 4 In this lesson we summarize what we have learned will about adding signed numbers. we have seen that when we add two negative numbers, the sum is a negative number. When we add two positive numbers, the sum is a From our practice on the number line -+- -\— positive number. -5 H -h 1 — i i— -2 "3 +2 +3 I + (-3) = -5 (-2) We have (+2) + (+3) = +5 when we add a positive number and number, the sum is positive or negative or zero also seen that a negative depending upon which, if either, of the numbers has the greater absolute value. -2 2 o -+- H -4- h +3 -wH H I 1 +5 l H h (+3) + (-5) = H -2 1 (-3) 1 h 1 H +3 1 — - 1 -h —— i -3 : i (+3) + (-3) = (- + (+5) = +2 1- 348 Math 87 We can summarize these observations with the following statements. 1. 2. The sum of The sum of two numbers with the same sign is the sum of their absolute values. The sign of the sum is the same as the sign of the numbers. two numbers with opposite signs is the difference of their absolute values. The sign of the difference is the same as the sign of the number with the greater absolute value. 3. The sum of two opposites is zero. We can use these observations to help us add signed numbers without drawing a number Example 1 Find each sum: (a) Solution line. (a) (-54) + (-78) (b) (+45) + (-67) (-92) + (+92) Since the signs are the same, we add the absolute values and use the same sign for the sum. -132 (-54) + (-78) = (b) (c) Since the signs are different, we find the difference of the absolute values and keep the sign of -67 because 67 is greater than 45. (+45) + (-67) = -22 (c) The difference of the absolute values of zero. Zero has no sign. The sum of two opposites (-92) + (+92) = Example 2 Solution Find the sum: (-3) + (-2) + We will show two methods. (+7) -92 and 92 + (-4) is is zero. 349 Adding Signed Numbers Method Add two numbers. Then add the third number. Then add the fourth number. 1. [(-5) 2. + (+7)} + (-4) added -3 and -2 (+2) + (-4) added -5 and +7 -2 added +2 and - 4 same sign all numbers with first. + (-2) + (-4)] + (+7) [(-3) rearranged (-9) + (+7) added -2 added Find each sum: ( (a) Solution problem Rearrange the terms and add the Example 3 first + (-2)] + (+7) + (-4) [(-3) Method the -2 + A (b) These numbers are not these signed numbers (+4.3) + (-7.24) integers, but the is the same as the method method for for adding adding integers. (a) The signs are both negative. absolute values We add and keep the same the sign. 2- = 22 6 + 3— = 3— iA -2- ( x + -3- 3 3 \ 6 6 6 (b) The signs are different. We find the difference of the absolute values keep the sign of -7.24. (+4.3) + (-7.24) = -2.94 and /?24 -4.3 2.94 350 Math 87 Practice Find each sum: a. (-56) + (+96) b. (-28) + (-145) c. (-3) + (-8) + (+15) d. (-5) ( *C\ V 6y Problem set 78 1. ( v f. + (+9) + (-3) (-1.6) + (-11.47) 3, Two trillion is how much more than seven hundred fifty answer in billion? Write the 2. (+7) A -3- + +5- e. + The scientific notation. taxi cost $2.25 for the first mile and additional tenth of a mile. For a 5.2-mile trip $10 and told the driver was the to each Eric paid 15tf for keep the change. How much driver's tip? 3. The product of x and 12 is 84. The product is 48. What is the product of x and y? 4. Three hundred twenty boys and four hundred crowded (a) of y and 12 girls into the assembly. What fraction of the students in the assembly were boys? (b) What was 5. What 6. Draw is the ratio of girls to boys in the assembly? the average of 1.74, 2.8, 3.4, 0.96, a diagram of this statement. 2, and 1.22? Then answer the questions that follow. The viceroy conscripted two fifths of the 1200 serfs in the province. (a) How many of the serfs in the province were conscripted? (b) What percent conscripted? of the serfs in the province were not 351 Adding Signed Numbers 7. Write an equation to solve this problem. is 8. What number | of 100? The temperature at the center of the sun 1.6 x 10" degrees Celsius. Use words about is to write that number. 9. 10. 11. 12. A red blood cell is about 7 x 10~ 6 meter in diameter. Use words to write that number. Divide 456 by 28 and write the answer mixed number. (a) as a (b) as a decimal (c) rounded rounded to to the nearest two decimal whole number. Find each sum: (a) (-63) + (-14) (b) (-16) + (+20) + (-32) Complete the Decimal Fraction table. 2.5 (a) 13. Find the next three numbers in 48, 24, 12, 6, 14. Evaluate: x + xy if . . —3 0.6 0.7 0.4 What is the perimeter of the hexagon? What = 0.4 angles. 16. (b) . 2 x = — and y questions 15 and 16. Dimensions are in meters. All angles are right Percent this sequence. Refer to this hexagon to answer 15. places. is the area of the hexagon? 352 Math 87 Solve: 4 — = y -1-7 17. 2 18. 9 Add, 25x = 10 9 subtract, multiply, or divide, as indicated: 19. 3 20. + 4' 5 gal 2 qt 1 pt 7 oz 1 gal 1 qt 1088 1 25. + 100 - [20 + 5(4 + 3(2 + 21. 23. 2 ft 1 ( 6 * 5 2m -800mm 24. 3- 1 min 2-ll 26. - mm 3 15 8 5 i- ( 1) 4 - 2- 6 27. 0.1 29. $15 + (0.06 x $15) (0.01 • 5 2) - = 9 oz 0.001) 3j 28. 0.1 + 0.2 + 0.3 + 0.4 30. 5.1 -f (5.1 t- 1.5) Area of a Triangle LESSON 79 22. 60 sec sec 2** 1 pt 1))] We have practiced finding the area of a rectangle and the area of a parallelogram. In this lesson the area of a triangle. We may we will practice finding cut out some us understand the concept of this lesson. triangles to help Area of a Triangle Project Fold a sheet of paper in draw half. Then use a pencil 353 and a few triangles on the paper that are large enough to cut out. Draw a right triangle, an acute triangle, and an obtuse triangle. ruler to a While the paper is still folded, cut out each triangle so that by cutting the folded paper you cut out a pair of identical (congruent) triangles at the same time. Then fit each pair of triangles together to form a quadrilateral with that pair. • What kind of quadrilateral is formed by the two is formed by the two is formed by the two right triangles? • What kind of quadrilateral acute triangles? • What kind of quadrilateral obtuse triangles? In all three cases above, to we can arrange the two triangles form a parallelogram. Thus, one triangle is half of a parallelogram. This fact gives us a clue for finding the area of a triangle. — 354 Math 87 We remember that the area of a parallelogram equals the product of the base and the height. Since the area of a triangle half the area of a parallelogram, to find the area of is a triangle we multiply the base by the height and then divide the product by 2. Area r , . base We notice = of a triangle base x height - base base may be one side the height may fall that in a right triangle the height of the triangle, whereas in other triangles inside or outside the triangle. In every case, the height is perpendicular to the base. Example Find the area of each triangle. Dimensions are in inches. (b) (a) (c) h__N Solution The base of each triangle triangle is 6 inches. and the height 8 inches, is Thus the area of each of each triangle is the same. Area r . , ot a triangle = Area = — base x height e 8 in. Area = 24 Practice Find the area of each a. triangle. b. in. • 6 in. 2 Dimensions are in centimeters. c. 355 Area of a Triangle Problem set Refer to this table to answer questions 1 and 7Q ^ 2. Number of Lincoln Head Pennies Minted Number Minted Date 1909-S 1,750,000 1909-S-VDB 1. 484,000 1914-D 1,193,000 1931-S 866,000 The 1909-S and 1909-S-VDB were both minted San in how many Lincoln head pennies were minted in San Francisco in 1909? Francisco. Together, 2. How many more 1914-D pennies were minted than 1931S pennies? 3. Gilbert wanted to buy packages of crackers and cheese from the vending machine. Each package cost 35$. Gilbert had 5 quarters, 3 dimes, and 2 nickels. How many packages of crackers and cheese could he buy? 4. The two prime numbers p and m are between 50 and Their difference is 6. What is their sum? 60. box to solve this problem. The ratio of flies to ants at the picnic was 2 to 15. If there were 60 flies, how many ants were there? 5. Use 6. Draw a ratio Then answer a diagram of this statement. the questions that follow. The survey found that only 2 out of 5 Lilliputians believe in giants. (a) According Lilliputians (b) If to the survey, what of of the do not believe in giants? 60 Lilliputians were selected many fraction them believe for the survey, in giants? how 356 Math 87 Write equations to solve Problems 7 and 7. Three fifths of 7- is 8. what number? 2 8. What number 9. Write each number in scientific notation, (a) 10. 0.95 of 3500? is 400,000,000 (b) 0.00000078 Write 1.7 x 10 9 in standard form. Then use words to write this number. 11. Use a unit multiplier 12. Complete the to convert Fraction table. to minutes. Decimal Percent 85% M.ss (a) 13. 2- hr Find the sum. (-48) + (+17) + (+31) 14. Write the prime factorization of 4900. 15. Evaluate: xy x if x = 0.2 and y = 2 y 16. In this figure the quadrilateral is a parallelogram. Dimensions are 12 in meters. (a) What is the area of the parallelogram? (b) What is the area of the shaded triangle? Solve: 17. 3.14 = lOr 18. 36 16 2j: 19. a 436 + w = 600 Percent of a Add, subtract, multiply, or divide, as indicated: 20. 200 - {100 - 21. 2 23. 357 Number L - 380 mL 720 mi = ? + 5) mL + 5(3 + 22. 4)]) yd 1 2 ft 2 ft / hr 1 24. 60 min hr 1 [3(4 8- + 3 in. 5j in. 7-6 6 8 \ 25. ,4| I30 1 (U H- I a 5 - 6- - 24 J - 27. 2.5 29. $37.50 * 0.75 (2.05 2.005) that percent - 28. [(0.6)(0.5)(0.4)] 30. $24 + (0.05 x $24) Percent of a We remember *4 2 3 LESSON 80 26. 0.3 Number means "by the hundred." 40 percent means 6 percent means 140 percent means 40 100 6 100 140 100 We can translate percent problems into equations the same way we translate fractional-part-of-a-number problems. We remember to replace is with = replace of with x replace percent with —percent 358 Math 87 Example 1 Solution What number We 40 percent of 75? is translate directly. What number is I \ 40 percent of 75? ill x N 100 Z5 question equation e To find the answer, we multiply. 40 x 75 100 ,.r N WN Example 2 Solution = 30 Eight percent of 36 We simplified what number? is translate directly. Eight percent of 36 is what number? 111 I WN x 36 = 100 To solve, we = Wfj N multiplied F WN simplified 2.88 = 80 equation multiply. 100 Problem set question I o Practice multiplied Write an equation to solve each problem. a. What number b. Three percent of 39 c. What number 1. is is 50 percent of 150? is what number? 25 percent of 64? Eight hundred seventy-six ten-thousandths is how much more than seventy-nine thousandths? Write the answer in words. Percent of a Number 359 2. Use a ratio box to solve this problem. The ratio of princes to knights at the tournament was 4 to 27. If 108 knights attended, how many princes were at the tournament? 3. One eighth of the possible outcomes were favorable, while the (a) outcomes were unfavorable. rest of the possible What fraction possible of the outcomes were unfavorable? (b) What was the outcomes of favorable ratio to unfavorable outcomes? 4. What is 12 the average 6 of 2-, 3-, 4, 2 5 and 4—? 6 3 5. Write an equation to solve this problem. is 6 percent of 350? 6. Draw a diagram of this statement. What number Then answer the questions that follow. Diane gave - of her 234 baseball cards to her brother. 7. 8. (a) What (b) How many percent of her baseball cards did Diane give to her brother? baseball cards did Diane have left? Write each number in scientific notation. (a) One hundred-thousandth (b) One hundred thousand Write 1.5 x 10~ 5 standard form. Then use words to write the number. 9. 10. Compare: - O 0.667 Evaluate: ab + a + b if a = —1 and 2 b = —1 4 360 Math 87 11. 12. 13. Divide 5 by 0.24 and write the answer decimal with a bar over the repetend. (a) as a (b) rounded to the nearest whole number. Find each sum: (a) (-148) + (-52) (b) (+7) + (-12) + Complete the (+6) + (-8) Decimal Fraction table. Percent 2 14. Find the missing number. F In 5 11 Find the area of each triangle. Out — U N 25 *" c 16 8 -*> 15. (b) (a) 9 T 1 64 -* o 121 N Dimensions are in meters. Solve: 16. 1.5c = 0.345 17. x = - 10 Add, 19. 18. °- 14 2 = ^ 35 subtract, multiply, or divide, as indicated: 10(10 + 10) - (10 10 • 10 + 10) centi- 361 Ratio Problems Involving Totals 20. 62 + 82 22. lday — 6 hr 15 1 23. 5 25. - 10 2 • min 45 36 • 2 m 3 A + 12 24. - ( 5-1 - 1- l 4J 6 2 6- + ( 6 + 1- 26. 6 2 - 25 cm = ?cm sec 60 min 1 hr mi min 4- 21. 3 5 I 27. 42.3 - 5.787 28. (0.7)(1.1)(1.3) 29. 1.02 + 0.12 30. 0.1 LESSON + '5 I + ; 0.0 2) Ratio Problems Involving Totals 81 problems require that we use the the problem. Consider the following problem. Some ratio The ratio of boys to girls students in the assembly, We row was number were 180 were there? 5 to 4. If there how many girls begin by making a ratio box. This time for the total total to solve we add a third of students. Ratio Actual Count Boys 5 B Girls 4 G Total 9 180 column we wrote boys and 4 for girls, then added these to get 9 for the total ratio number. We were given 180 as the actual count of students. This is a total. We can use two rows from this table to write a proportion. Since we were asked to find the number of girls, we will use the In the ratio 5 for 362 Math 87 "girls" row. Since we know also use the "total" row. both "total" numbers, Then we we will solve the proportion. Ratio Actual Count Boys 5 Girls 4 B G 4 Total 9 180 9 180 9G = 720 G= We find that there complete the Example The was were 80 girls. We can use 80 this answer ratio box. Ratio Actual Count Boys 5 Girls 4 Total 9 100 80 180 ratio of football players to soccer players in the 5 to 7. If to 48 players were in the room, room how many were football players? Solution We use the information in the problem to form a table. include a row for total. The total ratio number is 12. Ratio Actual Count Football players 5 F Soccer players 7 S 48 Total players 12 We "_5^ F_ ^12 48 12F 240 F 20 To find the number of football players, we wrote a proportion from the "football players" row and the "total players" row. We solved the proportion to find that there were 20 football players in the room. From this information we can complete the ratio box. Ratio Football players 5 Soccer players 7 Total players 12 Actual Count 20 28 48 363 Ratio Problems involving Totals Practice Problem set Solve these problems. Begin by drawing a ratio box. a. Acrobats and clowns converged on the center ring in the ratio of 3 to 5. If 72 entertainers performed in the center ring, how many were clowns? b. The was 1. If young men to young women at the prom 8 to 9. If 240 young men were in attendance, how many young people attended in all? 5 ratio of pounds of apples 81 2. cost $2.40, then (a) what is the price per pound? (b) what is the cost for 8 pounds of apples? What is the sum often million, fifty-five thousand, hundred one and six million, seven hundred thousand, four hundred eighty? 3. Use a ratio box 1320 4. The The fish in the pond, pond was how many car traveled 350 miles car averaged eight problem. The ratio of big to solve this fish to little fish in the two 4 to 11. If there were big fish were there? on 15 gallons of gasoline. how many miles per gallon? Round the answer to the nearest tenth. 5. If the average of three numbers is 1440, what is the sum of the three numbers? 6. Write twelve billion in scientific notation. 7. Draw a diagram of this statement. Then answer the questions that follow. One sixth of the five (a) How many (b) What was the ratio that dozen eggs were cracked. eggs were not cracked? of eggs that were cracked to eggs were not cracked? 364 Math 87 8. Draw segment AB. Draw segment DC parallel to segment AB but not the same length. Draw segments (a) between the endpoints of segments AB and DC to form a quadrilateral. What type (b) 9. of quadrilateral was formed in part (a)? Refer to the figure to answer the Dimensions questions. in are inches. Corners that look square are square. What (a) •? is ** the perimeter of the hexagon? What (b) 10. What is the area of the hexagon? the average of the two is numbers indicated by arrows on this number line? c 0.7 I I I I I I I 0.9 0.8 » I I I 1.0 I I I I I Write equations to solve Problems 11 and 12. 11. What number, is 75 percent of 64? 12. What number 0.3 of 7.4? 13. Find each sum: 14. is (a) (-3) + (-8) (b) (+3) (c) (-3) + (+8) + (-5) + (-8) Complete the table. Fraction (a) Decimal 0.35 Percent (b) - Ratio Problems Involving Totals 15. Use unit multipliers 16. Evaluate: ab + a 17. Find the missing number. + to convert 0.95 a -r ifa = answer number. the as a F In Divide 366 by 7 and write — Out 14 C T 17 ** 32 T -* O 11 20 N Solve: 19. 0.2 U N — 8 -** mixed to milliliters. and b = 5 5 18. L 365 21 - 9 + p = 20. 4.8 21. 10/7 Add, 7 = 240 subtract, multiply, or divide, as indicated: "7: 22. yd 1 2 + ft 23. 7 in. 1ft 9 in. A days phi 15 min - 1 days 7 hr 50 min 24. 26. 4.5 -s- (0.4 11 4 1 4-1 - f 1- 5 30. 3 l » + 4[(5 - 3 + 0.6 3 - 25. 0.5) 0.6 f 3 3- + 5- + 2- 5 28. + 27. 5- 6 2 36 &s 0^ 2" [ - 13 \ 29. 3 ) 2)(3 + 1)] 3 2 + a/4-7-3 366 Math 87 LESSON 82 Geometric Solids Geometric solids are shapes that take up space. Below show a few geometric solids. ^ we / 1 y Sphere Cone Cylinder Cube Triangular prism Pyramid v Polyhedrons Some geometric such as spheres, cylinders, and cones, have one or more curved surfaces. If a solid has only flat surfaces that are polygons, the solid is called a polyhedron. Cubes, triangular prisms, and pyramids are examples of polyhedrons. When describing a polyhedron, we may refer to its faces, edges, or vertices. A face is one of the flat surfaces. An edge is formed where two faces meet. A vertex (plural, vertices) is formed where three or more edges meet. solids, A prism is a special kind of polyhedron. A prism has a polygon of a constant size "running through" the prism. A rectangular prism has a rectangle of a constant size running through it. A triangular prism has a triangle of a constant size running through it. Thus, at least two faces of a prism are identical and are parallel. To draw a prism, we draw two identical, parallel polygons, as shown below. Then we draw lines connecting corresponding vertices. We may use broken lines to indicate edges hidden from view. Rectangular prism: Draw i the same size rectangle twice. Geometric Solids Then connect corresponding for vertices. (Use 367 broken lines hidden edges.) /I s* 1 • Triangular prism: Draw Connect corresponding Example 1 Example 2 Solution Example 3 Solution J. // the size triangle twice. solid to describe the shape of object. (a) Basketball (b) Shoe box (c) Can (a) Sphere (b) Rectangular prism (c) Cylinder A same vertices. Use the name of a geometric each Solution ^ of beans how many cube has (a) 6 faces (b) 12 edges (c) 8 vertices Draw (a) faces, (b) edges, a cube. A cube is a special kind of rectangular prism. A cube has a square running through it. All faces are squares. and (c) vertices? 368 Math 87 Practice Use the name of a geometric ^^\ a. solid to describe each shape. b. Box Cone Tent A d. triangular prism has Faces Draw g. how many of each? Edges e. f. Vertices a representation of each shape. Sphere Rectangular h. Cylinder prism Probiem set 82 1. 2. The bag contained 24 red marbles, 30 white marbles, and 40 blue marbles. What was the ratio of (a) red marbles to blue marbles? (b) white marbles marbles? When the product of - and of - 3. to red and - , what is - is subtracted from the sum the difference? the cost of calling Albuquerque 76 C for the first minute and 48 C for each additional minute, what is the If is cost of a 10-minute call? 4. On his first 5 tests Cliff averaged 92 points. 3 tests Cliff scored 94 points, 85 points, On and 85 (a) What was his average for his last 3 tests? (b) What was his average for all 8 tests? his next points. 5. Use a ratio box to solve this problem. The jeweler's tray was filled with diamonds and rubies in the ratio of 5 to 2. If 210 gems filled the tray, how many were diamonds? 6. Draw a diagram of this statement. Then answer questions that follow. Four fifths of the 360 dolls were sold during November. the 369 Geometric Solids (a) How many of the dolls were sold during November? (b) What percent of the dolls were not sold during November? 7. Draw a rectangular prism, and then answer these questions. A 8. rectangular prism has (a) edges? (b) faces? (c) vertices? how many Refer to these triangles to answer the questions. Dimensions are in meters. 9. (a) What (b) What is the perimeter of the isosceles triangle? the area of the scalene triangle? is Find the length of segments AC, and (b) 1 1 1 1 1 ' BC. 1 1 ' ' 1 1 ' ' 1 1 1 inches (c) C 8 A 10. AB, (a) 1 2 1 1 1 3 Write twenty-five ten-thousandths in scientific notation. Write equations to solve Problems 11 and 12. 11. What number is 24 percent of 75? 12. What number is 1.2 of 12? 13. Find each sum: (a) 14. (-2) + (-3) + (-4) Complete the table. (b) Fraction (a) (+2) + (-3) + (+4) Decimal (b) Percent 4% — 370 Math 87 15. Use a unit multiplier 16. Evaluate: a 2 17. Find the missing number. 18. - {a Round 7856.427 if to mm to centimeters. 700 to convert a = 9 U N 1 — — — 1 — O N 5 the Out F In nearest hundred. C T I *- — — 35 — 77 ~~* Solve: 19. 21. ^22 p - 5- = 7- 23. 2.5 15 w 12 3600 = 9y Add, 22. 20. subtract, multiply, or divide, as indicated: 9 + 8{7 1 • - 6 5 [4 + (3 - 2 • „„ yd 24. 1)]} .1, 2-hr • 2 1 25. 6.4 - ft (0.6 4 mi — hr : 1 3 in. - 26. 0.04) 3 + 0.6 (3X0.6) 27. 2 15 3 4 1- + 3- - 1( 29. 3- + 28. 6 - * 3- • 5- 5 5 52 - Vl6 + 3 ,\ 3 + 1- 30. 23 4 Weight LESSON 83 Physical objects are the earth composed of matter. The attraction between and a physical the metric system we object is the weight of the object. In use kilograms (kg), grams (g), and 371 Weight milligrams (mg) to measure the weight of an object.* The approximate weights of some familiar objects are shown here. D Math book 1 Paper kilogram = 1000 grams 1 To measure the weight of an a is known weight. The 112 A clip grain of salt 19 kg 1 1 mg gram = 1000 milligrams object, we compare its weight to apple balances 112 grams so its weight g. In the U.S. customary system and ounces 1 (oz) to we (t), pounds (lb), measure weight. ton = 2000 pounds The spring use tons 1 pound = 16 ounces scale indicates that the apple weighs 4 ounces. was placed on a grocer's scale, the scale would If a 1-kilogram weight read about 2.2 pounds. and milligrams to describe the masses of objects. A scientist uses a unit called a newton to describe weight. In the marketplaces of countries that use the metric system merchants use grams and kilograms as units of weight and the scientists in the same countries use grams and kilograms as units of mass. It can be confusing at times. *A scientist uses kilograms, grams, 372 Math 87 Example 1 Solution Use a unit multiplier to perform each conversion. (a) 1.2 (a) We kg to grams mg 250 (b) grams to form a unit multiplier from the equivalence 1 kg = 1000 g that has grams on top. Then we multiply to cancel kilograms. 1.2*.™?(b) We multiply by a unit multiplier that has grams Then we cancel milligrams. 250 Example 2 1200 g Simplify: lg jug" 1000 ;ng ~ + 3 lb (b) 5 lb 8 oz 3 lb 9 oz - 9 oz (16 oz a Solution (a) + 5 lb 8 oz 3 lb 9 oz (b) 51b -3 9 lb 1 oz 8 lb 17 oz top. 250 g = 0.25 g 1000 5 lb 8 oz (a) on lb >24 8 oz 9 oz 1 lb 15 oz Practice Simplify: a. it) lb + 4 lb 7 oz Problem set 83 - 12 oz Use unit multipliers to c. 32 oz to pounds e. 2.4 kg to 1. With the baby in d. f. oz 7 pounds 2 tons to 4000 mg L to grams arms Papa weighed 180 pounds. his arms Papa weighed 165 ^ pounds. his Without the baby in 1 lb perform each conversion. grams How much 9 lb 3 oz b. did the baby weigh? 373 Weight 2. Tim ran 15 miles (a) (b) 3. 4. If 3 What was in 2 hours. his average speed in miles per hour? What was the average number Tim to run each mile? of minutes it took notebooks cost $8.91, then (a) what is the cost of 1 notebook? (b) what is the cost of 5 notebooks? Use a box ratio to solve this problem. The ratio of black sheep to white sheep in the flock was 2 to 17. If there were 34 black sheep in the flock, how many sheep were in the flock in all? Write equations to solve Problems 5 and 5. What number is 40 percent of 65? 6. What number is 0.075 of 600? 7. Three the earth's surface (a) fifths of What percent is 6. covered with water. of the earth's surface is covered with water? (b) What is the ratio of the area of land to the area of water on the earth's surface? 8. Write 5.2 x 10 6 in standard form. 9. Use the name of a geometric of each object shown. (a) (b) solid to describe the shape (c) SouJ> 374 Math 87 10-12. millimeters. Corners Refer to the figure to answer questions Dimensions are -in that look square are square. 10. What is the perimeter of the hexagon? 11. What is the area of the hexagon? 12. What is the area of the triangle? 13. What the average of the two is numbers marked by arrows on this number line? 7.6 i 7.7 i i i i I 14. I I I I Complete the 7.9 7.8 I I I I I I I » I I I I I 1 1 I Fraction table. Decimal Percent i 15. Use a unit multiplier (b) (a) 25 to convert 10 kg to grams. Find each sum: 16. (-6) + (-2) + (+5) 18. Evaluate: X ~\~ XV — x 17 if x = (_ 7 ) . 7 + (_ 8 ) + (_ 6 ) and y = 8 Solve: 19. 245 - x = 179 20. 3n = 2.31 21. 4- + w 22. li = = 8 18 2 Add, 23. ^ 45 subtract, multiply, or divide, as indicated: 8 lb + 9 lb 8 oz 11 oz 24. 5 gal 'i qt 3 qt 1 pt + (-1) Circles 25. 16 + 0.8 - 0.04 27. * + 1 + 2 _ 1 6 5 3 2 29. 32 - 23 + A/2 • • 26. 0.4 + (0.5)(0.6) 28. 3- 5- • 4 30. - 0.7 111- 3 18 375 Investigating Circumference • 3 3 28 - 3[8 3 . LESSON 4 _ (4 (3 • + 2 - 1)] 4) Circles • Investigating Circumference 84 Circles A circle is a smooth curve, every point of which is the same distance from the center. The distance from the center to the circle is the radius. The plural of radius is radii. The distance across a circle through the center is the diameter. The distance around a circle is the circumference. We see that the diameter of a circle is twice the radius of the circle. Example 1 The diameter manvJ Solution of this circle centimeters The radius is its of a circle is is 2 m. How radius? half the length of the diameter. So the radius of this circle must be 1 m. We are asked to express the radius in centimeters. 100 cm. The radius equals 376 Math 87 Example 2 The radius of a circle 18 inches. is How many feet is its diameter? Solution The diameter of a circle twice the length of the radius. So is the diameter of the circle is = 36 2 x 18 in. We in. are asked to express the diameter in feet. = 3 ft of a circle is 36 The circumference in. related to the diameter The following of the circle in a special way. investigation will explore that relationship. Investigating circumference This investigation requires a tape measure (preferably metric) and a may number A of circular objects. calculator also be useful. and measure Select a circular object its circumference and its diameter as precisely as you can. To calculate the number of diameters that equal the circumference, divide the circumference by the Round diameter. Then repeat the quotient to two decimal places. the investigation with another circular object of a different size. Compare your results of other students in the class. results You will find that circumference for every circular object the with the is about 3.14 times as long as the diameter. Practice a. If the diameter of a circle many b. If c. meters, radius is how diameter is how its centimeters? the radius of a circle many is 3 is 30 inches, its feet? Find a circular object that you can measure, and record this information: 1. The name 2. Its 3. Its 4. The of the object diameter circumference when the circumference is divided by the diameter (round to two decimal places) result Circles Problem set 84 1. • 377 Investigating Circumference How Dan Spent According to this graph, His Income what percent of Dan's income was spent on items other than food and housing? If his income was $25,000, how much did he spend on food? 2. It is miles from Tim's 1^ house to school. How far does Tim travel in 5 days walking to school and back? 3. When sum the of 1.9 product of 1.9 and box to solve this problem. There was a total of 520 dimes and quarters in the soda machine. If the ratio of dimes to quarters was 5 to 8, how many dimes were there? 4. Use a 5. Saturn ratio is number 6. 2.2, and 2.2 is subtracted from the what is the difference? Draw 900 million miles from the sun. Write that in scientific notation. a diagram of this statement. Then answer the questions that follow. Three tenths of the 400 acres were planted with alfalfa. (a) What percent (b) How many of the land was planted with of the 400 acres alfalfa? were not planted with alfalfa? 7. 8. Forty percent of the 30 students earned an A on the How many students earned an A on the test? Draw a triangular prism so that the triangular bases are equilateral. 9. If test. The prism has how many the radius of a circle how many is centimeters? faces? 50 millimeters, its diameter is 378 Math 87 Refer to the figure to answer questions 10 and 11. Dimensions are in centimeters. 10. (a) What the area of the is parallelogram? (b) What is the area of each triangle? 11. Each triangle is isosceles. What is the perimeter of one of the triangles? Write equations to solve Problems 12 and 13. 12. What number is 90 percent of 3500? 13. What number is - of 6 14. Complete the 2-? 5 Decimal Fraction table. 0.45 (a) 15. (b) Find each sum: (a) (5) (b) 3 +,(-4] + (6) + (-1) + (-5) + (+4) + (-2) 16. Use unit multipliers 17. Find the missing number. to convert 1.4 kg to grams. In 26 7 16 -»• F U N -* C T -*I __». 18. Percent O N Out — — — 13 372 8 i 2 Estimate this product by rounding each number to one nonzero digit before multiplying. (2876)(513)(18) Circumference and 379 Pi Solve: 19. = 7x 5.6 0.9 o., 21. 12 — = - + 2 w 11 — 12 = 3 3 4 lb 12 oz + 25. 16 - 27. - • 24. n 7 oz iu 1 lb , (0.8 1- 8 • - 0.04) 4 -=- 1- 3 V9 + r A J 16 - a/9 12 ft 1 yd J 1 ft - 0.4[0.5 28. 6- + 2- + 1- 30. (0.6)(0.7)] 6 2 3 + 16 in. 26. 3 30 - 5 [4 + (3)(2) - 5] Circumference and Pi LESSON 85 00 22. subtract, multiply, or divide, as indicated: 23. 29. 654 - p = 456 n 1.5 Add. 20. In the preceding lesson a circle. we investigated the circumference of We measured both the circumference and the diameter Then we divided the circumference by the diameter the number of diameters that equal a circumference. of a circle. to find How many diameters equal a circumference? This question has been asked by people for thousands of years. They found that the answer did not depend on the size of the circle. The circumference of a circle is slightly more than three diameters. Another way to illustrate this fact is to cut a length of and find reach around the string equal to the diameter of a particular circle how many of these lengths circle. No matter what the diameters plus a little are needed to size of the circle, it takes three extra to equal the circumference. 380 Math 87 D "(o^ The Extra Extra amount needed is about, but not exactly, one seventh of a diameter. Thus the number of diameters needed extra to equal the circumference of a circle 22 or is about 3.14 or 7 They are approximations. There is no fraction or decimal number that exactly states the number of diameters in a circumference. (Some computers have calculated the number to more than 1 million decimal places.) We use the symbol n, which is the Greek letter pi Neither 3y nor 3.14 (pronounced is exact. number. "pie"), to stand for this The circumference the circle. This idea is of a circle is n times the diameter of expressed by the formula C = nd To perform calculations with k, we can use an approximation. The commonly used approximations for it are 3.14 and 22 7 For calculations that require great accuracy, more accurate approximations for n may be used, such as 3.14159265359 Sometimes the calculation is performed leaving tt as n. Unless directed to use another approximation, we will use 3.14 for k to perform the calculations in this book. Example 1 Solution The radius of a If the radius is circle is 10 cm. What 10 cm, the diameter is is the circumference? 20 cm. Circumference and Circumference = n • diameter « 3.14 « 62.8 The circumference Example 2 /""-""N Use 3.14 (a) for / lb) Use /r C = nd C « 3.14(30 C « 94.2 (b) in.) in. 22 -=- for = xd C - C - 44 d. # f (14 (c): (c) Leave ;ras ft) first for it 20 times 71.) 7rcf C = n (20 cm) /r, = 20/r cm then the unit of C Use 22 y for Leave /ras ^ - it what was the circumference of the circle? (Use 3.14 for .= it circle. Sylvia used a compass to draw a circle. If the point of the compass was 3 inches from the point of the pencil, C C ft b. Use 3.14 cm circle. (c) Find the circumference of each a. 20 cm \ C Note the form of answer measure. Practice • about 62.8 cm. Find the circumference of each (a) Solution is 381 Pi 382 Math 87 Problem set 1. 85 Hilda ran 8 laps of the track at a steady speed. If 4 1 minutes to run the first 3 laps, how long did her to run all 8 laps? it took it take 2. The average of three numbers is 2. If the greatest and the least is 1.5, what is the third number? 3. Use a ratio box to solve this problem. The ratio of princes to paupers in the kingdom was 1 to 24. If the total number in both categories was 4800, how many princes were in the kingdom? 4. How 5. James has read migrating duck fly in 8 hours average speed of 24 miles per hour? far will a many 6. If left to is 1 meter, its radius How is how (a) Draw (b) A triangular prism has how many more edges than a prism with bases that are right triangles. Find the circumference of each (a) /^~^\ Use 3.14 9. an millimeters? vertices? 8. at read? the diameter of a circle many 7. pages are of the 320 pages in the book. is 2.8 for (b) n Refer to the figure to answer the questions. Dimensions are in millimeters. (a) What is the area of the right triangle? (b) What is the area of the isosceles triangle? circle Use 22 -j for;r Circumference and 10. What is 383 Pi the perimeter of this equilateral triangle? 11. TTT Write 32,500,000,000 in I I scientific notation. inches 1 Write equations to solve Problems 12 and 13 12. What number is 5 13. What number is 14. Complete the percent of 1000? 0.015 of 600? Decimal Fraction table. Find each sum: (a) 6 + (-1) + (b) (-5) 16. Use 17. Evaluate: 18. 7.5% (b) (a) 15. Percent + (2) +(-?)+ (4) (+6) + (-3) a unit multiplier to convert 3 g to milligrams. —— — if xy x = — and y = — 4 ' 2 Divide 2.4 by 0.018 and write the quotient (a) as a decimal (b) rounded with a bar over the repetend. to the nearest whole number. Solve: 19. -3 + n = 1-1 4 21. f Add. 23. 25. 1 1.2 20. 14 2 - 479 = 563 22. 0.8 25m = 225 subtract, multiply, or divide, as indicated: kg - 350 g = 16 * 0.04 + 0.8 ? g 24. 2000 1 26. lb ton 16 oz 1 lb 10 - 0.1 - (0.01)(0.1) 384 Math 87 27. 4 4 2 - V5 2 - 3 2 2 30. 6 3 + 5 6 + 2-3 • 1 l2 The Opposite of the Opposite LESSON 86 • Algebraic Addition The opposite The of the 1^ 2 3- + 8- - 5- 28. 8 5 ,4 29. ( 3.1.5 graphs of -3 and 3 are the same distance from zero on the number line. The graphs are on the opposite sides of zero. 3,3, opposite -4-3-2-1 We 2 1 3 4 say that 3 and - 3 are the opposites of each other. 3 is the opposite of -3 We can read - 3 is the opposite of 3 way Then - (- 3) can be read This means that -(-3) is as the opposite of 3. as the opposite of the opposite of 3. another -3 to write 3. -(-3) -3-2-10123 Algebraic addition There are two ways to simplify this expression. 7 The first way is subtract 3 from _ 3 minus sign mean the answer is 4. to let the 7, 7-3 The second way is to we = 4 use the thought process of algebraic addition. To use algebraic addition, -3 to subtract. If we let the minus sign number and treat the problem an addition problem. This is what we think. mean that is a negative 7 + (-3) = 4 as The Opposite of the Opposite We get the same answer both ways. The in the way we We 385 Algebraic Addition • only difference is think. can also use algebraic addition to simplify this expression. 7 - (-3) We use an addition thought and think that 7 -(-3). This is what we think. 7 is added to + [-(-3)] But the opposite of the opposite of 3, so we can write 7 + [3] = 10 3 is another name for We will practice using the thought process of algebraic addition because algebraic addition can be used to simplify expressions that would be very difficult to simplify if we used the thought process of subtraction. Example 1 Solution Simplify: -3 - (-2) think addition. We think we are to add - 3 and - (- 2). This is what we think. (-3) + [-(-2)] We But the opposite of the opposite of have (-3) + Example 2 Solution Simplify: -(-2) - 5 We - [2] 2 is 2 itself. So = -1 (-6) We see three numbers. think addition. We think [-(-2)] + (-5) + [-(-6)] We simplify the first and third numbers and get [+2] + (-5) + [+6] = 3 Practice 1 Use algebraic addition to find these sums. a. (-3) - (+2) b. (-3) - (-2) c. (+3) - (+2) d. (-3) - e. (-8) + (-3) - (+2) (+2) - (-4) we 386 Math 87 Problem set 1. 86 The weight of the beaker and the liquid was 1037 g. The weight of the empty beaker was 350 g. What was the weight of the liquid? 2. Use a ratio box problem. Jenny's soccer ball is covered with a pattern of pentagons and hexagons in the ratio of 3 to 5. If there are 12 pentagons, how many to solve this hexagons are in the pattern? 3. 4. 5. When the and what 2, sum is 11 1 of ^ and % is divided the quotient? Pens were on sale 4 by the product of ^ for $1.24. (a) What was (b) How much would the price per pen? 100 pens cost? Christy rode her bike 60 miles in 5 hours. (a) What was her (b) What was average speed in miles per hour? the average number of minutes it took to ride each mile? 6. Sound travels about 331 meters per second in air. About how many seconds does it take sound to travel a kilometer? 7. The following were made on a scores 72, 80, 84, 88, 100, 88, Which (a) often? the scores were listed in order from the least to the greatest, what would be the middle score? (b) If What (c) 8. and 76 was made most score test: What is the average of is the scores? all two numbers marked by the average of the arrows on this number line? 8 I I | 10 9 I I » I I I I I | I I I I I I I » I 11 I | I I I | I I I I I | The Opposite 9. Opposite • -^>" S y (b) What the is name of this shape? /^-^\ Use 3.14 for y shape? Find the circumference of each (a) y / How many cubes were used to build this 387 Algebraic Addition This rectangular shape is two cubes high and two cubes deep. (a) 10. of the circle (b) n Leave n as n 11. Draw 12. Multiply twenty thousand and thirty thousand, and write the product in scientific notation. a right triangle that is also an isosceles triangle. Write equations to solve Problems 13 and 14. 13. What number is 75 percent of 400? 14. What number is 1- of 1-? 15. 1 3 2 Simplify: (a) 16. 1 (-4) - (-6) Complete the (b) (-4) - (+6) table. Fraction Decimal Percent 3 25 17. Use 18. Evaluate: (b) (a) a unit multiplier to convert 72 qt to gallons. x 2 + xy + y 2 if x = 4 and y = 5 388 Math 87 Solve: a + 3.7 = 4.09 19. 21. Add, 20. 290 - k = 29 22. 2.25 = 15w subtract, multiply, or divide, as indicated: 16 - 1 qt 1 pt 1 $12.00 1 hr 60 min 26. 10 - - (0.01X0.1)] 28. 4- 24. 1 oz hr • 25. 16 27. 5 (0.04 -5- 2 + 8 4 3 52 29. 3 . - 4- _ 0.8) 3 4 133- * 1- 4 3 • 2 24 30. LESSON [0.1 3 2 + 6[10 - (3 • 4 - 5)] Operations with Fractions and Decimals 87 Sometimes we encounter expressions that contain both fractions and decimals, such as this expression. 1 - 0.4 4 Before we numbers simplify, we rewrite the expression so that both are fractions or both numbers are decimals. Decimals Fractions - 3 - 0.4 9 ( 0.4 s 0.4 4 4 = 0.75 ) sj V 3 3 2 15 8 _7_ 4 5 20 20 20 0.75 / - 0.4 = 0.35 Operations with Fractions and Decimals Both answers are correct because The problems we — 20 7 389 equals 0.35. H book will see in this will ask for an answer in one form or the other form. Example Simplify each expression to the form indicated. (a) 3- + (b) 4.5 - 3.4 (decimal) - (fraction) 5 Solution (a) We 3 change 3— to the decimal number 3.75 and then 4 add. 3 3— + 3.4 3.75 + 3.4 problem 3changed 5 to 3.75 4 added 7.15 1 (b) We change 4.5 to the mixed number 4-. Then we subtract. problem 4.5 5 4 I_i changed 4.5 to 4-1 2 5 8 10 10 7 common denominator subtracted 10 Practice Simplify each problem to the form indicated. a. 3.8 + —3 1 (decimal) b. 5 - 0.45 0.5 (fraction) 3 8 c. - + (decimal) d. 2.3 - if 5 (fraction) 390 Math 87 Problem set 87 1. The awarded scores of 8.7, 8.2, 8.1, 8.5, and 8.5 to the contestant. The highest and lowest scores were not counted. What was the average of the 3 middle five judges scores? 2. Use a ratio box to solve this to ladies at the palace ball ladies attended, how many problem. The ratio of lords was 5 to 7. If lords were there? 3. Seven dictionaries were stacked on the shelf. If 3 of the dictionaries weigh a total of 96 ounces, all 7 dictionaries weigh how many pounds? 4. The diameter tire is 24 in. of Debbie's bicycle What is 420 lords and 96oz-< the circumference of the tire to the nearest inch? How many of 5. Five ninths of the 360 students were the students were boys? 6. Find the product of two thousandths and three thousandths. Then write the answer in scientific notation. 7. Use the name of girls. a geometric solid to describe each object. (b) (a) (c) S6ae6 Write equations to solve Problems 8 and 8. What number is 35 percent of 800? 9. What number is 2.5 of 40? 9. Operations with Fractions and Decimals 10. Find the circumference of each /~~^\ (a) Use 22 -=- for 391 circle (b) n Use 3.14 for n Refer to the figure to do Problems 11 and 12. Dimensions are in centimeters. 11. The larger is a rectangle. (a) What is the area of the rectangle? (b) What is the area of each triangle? 12. Use the answers shaded region. 13. Simplify: 14. shape (a) (-5) - (-5) (b) (-8) - (+7) Complete the to Problem 11 - (-6) + (-4) table. Fraction Decimal 1.75 (a) 15. Use a unit multiplier 16. Evaluate: 17. Round mn to find the area of the + Percent (b) to convert 0.25 {m - n 2 if m kg to grams. = 9 and n = 9 7.27 to the nearest thousandth. 392 Math 87 18. Find the missing number. [Hint: The rule is to multiply by a number and then add another number.) In 2 4 3 F U N — 5 C T — 9 -* 7 I O N 6 Out _ Solve: 19. y - 46 = 217 21. 4.54 + Add, m = 9.2 20. 3c = 5.64 oo 15 X 25 15 22. subtract, multiply, or divide, as indicated: i 6 lb 10 oz 23. + 9 oz 1 lb 1 24. hr 1 25. - 4 weeks 40 hr $12 week 1 + 0.58 1 month (decimal) 26. 4.5 28. 4.25 30. 4- 8 - 27. (7 29. 3- - 0.6) ( 8 -5- 0.05 A 1 3 3 4V 5- - 1V 5 *- ^ 5 (fraction) - 4 + ' (0.2)(5) 1 1 1- + 3- 2 A 3I The Addition Rule for 393 Equations The Addition Rule for Equations LESSON oo In this lesson we The method presented equations. method uses new method will begin using a is for solving used in algebra. This the addition rule for equations. Addition Rule for Equations same number added both sides of an equation, the solution (answer) is not changed. If Here's the how is to the addition rule works. Consider the following equation whose solution x + The addition is 5. = 8 3 (solution rule says that we add if is 5) the same number to both sides of this equation, the solution to the new equation will also be 5. To illustrate, we will add 2 to both sides of the equation. = 8 +2 = +2 x + 5 = 10 x + We original equation (solution 3 add new Adding both sides 2 to equation (solution see that the solution to the is 5) new equation is 5) is also 5. both sides of the equation did not change the solution, but it did not help us find the solution either. However, if we carefully select the number to add to both sides of the equation, the solution to the original equation will appear in the new equation. We will show this by adding - 3 to both sides of the equation. 2 to x + = 8 -3 = -3 x + original equation 3 = 5 add -3 new We see that the new equation, x + to the original equation. Since x + is x = 5, which is to both sides equation = 5, shows us the solution is x, the new equation the solution of the original equation. Solving an equation by the addition rule gets the variable 394 Math 87 by we itself on one side of the equals sign. To use this method follow these steps. 1. Find the variable in the equation and determine which side of the equals sign the variable occupies. 2. Find the number on the same side of the equals sign that is added to the variable. Add the opposite of this number to both sides 3. of the equals sign. Example 1 Solution Solve and check: m We want to get m all - 248 = 352 by itself on one side of the equals sign. The number on the same side as m is -248. To remove -248 we add +248 to both sides of the equation. m m - 248 = 352 +248 =+248 = 600 + m Now we Solution new to both sides equation solution check the solution. m — 248 = 352 original equation (600) - 248 = 352 substituted 600 for Solve and check: 263 = We want to The -47 add +248 = 600 352 = 352 Example 2 original equation x + 47 x all x is on the right-hand side with variable to simplified; solution checks get by itself on one side of the equals both sides of the equation, Now we m we will get 263 = x + 47 -47 = -47 add -47 216 = x + new 216 = x solution check the solution. 47. If x all we add by original equation to both sides equation sign. itself. The Addition Rule Example 3 Solution original equation 263 = (216) + 47 substituted 216 for 263 = 263 simplified; solution checks y- We want to all get y 4.7 by = 4.7 = 5.79 itself. y + can do this if we add +4.7 Now we original equation +4.7 add =10.49 new y = 10.49 4.7 to both sides equation solution check the solution. y - 4.7 = 5.79 original equation - 4.7 = 5.79 substituted 10.49 for (10.49) 5.79 = 5.79 w Solve and check: We want to to We 5.79 +4.7 Solution x both sides of the equation. y — Example 4 K 395 Equations 263=x+47 Solve and check: to for get + 7 simplified; solution checks —3 = —5 4 6 w all by itself. We can do this if we add — 3 both sides of the equation. — w = H 4 — original equation H 6 — — = w To add 10 12 — 3 , — to both sides 4 5 3 6 4 new we need to use a equation common denominator. 4 6 6 = + —5 and add 4 4 problem ^ 4 9 12 = — 12 1 used common denominators 396 Math 87 Now we can write the solution. w Now we 1 = 12 check our work by using — for w in the original equation. w + 5 —3 = — 4 ro + 12 y v 4 h 12 Practice Problem set 88 , 6 5 —3 = — 19— . original equation substituted 6 = —5 12 6 10 12 5 5 5 6 6 common — 12 1 for w denominators simplified 6 simplified; solution checks Use the addition rule to solve each equation. Show your work. Check each answer. a. 463 = c. n - 1. 2. 1.3 - 281 b. p + 56 = 203 = 12.28 d. 5.7 = x + 1.35 Fully dressed, O'Riley weighs 123 lb. How much does O'Riley actually weigh if the weight of his clothes is 80 oz? When the sum and 3. m - , what is of - and - is divided by the product of - the quotient? Nine seconds elapsed from the time Mark saw the lightning until he heard the thunder. The lightning was about how many kilometers from Mark? (Sound travels about 331 meters per second in air.) The Addition Rule for Equations 397 4. Use a ratio box to solve this problem. The ratio of lefthanded students to right-handed students in the math class was 2 to 3. If 18 of the students were right-handed, how many students were there in the math class? 5. It was estimated route. Write that 6. that 1^ million people lined the parade number in Five out of every (a) fifty What percent scientific notation. fans cheered for the visiting team. of the fans cheered for the visiting team? (b) What fraction of the fans did not cheer for the visiting 7. 8. team? The average weight of the three big men was 288 pounds. Two of the big men weighed 252 and 261 pounds, respectively. What was the weight of the other big man? How many cubes were used to build this rectangular prism? ' ^S ss s s s s s s^ ^ 9. Find the circumference of each (a) S~~^\ Leave n as n circle M Use 3.14 for n 398 Math 87 10. Refer to the figure to answer the questions. Dimensions are in centimeters. (a) What the area of the is rectangle? (b) What is (c) What fraction of the rectangle the area of the shaded triangle? is shaded? Write equations to solve Problems 11 and 12. 11. What number is 80 percent of 500? 12. What number is - of 100? 3 13. 14. Simplify: (a) (-8) + (-5) - (-9) (b) (+2) - + (-3) (-7) Complete the table. - (+4) Decimal Fraction 15. Use a unit multiplier 16. Evaluate: 17. Find the next three numbers in x = 10000, 1000, 100, Solve and check. 18. Show your x - 47 = 360 2- tons to convert if 1% (b) (a) x - y x - 12 and to y = ± = i 6 n pounds. 3 this sequence. , , work. = 19. y+ 21. lOp = 12.5 1.4 Solve: 20. Percent 5.17 More on Add, subtract, multipy, or divide, as indicated: min 7 sec hr 20 min 15 sec 16 2 hr 22. - 24. 1 - - 0.025 2 pt 1 gal 1 qt 16 oz (decimal) 2- x 25. 0.9 1 pt (fraction) kj - 26. 7 2». ~5 3+ 27. (0.6)(0.05) 8 f 3 -- 5l 1- 4.25 - 3- 29. + [4 -s- (0.2 -*- 5)] 1 6J I 20 + {45 - 5 [8 - 2(8 5 2 - 4 3 • 2)]} 2 More on Scientific Notation LESSON 89 4 qt 23. 3 30. 399 Scientific Notation When we write a number in scientific notation, we usually put the decimal point just to the right of the first digit that is not zero. 4600 x 10 5 To write in scientific notation, we will use two write 4600 in scientific notation. In place of write 4.6 x 10 3 . we 4600 we steps. First will will Now we have (4.6 For the second step x 10 3 ) x 10 5 we change the two powers We recall that of 10 into one 3 10 means the decimal point is 3 places to the right and 10 means the decimal point is 5 places to the right. Since 3 places to the right and 5 places to the 8 right is 8 places to the right, the power of 10 is 10 power of 10. 5 . 4.6 x 10 8 400 Math 87 To perform the exercises in this lesson, first change the decimal number to scientific notation. Then change the two powers of 10 to one power of 10. Example 1 Solution Write 25 x 10~ 5 in scientific notation. We write 25 in scientific notation. (2.5 x 10 1 ) x 10~ 5 We combine the powers of 10 by remembering that 1 the right plus 5 places to the place to equals 4 places to the left. left 2.5 x 10 4 Example 2 Solution Write 0.25 x 10 4 in scientific notation. First we write 0.25 in scientific notation. (2.5 Since 1 place to the places to the right x 10" 1 x 10 4 ) plus 4 places to the right equals 3 can write left we 2.5 x 10 3 With practice you will soon be able to perform these exercises mentally. Practice Problem set 89 Write each number in scientific notation. a. 0.16 x 10 6 b. 24 x 10" 7 c. 30 x 10 5 d. 0.75 x 10" 8 1. The following is a list of scores Jan received in a diving competition. 7.0 6.5 6.5 7.4 7.0 6.5 6.0 (a) Which (b) the scores were arranged in order from the least to the greatest, which score would be the middle If score? score was received most often? More on (c) What is (d) What is the average of all Scientific Notation 401 the scores? the difference between the highest score and the lowest score? 2. 3. Use a ratio box to solve this problem. The team won 15 games and lost the rest. If the team's won-lost ratio was 5 to 3, how many games were played? Brian swam 4 minutes will 4. swim 10 take Brian to it how many laps? Write each number in scientific notation, (a) 5. laps in 6 minutes. At that rate, 15 x 10 5 Draw 0.15 x 10 5 (b) Two fifths of the 70 How many barges were a diagram of this statement: barges were loaded with coal. loaded with coal? stump was 40 cm. Find the circumference of the tree stump to the nearest centimeter. 6. The diameter 7. Use the name of a geometric of the tree solid to describe the shape of these objects. (a) 8. A volleyball A water pipe Find the circumference of each (a) S~^\ Use 3.14 9. (b) for Use Simplify: (-4) + (-5) (b) (-2) circle, (b) n (a) A tepee (c) - (-6) + (-3) - (-4) - (+5) ^7 for /r 402 Math 87 Refer to the figure to do Problems 10 and 11. Dimensions are in millimeters. Corners that look square are square. 10. (a) What (b) What is the area of the hexagon? the area of the shaded is triangle? 11. What fraction of the hexagon is shaded? Write equations to solve Problems 12 and 13. 12. What number is 50 percent of 200? 13. What number is 2.5 of 4.2? 14. Complete the table. Fraction Decimal Percent 3 (b) (a) 20 15. Use a unit multiplier 16. Evaluate: a 2 17. Find the missing number. - Va + ab In 8 6 10 4 18. to convert 16 if F I O N to ounces. a = 4 and b = 0.5 Out U N -+ C T pounds 15 -*- 11 — _ 19 Divide 144 by 11 and write the answer (a) as a decimal with a bar over the repetend. Qb) rounded to the nearest whole number. Multiplication Rule for Equations Solve and check. 19. Show your work p - 49 75 = 403 20. t + - = 15 8 16 Solve: 21. 7q = 357 22. °=^ 8 2 Add, subtract, multiply, or divide, as indicated: 23. 24. 5 ft 7 in. + 6 ft 8 in. 350 m 60 sec 1 sec 25. 2- + 27. 6 ( 29. v - 1 min 0.15 + (0.5 1 km 1000 (decimal) 4) 3 26. - + 28. $7.50 - 0.075 30. 1 4- + 5+ 1- 2 y LESSON 6.5 (fraction) ' 3 2> 3- 1- 4 m 6 V } Multiplication Rule 90 for Equations we learned an algebraic method for solving equations that used the addition rule. In this lesson we will discuss another rule used in algebra to solve equations. We remember that the product of any number and the number 1 is the number itself. In Lesson 88 1-4 = 4 We remember that the 1-6 = 6 reciprocal of a 1 • x = x number is the number 404 Math 87 with the top and the bottom terms interchanged. 4 — is 3 —3 is 4 —\ is the reciprocal of F the reciprocal of F —3 4 4 — 3 the reciprocal of 5 5 5 is the reciprocal of If we We multiply a number by also answer reciprocal, the 4 3 = 12 _ 1 3 4 12 5 remember is its also that if we —\ divide a = 5 answer is 1. = 5 number by itself, the 1. 3 — = 4 1 —43 1 2 = „ 1 2 = 1 We can solve some equations by remembering these facts and using either the multiplication rule for equations or the division rule for equations. Multiplication Rule for Equations both sides of an equation are multiplied by the same number (but not zero), the solution (answer) is not changed. If Division Rule for Equations both sides of an equation are divided by the same number (but not zero), the solution (answer) is not changed. If Example 1 Solve: 4x = 15 — 405 Multiplication Rule for Equations Solution by itself. We have a choice. We can use the division rule and divide both sides of the equation by 4, or we can use the multiplication rule and multiply both sides u ~* """ " 4' "" We want x all * * * t_» • ing the division rule: 4x = 15 4x 15 4 4 original equation divided both sides by 4 lx = 3- simplified both sides 4 x=3*4 lx = X Using the multiplication 11 4x = 15 — • 4x = 15 4 • — 4 — 15 lx = Solution 2 Solve: original equation multiplied both sides by F J —1 4 simplified both sides 3 x - 3— Example 2 rule: simplified x = 150 To change |x to lx, we either divide both sides by \ or multiply both sides by -. With fractions it is usually easier to multiply by the reciprocal, so we will multiply both sides by |. 2 x = 150 original equation / f 3 2 ^ 2 v3 , —x lx = - -(150) multiplied by — simplified both sides x = 225 simplified 406 Math 87 Now we check our answer. 2 -x = 150 original equation -(225) = 150 substituted 225 for x 3 150 = 150 Example 3 Solution check Solve: 3.5y = 280 To change 3.5y to ly we either divide both sides of the equation by 3.5 or multiply both sides by With decimal — numbers it is . usually easier to divide. 3.5y = 280 original equation ' = 3.5 3.5 ly = 80 y = 80 Now we divided both sides by y 3.5 simplified both sides ly = y check our answer. 3.5y = 280 3.5(80) = 280 280 = 280 Practice original equation substituted 80 for check Solve each equation by dividing or multiplying. step. Then check each solution. a. 8m y = 416 b. -x Show every = 72 4 c. -w 3 Problem set 90 1. = - d. 0.4n = 1.84 3 problem. Four hundred fifty students attended the assembly. If the ratio of boys to girls in the assembly was 4 to 5, how many girls attended the assembly? Use a ratio box to solve this . 407 Multiplication Rule for Equations What 2. numbers marked by the average of the three is arrows on this number line? 3.2 I | 3.4 3.3 I I 1 1 I I I 1 I 1 | I » I I 1 I I I | 3.6 3.5 I I 1 I » I i 1 I | t » I I 1 i I I I I | t Refer to this figure to do Problems 3-5. Dimensions are in centimeters. Corners 4 that look square are square. 3 What is the perimeter of the hexagon? 4 What is the area of the hexagon? 5. What 6. Use 7. Draw is the area of the shaded triangle? a unit multiplier to convert 3.5 g to milligrams. a diagram of this statement: In the first the season the Madrigals played 18 games. games will the Madrigals play during the third of How many whole season? Write equations to solve Problems 8-11. 8. What number is - of 72? 6 9. Forty-eight percent of 25 is what number? 10. What number 11. What 12. A 13. Find the circumference of each (a) is is 0.1 of 110? 75 percent of $40? cube has how many more X"~^X Leave it as n g* edges than faces? circle, (b) Use 3.14 for n 408 Math 87 14. Complete the table. Decimal Fraction 0.75 (a) Percent (b) 15. What number is 16. How much 30 percent of $5.40? 17. Sketch a picture of a triangular prism. has how many edges? 18. Find the missing number. is 42 percent of 300? F In -2 U N 4 C T 10 I O N 16 A triangular prism Out — — — _ 4 10 16 19. Write 37,500,000,000 in scientific notation. 20. Evaluate: my - y2 if m Show each Solve and check. = 12 and y = step. 21. 8x = 31.2 22. -y 23. m 24. ^34 Add, 3 + 3.4 = 7 = 24 23 p - - = — subtract, multiply, or divide, as indicated: 25. - 26. 1 oz 3 qt 1 pt 5 1 qt 1 pt $300 1 week 5 days week 7 oz 1 day 8 hr 409 Volume A 3- - 1- ( 27. 28. 7- 29. 3- + 5 0.65 4 3 5 3 (decimal) 5 30. - (-8) (a) (-7) + (-12) (b) (-24) + (-18) - (+32) Volume LESSON 91 Geometric solids are shapes that take up space. We use the word volume to describe the space occupied by a shape. To measure volume, we must use units that occupy space. The units that we use to measure volume are cubes of certain sizes. We can use sugar cubes to help us think of volume. Example 1 This rectangular prism was constructed of sugar cubes. is Solution how many Its volume cubes? To find the volume of the prism, number of cubes it we must calculate the We see that there are contains. 3 layers of cubes. Each layer contains 3 rows of cubes with 4 cubes in each row, or 12 cubes. Three layers with 12 cubes in each layer means that the volume of the prism is 36 cubes. Volumes are measured by using cubes of a standard size. A cube whose edges are 1 centimeter long has a volume of 1 cubic centimeter, which we abbreviate by writing 1 cm 3 . ^-^" 1 ^0*" cm H h1 Similarly, A 1 1 cubic centimeter = 1 cm 3 cm cm each of the edges is 1 foot long, the volume is 1 If each of the edges is 1 meter long, the volume is if cubic foot. 1 cubic meter. 1 cubic foot = 1 ft 3 1 cubic meter = 1 m' 410 Math 87 To calculate the volume of a solid, we can imagine constructing the solid out of sugar cubes of the same size. We would begin by constructing the base and then building up the layers to the specified height. Example 2 Find the number of 1-cm cubes that can be placed inside a rectangular box with the dimensions shown. 3cm 5 Solution The base of the box is 5 cm cm by 4 cm. So we can place 4 rows of 5 cubes on the base. Thus there are 20 cubes on the first layer. Since the box is 3 can fit 3 layers of cubes in the box. 20 cubes llayer We Example 3 cm high, we /S sS sS s/ s S* '*>S- s*> S /y x 3 layers" = 60 cubes find that 60 cubes can be placed in the box. What is the volume of this cube? Dimensions are in inches. 4 Solution The base 4 in. by 4 in. Thus, 16 cubes can be placed on the base. is Since the big cube is 4 in. high, there are 4 layers of small cubes. 16 cubes liayer x 4 -Layers" a ,s aa .^'^W -PPT^ = 64 cubes Each little cube has a volume of 1 cubic inch. Thus, volume of the big cube is 64 cubic inches (64 in. 3 ). the 411 Volume Practice a. y ; s ; / / / y / y / / y y y / / y / / / // y This rectangular prism was constructed of sugar cubes. Its volume is s' how many sugar cubes? b. V Find the number of 1-cm cubes that can be placed inside a box with dimensions 10 cm as illustrated. 10 10 c. What cm cm volume of this rectangular prism? Dimenis the sions are in feet. 10 Problem set 1. It was 38 kilometers from the encampment to the castle. Milton galloped to the castle and cantered back. If the round trip took 4 hours, what was his average speed in kilometers per hour? 91 2. box to solve this problem. The ratio of dogs to cats in the neighborhood was 4 to 7. If there were 56 dogs in the neighborhood, how many cats were in the Use a ratio neighborhood? 3. tape measure, Gretchen found that the circumference of the great oak was 600 cm. She estimated that its diameter was 200 cm. Was her estimate for the diameter a little too large or a little too small? Why? 4. Grapes were priced Using a at 3 (a) What was (b) How much would pounds the price per for $1.29. pound? 10 pounds of grapes cost? 41 2 Math 87 product of nine tenths and eight tenths is subtracted from the sum of seven tenths and six tenths, what is the If the 5. difference? Three fourths of the 6. (a) How many (b) What 188 were singles. singles? percent of the batter's hits were not singles? Q 2 + (~ 5 ) ^ Find the number of 1-cm cubes that can be placed in 8. were hits of the batter's hits Compare: 2 - 5 7. batter's this box. „ 3 yS _ cm /A cm 5 Find the circumference of each 9. ^—\ (a) cm circle, (b) Use 3.14 Leave n as n for n Write each number in scientific notation. 10. 12 x 10" 6 (a) What 11. is 0.12 x lO" 6 (b) the average of the three numbers marked by arrows on this number line? 0.7 I 12. 0.8 I I I » I I I Use a unit multiplier 1.0 0.9 to convert I I I I 2000 g I 1 I I to kilograms. Volume 413 Write equations to solve Problems 13 and 14. 13. What number is 15 percent of 2400? 14. What number is - of 100? 6 15. Complete the table. Decimal Fraction M (a) 16. 14% Simplify: (a) (-6) - (b) (-5) + (-2) - (-7) - (-4) + (+2) 17. Evaluate: ab 18. Round - [a - b) if (+9) a = 0.4 and 6 = 0.3 29,374.65 to the nearest whole number. Show your Solve and check. 19. Percent work. q + 36 = 41.5 20. 4.3 = x - 0.8 Solve: 21. 5n = 24 1.2 22. 1.5 Add, subtract, multipy, or divide, as indicated: 23. 10 lb - 6 $5.25 1 9.2 x 27. (0.06 29. 3- hr 8 hr 1 7 oz lb 25. 4 24. 94- 5) ' 2 -s- V 13 days 1 week 1— (decimal) 26. 11.5 - 0.004 28. $15 + (0.06)($15) 30. 4- + 5- - 1- -*• > • day 5 3 (fraction) 111 2 6 3 41 4 Math 87 Finding the Whole Group When a Fraction Is Known LESSON 92 Drawing diagrams of fraction problems can help us understand problems such as the following. Three fifths of the fish in the there were 45 blue gill pond were blue in the pond, gill. If how many fish were in the pond? fish The 45 blue We parts. gill in are 3 of the 5 divide 45 by 3 and were blue find that there are 15 fish in (45) gill Since each of the 5 15 fish, there were 75 each part. parts is fish in all. were the pond 15 fish 15 fish 15 fish 15 fish ' r | b were not blue gill 1 5 fish 1 pages Example When Sean finished page 51, he was - of the way through his book. His book had how many Solution Sean read 51 pages. This is < 1 7 pages 1 7 pages 1 7 pages 1 7 pages (51) 3 of 51-^3 each part is 17 pages. Thus the whole book, all 8 parts, total 8x17, which is 136 pages. is 17, Draw read pages? 8 parts of the book. Since Practice | 8 in all 17 pages f- not yet b read < 17 pages 1 7 pages 1 7 pages a diagram to solve this problem. Three fifths of the students in the class are boys. If there are 15 boys in the class, how many students are there in all? Problem set 1. George bought 92 5 pounds of grapes for $1.20. (a) What was (b) What would be the cost of 12 pounds of grapes? the price per pound? 5 Finding the Whole Group 2. When a Fraction Is Known 41 problem. The clipper sailed for 48 hours at an average speed of 6 nautical miles per hour. How far did the ship sail? Use box a ratio to solve this On his first 2 tests Nate's average score was On his next 3 tests Nate's average score was 3. 80 percent. 90 percent. What was his average score for all 5 tests? [Hint: Find the total number of points scored on all tests, then divide by 5.) 4. Twenty billion is how much more than nine billion? Write the answer in scientific notation. sum 5. What 6. Use a ratio 7. Draw a diagram of this statement. is the of the first five prime numbers? box to solve this problem. The ratio of new ones to used ones in the box was 4 to 7. In all there were 242 in the box. How many new ones were in the box? Then answer questions that follow. When Debbie finished page 78, she was | way through of the her book. (a) How many pages are in her book? (b) How many pages does she have read? what number? 8. Three fourths of 24 9. Find the number of 1-inch is left to cubes that can be placed in this box. Dimensions are in 4 inches. 10. Find the circumference of each (a) ^—\ Use 3.14 for circle (b) n Use 22 y for n the 41 6 Math 87 11. Write each number in scientific notation, 25 x 10 6 (a) 12. Complete the KT 6 25 x (b) table. Fraction Percent Decimal 0.1 (a) (b) 13. Write an equation to solve this problem. What number is 35 percent of 80? 14. Find the missing number. F In 3 U N — C ^ T 5 I 7 N 15. A 16. (a) Out rectangular prism has 10 -** 12 — 14 how many vertices? Find the perimeter of the trapezoid. Dimensions are in centimeters. (b) Find the area of the shaded isosceles triangle. OA of 84 17. Compare: 0.03 of 112 18. Divide 256 by 24 and write the quotient as a mixed number. 19. Use a unit multiplier to convert 340 cm to meters. 417 Implied Ratios 20. Evaluate: y - xy if x = 0.1 and y = 0.01 22. 4x = 4.56 Solve: + 5.75 = 26.4 21. in 23. 20 24 Add, 55 y subtract, multiply, or divide, as indicated: 4 2 + {20 - 2 [6 - 24. (5 lyd^ 25. - 26. 2)]) 60 3.5 hr 1 1 ft ( 27. 6- 4-1 v 29. 2- - 30. (a) 60 sec hr 1 min 1 in. • 2 2- 2 3 1.5 ill A 7- - 5- + 1- 28. 6 2 ; 3 (fraction) (-5) + (-6) - (-7) (b) LESSON 93 min (-15) - (-24) - (+8) Implied Ratios Many rate problems can be solved by completing a proportion. Consider the following problem. If 12 books weigh 20 pounds, how much would 30 books weigh? We will illustrate First we pounds, two methods will use the rate method. we can If write two rates. 20 pounds 12 books (a) problem. 12 books weigh 20 for solving this 20 pounds (b) 12 books 418 Math 87 To find the weight of 30 books, by rate (b). 30-boeks- x We 19 we could multiply 30 books r~ = i 50 pounds would weigh 50 pounds. find that 30 books Now we will solve the proportion. We will record same problem by completing a the information in a ratio box. Instead of using the words "ratio" and "actual count," we will write "case 1" and "case 2." We will use p to stand for pounds. Case 1 Case 2 12 30 20 P Books Pounds From the table we write a proportion and solve 30 12 20 it. proportion P 12p = 20 30 cross multiplied 30 VZp _ 20 yz 12 • divided by 12 p = 50 We Example 1 Solution simplified find that 30 books would weigh 50 pounds. pounds of grapes cost $1.20, how much would 12 pounds of grapes cost? Use a ratio box to solve the problem. If 5 First we draw Now we the ratio box. We use d for dollars. Case 1 Case 2 Pounds 5 12 Dollars 1.2 d write the proportion and cross multiply. 5 12 1.2 d 5d = 12(1.2) 9 Implied Ratios Now we solve by dividing both sides by Example 2 Solution _ nî) id = d= 41 5. i|4 = $2 88 . Mrs. C can tie 25 bows in 3 minutes. How many bows can she tie in 1 hour at that rate? Work the problem (a) using rates and (b) using a ratio box. We can use either minutes or hours but not both. The units must be the same everywhere in a problem. Since there are 60 minutes in 1 hour, we will use 60 minutes instead of 1 hour. —boWS — , 25 fa] -7: , : __ -nni. 60 jam" = 500 bows . x So Mrs. C can (b) tie 500 bows in Case 1 Case 2 25 b Bows Minutes 1 hour. 60 3 Next we write the proportion, cross multiply, and solve by dividing by 3. 25 proportion 60 3 25 60 = 3b • = 3 -5- P 500 = b Practice a. m^ simplified ratio ride 75 b. divided by 3 box to solve this problem. Kevin rode 30 km hours. At that rate, how long would it take him to Use a in 2 cross multiplied km? needed to feed 40 head of cattle, how many bales are needed to feed 50 head of cattle? Use the rate method and then use a ratio box to solve this problem. If 6 bales are 420 Math 87 Problem set 1. Napoleon Bonaparte was born in 1769 and died For how many years did he live? 2. In her 93 4 games in 1821. averaged 4 points per game. In the next 6 games Jill averaged 9 points per game. What was her average number of points per game after 10 games? [Hint: Find the total number of points scored first Jill before dividing.) 3. Use a unit multiplier 4. If the product of and 5. -, what is - to convert 2.5 and L - is subtracted to milliliters. from the sum of - the difference? Use a ratio box to solve this problem. The ratio of carnivores to herbivores in the jungle was 2 to 7. If there were 126 carnivores in the jungle, how many herbivores were there? 6. Use a ratio pounds, 7. how many Two fifths of The diameter Draw of a bicycle tire tire to is 20 in. Find the distance the nearest inch. a diagram of this statement. Then answer the questions that follow. Edmund received 150 votes. This was two thirds of the votes cast. (a) (b) 10. 60 what number? around the 9. problem. If 4 books weigh 9 pounds would 14 books weigh? to solve this Write an equation to solve this problem. is 8. box How many votes How many votes were cast? were not for Edmund? What is the volume of a block of ice with the dimensions 6 shown? 10 in. in. 421 Implied Ratios 11. in scientific notation, 0.6 x 10 6 (a) 12. number Write each What is 0.6 x lO" 6 (b) numbers marked by the average of the three arrows on this number line? 13. | I I I + I I I Complete the I I | 1.6 1.5 1.4 1.3 I I I » I I I I I I | I I I | I I I » | I I table. Decimal Fraction Percent 3 (b) (a) 5 14. Write an equation to solve this problem. 75 percent of $24? 15. Write the prime factorization of 540. 16. (a) How much Find the area of the parallelogram. (b) 17. Name (a) 18. 6 Find the area of the shaded triangle. ^ each geometric solid, =L ~> f (b) (c) Find the next three numbers in 1, 4, 9, 16, 19. Compare: - 20. Evaluate: n O 25, 36, this sequence: , 0-667 mn if m = 3.6 and n = 0.9 in. is 422 Math 87 Solve: 21. 2 - = m 22. 1 lit = 1760 3 2 23. £ 3 3 2 Add, nAX tad subtract, multiply, or divide, as indicated: [30 - 4(5 -2)] + 5(3 3 - 52) • V9 + Vl6 to - mi 1 gal 1 qt 1 I + f4- 2 CO l 5i + • 1 yd — — *=ft • • 2 to 29. 26. 2 gal 1 qt - 27. 5280 1 25. 1 mi 3 ft 1 pt 2^ 6- 3 28. 7i- r5-i ii + 1- 6 *) 2 V l (dec ;imal) 1.9 4 30. (a) (-7) + (+5) + (-9) (b) (16) + (-24) - (-18) Fractional Part of a Number and Decimal Part of a Number, Part 2 LESSON 94 some fractional-part-of-a-number problems the fraction is unknown. In some fractional-part-of-a-number problems the total is unknown. As we discussed in Lesson 74, we In can translate these problems to equations by replacing the word of with a multiplication sign and by replacing the word is with an equals sign. Example 1 What fraction of 56 is 42? Fractional Part of a Solution We Number and Decimal Part of a Number, Part 2 423 an equation by replacing what fraction with F replacing of with a multiplication symbol, and replacing is with an equals sign. translate this statement directly into W What fraction of 56 is 42? WF To solve, we , question x 56 = 42 equation divide both sides by 56. WF x 56 42 divided by 56 56 56 WF 3 — = simplified "What decimal part of 56 is 42?" the procedure would have been the same. As the last step we would have written ^ as the decimal number 0.75. If the question had been, WD = 0.75 Example 2 Solution Seventy-five We make is what decimal a direct translation. Seventy-five is II what decimal we part of 20? question II I WD = 75 To solve, part of 20? x 20 equation divide both sides by 20. 75 = WD 20 X20 y 20 3.75 = WD simplified had begun "What fractional part," we would have written the answer as a fraction or as a mixed number. If the question — 20 —4 WFK fraction = Wj? F reduced = 3 3— = 4 WF v mixed number 424 Math 87 Example 3 Solution Three fourths of what number In this direct 60? is problem the total is the unknown. We can still do translation from the question to the equation. Three fourths of what number I I 1 3 To solve, 60? question i = 60 equation multiply both sides by -x-xC N 4 4 i WN we is a 3 = 60x-4 3 WN multiplied by 3 = 80 simplified Had the question been phrased by using 0.75 instead of ^ the , procedure would have been the same. Seventy-five hundredths of what To T T 0.75 x solve, we can _ WN T 60? question T =60 equation 60 divided by 0.75 0.75 0.75 WN = Problem set 94 T is divide both sides by 0.75. 0.75 x WrN Practice number 80 simplified a. What b. Seventy-five c. Eighty 1. During the first 3 days of the week, Mike read an average of 28 pages per day. During the next 4 days, Mike averaged 42 pages per day. For the whole week, Mike read an average of how many pages per day? fraction of 130 is is 0.4 of is 80? what decimal part of 300? what number? Fractional Part of a 2. Number and Decimal Part of a Number, Part 2 425 Twelve ounces of Brand X costs $1.14. Sixteen ounces of Brand Y costs $1.28. Brand X costs how much more per ounce than Brand Y? 3. Use a unit multiplier to convert 4— 4. The 5. Use a ft to inches. and hazelnuts in its cache in the ratio of 7 to 3. If it had a total of 2100 nuts in its cache, how many hazelnuts had the squirrel saved? Use a ratio box to solve the problem. squirrel saved acorns ratio box pounds of how much would 8 pounds of apples to solve this apples cost $1.40, problem. If 5 cost? 6. Draw a diagram of this statement. Then answer the questions that follow. Five sixths of the 300 triathletes completed the course. (a) How many (b) What was triathletes completed the course? the ratio of triathletes the course to those who did not complete the course? Write equations to solve Problems 7-10. 7. Fifteen is —3 of what number? 8 8. Seventy 9. Two is what decimal fifths of What number 11. What is the part of 200? what number 10. is is 120? 60 percent of 180? volume who completed of this cube? 3 in. ^ 426 Math 87 12. Find the circumference of each X Use 13. 22 y (b) n for Complete the Leave n as n table. Decimal Fraction 3I What number 15. Find the missing number. is Percent (b) (a) 14. 20 percent of $35? In 2 F Out U N — — — 60 C T 16. circle 5 I 3 O N 24 __ Write four hundred twenty-five million in scientific notation. 17. Refer to the figure to answer these questions. AB (a) II (b) 18. II BC and EC. What type is AE figure of quadrilateral DC ABCE? What type of quadrilateral is figure Arrange these numbers in order from b ABDE? least to greatest: 0.013, 0.1023, 0.0103, 0.021 — Multiplying 19. What number 20. Evaluate: — 12 is [m + n) 427 and Dividing Signed Numbers of 108? - mn if 2 1 = 1— and n = 2- m Solve: 21. p + 3- = 7F 2 5 23. 17 Add, 24. 25. - 22. 3n = 0.138 4.8 subtract, multiply, or divide, as indicated: V49 + {5[3 2 - 4hr - ,, 1 hr mm ,^ 15 mm . 3- + iY 8 2 v 29. 8- + 30. (a) (2 3 - a/25)] 15 sec on 30 sec 5 ( 27. = 0.36 *> 26. 7.5 ;v 52} 24 hr — — day ] 1 28. • J qA l 2 - 5^- 4; — 60 min r -— 1 nr -60 sec mm ; 1 + 1- + 1- 6 3 4 (fraction) - (-9) + (-11) (+14) (b) (26) + (-43) - (-36) Multiplying and Dividing Signed Numbers LESSON 95 We can develop the rules for the multiplication and division of signed numbers if we remember that multiplication is a shorthand notation for repeated addition. We remember that 2 times 3 means 3 + 3 and that 2 times -3 means (-3) + (-3), so 2(3) _ = 6 and 2(-3) = -6 428 Math 87 We remember that division undoes multiplication, so the following must be true. 2(3) If = 6 then - = 3 - = and 2 and if 2(-3) = -6 then 2 3 — = -3 and 2 = +2 — -3 We use these examples to illustrate the fact that when we multiply or divide two positive numbers the answer is a positive number. Also, when we multiply or divide two numbers whose signs are different, the answer is a negative number. But what happens if we multiply two negative numbers? Since 2 times - 3 equals - 6 2(-3) = -6 then the opposite of 2 times -3 should equal the opposite of -6, which is 6. (-2H-3) = +6 And since division undoes multiplication, these division examples must also be true. ±6 = -3 -2 and ±* = -2 -3 This gives us the rules for the multiplication and division of signed numbers. Rules for Multiplication and Division 1. numbers that are multiplied or divided have the same sign, the answer is a If the two positive number. 2. If the two numbers that are multiplied or divided have different signs, the answer negative number. Here are some examples. Multiplication (+6)(+2) = +12 DnasiON — = +3 +2 is a Multiplying and Dividing Signed Numbers "6 (-6X-2) = +12 -2 "6 (-6H+2) = -12 (+6)(-2) Example +6 -2 = -3 (c) (d) (6)(-3) (-6)(-4) We divide or multiply as indicated. If both signs are the same, the answer is positive. If one sign is positive and the (a) is negative, the -3 answer +4 (b) is negative. (c) -18 (d) +24 Divide or multiply: a. (-7)(3) b. (+4)(-8) d. (-6H-4) e. ^| -28 Problem set =-3 +2 = -12 ^ (b) =jj^ other Practice = +3 Divide or multiply: (a) Solution 429 , c. f. (8)(+5) ^ +30 1. Use a ratio box to solve this problem. If Mrs. C can wrap 12 packages in 5 minutes, how many packages can she wrap in 1 hour? 2. Lydia walked for 30 minutes a day for 5 days. The next 3 days she walked for an average of 46 minutes per day. What was the average amount of time she spent walking during those 8 days? 3. If 95 the and 4. sum 0.5, of 0.2 what is and 0.5 is divided by the product of 0.2 the quotient? Use a unit multiplier to convert 23 cm to millimeters. 430 Math 87 5. Use a ratio box to solve this problem. The ratio of paperback books to hardbound books in the school library was 3 to 11. If there were 9240 hardbound books in the library, how many books were there in all? 6. Write each number in scientific notation, (a) 7. 24 x 10- 5 Draw 24 x 10 7 (b) a diagram of this statement. Then answer the questions that follow. The 30 test's (a) (b) true-false questions amounted to ^ of the questions. How many How many questions were on the test? of the questions were not true-false? Write equations to solve Problems 8-11. 8. Forty-five is —5 of 9. Twenty-four is 0.4 of what number? what number? 10. What number is 11. What decimal part of 30 12. Divide or multiply: 80 percent of 760? is 21? (b) -36 -6 (d) (+8)(+7) -36 (a) 9 (c) 13. 9(-3) Find the number of 1-ft cubes that will fit inside a closet with dimensions as shown. 8ft /3 4ft ft Multiplying 14. Find the circumference of each ^~^\ (a) Use 3.14 15. and Dividing Signed Numbers for Complete the 431 circle (b) 22 n Use -y for k table. Decimal Fraction Percent 2.5 (a) Use angle measurements (b) each to classify triangle. Then find the area of each triangle. Dimensions are in centimeters. 8 16 A. / • i\ 18. "N. 8 19. Name (a) each three-dimensional A\ figure, (b) 20. Compare: - of 96 21. Evaluate: ab - (a O - -of 84 b) if a = 5 — and 22. a + - = 14 24. 20w = 8 5.6 Show your —3 4 6 Solve and check. b = work. 23. b - 1.6 = 0.16 432 Math 87 Add, subtract, multiply, or divide, as indicated: 25. 2 + 27. 1 yd yd 7 in. 2 8 in. 26. 12- • 4- • 5 29. 2.25 x 30. (a) 0.5 28. 7- * 2 10 m A 23 1- cm 100 m 1 ft 112 2 1 mm cm A 6- • 15 3 (fraction) (-8) + (-7) - (-15) (b) (-15) + (+11) - (+24) Area of a Complex Figure • Area of a Trapezoid LESSON 96 Area Of a Complex figure Example 1 ft 1 We have practiced finding the areas of figures that can be divided into two or more rectangles. In this lesson we will begin finding the areas of figures that include triangular regions as well. Find the area of this figure. Corners that look square are square. Dimen- 4 10 sions are in millimeters. 16 Solution We draw 7 a dotted line that divides the figure into a rectangle and a 4 10 triangle. X. 6 Area of rectangle =7x10 Area of triangle = Total area 70 mm 2 = + 27 mm 2 97 mm 2 > Area of a Complex Figure Example 2 Find the area of this figure. Corners that look square are square. Dimen- • Area of a Trapezoid sions are in centimeters. Solution many ways There are 433 10 to divide this 10 figure. 6 6 4 6 i\ l\ 7 10 10 \ i !\\ i i 4 — i 3 4 \ 10 3 i 4 6 10 10 (b) (a) (c) We decide to use (c). We will find the area of the big rectangle and subtract from it the area of the triangle. Area of rectangle =10x10= Area of triangle = 4x7 100 cm 2 = - 14 cm 2 86 cm 2 Area of figure Area of a We remember that a quadrilateral with just one pair of parallel trapezoid sides is is Solution One way to find the area of a trapezoid to divide the trapezoid into find the Example 3 a trapezoid. combined area two triangular regions and of the triangles. 10 Find the area of this trapezoid. Dimensions are in centimeters. We can divide the trapezoid into two triangles by drawing either diagonal. We show both ways. 10 B (a) (b) 434 Math 87 Both figures have an upper triangle [A] and a lower triangle [B). The height of all four triangles is 6 cm. Area of triangle A = Area of triangle B = 10 x 6 7x6 30 cm 2 = + 21 cm 2 51 cm 2 Total area Practice Find the area of each figure. Dimensions are in centimeters. Corners that look square are square. 12 a. 20 C. 10 15 8 Problem set 1. 96 Pablo ran an 8-lap race. For the first 5 laps he averaged 72 seconds per lap. For the rest of the race he averaged 80 seconds per lap. What was his average lap time for the 2. 12 If whole race? 30 ounces of cereal cost $2.49, what is the cost per ounce? 3. One thousand five hundred meters is how many kilometers? 4. The sum of - and of I and |? 5. Use a ratio - is how much greater than the product box to solve this problem. The ratio of Marci's age to Chelsea's age is 3 to 2. If Marci is 60 years old, she is how many years older than Chelsea? 6. Write each (a) number 12.5 x 10 4 in scientific notation, (b) 12.5 x 10~ 4 Area of a Complex Figure 435 Area of a Trapezoid • 7. Use a ratio 8. Draw a diagram of this statement. problem. Martha rode 40 miles in 3 hours. At this rate, how long would it take Martha to ride 100 miles? box to solve this Then answer questions that follow. Two fifths of the library's 21,000 books were checked out during the school (a) How many books (b) How many year. were checked out? books were not checked out? Write equations to solve Problems 9-12. 9. Sixty y — 12 5 is of what number? 10. Seventy percent of what number 11. Thirty-five is 12. Fifty-six is what decimal 13. Simplify: what is 35? fraction of 80? part of 70? -120 (a) (b) (-12X11) (d) 12(+20) 4 -120 (c) 14. -5 Find the volume of this rectangular prism. Dimensions are in centimeters. / / A 20 15. The diameter of the plate 11 inches. Find its was circumference to the nearest half inch. 12 U 16. Find the area of this trapezoid. Dimensions are in inches. h. 16 10 the 436 Math 87 17. A 12 corner was trimmed from a square sheet of paper to leave the paper in this shape. Dimensions are in centimeters. 10. 20 14 Find the perimeter of the (a) figure. Find the area of the (b) 18. Complete the table. Decimal Fraction What 20. Evaluate: is Percent 125% (b) (a) 19. figure. 20 percent of $12.50? X x 3 - xy if x= 2 and y = 0.5 y Solve and check. 21. -x Show each step. = 40 22. 1.2 8 23. y + Add, 24. = 8.47 3.6 92 - [3 3 - (9 • 3 - V9)] 48 min 20 sec hr 23 min 48 sec 2 hr - 1 f 5 A V 29. -- ,0 1- 26. 100 yd 28. 1- + 0.47 (decimal) (b) (-5) + (-4) - - (-40) (-3) - (+2) 2 12 in. lft 111 5 (-26) + (-15) -—-r 3- + 2- - 1- V (a) • y 5 30. = 26.4 subtract, multiply, or divide, as indicated: 25. 27. W 4 Inverting the Divisor LESSON 97 437 Inverting the Divisor Hidden Ones Every number has a coefficient of 1. Every number has a divisor of 1. Every number has an exponent of 1. 4 = 1-4 4 = 1 4 = 41 In this lesson fractions we will be concerned with divisors of have a divisor of 1. Even 1. 4 _9^ 3. -- i 1 3 10 10 we have been dividing fractions by inverting and multiplying. Now we will show the reason Since Lesson 29 the divisor for this rule. We remember that we can multiply a number by a number that equals 1 without changing the value of the ' 2 3 number. _6 x 15 5 Because 3 over 3 equals 1, we have multiplied - by 1. This lets us see why - and — have the same value. We can use this fact to simplify complex fractions. Example 1 Solution Simplify: f The product of a number and its reciprocal is 1. The product of | and - is 1. If we multiply the denominator by -, we have to multiply the numerator by 3 _9^ 5 2 10 _9_ 1 10 3 - 438 Math 87 We could have inverted the divisor and multiplied as show we here. 3 5. 3 3 _9_ 2 5 2 10 3 If we multiply above and below by instead -, we can simplify the expression without using a rote rule. —l 25 Example 2 Solution Simplify: First I we write both numerator and denominator as fractions. 77 100 Now we multiply y f y above and below by 77 Example 3 1 10Q 1 77 100 300 77 100 1 1 300 1 100 — Simplify: 73 Solution We begin by writing both numerator and denominator as fractions. 15 22 3 Now we multiply above and below by 15 _3_ 1 22_ 22 _3_ 3 22 — 3 45 .. 22_ 1 . 2 J_ 22 Inverting the Divisor Example 4 Solution 1 Change 83- percent A percent is 83^% 3 439 to a fraction. a fraction that has a denominator of 100. Thus is 1 100 Next we write both numerator and denominator as fractions. 250 3 100 1 Now we multiply above and below by 250 3 Practice 1 100 250 100 _ 300 _ 5 1 100 1 1 100 6 1 Simplify: K 12 100 6 Change each percent c. 66-% 3 Problem set 97 to a fraction. d. 8-% 3 1. Nestor finished a 42-kilometer bicycle race in 1 hour 45 minutes. What was his average speed in kilometers per hour? 2. Kim's scores in the diving competition were 7.9, 8.3, 8.1, 7.8, &v4, 8.1, and 8.2. The highest and lowest scores were not counted. What was the average of the remaining scores? 440 Math 87 3. box to solve this problem. The ratio of good guys to bad guys in the movie was 2 to 5. If there were 35 guys in the movie, how many of them were good? 4. Use unit multipliers to convert 2 kilometers to centimeters. 5. Change 16- percent 6. Use a Use a ratio 2 24 7. One box to solve what number? ratio to is to a fraction. this problem. Thirty is to mass was quartz. If the weight of the rock was 144 grams, what was the weight of the sixth of the rock's quartz in the rock? a = what does V2a equal? 8. If 9. Simplify: 10. 8, (a) -60 -12 (b) (-8X6) (c) 40 -8 (d) (-5H-15) What is the circumference of this circle? Leave n as n 11. 80 as The shows a pyramid with a square base. Copy the figure and find the number of its (a) faces, (b) edges, and (c) vertices. figure Write equations to solve Problems 12 -16. 12. What 13. What number is 10 percent of $37.50? is 8 of 72? Inverting the Divisor what 14. Twenty-five 15. Sixty 16. Twenty percent of 30 17. Complete the is is 441 fraction of 60? what decimal part of 80? is what number? table. Fraction Decimal Percent 5 18. (b) (a) 6 A square sheet of paper with an area of 81 in. 2 has a corner cut off, forming a pentagon as shown. (a) What is the perimeter of the pentagon? \c (b) 19. What What kind is the area of the pentagon? of angle is made by the hands of the clock at 3 o'clock? 20. Multiply two thousand by three thousand and write the product in scientific notation. Solve each equation. 21. x - 25 = 96 Show each step. 22. -m = 12 3 23. 2.5p = 6.25 Add, 24. subtract, multiply, or divide, as indicated: 25. Vl3 2 - 27. 3- x 4- x 0.16 4 28. 10 = / + 3 3- + 8 5 2 26. (fraction) 6 6.7 + 84 (decimal) 1 ton - 400 lb — 442 Math 87 - 29. (-3) + (-5) 30. (-73) + (-24) (-3) - (+5) - (-50) More on Percent LESSON 98 We remember 40 percent means 40 hundredths. that 40% If what a problem asks 40 100 means percent, we will write —W— p . We write our answer with a percent sign. Example 1 Solution What percent We of 40 is 25? can translate the question What percent I —w p to an equation and MM question x 40 = 25 equation H of 40 is 25? 100 To solve, we divide both sides by 40. first MS -ir- Wp 25 100 40 w i ' P t^£ = i©cr Wp = divided by 40 40 Then we multiply both oecr solve. 25 tt: 40 • sides ioo —r~ l simplified by . ,, u multiplied by y v , .. simplified 100 i will 443 More on Percent W = 62.5% p Since is Example 2 Solution Wp stands for "what percent," the answer to the question 62.5%. Fifteen percent of We what number translate the question to Fifteen percent of \ solve, ^ ys Solution we * iqg \ 600? question \ equation multiply both sides by 100 over 15. ^,= N 600.*^ 15 multiplied by WN simplified is 100 15 = 4000 what number? write the equation and solve. 20 100 Problem set 98 solve. = 600 W,N Twenty percent of 40 Practice an equation and \ \ Twenty percent of 40 We 600? is 100 To is what number 15 Example 3 simplified x Twenty-four b. What c. Fifteen percent of 1. Use percent of 6 a ratio what number? box question 40 = W;N equation 8 = w,N simplified what percent a. is is is of 40? 2? what number to solve this is 45? problem. Tammy saved and pennies in a jar. The ratio of nickels to pennies was 2 to 5. If there were 70 nickels in the jar, how many coins were there in all? nickels 444 Math 87 Refer to the line graph to answer questions 2-4. Jeremy's Test Scores 100 90 o o iS 80 CD — -^ O o p- 70 Test Test 2 1 If there 3. What was 4. (a) Which (b) What was and Name (a) (b) (c) 6. Test 4 A A A Use a 7. Draw 1, Test 6 how many questions Jeremy's average score? score did Jeremy make most often? the difference between his highest score his lowest score? the shape of each object. marble length of pipe box of ratio tissue box to solve this inches equal 254 centimeters. equal 250 inches? / Test 5 were 50 questions on Test did Jeremy answer correctly? 2. 5. Test 3 One hundred How many centimeters problem. a diagram of this statement. Then answer questions that follow. Three fifths of those present agreed, but 12 disagreed. (a) What fraction of those present disagreed? the 445 More on Percent (b) How many were present? (c) How many of those present agreed? Write equations to solve Problems 8-11. — 25 4 what number? 8. Forty J 9. Twenty-four percent of 10,000 is of is 10. Twelve percent of what number 11. Twenty 12. Simplify: (a) is what percent (b) -250 13. -5 15 table. Decimal 0.2 (a) is 4.5 percent of Simplify: Percent (b) $20? 14 15. -15(-5) -225 Complete the What 240? (d) Fraction 14. is of 25? 25(-5) (c) what number? 60 (b) (a) 100 2 3 16. Find the area of this figure. Dimensions are in feet. Corners that look square are square. 10 4 4 A / \ h 18 17. Sketch a picture of a cube with edges 2 cm long. What is the volume of the cube? 18. Write twelve billion in scientific notation. \ 446 Math 87 19. Find the circumference of each ^~^\ (a) Leave it as (b) Use 3.14 it Solve each equation. 20. 3x = 26.7 22. -x circle Show each for it step. 21. y - 3- = 7 = 48 3 23. Find the missing number. In F 3 C T -1 I 4 Add, 24. N - {2 -2 - -3 — 2 + 3[4 2 - (4)(V9)]} 3 4 gal 3 qt 1 pt + -*- subtract, multiply, or divide, as indicated: 52 25. Out U N — 26. 1 ft 2 1ft 1ft 2 qt 1 pt 1 gal 7^ 27. 5 U 3 29. 3- - 30. (a) r 1 1^ 3 2.5 -5- 3 ) 28. si5 J „1 „1 2+ 1— 2 4 (fraction) (-3) + (-4) - (+5) (b) (-6) - (-16) - (+30) 447 Graphing Inequalities Graphing Inequalities LESSON 99 The symbols We have used the symbols >, <, and = to compare two numbers. > and < In this lesson we will introduce the symbols > and <. We will also practice graphing on the number The symbols > and < combine the sign with the equals sign. line. greater than/less than We read the first end of the symbol and do not read the other end. Thus, reading from right, the symbol left to > is read, "greater than or equal to," and, reading right, the from left to symbol < is Graphing on the number line read, "less than or equal to." To graph a number on the number line, we draw a dot at the point that represents the number. Thus, when we graph 4 on the number line, it looks like this: -10123456789 This time we will graph all the numbers that are greater than or equal to 4. We might think the graph should look like this: It is true that all the dots mark points that represent numbers However, we did not graph all the numbers that are greater than 4. For instance, we did not graph 10, 11, 12, and so on. Also, we did not graph 4|, 5|, 6|, and so on. If we graph all of these numbers, the dots are so close together that we end up with a solid line that goes on and on. Thus a graph of all the numbers greater than or that are greater than or equal to 4. equal to 4 looks like -1 this: 448 Math 87 marks the 4. The shaded line marks the numbers greater than 4. The arrowhead shows that this shaded line goes on without end. The Graphing large dot Expressions such as the following are called inequalities. inequalities x < 4 (a) We read is x > 4 (b) as "x is less than or equal to 4." greater than 4." (a) We read (b) as "x We can graph inequalities on the number line by graphing all Example 1 Solution the numbers Graph on We number make line: are told to graph all to 4. we a that We draw shade shows all the inequality a true inequality. x < 4 numbers that are less than or equal and then The arrowhead a dot at the point that represents 4, the points to the left of the dot. that the shading continues without end. -10123456789 Example 2 Solution Graph on a number line: x > 4 We are told to graph all numbers greater than 4 but not including 4. We do not start the graph at 5 because we also need to graph numbers like 4| and 4.001. To show that the graph does not include 4, we draw an empty circle at 4 and then shade the number line to the right of the circle. Practice a. On b. On a number line, a number equal to line, graph graph all all the the x < -1 numbers 1. Graph each inequality on a number c. numbers d. line. x > -1 less than 2. greater than or 449 Graphing Inequalities Problem set 1. Use a ratio box to solve this problem. If 4 cartons are needed to feed 30 hungry children, how many cartons are needed to feed 75 hungry children? 2. Gabriel's average score after 4 tests 99 must Gabriel average on the next was What score have a 6-test 88. 2 tests to average of 90? 3. - 4. the If , sum what is of - and - is divided by the product of - and the quotient? problem. The ratio of monocotyledons to dicotyledons in the nursery was 3 to 4. If there were 84 dicotyledons in the nursery, how many Use a ratio box to solve this monocotyledons were there? 5. The diameter of a nickel is mm. Find the circumference 21 of a nickel to the nearest millimeter. 6. 7. 8. Graph each inequality on a separate number x < (a) x > Use a unit multiplier to convert 1.5 kg to grams. 2 [b) 1 Five sixths of 30 people who participated in the taste test preferred the taste of Brand X. The rest preferred Brand Y. How many more people preferred Brand X than preferred Brand Y? Write equations to .solve Problems 9-12. 9. line, Forty-two is seven tenths of what number? what number 600? 10. Sixty percent of 11. Forty percent of 50 12. Forty 13. Write 1.5 x 10 -3 in standard form. is is what percent is what number? of 50? 450 Math 87 14. Simplify: -450 (b) (a) 15 9 (c) 15. 15(-20) Complete the -15(-12) (d) table. Fraction Decimal Percent 50% (b) (a) 83 16. Simplify: 100 40 17. Find the area of this trapezoid. Dimensions are in millimeters. 18. A box 12 cm 24 cm long, wide, and 10 cm high. of tissues is 24 Sketch a picture of the box and find its volume. 19. Find the missing number. In 2 3 4 5 20. How much is m U N — (J — 1 T — 9 13 *" O -* N 21 30 percent of $18.50? Solve each equation. 21. - Out F Show each + 8.7 = 10.25 step. 22. -w 3 23. 0.7y = 48.3 = 36 Insufficient Information Add, 2 451 Comparisons + 10[2 3 - (3)(\4)]} - V36 {4 25. 1 yd - 1 ft 3 in. 26. m 100 1 2 15 7- • 3 • 100 1 cm m - 1- 1 O i 3- - 2- - 1( - 28. 9 2 7.2 cm m 1 29. Quantitative subtract, multiply, or divide, as indicated: 24. 27. • 5 V 47 2 (decimal) 5 30. - (-8) (a) (-10) (b) (+10) + (-20) - (+6) - (-30) • [LESSON Insufficient Information 100 Quantitative Comparisons Insufficient Sometimes we encounter problems information enough) information to determine the answer. The following problem provides insufficient information to answer the question. for which there is insufficient (not A 10-pound bag of potatoes costs $1.49. What is the average price of each potato? > Since we do not know the number of potatoes in the bag, we do not have enough information to find the average price of each potato. We will practice recognizing problems with insufficient information as we answer quantitative comparison problems like the examples in this lesson. 452 Math 87 Quantitative comparisons We have practiced comparing numbers using the symbols >, <, and =. In this lesson we will begin considering comparison problems in which insufficient information has been provided determine the comparison. to Example 1 The numbers x and y are whole numbers. Compare: *Oy Solution I We are told that to determine the comparison, as our answer x and y are whole numbers, but we are not given information that will let us determine which is greater, or if x and y are equal. Since we do not have enough information we write insufficient information. Example 2 The number x is positive and y is negative. Compare: *Oy Solution We I enough information to determine what However, we are given enough information are not given each number is. to determine the comparison. Any positive number than any negative number. Thus, the answer is is greater x > y Example 3 a - b - Compare: aQb Solution The equation does not provide enough information to detennine the value of either number. However, since their difference is zero, the two numbers must be equal. a = b Practice Answer each comparison by writing >, <, =, or insufficient information. a. x— y= b. 5i = n i 1 O7 mQ n Compare: x Compare: I Insufficient Information c. d. a x • is b = Ob Compare: a 1 not positive, y is Comparisons 453 % not negative. Compare: Problem set Quantitative • xQ y 1. The average number of students in 4 classrooms was 33.5. If the students were regrouped into 5 classrooms, what would be the average number of students in each room? 2. Nelda drove 315 kilometers and used 35 liters of gasoline. Her car averaged how many kilometers per liter of gas? 3. Use a ratio box to solve this problem. The ratio of winners to losers was 7 to 5. If the total number of winners and losers was 1260, how many more winners were there 100 than losers? 4. Write each number in scientific notation, (a) 37.5 x 10- 6 Q (b) — = 37.5 x 10 6 5. Compare: x 6. Graph each inequality on a separate number 7. (a) x < Use a ratio snow fell Draw if 1 2 (b) box line, x > -1 problem. Four inches of in 3 hours. At that rate, how long would it take for 1 foot of 8. y to solve this snow to fall? a diagram of this statement. Then answer questions that follow. Twelve students earned A's. This was —3o of the students in the class. i\ (a) How many (b) What percent of the students did not earn A's? students did not earn A's? the ) 454 Math 87 Write equations to solve Problems 9—12. 9. Thirty-five j — 7 is of what number? 11 10. What 11. What number 12. Three hundred sixty 13. Simplify: percent of 20 is 17? is 5 percent of 360? is 75 percent of what number? 144 -144 (b) (a) +6 -8 (c) 14. -12(12) (d) Complete the -16(-9) table. Fraction Decimal Percent i 25 15. What 16. Simplify: y is , 17. (a) (b) 60 percent of $8.40? , * — 62^2 - 100 A square sheet of paper with an area of 100 in. 2 has a corner cut off as shown in the figure. 10 Dimensions are in inches. (a) What is the perimeter of the shape? (b 18. What is the area of the shape? In the figure, each small cube yy y y / / y y / s / y y y / v" cubic centimeter. What is the volume of this rectangular prism? is 1 ,7 ' / / / \y Insufficient Information 19. • Quantitative Find the circumference of each 455 Comparisons circle (b) Use 3.14 20. for Leave n as n n Identify each angle as acute, right, or obtuse. (b) (a) Solve each equation. 21. 1.2x = 2.88 23. -w = 2 Add, 1 22. 3 - 2 min 24 hr 45 min 30 hr 15 3 7- - 28. 9J 1.5 - - 3 yd 2 3 ft yd 1 11 5 4- + 3- + 7- 6 (decimal) (+20) 1 1 4 (-10) 26. sec 1 V 3- + sec 3)] 5 ( 30. 3- = x + — 10 100 + 5[3 3 - 2(3 2 + 25. 29. every step. subtract, multiply, or divide, as indicated: 24. 27. Show (c) (-30) 3 4 ft yd 456 Math 87 LESSON Measuring Angles with a Protractor 101 acute, right, we discussed angles and classified angles as or obtuse. In this lesson we will begin measuring In Lesson 18 angles. Angles are commonly measured in units called degrees. The abbreviation for degrees is a small circle written above and to the right of the number. One full rotation, a full circle, measures 360 degrees. A A full circle half circle measures half of 360°, measures 360°. which A half circle One fourth of a full rotation measures one fourth of 360°, which is 180°. measures 180 a right angle. is A right is A c right angle 90°. angle measures 90 c Thus, the measure of an acute angle is less than 90°, and the measure of an obtuse angle is greater than 90° but less than 180°. An angle that measures 180° is a straight angle. The chart below summarizes the types of angles and their measures. Measure Type of Angle Acute angle Greater than 0° but less than 90° Right angle Exactly 90° Obtuse angle Greater than 90° but less than 180° Straight angle Exactly 180° Measuring Angles with a Protractor 457 A protractor may be used to measure angles. The protractor placed on the angle to be measured so that the vertex is under the dot or circle or cross-mark of the protractor, and is one side of the angle is under the line at either end of the scale of the protractor. The measures ZAOB of the three angles = 30° ZAOC shown = 110° are as follows. ZAOD = 150° Notice that there are two scales on a protractor, one starting from the left side, the other from the right. One way to check whether you are reading from the correct scale is to consider whether you are measuring an acute angle or an obtuse angle. Example Find the measure of each angle. O S3 (a) ZAOB (b) ZAOD TC (c) ZEOD 458 Math 87 Solution (a) ZAOB an acute angle. We use the scale with the numbers less than 90. Ray OB passes through the mark halfway between 40 and 50. Thus the measure of ZAOB is (b) (c) Practice Problem set is 45°. ZAOD is an obtuse angle. We read from the scale with numbers greater than 90. The measure of ZAOD is 140°. ZEOD is an acute angle. The measure of ZEOD is 40°. Find the measure of each angle. a. ZAOD b. ZAOC c. ZAOE d. ZFOE e. ZFOC f. ZAOB 1. Tina 2. Evaluate: 3. Compare: a Qb 4. Use box 101 mowed lawns 4 hours and earned $5.25 per hour.i Then she washed windows for 3 hours and earned $3.50' per hour. What was Tina's average hourly pay for the 7hour period? x + a ratio [x2 for - xy) if - y ab = if x = 4 and y = 3 2 to solve this problem. When Nelson cleaned his room, he found that the ratio of clean clothes to dirty clothes was 2 to 3. If 30 articles of clothing were discovered, how many were clean? J 459 Measuring Angles with a Protractor 5. The diameter of a half dollar is 3 cm. Find the circumference of a half dollar to the nearest millimeter. 6. Use 7. Graph each inequality on (a) 8. a unit multiplier to convert a separate x > -2 (b) Use a ratio box to solve 1 1- quarts this to pints. number line, x < problem. In 25 minutes, 400 customers entered the attraction. At this rate, how customers would enter the attraction in 1 hour? 9. Draw a diagram of this statement. many Then answer questions that follow. Nathan found that knee joint to his hip was 18 inches from his joint. This was - of his total it height. (a) What was Nathan's (b) What was Nathan's total height in inches? total height in feet? Write equations to solve Problems 10 -13. 10. Six hundred is —5 of what number? 9 11. Two hundred eighty 12. What number is 13. Sixty 14. Simplify: what percent of 400? 4 percent of 400? 60 percent of what number? 600 -15 n (a) (bj -600 -12 (c) 20(-30) (d) +15(40) r 15. is is i What is , 80 percent of $12.50? the 460 Math 87 16. Complete the table. Fraction Decimal 0.3 (a) 17. Percent (b) Express in scientific notation. (a) 30 x 10 6 30 x 10~ 6 (b) 18. Find the area of this trapezoid. Dimensions are in meters. 19. each edge of a cube is 5 inches, what is the volume of the cube? 20. x If D_ Find the measure of each angle. ****" "*"*^ JK % R (a) ZROS (b) Solve each equation. 21. ZVOS Show each 17a = 408 (c) step. 22. -m = 48 8 23. 1.4 Add, 24. = x - 0.41 subtract, multiply, or divide, as indicated: 23 + 4 • V25 - 5 • 2 V4 • 32 ZVOU 461 Using Proportions to Solve Percent Problems 25. 10 lb 6 oz 7 lb 11 oz 1 27. 7- * 2 29. ( 3 • 1 10 cnr 1 -5^ 28. mm cm mm 10 1 cm ^11 4- + 3- - 76 9y I 7- x 26. 3 4 1.4 (fraction) 7 30. (-3) - (+7) LESSON + (-10) Using Proportions to Solve Percent Problems 102 A percent is a ratio. Thus, percent problems can be solved using the same method we use to solve ratio problems. Consider the following problem and explanation. Thirty percent of the class passed the test. If 21 students did not pass the test, how many students were in the class? about 2 parts of a whole class. We recognize this as a part-part-whole problem. One part of the class passed the test; the other part of the class did not pass the test. The whole class is 100 percent. The part that passed was 30 percent. Thus, the part that did not pass must have been 70 percent. We will record these numbers in a ratio box just as we do with ratio problems. The problem is Percent Actual Count Passed 30 70 100 Did not pass Whole As we read class we find an actual count as well. who did not pass the test. We record the problem, There were 21 students 462 Math 87 21 in the appropriate place of the table and use letters in the remaining places of the table. Percent Actual Count Passed Did not pass Whole We class 30 70 100 P 21 W will use this table to help us write a proportion so that we can solve the problem. We will use the numbers in two of the three rows to write a proportion. This time we will use the numbers in the second row because we know both number of we will also use the third row. Then we numbers. Since the problem asks for the students in the class, solve the proportion. total Percent Actual Count Passed Did not pass Whole class 30 70 100 P 21 70 21 W Too W 70W 2100 w By solving the proportion, we find that there were 30 in the whole class. Example 1 Solution 30 students Forty percent of the leprechauns had never seen the pot of gold. If 480 leprechauns had seen the pot of gold, how many of the leprechauns had not seen it? We may solve this problem just as we solve a ratio problem. We use the percents to fill the ratio column of the table. All was 100 percent. The part that had never seen the pot of gold was 40 percent. Therefore, the part that had seen the pot of gold was 60 percent. The number 480 was the actual count of the leprechauns who had seen the gold. We write these numbers in the table. the leprechauns Percent Actual Count Had not seen Had seen Total 40 60 100 N 480 T 463 Using Proportions to Solve Percent Problems Now we use the table to write a proportion. Since we know both numbers in the second row, we will also use that row in the proportion. Since the problem asks us to find the actual count of leprechauns who had not seen the pot of gold, we will also use the first row in the proportion. Percent Actual Count Had not seen Had seen 40 60 100 Total N 40 N 480 60 480 T 60N = 19,200 N= We Example 2 find that 320 leprechauns Twenty-seven of the 45 elves had to work the night work the night shift? Solution We make shift. had not seen the pot who worked What percent 320 of gold. in the toy factory of the elves had to box and write in the numbers. The total number of elves was 45, so 18 elves worked the day shift. a ratio Percent Actual Count Night Day shift shift use shift. Pd 100 Total We Pn PN 27 18 45 to stand for the percent who worked the night We use this row and the total row to write the proportion. Percent Actual Count Night Day shift shift Pn Pd 100 Total 27 18 45 N 27 100 45 45PN = 2700 N We Practice find that 60 percent of the elves Use a a. ratio box to solve worked the night shift. each problem. Twenty-one of the 70 acres were planted in alfalfa. percent of the acres was not planted in alfalfa? ... 60 What 464 Math 87 b. Problem set has 60 percent of the book to read. If she has read 120 pages, how many pages does she still have to read? Lori figures she still Use the information given 102 On to answer questions 1-3. his first 15 tests, Paul earned these scores: 70, 85, 80, 85, 90, 80, 85, 80, 90, 95, 85, 90, 100, 85, 90. 1. What was 2. If 3. (a) Paul's average test score? were arranged in order from lowest to highest, what would be the middle score? This number is called the median. Paul's scores Which is (b) score did Paul earn most often? This score called the What was mode. the difference between Paul's highest score and his lowest score? This number is called tall. Danny the range. 4. Danny is is 5. how many Use a His sister is 5 6 " inches taller than his sister? 6' 1 " (6 ft 1 in.) tall. ratio box ' to solve this problem. Michelle pencils for 75$. At this rate, 7. how much would Graph each inequality on a separate number (a) bought 5 she pay dozen pencils? for a 6. • x < 4 Draw (b) line, x > -2 a diagram of this statement. Then answer the questions that follow. Gilbert answered 48 questions correctly. This was - of the questions on the test. (a) How many questions were on the test? (b) What was the ratio of Gilbert's correct answers to his incorrect answers? 1 465 Using Proportions to Solve Percent Problems 8. B is located halfway between points A and C, what length of segment AB? If point is the i i I i i | i i I I i l | | " | l " I I " I I " I I I inches 9. Use a ratio box to solve this problem. The ratio of gleeps to blobbles was 9 to 5. If the total number of gleeps and blobbles was 2800, then how many gleeps were there? x = what does x2 + Vx equal? 10. If 11. Compare: 12. Complete the 9, m Qn if — = 0.5 table. Decimal Fraction 2- (a) Percent (b) Write equations to solve Problems 13 and 14. 13. What percent 14. Fifty percent of 15. Simplify: of 40 is 12? what number is 0.4? 1M 100 Use a ratio box to solve Problems 16 and 17. 16. Nathan correctly answered 21 of the 25 questions. What percent of the questions did he answer correctly? 17. Twenty percent of the 4000 acres were many acres were not left fallow? left fallow. How 466 Math 87 18. Find the measure of each angle. (fliiiiiinii|iiii|iiiiiff*^ ZAOD (a) ZAOB 19. Which 20. What is the area of this trapezoid? (b) angle ZFOB shown in (c) (d) Problem 18 measures 55°? 10 Dimensions are in inches. 21. U N -1 C 2 56 x 10 7 Solve each equation. 12 13 in. 20 in. Out -5 T 2 I O N -2 (b) Show 5x = 16.5 Add, 25. in in. Write each number in scientific notation, (a) 23. F 4 6 22. 13 Find the missing number. In 56 x lO" 7 each step 24. 1 3 2 8 3- + a = 5- subtract, multiply, or divide, as indicated: 32 + 5[6 - (10 - 2 3 )] ZFOC in. Area of a Circle 26. 1 + 2 day 8 hr 15 min day 15 hr 45 min 28. 8- - 29. (a) (-12X-9) 30. (a) (-3) + (-4) (5 - 2- x 27. 4.5 - 6 467 (fraction) 3.4) (decimal) -100 (b) - (-5) (b) - (+20) + (-7) Area of a Circle LESSON 103 (-18) We can find the areas of some polygons by multiplying the two perpendicular dimensions. • We find the area of a rectangle by multiplying the length times Width the width. Length We find the area of a parallelogram by multiplying the base times the Height height. Base We find the area of a triangle by multiplying the base times the height, then dividing by 2. Height Base To find the area of a circle, we again begin by multiplying two perpendicular dimensions. We multiply the radius times the radius. This gives us the area of a square built on the radius. 468 Math 87 If the radius of the circle is 9. If We is 3, the area of the square the radius of the circle is 32 , which the area of the square r, see that the area of the circle is is less is r 2 . than the area of four of these squares. However, the area of three of these squares does not equal the area of the The number area of the circle quite circle. whose area exactly equals the number between 3 and 4. The exact of squares a is Thus, to find the area of a circle, we first find the area of a square built on the radius. Then we multiply that area by n. This is summarized by the equation number is n. Area of Example Find the area of each (a) X^X circle = nr 2 circle (b) (c) 22 Use 3.14 Solution (a) for Use -y it for it The area of a square built on the radius 2 is 100 cm We multiply this by n. . A A A (b) The = nr 2 « (3.14)(100 cm 2 ) cm2 « 314 area of a square built on the radius is 49 A - nr 2 A - A « 154 in. We multiply this by n. 7 22 y 2 • ^Tin. 2 in. 2 Leave it as % 469 Area of a Circle (c) Since the diameter is 6 ft. The the radius by is 12 ft, the radius area of a square built is 36 ft 2 . We on multiply this n. A A A = KT 2 = n • 36ft 2 36;rft 2 = Notice the answer is written as 36 times n followed by the unit. Practice Find the area of each circle. b. a. 7 t 73 Use 3.14 for n Leave n as n 7t" 15 22 Use -y for n 17 Problem set 103 Find the volume of this rectangular prism. Dimensions sions are in 2. The heights 11", 6' of the 5 basket- were 6' 2", and 6' ball starters 5' 2.5 feet. 3", 6' 5", 1". Find the average height of the 5 {Hint Change all measures to inches before starters. dividing.) 3. problem. The student/teacher ratio at the high school was 20 to 1. If there were 48 high school teachers, then how many students were there? 4. Use a unit multiplier 5. Graph each inequality on Use a ratio box to solve (a) x < -2 this to convert 66 inches to a separate (b) x > number feet. line, 470 Math 87 Use box 6. problem. Don's heart beats 225 times in 3 minutes. At that rate, how many times will his heart beat in 5 minutes? 7. Draw a ratio to solve this a diagram of this statement. Then answer the questions that follow. Two of the students in the class were fifths boys. There were 15 girls in the class. (a) How many (b) What was 8. Evaluate: 9. What students were in the class? the ratio of girls to boys in the class? y + xy - x ifx=3 percent of this circle Qb shaded? is a - b 10. Compare: a 11. Find the circumference of each (a) if / X Use 3.14 for Find the area of each 13. Complete the 14. circle, (b) 22 Use -y circle in for Problem n 11. table. Fraction (a) negative is n 12. and y = 4 Decimal 1.6 Percent (b) Write an equation to solve this problem. What is 8 percent of $25? 15. Express in scientific notation. (a) 12 x 10 5 (b) 12 x 10- 5 . . Area of a Circle box to solve 16. Use a 17. Use a 18 Find the area of this figure. Dimensions are in inches. Corners that look square 471 problem. Sixty-four percent of the 175 students correctly described the process of photosynthesis. How many students correctly described this process? ratio this box to solve this problem. Ginger still has 40 percent of her book to read. If she has read 180 pages, how many pages does she still have to read? ratio 5 are square. 3 19. (a) Find the measure of ZAOE. (b) Find the measure of ZFOC. (c) Find two angles that each measure one half of a right angle. 20. Write the prime factorization of 816. 21. Find the next two numbers in z/ 22. y o , . , Write one hundred million in scientific notation. Solve each equation. 23. , this sequence. -x 4 = 36 Show each step. 24. 3.2 + a 3.46 472 Math 87 Add, subtract, multiply, or divide, as indicated: h2 v 25. + 42 4 gal 2 qt 26. c - 27. 3- 28. 4.5 29. (a) 30. LESSON 104 Multiplying powers (a) (7 -*• * ^ 3 qt 1 pt 0.2) (decimal) + 2- - ( 1 gal 3 (fraction) 4) (b) (-2)(+3)(-4) (b) (-20) + (+30) } - (-3) + (-4) (-2) Multiplying Powers of 10 - (-40) Multiplying • Numbers in Scientific Notation Here we show two powers of 10 of 10. 1Q3 We remember 1Q4 that 10 3 means 10 • 10 • 10 • 10 • 10 and 10 4 means 10 We can multiply powers of 10 3 10 \ We We • 10 10. 10 4 • I • 10 see that 10 3 times 10 4 can write this as 10 7 . ^\_ f • 10 • 10 N • means 10 • 10 • 10 that 7 tens are multiplied. Multiplying Powers of 10 • Multiplying 10 3 As we • Numbers in Scientific 473 10 4 = 10 7 focus our attention on the exponents, 3 Notation + 4 = we see that 7 This example illustrates an important rule of mathematics. When we multiply powers of 10, we add the exponents. Multiplying numbers in scientific notation Example 1 Solution To multiply numbers that are written in scientific notation, we multiply the decimal numbers to find the decimal number Then we multiply powers of 10 to find tne power-of-10 part of the product. We remember that when we multiply powers of 10, we add the exponents. part of the product. the Multiply: (1.2 x 10 5 )(3 x 10 7 ) We multiply 1.2 by 3 and get 3.6. 10 7 and get 10 12 The product is Then we multiply 10 5 by . 3.6 x 10 12 Example 2 Solution Multiply: x 10 6 )(3 x 10 5 ) (4 We multiply 4 by and get 10 11 . 3 and get 12. The product Then we multiply 10 6 by 10 5 is 12 x 10 11 We simplify this expression by writing (1.2 Example 3 Solution Multiply: (2 x 10 1 ) x 10 11 = 1.2 x 10 12 x 10- 5 )(3 x 10~ 7 ) We multiply 2 by 3 and get 6. To multiply 10~ 5 by 10~ 7 add the exponents and get 10~ 12 Thus, the product is . 6 x 10" 12 Example 4 Multiply: (5 x 10 3 )(7 x lO" 8 ) we 474 Math 87 Solution We multiply get 10~ 5 . 5 by 7 and The product get 35. We multiply is 35 x 10" 5 (3.5 product x 10 1 ) x 10- 5 3.5 x 10" 4 PractiC6 Problem set 104 10 3 by 10~ 8 and simplified Multiply and write each product in scientific notation. x 10 6 )(1.4 x 10 3 ) a. (4.2 b. (5 x 10 5 )(3 x 10 7 ) c. (4 x 10- 3 )(2.1 x 10- 7 ) d. (6 x 10- 2 )(7 x 10" 5 ) 1. The 16-ounce box cost $1.12. The 24-ounce box cost $1.32. The smaller box cost how much more per ounce than the larger box? 2. Use a box to solve this problem. The bad apples in the basket was 5 ratio apples to were 70 apples in the basket, how many ratio of good to 2. If there of them were good? 3. Jan's average score after 15 tests score on the next 5 tests score for 4. all was 90. average average 20 tests? Jackson earns $6 per hour at a part-time job. How much does he earn if he works for 2 hours 30 minutes? 5. Convert 24 shillings to pence. 6. Graph x < - 1 on a number 7. was 82. Her What was her Use a ratio box to solve is to what number? this (1 shilling = 12 pence) line. problem. Five is to 12 as 20 I Multiplying a = Powers of 10 475 Multiplying in Scientific Notation then what does 4a + 5 equal? 8. If 9. Four fifths of the football team's 30 points were scored on pass plays. How many points did the team score on 3, pass plays? Qy 10. Compare: x 11. Find the circumference of each if x + y = 10 /^X (a) circle, (b) 22 Use -j Leave n as n 12. Find the area of each 13. What 14. Complete the is the volume circle in for Problem n 11. of a cube with edges 10 cm long? table. Fraction (a) Decimal (b) Percent 250% What is 6 15. Write an equation to solve this problem. of $8.50? Use a ratio 16. Judy found that there were 12 minutes of commercials during every hour of prime time programming. Commercials were shown for what percent of each hour? 17. Thirty percent of the boats that traveled box to solve Problems 16 and Monday were steam-powered. percent 17. up the river on 42 of the boats that traveled up the river were not steam-powered, then how many boats were there in all? If 476 Math 87 33 18. Simplify: - 00 19. 40 Find the area of this trapezoid. m m 24 25 m b_ 35 20. (a) Sketch a right angle. many (b) A right angle m measures degrees? Sketch an acute angle that is half of a right angle. An angle that is half of a right angle measures many degrees? 21. F U 5 7 —+ N — C T — 2 -*• . O N x 10 4 )(6 x 10 5 ) (3 Solve each equation. — 21 -* 11 — 15 -*> 5 Show (b) Add, (1.2 24. 0.4y = 1.44 2 subtract, multiply, or divide, as indicated: 52 + 3[4 2 - 2(5 - 2)] 26. 3 27. x lQ- 3 )(4 x lO" 6 ) every step. b - 1- = 43 25. I Out Multiply and write each product in scientific notation, (a) 23. how Find the missing number in each diagram. In 22 how 0.6 x 3- -s- 2 (fraction) 1 yd yd 2 ft 7 in. 2 ft 8 in. Mean, Median, Mode, and Range 28. 4-1- 5.63 5 V 29. (a) 30. (a) 477 (decimal) y [zfypfy - (-3) + (-4) (-5) M (-3)(-4)(-5) (b) (-15) - (+14) + (+10) Mean, Median, Mode, and Range LESSON 105 The words mean, median, mode, and range An example study of statistics. is the best are used in the way to explain the meanings of these words. On his 100 If we Paul earned these scores. first 5 tests 80 70 70 90 arrange these scores in order from least to greatest, get — — Range 70 70 80 90 100 ' v ' t Mode The low of 70 to a high The difference between 70 and 100 is scores ranged from a of 100. 30. Median We say that the range of this group of numbers is 30. English the word "mode" means the customary fashion or style. The number 70 occurs more than any other number, and it is In called the mode of this group of numbers. the "fashionable number." It is we 478 Math 87 3. 4. The middle highway is called the "median" of the highway. The median of a group of numbers is the middle number when the numbers are arranged in order. The median of this group of numbers is 80. The mean of a group of numbers is the average of the group. The mean of these numbers is of a divided 82. .. Mean = 70 + 70 + 80 + 90 + 100 average = = *™ = 82 I 5 The nonsense phrase "mean old average" can be used as a mnemonic device to associate the words "mean" and "average." The median not always one of the numbers listed. If there are an even number of numbers, the median is the number is halfway between the middle two numbers. Thus the median is the average of the middle two numbers. For example, the middle of the following list of numbers is between 70 and 80. The median is 75, which is the number halfway between 70 and 80. 65 65 [70 80] 80 4 100 ,! There are two modes, 65 and 80. Example Amanda is in the Strikers bowling league. Her bowling scores for the first 103, 116, Solution 10 games were 95, 103, 96, 110, 103, 109, 110, and 105. (a) What was her mean (b) What was her median (c) What was the (d) What was the range of her scores? (a) To find the "mean old average," we add the 10 numbers and divide by 10. The sum is 1050. mode score? score? of her scores? ™™ 10 =105 , 479 Mean, Median, Mode, and Range (b) To find the median, we arrange the numbers in order and select the middle number. Since there are two middle numbers, we find their average. 96 95 103 103 109 105] [103 110 110 116 t 104 (c) The mode is (d) is the number that appears most often, which 103. The range is the difference between the and the smallest number. largest number 116 - 95 = 21 PractiCG Find the following given this list of measurements: 64 cm, 72 cm, 68 cm, 72 cm, 59 cm, 67 cm, 74 a. Problem set 105 Mode b. Mean c. Median d. cm Range Refer to this graph to answer questions 1-3. A Daily High Mon. 1. What was the range Monday to Friday? 2. Which day had Tue. - Temperature Wed. Thu. in the daily high temperature from the greatest increase in temperature from the previous day? 3. Wednesday's high temperature was how much lower than the average high temperature for theSeNljdays? 480 Math 87 4. Frank's scores on 10 tests were as follows: 90, 90, 100, 95, 95, 85, 100, 100, 80, For this set of scores find the (a) mode, and mean, 100 (b) median, (c) (d) range. 5. Use a ratio box to solve this problem. The ratio of rowboats to sailboats in the bay was 3 to 7. If the total number of rowboats and sailboats in the bay was 210, how many sailboats were in the bay? 6. Graph x < 7. Write a proportion to solve this problem. how much would 10 cost? 8. Five eighths of the 3 on a number line. If 4 cost $1.40, members supported the treaty, whereas 36 opposed the treaty. How many members supported the treaty? 9. Evaluate: a - {ab 10. Compare: f Q) g 11. (a) -a if 2 if ) —f a = 3 and b = 4 * = 1 Find the circumference of this circle. (b) 12. Find the area of Use unit multipliers this circle. Use 3.14 to convert for n 4.8 meters to centimeters. 13. A rectangular prism has 14. Complete the table. Fraction i| how many Decimal (a) faces? Percent (b) J 481 Mean, Median, Mode, and Range 15. Write an equation to solve this problem. percent of $18? 16. Simplify: What is 30 100 Use a ratio box to solve Problems 17 and 17. When the 18. Sixty percent of the door was left open, 36 pigeons flew the coop. If this was 40 percent of all the pigeons, how many pigeons were there originally? there gnomes were 3 feet tall or less. If were 300 gnomes in all, how many were more than 3 19. 18. feet tall? A square sheet of paper with a perimeter of 48 in. has a corner cut off, (a) forming a pentagon as shown. What is the perimeter of the pentagon? (b) What is the area of the pentagon? 20. Find the measure of each angle, (a) ZAOC (b) ZEOC (c) ZAOD 482 Math 87 21. Find the next two numbers in -8,-6,-4, 22. , Multiply and write the product in scientific notation. (1.5 x 10" 3 )(3 x 10 6 ) Show each Solve each equation. 23. this sequence: b - 4.75 = 5.2 Add, step. 24. -y = 36 subtract, multiply, or divide, as indicated: 25. V5 2 - 4 2 + 27. 0.9 *- 2- • 23 26. 1 m - 45 mm 24 (decimal) 4 28. 7.8 - ( 5-1 + V LESSON 106 29. (a) 30. (+30) \ 0.2 3 (fraction) J M [zfj^fy - (-50) - (+3)(-5)(+2) (+20) Order of Operations with Signed Numbers • Functions, Part 2 Order Of To simplify expressions that involve several operations, we Operations With Signed perform the operations in a prescribed order. We have practiced simplifying expressions with whole numbers. In this lesson we W1U begin simplifying expressions that contain both whole numbers and negative numbers. numbers Example 1 Simplify: (-2) + (-2)(-2) - t=?l J Order of Operations with Signed Numbers Solution First we • multiply and divide in order from + (-2)(-2) - (-2) 483 Functions, Part 2 left to right. (-2) (+2) + (-2) Then we add and - (+4) (-1) subtract in order from (-2) + - (-1) ?2) - (-1) (+ 4) left to right. +3 Example 2 Solution - Simplify: (-2) We [(-3) - (-4)(-5)] simplify within brackets first. Within the brackets we follow the order of operations, multiplying and dividing before adding and subtracting. (-2) - [(-3) (-2) - [(-3) (-2) - (-23) - (-4X-5)] - (+20)] +21 Functions We remember between two sets of numbers. We have practiced finding missing numbers in functions when some number pairs have been given. For instance, the missing numbers in the functions below are 14 and 7. In 3 4 7 We have that a function is a relationship F OUT^ U N — C — 6 -*- 8 T I O N _ In 14 — F U N Out 7 c 10 -*> 5 T I 3 -* O -2 N found the missing numbers by first finding the rule of the function. The rule of the function on the left is, "Multiply the In number by 2 to find the Out number." The rule of the 484 Math 87 function on the right is, "Subtract 7 from the In number to Out number." find the Often the rule of a function is expressed as an equation with x standing for the In number and y standing for the Out number. If we write the rule of the function on the left as an we equation, get y = 2x we write the rule of the function on the right as an equation, we get y = x - 7 If we will practice finding missing when the rule is given as an equation. Beginning with this lesson numbers Example 3 in functions y = 2x + Find the missing numbers in this function. X 1 y 4 7 Solution The letter y stands for the Out number. The letter x stands for We numbers and are asked to find the Out number for each by using the rule of the function. The expression 2x + 1 shows us what to do to find y, the Out number. It shows us we should multiply the x number by 2 and then add 1. The first x number is 4. We multiply by 2 and add 1. the In number. We y = 2(4) y = 8 + 1 multiplied y= 9 added find that the number is 7. are given three In We + substituted 1 y number is 9 when x multiply by 2 and add is 4. 1. y = 2(7) y = 14 + 1 multiplied y- 15 added + 1 The next x substituted J Order of Operations with Signed Numbers The third x number y = 2(0) y = + y = + 485 Functions, Part 2 multiply by 2 and add substituted 1 multiplied 1 are 9, 15, and 1. X y 4 9 7 added 1 The missing numbers Practice We is 0. • s 1. Simplify: a. + (-3X-3) - (-3) (-3) (+3) - - (-5H-6)] b. (-3) c. (-2)[(-3) - (-4K-5)] [(-4) Find the missing numbers in each function. d. y = 3x X 1 1 e. y = ^ X 7 y f . y = 6 3 - x 7 7 1 1 4 Problem set X 8 4 1. Use a ratio box to solve this problem. The team's ratio of games won to games played was 3 to 4. If the team played 24 games, how many games did the team fail to win? 2. median, (c) mode, and range of the following group of scores: 106 Find the (a) mean, (b) 70, 80, 90, 80, 70, 90, 75, 95, 100, 3. (d) 90 problem. Mary was chagrined to find that the ratio of dandelions to marigolds in the garden was 11 to 4. If there were 44 marigolds in the garden, how many dandelions were Use a ratio box to solve the there? 4. Use unit multipliers to convert 0.98 liter to milliliters. . 486 Math 87 5. Graph x > 6. Use a ratio on a number box line. to solve this miles in 10 seconds, how problem. If sound travels far does sound travel in 2 1 minute? 7. Draw a diagram of this statement. Then answer the questions that follow. Thirty-five thousand dollars charity drive. This The (a) was raised in the was seven tenths goal of the charity drive of the goal. was to raise how much money? The drive fell short of the goal by what percent? (b) 8. Compare: 2a 9. What is O° 2 if o is a whole number the circumference of a circle whose radius is 4 meters? 10. What is the area of a circle whose radius is 4 meters? i. 11 What fraction of this circle is shaded? 12. A certain rectangular prism is 5 inches and 3 inches high. Sketch the figure and find its long, 4 inches wide, volume. 13. Complete the table. Fraction Decimal Percent i 40 (a) (b) 14. Write an equation to solve this problem. percent of $1.20? 15. Ten is 20 percent of what number? What is 5 Order of Operations with Signed Numbers 487 Functions, Part 2 • 8 16. Simplify: 100 Use a 17. ratio box Max was to solve delighted Problems 17 and when he found 18. that he had correctly answered 38 of the 40 questions. What percent of the questions had he answered correctly? 18. Before the clowns arrived, only 35 percent of the children wore happy faces, 19. 91 children did not wear happy children were there in all? faces. If how many (a) Name (b) Find its perimeter. (c) Find its area. 16 this shape. 12.5 cm \ i 12 cm cm 12.5 16 cm cm 20. 21. (a) Find mZTOS. (b) Find mZQOT. (c) ZQOR (d) Find is (The m means "the measure one third of a right angle. Find Find the missing numbers in this y = 2x X Multiply and write the product in 5 scientific notation. 3 (5 mZQOR. mZTOR. function. 22. of.") x 10- 3 )(6 x 10 8 ) 1 y 1 488 Math 87 Solve each equation. 13.2 = 23. Show each 1.2W step. 24. - = 1- c + 4 6 Add, subtract, multiply, or divide, as indicated: 3(20 - 25. [6 - 3(10 - 2 26. 4)]| 27. 0.6 x 5- 28. 6- + 4.9 -s- min 25 hr 45 min 30 3 hr 15 sec 2 sec 4 (fraction) + 12.25 (decimal) 5 29. (-2) + (-2)(+2) - |=|i 30. (-3) - [(-2) - (+4)(-5)] Number Families LESSON 107 In mathematics we give special names to certain sets of numbers. Some of these sets or families of numbers are the counting numbers, the whole numbers, the integers, and the rational numbers. In this lesson we will describe each of these • number families and discuss how they The Counting Numbers. Counting numbers are the numbers we say when we count. The first counting number is 1, the next is 2, then 3, and so on. Counting numbers: 1,2, 3,4, The three dots mean • are related. that the list 5, ... goes on and on. The Whole Numbers. The whole number family includes all of the counting number family and has one more member, which is zero. Whole numbers: 0, 1, 2, 3, 4, 5, . . . - Number 489 Families we use a dot to mark each of the whole numbers on the number line, the graph looks like this. If -5-4-3-2-1 —I 1 1 2 1 4 3 5 -— 1 1 Notice that there are no dots on the negative side of the number line, because no whole number is a negative number. Also notice that there are no dots between the whole numbers because numbers between whole numbers are not "whole." We put an arrowhead on the right end of the number line to indicate that the whole numbers continue without end. The The Integers. integer family includes all of the whole numbers. The integer family also includes the opposites (negatives) of the positive whole numbers. The list of integers goes on and on in both directions as indicated by the dots. Integers: A graph . . . , -4, -3, -2, -1, 0, 1, 2, 3, 4, . . . of the integers looks like this. -5-4-3-2-1 1 2 4 3 -*- — 5 includes the numbers -2,-1, 0, 1, 2, etc. ., but does not include -, -, and other fractions. Note the arrowheads on both ends of the number line to indicate that the integers continue without end in both directions. Thus the set of integers . The Rational Numbers. The rational number family includes all numbers that can be written as a fraction of two integers. Thus all fractions name a rational number. Here are some examples of rational numbers. 1 5 -3 -4 2 3 2 1 3 2 1 Notice that the family of rational numbers includes all the integers, because every integer can be written as a fraction whose denominator is the number 1. For example, we can write -4 as a fraction by writing -4 The set of rational numbers also includes all the positive and negative mixed numbers, because these numbers can 490 Math 87 we be written as fractions. For example, can write 4~ as 21 5 Sometimes rational numbers which case the decimal in 1 are written in decimal form, will either terminate = 0.125 8 or it will repeat. 5 = 0.8333 6 The following diagram may be helpful relationships shows between these families of numbers. The diagram that the set of rational number numbers includes Rational 2 o Numbers Whole Numbers Counting Numbers 1,2, 3, Solution the other Integers -2 1 all families described in this lesson. 1 Example in visualizing the Graph the . . . integers that are less than 4. We sketch a number line and draw a dot at every integer that Since the set of integers includes whole numbers, we draw dots at 3, 2, 1, and 0. Since the set of integers also includes the negatives of the positive whole numbers, we continue to draw dots at -1, -2, -3, and so on. We draw an is less than 4. arrowhead less graph of integers which than 4 continues without end. -5 —Example 2 to indicate that the Answer -4 -3 -2 —- —- —- -4- true or false. whole numbers (a) All (b) All rational are integers. numbers are integers. 2 3 4 -- -- H- are Number Solution (a) True. Every whole number is 491 Families included in the family of integers. (b) False. Although every integer not true that every rational are some rational is a rational number, number numbers, such as is - an and integer. -, it is There that are not integers. Practice Graph the b. Graph the whole numbers Answer Problem set 1 07 integers that are greater than -4. a. that are less than 4. true or false. whole number. c. Every integer is a d. Every integer is a rational number. 1. Heavenly Scent was priced at $28.50 for 3 ounces, while Eau de Rue cost only $4.96 for 8 ounces. Heavenly Scent cost how much more per ounce than Eau de Rue? 2. Use a ratio box to solve to veterans in the camp were 252 in the camp. problem. The ratio of rookies was 2 to 7. Altogether there this How many of them were rookies? 3. The seven linemen weighed 197 lb, 213 lb, 246 lb, 205 lb, 238 lb, 213 lb, and 207 lb. Find the (a) mode, (b) median, (c) mean, and (d) range of this group of measures. 4. Convert 12 bushels to pecks. 5. The Martins drove the (1 bushel = 4 pecks) car from 7 a.m. to 4 p.m. traveled 468 miles. Their average speed and was how many miles per hour? 6. Graph the 7. Use a ratio box what number is integers that are less than or equal to to solve this to 30? problem. Nine is 3. to 6 as 492 Math 87 8. Nine tenths of the school's 1800 students attended the homecoming game. (a) How many of the school's students attended the homecoming game? (b) What percent of the school's students did not attend the homecoming game? Evaluate: b 2 - 4ac 10. Compare: a 2 Qa 11. (a) 9. a = if if a is 1, b = 5, and c = 6 positive Find the circumference of this circle. (b) 12. Find the area of this Simplify: circle. — Leave ;ras n 22 13. shown is a triangular prism. Copy the shape on your paper and find the number of its (a) faces, (b) edges, and (c) The figure vertices. 14. Complete the table. Fraction Decimal 0.9 (a) 15. 17. (b) Write an equation to solve this problem. percent of $6.50? Use a 16. Percent ratio box to solve The sale price What was the Problems 16 and of $24 What is 80 17. was 60 percent of the regular price. regular price? Forty-eight corn seeds sprouted. This was 75 percent of the seeds that were planted. How many of the planted seeds did not sprout? Number 18. Thirty 19. (a) 493 Families 40 percent of what number? is 15 Classify this quadrilateral. mm P~ Find (b) its perimeter. 20 Find (c) its 25 mm mm area. 30 20. mm Dn 80 » A' Find the measure of these angles. ZCOF (a) 21. 22. Find the missing numbers in y = 3x + this function. X Multiply and write this product in 4 scientific notation. 7 (1.2 5 x 10 )(1.2 Solve each equation. 23. ZAOE (b) 56 = x 10- 8 ) Show each -w Add, 25. 24. 4.8 2 - 62 - a/10 2 / 27. 12 82 \ 5- 6 1.75 (fraction) m m + c = 7.34 subtract, multiply, or divide, as indicated: a/10 y step. 8 26. oz + 4 lb 7 oz 5 lb 9 1 S -6 . 494 Math 87 28. 1.4 + 3- x 10 3 (decimal) 2 29. (-4H-5) - (-4)(+3) 30. Memorizing LESSON 108 - (-2)[(-3) (-4)(+5)] Common Fraction- Percent Equivalents Certain percents are encountered so frequently that worth the is memorize some of the more common equivalents. The list below includes percent effort to fraction-percent equivalents for halves, thirds, fourths, tenths, it and fifths, sixths, eighths, twelfths. Also included are 1 8 2 8 3 , - 6 37±% 7 50% and 1UU 2 = 75 % - f = 87§% = 1 = 58|o/o z ? = 66-% 8 8 1 = — 1 , 62-% 8 25 % \ = 4 8 , 5 12§% 8 12*%. 111 — — — zU zb oU 8 = 100% 8 8)l00% 12 2_ 7 8§% 12 - = 16-% 12 6 _3_ = 25% T 4 12 _4_ 12 1 12 3 1 = 33 3 3 8 % 9 = 12 10 12 12 8^% _6_ 12)l00% 12 | = 50 % - = 75% 4 = - = 83-% 6 11 4lfo/o = 3 91-% 12 3 12 = 12 3 3 1 = 100% Memorizing Common 495 Fraction-Percent Equivalents The upper circle shows the division of 100 percent into eighths. The lower circle shows the division of 100 percent into twelfths. Use of these circles may make the following fraction-percent equivalents easier to memorize. Fraction-Percent Equivalents Fraction 1 2 1 Percent Fraction 1 50% 6 5 33-% 3 1 4 3 4 1 5 2 5 3 5 4 5 Practice Problem set 108 66 - Fraction 9 ie|% 10 1 % 8 12 37 % l 62 % l 8 5 75% 8 12 7 12 11 12 7 20% 8 1 40% 10 3 60% 10 7 80% To 90% ef% 5 ll\% 3 25% Percent 1 83^% 6 3 2 Percent 1 87§% 20 1 10% 25 1 30% 50 1 70% 100 4lf% 58-% 9l|% 5% 4% 2% 1% and tables in the lesson. Practice remembering percents for fractions and practice remembering fractions for percents. Timed written tests give valuable practice for helping you to remember these commonly encountered percents. Study the 1. How lists average speed 2. hours 30 minutes 450 miles per hour? far will the jet travel in 2 is Use a unit multiplier to convert 3 yd to feet. if its 496 Math 87 3. Use a ratio box to solve this problem. If 240 of the 420 students in the auditorium were girls, what ratio of boys to girls in the auditorium? was the 4. Geoff and his two brothers are very tall. Geoff's height is 18' 3". The heights of his two brothers are 17' 10" and 17' 11". What is the average height of Geoff and his brother giraffes? 5. The Martins' car traveled 468 miles Their car averaged how many on 18 gallons of gas. miles per gallon? 6. Graph the whole numbers 7. Use a ratio 8. Draw a diagram of this statement. that are less than or equal to problem. The road was steep. Every 100 yards the elevation increased 36 feet. How many feet did the elevation increase in 1500 yards? box to solve this Then answer questions that follow. Fifty-six antelope This was were seen in the clearing. 7 - of the herd. o (a) How many (b) What percent antelope were in the herd? of the herd was not seen in the clearing? 9. 10. If x = 4 and y = 3x - 1, Find the circumference of y equals what number? this circle. 11. 5. Find the area of this circle. Use 22 -j for n the Memorizing 12. Common -ZL The shape shown was built of 1-inch cubes. What is the volume ^zzz. ^^ of the shape? 13. Complete the / vy table. Decimal Fraction (b) (a) 14. 15. Percent Yl\% Write an equation to solve this problem. is 25 percent of 4? Use a ratio The box to solve Problems 15 and sale price of $24 What was 497 Fraction-Percent Equivalents was 80 percent What number 16. of the regular price. the regular price? 16. David had finished 60 percent of the race, but he had 2000 meters to run. How long was his race? 17. Seventy 18. Find the area of this figure. Dimensions are in centimeters. is what percent of 80? 6 8 / 10 19. In the figure, angle and ZAOE is a straight mZAOB = mZBOC = mZCOD. (a) Find mZAOB. (b) Find mZAOC. (c) Find mZEOC. ) still . 498 Math 87 66f 100 20. Simplify: 21 Find the missing numbers in this 24 y = x X y function. 22. Multiply and write the product 3 in scientific notation. 4 (4 -v Solve each equation. 23. d - 8.47 = Add, 25. 12 x 10~ 5 )(2.1 x 10" 7 ) Show each 9.1 step. 24. 0.25m = 3.6 subtract, multiply, or divide, as indicated: 3 5.2-1 + 4-3 27. 6- - 28. 3.7 + 26. 1 2 kg - 75 g 0.02 x 12 (fraction) + 2- + 15 (decimal) 8 29. + (-3)(+4) (+3) + (-4) (-3) LESSON 109 Multiple unit multipliers 30. (-5) - (-2)[(-3) - (+4)] Multiple Unit Multipliers • Conversion of Units of Area We may multiply a number by 1 repeatedly without changing the number. 5-1 5-1-1 5-1-1-1 = 5 = 5 = 5 Multiple Unit Multipliers 499 Conversion of Units of Area • Since unit multipliers are forms of 1 , we may also multiply a measure by several unit multipliers without changing the measure. ._ 12 3irr j, 10 __ in. ^-T^'Tir same distance = n 360m . - We unit multiplier to change from yards to feet used one and a second unit multiplier to change from feet to inches. Of course, we Ten yards is the as 360 inches. did not need to use two unit multipliers. Instead we could have changed from yards to inches using the unit multiplier 36 in. lyd However, sometimes it prevents mistakes if we use more than one unit multiplier to perform this conversion. Example 1 Solution Use two unit multipliers We are changing units to convert 5 from hours — hours *- hours to seconds. to seconds. seconds We will perform the conversion in two steps. We will change from hours to minutes with one unit multiplier and from minutes to seconds with a second unit multiplier. hours n 5hr , Conversion Of units of • —* —*• minutes 60j»«t — -r— —— 60 sec • - To convert from one unit of area use two unit multipliers. seconds ..„„«« = 18,000 sec to another, it is helpful to Consider this rectangle, which has an area of 6 3ft 2 3 ft • 2 ft = 6 ft 2 ft ft 2 . 500 Math 87 Recall that the expression 6 6 ft 2 to in. 2 , we ~ 6 r.9 2 ft 2 means 6 ft Thus, to convert ft. • convert from ft To cancel ft • to ft feet twice, we — if • • in. use two unit multipliers. —y 12 r„ c^ r, 6 if in. • 12 in. • in. ~YW __. = 864 -*• Example 2 Solution Convert 5 yd 2 to square Since yd 2 means yd cancel yards twice. • ^ / we yd, -, o 1.2 m 2 100 cm _^ — jh^ • Use two unit multipliers use two unit multipliers to 45 ft 2 100 cm — = • ^ n nnn 12,000 cm 2 lift to perform each conversion. 1— hr 5 yd to inches J b. c. 15 yd 2 to square feet d. 1. Jackson earns $6 per hour at a part-time job. does he earn working 3 hours 15 minutes? 2. Mikki was not happy with her test average after 6 tests. On the next 4 tests, Mikki 's average score was 93, which raised her average score for all 10 tests to 84. What was Mikki 's average score on the first 6 tests? 3. Use two unit multipliers a. Problem set 2 to square centimeters. ljfT Practice in. - 5 Convert 1.2 *• 864 in * - feet. -* Solution . - ^.^.^.^=45ft.ft 5yd 2 Example 3 m 109 to seconds 2 20 cm2 to square millimeters to convert 6 ft 2 to How much square inches. Multiple Unit Multipliers 4. • 501 Conversion of Units of Area problem. The ratio of woodwinds to brass instruments in the orchestra was 3 Use box a ratio to solve this were 15 woodwinds, instruments were there? to 2. If there how many brass 5. Graph the counting numbers 6. Use a ratio box to solve this problem. Artichokes were on sale 8 for $2. At that price, how much would 3 dozen that are less than 4. artichokes cost? 7. Draw a diagram of this statement. Then answer the questions that follow. When Sandra walked through the house, she saw that 18 lights were on and only - of the lights were off. (a) How many (b) What percent — 8. Evaluate: a 9. Compare: x 10. 11. lights [b Q — y were 3- off? <X of the lights were on? [a if — b)] if a = 5 and b = 3 x and y are negative and — = 2 A horse was tied to a stake by a rope that was 30 feet long so that the horse could move about in a circle. (a) What is the circumference of the circle? (b) What is the area of the circle? What percent of this circle is shaded? 12. Sketch a cube with edges 3 long. What is the volume cm of the cube? 13. Write an equation to solve this problem. is 75 percent of 100? What number 502 Math 87 14. Complete the table. Decimal Fraction 0.125 (a) 15. Simplify: Use a 16. ratio The (b) — box to solve Problems 16 and 17. was $24. The sale price was $18. The was what percent of the regular price? regular price sale price 17. Percent The auditorium seated was enough 375, but this only 30 percent of those who wanted a wanted a seat but could not get one? seat. for How many 18. Write an equation to solve this problem. Twenty-four 25 percent of what number? 19. (a) Classify this quadrilateral. (b) Find the perimeter of the 10 50 mm mm 30 quadrilateral. (c) Find the area of the quad- 50 mm rilateral. 20. y - x - Find the missing numbers in this function. 21. X 10 Multiply and write this product 7 in scientific notation. (9 Solve each equation. 22. 8- = d - 56 5 x 10- 6 )(4 x 10" 8 ) 2 Show each step. 23. -m 6 = 90 y 5 mm is Multiple Unit Multipliers • Conversion of Units of Area 24. Find the measure of these angles, ZGOF (a) Add, 6{5 26. 1 4 - 3 [6 - • yd 1 ft 3^ in. 2 6^ in. ft 30. - 1)]) 9.5 2 - ( 5 5 - (~3)(-4) 1 (fraction) 9J I 29. (3 7- + 12- * 10 2 (decimal) 2 28. ZAOE subtract, multiply, or divide, as indicated: 25. 27. (b) - (-3) (-3) - (+4)(+3) (+5) + (-2)[(+3) - (-4)] 503 504 Math 87 Sum of the Angle Measures LESSON 110 of a Triangle Project • Straight Angles Materials needed: Paper (preferably construction paper) Straightedge Scissors Using a pencil and straightedge, draw a triangle on a sheet of paper. Then cut the triangle from the paper. Next label the angles 1,2, and 3 as shown below. After labeling the angles, cut each angle from the triangle as shown by the dashed lines in the illustration. Then arrange the pieces to meet at one point as illustrated. You will find that the angles form a semicircle. Compare your results with the results of others in the class. The triangles all have different sizes and shapes. We note that in every example the angles form half of a circle. Sum Of the angle measures Of a triangle The project in this lesson illustrates that the sum of the measures of the angles of a triangle equals 180° (half of a we we know the measures of two angles of a triangle, can calculate the measure of the third angle. circle). If ! Sum Example 1 What is of the Angle Measures of a Triangle • Straight Angles 505 the measure of ZB in this triangle? Solution The sum of the angle measures We 180° for any triangle. the measure of ZA is is see that The 41°. ZC shows that the measure of ZC is 90°. Thus the sum of the measures of ZA and ZC is 131°. The measure of ZB plus the measures of the other two angles must total 180°. We find the measure of ZB square ' at by subtracting 131° from mZA + 180°. mZC = 131* 180° -131° 49° The measure Straight angles of ZB is 49°. The angle below forms a straight straight angle and its measure is line. This angle is called a 180°. ,180°^ » — B When two or more angles together form a straight angle, then the Example 2 sum What is of their measures the measure of is 180°. ZAOB? o Solution ZAOB and ZBOC combine to form a straight angle, their measures must total 180°. We can find the measure of ZAOB by subtracting 60° from 180°. Since Measure of ZAOB = 180° - 60° = 120° The measure of ZAOB is 120°. 506 Math 87 Example 3 Solution What is the measure of Zx? We see that three angles combine to form a straight angle. We are given the measures of two of the angles The sum of the measures of these angles measure of (50° and 90°). 140°, so the is Zx is 180° - 140° = 40° Example 4 Solution Find the measure of ZA. Finding the measure of ZA requires several steps. We begin with what we know and gradually work our way to angle A. First at Practice we copy the figure, C with numbers to make naming three of the four angles it easier to discuss them. (a) Angle 1 + 90° + 40° = 180°, so angle (b) Angle 1 + angle 2 = 180°, so angle 2 = 130°. (c) Angle 2 + angle 3 (d) Angle 3 = 180°, so angle + 70° + angle Use this figure to find the measure of each angle. a. ZX b. c. ZK d. ZY ZD A 3 1 = = 50°. 50°. = 180°, so angle A = 60°. Sum Problem set of the Angle Measures of a Triangle box • 507 Straight Angles 1. problem. Jason's remote control car traveled 440 feet in 10 seconds. At that rate, how long would it take the car to travel a mile? 2. In the forest there 110 Use a ratio of lions to tigers to solve this were was lions, tigers, 3 to 2. The and bears. The ratio ratio of tigers to bears were 18 lions, how many bears were there? Use a ratio box to find how many tigers there were. Then use another ratio box to find the number of was 3 to 4. If there bears. 3. measured the shoe box and found that it was 30 cm long, 15 cm wide, and 12 cm high. What was the volume Bill of the shoe box? 4. A baseball found by dividing the number of hits by the number of at-bats and writing the result as a decimal number rounded to the nearest player's batting average is thousandth. If Erika had 24 hits in 61 at-bats, what was her batting average? \ 5. Use two unit multipliers to convert 18 square feet to square yards. 6. Graph the 7. Draw integers greater than -4. a diagram of this statement. Then answer the questions that follow. Jimmy bought the shirt for $12. This was - of the regular price. (a) What was (b) Jimmy bought the regular price of the shirt? the shirt for regular price? 8. Use this figure to find the measure of each angle. (a) Aa (b) Ah (c) Ac (d) Ad what percent of the 508 Math 87 9. What is the circumference of this circle? 10. Simplify: 91| 22 100 Use -y ab + a a + b 11. Evaluate: 12. Compare: a 2 13. Complete the if Qa for k a = 10 and b = 5 if o = 0.5 table. Fraction Decimal Percent 7 (b) (a) 8 14. Write an equation to solve this problem. is 100 percent of 50? Use a ratio 15. Forty-five percent of the box to solve a hamburger. How Problems 15 and What number 16. 3000 fast-food customers ordered many of the customers ordered a hamburgert 16. The The $24 was 75 percent of the regular sale price of sale price was how many price. dollars less than the regular price? 17. Write an equation to solve this problem. Twenty percent of 200? 18. Find the area of this trapezoid. 30 26 what 26 mm mm mm 10 is mm Sum 19. of the Angle Measures of a Triangle A full circle contains 360°. • 509 Straight Angles How many degrees is — of a full circle? 20. y=3 x Find the missing numbers in this function. X y 12 21. Multiply and write the product in scientific notation. 9 6 (1.25 x 10- 3 )(8 22. -p 9^ Add, 25. step. 24. 12.3 = 4.56 + / subtract, multiply, or divide, as indicated: 9-8-7-6 27. 3.6 x 28. 13- 3 26. 1 5 — x 20 10 2 (decimal)} f 30. Show each = 72 6 29. ) The lengths of two sides of an isosceles triangle are 4 cm and 10 cm. Sketch the triangle and find its perimeter. Solve each equation. 23. x 10~ 5 ,\ 4.75 + (fraction) V (+3) + (-4)(-6) - (-3) + (-4) (-5) - (+6)(-2) + (-2)(-3)(-l) (-6) hr - 15 min 45 sec . 510 . Math 87 Equations with Mixed Numbers LESSON 111 We have been solving equations *x one like this = 7 5 by multiplying both sides of the equation by the reciprocal of the coefficient of x. Here the coefficient of x is -, so we 5 multiply both sides by the reciprocal of - which is , 5 i i r i " : multiplied by 7 4 X 35 4 X = 8 - simplified 3 mixed number 4 When solving an equation that has a convert the mixed first step. Then we number mixed number, we an improper fraction as the multiply both sides by the reciprocal of to the improper fraction. Example 1 Solution Solve: First 3-x = we write 5 3- as an improper —x = fraction. 5 3 Then we multiply both the reciprocal of sides of the equation — V6' t X ~ 10 " by — which is 10 , b 3 x = — 2 we usually convert an improper fraction such as - to a mixed number. In algebra, we usually leave improper fractions in fraction form, which we will do beginning with In arithmetic, this lesson. 1 " Equations with Mixed Numbers Example 2 Solution = 2-y 2 Solve: \ 51 -7 8 Since we will be multiplying on both sides to find y convert both mixed numbers to improper fractions. we first 15 y J 2 Then we multiply both sides 8 by 2 5 2 y y = Practice which -, 2 15 5 8 the reciprocal of -. 3 4 Solve: a. 1-x = 36 b. 8 c. 2-w 1. = 6- d. The sum 2. = 12-v 3' 5 5 and 0.9 is how much greater than the and 0.9? Use words to write the answer. of 0.8 product of 0.8 111 3-a = 490 2 4 Problem set is For this set of scores find the mode, and (a) mean, median, (c) (d) range. 8, 6, 9, 10, 8, 7, 9, 10, 8, 10, 9, 3. (b) 8 The 24-ounce container was priced at $1.20. This container costs how much more per ounce than the 32-ounce container priced at $1.44? 4. Twenty-two of the ninety 2-digit numbers are prime numbers. What is the ratio of the number of prime numbers to the number of composite 2-digit numbers? Use a 5. ratio box to solve Twentv-seven is Problems 5-7. to 36 as 36 is to what number? 512 Math 87 6. 7. The sale price What was the of $36 was 90 percent Seventy-five percent of the citizens voted for Graham. for how many of them If did not vote Graham? what percent of 30? 8. Twenty-four 9. Use two unit multipliers Draw is to convert a diagram of this statement. 100 yd to inches. Then answer the questions that follow. Three hundred doctors recommended Brand X. This was - of the doctors surveyed. (a) How many doctors were surveyed? (b) How many doctors surveyed did not recommend Brand X? x = and y = 2x + 11. If 12. Compare: a 13. If 4.5 Q ab if 1, equals what number? and b > a < the perimeter of a square y is 1 foot, 1 what is the area of the square in square inches? 14. Complete the table. Fraction Decimal 1.75 (a) 15. Percent (b) Write an equation to solve this problem. What is 6 percent of $325? 16. I regular price? there were 800 citizens, 10. of the regular price. Multiply and write the product in scientific notation. (6 x 10 4 )(8 x 10- 7 ) , 3 51 Equations with Mixed Numbers 17 . What is the volume of a cereal box with dimensions as shown? Dimensions are in inches. 12 18. Find the circumference of this 19. Find the area of circle. this circle. Use 3.14 20. 21. (a) Find mZAOB. (b) Find mZEOD. (c) Find mZCOA. What is - of 20? 3 22. Sixteen 23. Graph x < is 40 percent of what number? 4. Solve: 24. 2-w 3 = 24 25. x + 3.5 = 4.28 for n 51 4 Math 87 Add, subtract, multiply, or divide, as indicated: 26. 10 4 - (10 3 - 10 2 ) 28. 1 3- - 2- - ( 5 V 1 ton - 100 pounds \ (decimal) 1.2 2 J 29. (-3) + (-4K-5) 30. [(-3) - 27. - (+2)][(+2) (-6) - (-3)] Evaluations with Signed Numbers • Signed Numbers without Parentheses LESSON 112 Evaluation with signed We have practiced evaluating expressions such as x — xy — y numbers by using positive numbers in place of x and y. In this lesson we will practice evaluating such expressions by using negative numbers as well. When evaluating expressions that contain signed numbers, it is helpful to replace each letter with parentheses as the first step. Doing this will help prevent making mistakes in signs. Example 1 Solution Evaluate: We x - xy - y if x = -2 and y = -3 write parentheses for each variable. ()-()()-() Now we write the proper (-2) We multiply parentheses number within - (-2)(-3) - the parentheses. numbers (-3) insert (-3) multiplied first. (-2) - (+6) - 5 Signed Numbers without Parentheses Then we add algebraically from left to right. (-8) Signed numbers 51 - (-3) added -2 and -6 -5 added -8 and +3 Signed numbers are often written without parentheses. To simplify an expression such as Without parentheses _3 + 4 _ 5 _ 2 we may mentally insert parentheses and then add algebraically. -3 -2 -5 +4 (we (-3) + (+4) + (-5) + (-2) = Example 2 Solution Simplify: We may +3-4 + -6 see) (we think) 2-1 think of this expression as (pos 3) plus (neg 4) plus (pos 2) plus (neg 1) + (+3) Then we add algebraically (+3) Example 3 Solution Simplify: We may -2 + + (-4) from + (+2) (-1) left to right. + (-4) + (+2) + (-1) = 3(-2) - 2(+4) think of this expression as (neg 2) plus (pos 3) times (neg 2) plus (neg 2) times (pos 4) (-2) We + multiply (+3) (-2) + (-2)(+4) first. (-2) + (-6) + (-8) Then we add algebraically from left to right. (-2) + (-6) + (-8) = -16 51 6 Math 87 Practice Evaluate each expression. Write parentheses as the a. x + xyy - x = if \i first step. and y = -2 3 / b. -JB + it --ffrih if m = — 2 and n = -5 Simplify: c. -3 + 4 - - 5 2 d. -2 + 3(-4) - 5(-2) i e. Problem set 1. -3(-2) - For his next 4 5(2) first + 3(-4) was 6 tests, Ted's average score average score was 94. average score for his first 10 tests? 112 2. 3. Use 4. tests, his 86. For his What was his Ten billion is how much more than nine hundred eighty million? Use words to write the answer. The Martins completed the 130-mile trip in 2| hours. What was their average speed in miles per hour? a ratio box to solve Problems The to 5. 4-7. ratio of laborers to supervisors at the job site Of the 120 individuals at the job site, was 3 how many were laborers? 5. Vera bought 3 notebooks for $8.55. At this much would 5 notebooks cost? 6. The rate, was 90 percent of the regular price. was $36, what was the sale price? sale price regular price how If the 7. Forty people came to the party. This was 80 percent of those who were invited. How many were invited? 8. Write an equation to solve this problem. Twenty percent of what number? 9. Use two unit multipliers feet. to convert 3600 in. 2 is 40 to square 7 51 Signed Numbers without Parentheses 10. Draw Then answer a diagram of this statement. the questions that follow. Three fourths of the questions on the test were multiple-choice. There were 60 multiple-choice questions. (a) How many questions were on the test? (b) What percent of the questions on the test were not multiple-choice? 11. Evaluate: x - y - xy 12. Compare: m Qn if if x = -3 and y = — 2 m is an integer and n is a whole number 13. (a) Classify this quadrilateral. (b) Find the perimeter of this 15 13 mm mm n 12 y mm figure. 20 Find the area of this (c) 14. Complete the table. Fraction What number is Decimal (a) if 15. mm figure. Percent (b) 33^ percent of 30? [Hint: Change 33^ percent to a fraction.) 16. Multiply and write the product in scientific notation. (2.4 17. A x 10- 4 )(5 x 10- 7 ) pyramid with a square base has how many (a) faces? (b) edges? (c) vertices? . 518 Math 87 18. Find the circumference of (a) this circle. Find the area of this (b) circle. Use 3.14 19. for n y = 2x - Find the missing numbers in this function. X 5 y 2 3 5 20. Use the information in the to answer these questions. (a) What is (b) What is (c) What is figure mZX? mZY? mZA? what 21. Write an equation to solve this problem. Twenty percent of 30? 22. Graph x > -3. 23 Segment AB is how many millimeters longer than segment is BC? A cm C B •- 2 1 3 4 5 6 7 9 8 Solve: 24. 5=y- 4.75 25. 1 1 = 73-y 3^ 2 Add, 26. subtract, multiply, or divide, as indicated: V3 2 + 24 32 1 ft sec 60 sec 1 min 10 11 = Sales Tax 28. 5- + 2.5 + - (fraction) 6 3 29. 2§ + 3.5 — —2- 30. (a) ( (decimal) 3) r} ^ +5] (b) -3 + 4-5 + 6-7 Sales Tax LESSON 113 To find the amount of sales tax on a purchase, the full price of the purchase by the tax rate. Example 519 1 Solution A bicycle is on (a) What is the tax on the bicycle? (b) What is the total price including tax? (a) sale for $119.95. The tax rate is we multiply 6 percent. To find the tax, we change 6 percent to the decimal 0.06 and multiply $119.95 by 0.06. We round the result to the nearest cent. 119.95 x ,06 7.1970 (b) To find the total price, — $7.20 including tax, we add the tax to the price. 119.95 + 7.20 $127.15 Example 2 price tax total Find the total price, including tax, of an $18.95 book, a $1.89 pen, and a $2.29 pad of paper. The tax rate is 5 percent. 520 Math 87 Solution We begin by finding the combined price of the items. 1.89 book pen 2.29 paper $18.95 + $23.13 Next we multiply the combined price by 0.05 round the product to the nearest cent. (5 percent) and $23.13 x 0.05 $1.1565 Then we add — $1.16 tax the tax to the price to find the total. $23.13 + 1.16 $24.29 Practice a. total Find the sales tax on a $36.89 radio if the tax rate is 7 percent. b. Find the total price of the radio, c. Find the total price, dinner, a 95(2 Problem set 1. 113 are Darren's 100-meter dash times, in seconds, during track season. (c) tax. including 6 percent tax, for a S6.95 beverage, and a $2.45 dessert. The following marks mode, and including Find the (a) median, (b) range of these times. 12.3, 11.8, 11.9, 11.7, 12.0, 11.9, 12.1, 11.6, 11.8 2. How much money hours 45 minutes Use a 3. ratio box The recipe at to solve does Jackson earn working $6 per hour? Problems 3-6. and 2 eggs to make were used to make more called for 3 cups of flour 15 cups of flour servings, how many eggs should be used? 6 servings. If for 3 521 Sales Tax how 4. words per minute. At many words can he type in 90 seconds? 5. Ten students scored 100 percent. This was 40 percent of the class. How many students were in the class? 6. The Lester can type 48 If dress was on sale for the regular price that rate, 60 percent of the regular price. was $24, what was the 7. Write an equation to solve this problem. of 60 is 24? 8. Use two unit multipliers 9. Draw sale price? What percent to convert 3 gal to pints. a diagram of this statement. Then answer the questions that follow. The Trotters won - of their games. They won 20 games and lost the rest. (a) How many (b) What was games did they play? the Trotters' ratio of games won to games lost? x = -2 and y = 2x + 10. If 11. Compare: 12. The w Qm figure at right 1, w is of w if is a y equals what number? 0.5 and m is the reciprocal hexagon with dimensions given in centimeters. Corners that look square are square. Find the area of the hexagon. 13. A television was on sale for $325. The tax rate percent. (a) What was the tax (b) What was the total price, including tax? on the television? was 6 — 522 Math 87 14. Multiply and write the product in scientific notation. (8 15. Complete the x 10 8 )(4 x 10- 2 ) table. Fraction Decimal 0.02 (a) (b) 16. What 17. Sketch this cube. Then find its volume. Dimensions are in is Percent 65 percent of $24? ^ y 4 A inches. 4 4 -^- 18. Find the circumference of this 9cm\ circle. . 19. Find the area of this circle. Leave n as n 20. Find the missing numbers in this function. y = 3x + X 5 21. Divide 1.23 by 9 and write the 3 quotient (a) with a bar over the repetend. (b) rounded to three decimal places. 22. The ratio B to to 3. of the measure of angle the measure of angle C is 2 What is the measure of angle y 1 Percents Greater than 100, Part 23. Graph the counting numbers 523 1 greater than -2. This will be the graph of the positive integers. 24. Draw a pair of parallel lines. Draw another pair of parallel lines that are perpendicular to the first pair. What kind of quadrilateral is formed? Solve: 25. 3-d = 88 26. n + 1.61 = 10.6 7 Add, 27. subtract, multiply, or divide, as indicated: 52 + 3 (3 - V8l) 28. l — 29. '4* (2.7) 9y V LESSON 114 30. 3 v min 30 48 min 45 hr 15 (-2)(-3) sec - (-4H-5) ; Percents Greater than 100, Part 1 The methods we have used to work with percents less than 100 may also be used to work with percents greater than 100. In this lesson we will practice writing equations to solve problems that include percents greater than 100. Example sec 1 Solution Fifty is We what percent of 40? translate directly to an equation. Ml Fifty is 50 = what percent Wp 100 of 40? question M x 40 equation . 524 Math 87 We 100 -ttt cn = 50 • 40 Wp^ __.Mf — VMS —rz&Q - • JtOCT 125 = Since 125 Solution We . .. ., 100 40 simplified we can write Wp solution This answer makes sense because 50 is more than 100 percent of 40. What . multiplied by F y Wp a percent, is 125% = Example 2 —— solve by multiplying both sides by is greater than 40, so 50 106 percent of $12.40? is We remember that translate directly. 106 percent means 106 over 100. What i W = Solution Sixty We question 1 I x 12.40 equation H 13.144 $13.14 solution — 150 percent of what number? is translate directly. Sixty 60 We of $12.40? 100 W= Example 3 106% M is is 150% of 150 = -^j^ 100 what number? WN x solve by multiplying both sides 100 T50 • 60 = 40 = 1ZQ m WN • question W" ±M xm ' equation by 100 multi P hed b y solution ioo ^ Percents Greater than 100, Part Practice a. What is 150 percent of 56? b. Sixty is 250 percent of what number? 1 525 > tf Or) Problem set 1 . 114 The mean of these numbers median? 3, 12, ,7, is how much greater than the 5, 18, Q, 9, 28 I 2. What is the quotient by the product 3. of ^ when the sum and of - and - is ( divided --? The lengths of two sides of an isosceles triangle are 5 in. and 1 ft. Sketch the triangle and find its perimeter in inches. Use a 4. ratio The 7. If box to solve Problems 4-7. youths to adults at the convocation was 3 to 4500 attended the convocation, how many adults ratio of were present? 5. knight went over 6. 7. he went up did he go up? Every time the knight went over 8, how far 2, 1. If the Eighty percent of those who were invited came to the party. If 40 people were invited to the party, how many did not come? The If dress the sale was on sale for 60 percent of the regular price. price was $24, what was the regular price? One hundred 8. Write an equation to solve this problem. forty percent of what number is 70? 9. Use two unit multipliers to convert 1 ,000,000 cm2 to square meters. 526 Math 87 10. Draw a diagram of this statement. Then answer the questions that follow. showed that 7 out of 10 voters cast their ballot for the incumbent. The incumbent Exit polls received 1400 votes. (a) How many (b) What percent voters cast their ballots? of the voters did not vote for the incumbent? x + xy - xy 11. Evaluate: 12. Compare: a 13. If Qa- a if x = 3 and y = — 2 ifa<0 the perimeter of a square is 1 meter, what is the area of the square in square centimeters? c 14. Find the 15. Multiply and write the product in scientific notation. including tax, of a $12.95 bat, a $7.85 baseball, and a $49.50 glove. The tax rate is 7 percent. total price, (3.5 16. Complete the ) table. Fraction (a) x 10 5 )(3 x 10 6 Decimal (b) Percent 33^% Write an equation to solve Problems 17 and 18. 17. What is 125 percent of 84? 18. What is 106 percent of $180? 19. What is volume of this rectangular prism? Dimensions are in feet. the Percents Greater than 100, Part 20. 527 1 Find the circumference of (a) this circle. Find the area of this (b) circle. 22 Use -y 21. for n y = 3x Find the missing numbers in this function. X y 2 -1 15 22. Polygon ZWXy What the measure of each is is a rectangle. angle? Aa (a) Zb (b) Zc (c) 23. Graph x > -2. 24. What numbers correspond to on this number line? the points marked A and B A -- 0.1 -\ 1 1 H Solve: 25. p + Add, 7 = 50.2 26. = 51-/ y 4 4 subtract, multiply, or divide, as indicated: 27. V100 - 64 - 28. 8 lb 7 oz | + 2 lb 9 oz (0.2)(10 2 ) 29. v 2 y 3o. HIMi - (-i) B -- 528 Math 87 LESSON 115 Percents Greater than 100, Part 2 we have considered until now have used a percent to describe part of a whole. In this lesson we will consider percent problems that use a percent to describe an amount of change. The change may be an The percent problems that increase or a decrease. Original Increase: number + Original Decrease: number amount change amount new of number new of change number We may use a ratio box to help us with increase-decrease problems just as we have with ratio problems and part-partwhole problems. However, there are some differences in the way we set up the ratio box. When we make part-part-whole problem, the bottom column is number a table for a in the percent 100 percent. Percent Actual Count Part Part Whole 100 When we set up a ratio box for an increase-decrease problem, we also have three rows. The three rows represent amount of change, and the new use the words original-change-new on the the original number, the number. We will Most increase-decrease problems consider the original amount to be 100 percent. So the top number in the percent left side. column will be 100 percent. Percent Actual Count Original Change New 100 529 Percents Greater than 100, Part 2 an increase, we add it to the original amount to get the new amount. If the change is a decrease, we subtract it from the original amount to get the new amount. If Example 1 the change is The county's population increased 15 percent from 1980 to 1990. If the population in 1980 was 120,000, what was the population in 1990? Solution we The percent describes an amount of change. This is an increase problem. We make a table and write the words "original," "change," and "new" down the side. Since the change was an increase, we write a plus sign in front of change. In the percent column we write First identify the type of problem. 100 percent for the original (1980 population), 15 percent for the change, and add to get 115 percent for the new (1990 population). Percent Actual Count Original 100 15 C 115 N + Change New 120,000 100 120,000 115 N column we write 120,000 for "original," and use letters for "change" and "new." We are asked for the new number. Since we know both numbers in the first row, we use the first and third rows to write the proportion. In the actual count 100 115 WON N The county's population Example 2 Solution The price First we 120,000 N = 13,800,000 = 138,000 in 1990 was 138,000. was reduced 30 percent. $24.50, what was the original price? We make identify the problem. This If is the sale price was a decrease problem. and write original, change, and new down the side with a minus sign in front of change. In the percent a table . 530 Math 87 column we write 100 percent for original, 30 percent for change, and 70 percent for new. The sale price is the new actual count. We are asked to find the original price. Percent Actual Count R 100 30 Original Change New C 24.50 70 100 R 70 24.50 70R = 2450 R = The Practice original price before Use a a. ratio b. to solve was reduced was $35. each problem. was $24.50, but the item was on 30 percent off. What was the sale price? The for box it 35 regular price The number sale of students taking algebra increased 20 percent in one year. If 60 students are taking algebra this year, how many took algebra last year? c. Problem set Bikes were on sale for 20 percent off. Tom bought one for $120. How much money did he save by buying the bike at the sale price instead of at the regular price? 1 The product of the first three prime numbers is how much less than the sum of the next three prime numbers? 2. After 5 tests 115 Amanda's average score was 88. What score must she average on the next 2 tests to have a 7-test average of 90? 3. Jenna finished a 2-mile race in 15 minutes. her average speed in miles per hour? Use a 4. ratio box to solve What was Problems 4-7. Forty-five of the 80 students in the club were girls. was the ratio of boys to girls in the club? What 531 Percents Greater than 100, Part 2 5. Two dozen would 60 sparklers cost $3.60. At that rate, how much sparklers cost? 6. The county's population increased 20 percent from 1980 to 1990. If the county's population in 1980 was 340,000, what was the county's population in 1990? 7. Because of unexpected cold weather, the cost of tomatoes increased 50 percent in one month. If the cost after the increase was 96(2 per pound, what was the cost before the increase? 8. Write an equation to solve this problem. Sixty percent of 75? 9. Use two unit multipliers to convert 100 cm 2 is what to square millimeters. 10. Draw a diagram of this statement. Then answer the questions that follow. Five eighths of the trees in the grove were deciduous. There were 160 deciduous trees in the grove. (a) (b) How many trees were in the How many of the trees in grove? the grove were not deciduous? x = — 5 and y = 3x — 11. If 12. Compare: 13. Find the area of 14. The 30% of 20 O 1, then y equals what number? 20% of 30 10cm this figure. price of the stereo $179.50. The tax rate 6 percent. was was 6.5 cm 6 5 cm cm (a) What was the tax on the stereo? (b) What was the total price, including tax? /6.5 cm 532 Math 87 15. Multiply and write the product in scientific notation. (8 16. Complete the x 10- 5 )(3 x 10 12 ) table. Fraction 2± Decimal (a) Percent (b) 17. Write an equation to solve this problem. percent of $14,500? 18. What number 19. A is What is 6.5 250 percent of 60? triangular prism has how many 20. (a) triangular faces? (b) rectangular faces? John measured the diameter of his bicycle tire and found that it was 24 inches. What is the distance around the tire to 21. the nearest inch? (Use 3.14 for Find the missing numbers in this function. 22. The ratio of the y = 7x + X measures of the two angles was 4 to the sum of their measures 180°, n.) 1 V 6 5. If 3 was -2 what was the measure of the smaller angle? 23. Graph the whole numbers 24. Draw greater than -5. a pair of parallel lines. Draw a second pair of parallel lines that intersect but are not perpendicular to the first pair. What kind of quadrilateral is formed? Solving Two-Step Equations 533 Solve: 25. 3-x = 66 26. w - 0.15 = 4.9 7 Add, subtract, multiply, or divide, as indicated: 27. (2 29. 5 • 3) 2 - 2(3 2 28. ) A - ( I LESSON 116 1 3- -- 30. 1.5 a 1 L - 100 mL (-8X-6X-5) (-4)(-3)(-2) ) Solving Two-Step Equations We have solved one-step equations by adding or subtracting. We have also solved one-step equations by multiplying or we will begin solving two-step equations. equations such as 3x + 2 = 7, we work to dividing. In this lesson When we solve get the equation in this form: = a particular number lx + Then we can write x = a particular number Thus, to solve 3x + 2 = 7, we take one step to change 2 to and a second step to change 3 to 1. The order of the steps is important. We use the addition rule first. Then we use the multiplication rule. To change +2 we add -2 to 0, to (or subtract 2 from) sides of the equation. 3x + 2 = -2 3x + y 7 -2 equation add -2 = 5 added 3x = 5 3x + = 3x both 534 Math 87 Now we change 3xto lxby dividing both sides of the equation by 3 or by multiplying both sides by -. Multiply by Divide by 3 -1 3 3x 5 1 3 3 3 • Example 1 Solution Solve: First 5 lx = - 3 3 5 x = — 5 x = — 3 3 5x - 9 = 36 we change -9 by adding 9 to added 5x + will divide both sides —=— 5 by = 5x 5. divided by 5 5 lx = 9 x = -x both sides. add 9 5x = 45 Now we to equation = 45 5x + Solve: 3 5 lx = - 5x - 9 = 36 +9 +9 Example 2 3x = I simplified lx = x 9 + 6 = 18 4 Solution First we add -6 to or subtract 6 3 —x + 6 = 18 -6 ^x + 4 7X 4 -6 from both sides. equation add -6 = 12 added = 12 -x 44 + = -x . Solving Two-Step Equations Then we multiply both 4 — 3 —x • = 4 — 4 3 sides by 535 — multiplied by 12 3 48 lx multiplied 3 x = 16 Example 3 Solution 0.2x - 1.4 = 3 Solve: First simplified we add +1.4 to both sides. 0.2x - 1.4 = +1.4 0.2x + simplified 3 add +1.4 added = 4.4 0.2x = 4.4 Then we divide by Practice Solve. a. 3x - 2 = c. -x - 5 = simplified 0.2. 0.2x 4.4 0.2 0.2 divided by 0.2 lx = 22 divided x = 22 lx = X Show each 1.4 step. 7 7 b. 2x + d. -x = 15 3 + 7 = 37 6 Problem set 116 e. 0.4x + 1.4 = 3 1. From Don's house the round trip was on f. 1.2x to the lake is 30 - 0.7 = 8.9 km. If he completed his bike in 2 hours 30 minutes, his average speed in kilometers per hour? what 536 Math 87 2. Find (a) the mean and numbers. ratio box the range for this set of 40 10, 4, 5, 8, 5, 4, 8, 3, 9, 7, 5, Use a (b) Problems 3-6. to solve marbles to blue marbles in the bag was 7 to 5. If there was a total of 600 red and blue marbles in the bag, how many marbles were blue? 3. The 4. The machine could punch out 500 ratio of red in 20 minutes. At that rate, plastic pterodactyls how many could it punch out in l| hours? 5. The price was reduced by 25 percent. was $24, what was the sale price? 6. The price was reduced by 25 percent. was $24, what was the regular price? 7. Write an equation to solve this problem. Seventy-five what percent of 60? 8. Use two unit multipliers 9. Draw If to convert 7 a diagram of this statement. questions that follow. the regular price If the sale price is days to minutes. Then answer the Five ninths of the 45 cars pulled by the locomotive were not cattle cars. (a) How many cattle cars were pulled by the locomotive? (b) What percent were not 10. Compare: 3 11. Evaluate: ab 12. Find the of the cars pulled by the locomotive cattle cars? O 33% - a - b if a = -3 and b = -1 including 5 percent tax, for a $7.95 dinner, a 90tf beverage, and a $2.35 dessert. total price, Solving Two-Step Equations 13. 537 A corner was cut from a square sheet of paper, resulting in this pentagon. Dimensions are in inches. (a) What pentagon? this (b) the perimeter of is What the area of this is pentagon? 14. Complete the table. Decimal Fraction 0.08 (a) Percent (b) What number 15. Write an equation to solve this problem. is 120 percent of 360? 16. Multiply and write the product in scientific notation. (8 x 10- 3 )(6 x 10 7 ) 17. Each edge of the picture cube was 10 cm. What was the volume of the cube? 18. What 19. What is the area of this circle? is the circumference of this circle? Use 3.14 20. Find the missing numbers in this function. 21. number? 22. 1 -2 Graph the integers greater than - 2 and that less are than 2. n y = 2x + X Write an equation to solve this problem. Sixty is - of what for y 3 538 Math 87 23. Find the measure of each angle. ° Z.X (a) ^ (b) ZZ (c) 24. D What is the sum number line? » 1 1 numbers labeled B A 0.4 _l of the 1 1 A and B on this 0.5 * 1 1 1 1 1 1 1 h Solve: 25. 3x + 2 = 9 26. -w + 4 = 14 3 27. 0.2y Add, 28. -1 = 7 subtract, multiply, or divide, as indicated: 3|2 3 a/36 29. 5- + 3- + 4 30. 9 [+6] (a) [ \ (-4) - LESSON 117 5 l [b) } 2.5 6 -3(4) + 2(3) - 1 (-1) Simple Probability A The singular of dice is die. die has six faces. If we roll a die, the number of dots on the top face will be 1, 2, 3, 4, 5, or 6. Each face is equally likely to occur, and each roll of the die is called an outcome. Probability is the likelihood that a particular event will occur. Probability Probabilitv We is written as a number of favorable outcomes number of possible outcomes will consider three examples. ratio. Simple Probability Example 1 A single die is rolled. What is the probability that Solution 539 it will come up (a) a 4? (b) a number greater than 4? (c) a number greater than 6? (d) a number less than 7? There are 6 different faces on a die, so there are 6 equally likely outcomes. Thus, 6 will be the bottom number of each of these ratios. There (a) is only one way to roll a 4 probability of rolling a 4 (b) (c) with one die, so the is -. The numbers greater than 4 on a die are 5 and 6, so there are 2 ways to roll a number greater than 4. Thus the probability is -, which we reduce to -. On a die, there are no numbers greater than 6, so there is no way to roll a number greater than 6. Thus the probability is -, which is 0. An event that cannot happen has a probability of zero. (d) On numbers less than 7, so there are 6 ways to roll a number less than 7. Thus the probability is -, which is 1. An event that is certain to happen has a die, there are 6 a probability of Example 2 If 1. this spinner is spun, what is the probability of the spinner Solution (a) stopping on 3? (b) not stopping on 3? There are number (a) 5 equally likely outcomes, so 5 is the bottom of each ratio. There is one way probability is -. for the spinner to stop on 3, so the 540 Math 87 (b) There are 4 ways probability is for the spinner to not stop on 3, so the -. Notice that the probability of an event happening plus the probability of an event not happening is 1. Example 3 What is the probability of this spinner stopping on 1? Solution There are 3 possible outcomes, but the outcomes are not equally likely. Since region 1 occupies half the area, the probability of the spinner stopping in region 1 is -. Regions 2 and 3 each occupy - of the area, so the probability of the spinner stopping Practice a. What is the probability of rolling a with one b. on 2 is \ and on number less 3 is \. than 4 roll of a die? What is the probability of this spinner stopping on 3? c. What is the probability of this spinner stopping on 5? d. What is the probability number less than 6? e. What is the probability of this spinner of this spinner stopping on a stopping on C? f. What is the probability of this spinner not stopping on B? Problem set 1. 117 Twenty-one billion is how much more than nine billion, eight hundred million? Write the answer in scientific notation. 2. The an average speed of 48 miles per hour for the first 2 hours and at 60 miles per hour for the next 4 hours. What was the train's average speed for the 6-hour trip? (Average equals total miles divided by total train traveled at time.) Simple Probability 3. 4. 541 10-pound box of detergent costs $8.40. A 15-pound box costs $10.50. Which costs the most per pound? How much more? A In a rectangular prism, what the ratio of faces to is edges? Use a 5. box ratio to solve Problems 5-8. The team's won-lost ratio was 3 to 2. If the team won 12 games and did not tie any games, then how many games did the team play? 36 as 42 what number? 6. Twenty-four 7. What number 8. During his slump, Matt's batting average dropped by 20 percent to .260. What was Matt's batting average before his slump? 9. Use two unit multipliers 10. (a) 12 (b) 1 Draw ft 2 km is to is to 20 percent less than 360? is to perform each conversion. to square inches to millimeters a diagram of this statement. Then answer the questions that follow. The duke conscripted two serfs in his in (a) (b) fifths of the male dominion. He conscripted 120 serfs all. How many male serfs were in the duke's dominion? How many male serfs in his dominion were not conscripted? 11. If a die number is tossed once, what rolled (a) a prime (b) a is is number? number greater than 6? the probability that the 542 Math 87 y = Ax - 3andx = -2, then y equals what number? 12. If 13. The perimeter of a certain square area of the square in square 14. The price of the new car is 4 yards. Find the feet. was $14,500. The tax rate was 6.5 percent. 15. (a) What was the sales tax on the car? (b) What was the total price including tax? Complete the table. Decimal Fraction Percent 66|% (b) (a) 16. What 17. Multiply and write the product in scientific notation. is 200 percent of $7.50? (2 18. x 10 8 )(8 x 10 2 ) Robbie stores his 1-inch blocks in a box with inside dimensions as shown. many blocks will fit How 2 in. 6 8 in. in. in this box? 19. The length of each side of 14 in. the square equals the diameter of the circle. The area of the square is how much greater than the area of the circle? 20. Use 22 -j for n Divide 7.2 by 0.11 and write the quotient with a bar over the repetend. — 1 543 Simple Probability 21. y = 5x - 12 Find the missing numbers in this function. 22. Graph -1 2. the measure In the figure, of 3 numbers greater than - 2 and are also less than 23. y of the all that are X ZAOC is half the measure ZAOD. The measure of ZAOB •>/ of measure of ZAOD. r -«-• 24. ^* one third the is (a) Find mZAOB. [b) Find mZEOC. The length — •-»- o of segment length of segment /" #C is how much than the less AB? B • * 1 1 | 1 1 | 1 1 1 1 ' 1 inches 1 2 1 1 1 3 Solve: 25. 27. 3x - 2 = 9 1.2p + 4 Add, yd 6 1 29. 4-7 - 2 = 1- 3 9 6-m 1 = 28 in. 2.5 - (b) ft 7 in. 4 (-8) (a) 1 1 - - 8 30. 26. subtract, multiply, or divide, as indicated: 28. ' 1 ' ' 1 - (-6) - (4) -3 -5(-4) - 3(-2) - 1 544 Math 87 Volume of a Right Solid LESSON 118 A right solid is a geometric solid whose sides are perpendicular to the base. The volume of a right solid is the area of the base times the height. This rectangular solid is a right solid. It 2 deep, so the area of the base is 10 5 long and 2 m m m ////// is . / / / / / // One cube will fit on each square meter of the base and the cubes are stacked 3 high, so Volume = The volume of any the height. If area of base x height m 30 m = 10 2 = 3 x 3 m right solid is the area of the base times the base of a right solid called a right circular cylinder. polygon, the solid If is a circle, the solid the base of the solid is is a called a prism. is ^_7 Example 1 Right square Right triangular Right circular prism prism cylinder Find the volume of are in centimeters. this right triangular prism. We show two Dimensions views of the prism. 3^ 4 Volume Solution The area of the base is the area of the triangle. Area of base = (4 The volume equals the area Volume = Example 2 The diameter (6 cm)(3 cm) cm 2 )(6 cm) we What find the area of the base. diameter of the circular base the radius is 10 cm. Area = kt 2 = The volume equals the Volume = Practice cm 2 = 36 cm 3 of this right circular cylinder 20 cm. Its height is 25 cm. volume? Leave n as n. First = 6 of the base times the height. is Solution 545 of a Right Solid is 7r(10 is its The 20 cm, so cm) 2 = 100/r cm 2 area of the base times (100/r cm 2 )(25 cm) its height. = 2500/r cm 3 Find the volume of each right solid shown. Dimensions are in centimeters. c. 10 Leave n as n d. 7 Base Problem set 118 1. The Prism each tenth of a mile. What was the average cost per mile for a 4-mile taxi taxi ride cost $1.40 plus 35<2 for ride? 546 Math 87 2. The product of the first numbers four prime how is much greater than the sum of the next four prime numbers? 3. What is the average of these fractions? 1 I JL 4' 6' 12 Jackson is paid $6 per hour, 4 hours 20 minutes? 4. If 5. What is how much will he earn in the ratio of the shaded area to the unshaded area of this rectangle? Use a ratio box to solve Problems 6-8. 6. 600 pounds of sand cost $7.20, what would be the cost of 1 ton of sand at the same price per If pound? 7. What 8. The is 30 percent more than $3.90? cost of production rose 30 percent. $3.90 per unit, what 9. was the old Use two unit multipliers If the new cost is cost per unit? to convert 1000 mm 2 to square centimeters. 10. Draw a diagram of this statement. questions that follow. Three fifths Then answer the of the Lilliputians believed in giants. The other 60 Lilliputians did not believe in giants. 11. (a) How many Lilliputians were there? (b) How many Compare: a Lilliputians believed in giants? Qb a a counting integer if is number and b is an Volume 12. Evaluate: 13. If a die is number 14. m{m + n) if m = -2 and n = tossed once, what rolled —3 the probability that the is (a) an even number? (b) a number is 547 of a Right Solid than 7? less Find the volume of this triangular prism. Dimensions are in millimeters. 15. was 6 cm. Its height was 10 cm. What was the volume of the soup The diameter of a soup can can? 16. Find the including 6 percent tax, of 3 tacos at $1.25 each, 2 soft drinks at 95<£ each, and a shake at total cost, $1.30. 17. Complete the table. Fraction 2- Decimal (a) Percent (b) 18. What 19. Multiply and write the product in scientific notation. is 6.5 percent of (8 20. $36? x 10- 6 )(4 x 10 4 ) Find the missing numbers in this function. y = ,x + X 6 21. Divide 1000 by 48 and write the quotient as a mixed number. 4 -2 y 1 548 Math 87 22. Find the measures of the following angles. D ZX (a) 23. Graph 24. What is the average this number line? all the negative of the that are greater than -2. numbers labeled B A 1.5 h H numbers H h A and B on 1.6 -- h H Solve: 25. 5w + 11 = 51 26. -x - 2 = 14 3 27. 0.9m + Add, 28. 1.2 = 3 subtract, multiply, or divide, as indicated: Vl 3 + 2 3 + (1 + 2) 3 29. 5-2- ( ^\ v'4y 30. (a) (-10) + (-8) - (-6) (-2)(+3) (b) -8 + LESSON 119 3(-2) - 6 Rectangular Coordinates By drawing two perpendicular number the marks, Then we we can create a grid or graph lines and extending over an entire plane. can identify any point on the coordinate plane with two numbers. — Rectangular Coordinates 549 y -3,2 (3 L 2) • — -»6 -5 -4 -3 -2 (-3, -1 1 2 3 4 5 x 6 (3,-2) -2) The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The point at which the x-axis and the y-axis intersect is called the origin. The two numbers that indicate the location of a point are the coordinates of the numbers in parentheses, such as (3, 2). The first number shows the horizontal (<h>) direction and distance from the origin. The second number shows the vertical (J) direction and distance from the origin. The sign of the number indicates the direction. point. The coordinates are written as a pair of Positive coordinates are to the right or up. Negative coordinates are to the left or down. The two axes divide the plane into four regions called quadrants, which are numbered counterclockwise beginning with the upper right as first, second, third, and fourth. The signs of the coordinates of each quadrant are shown below. y ii Second Quadrant I First Quadrant (++) III Third Quadrant IV Fourth Quadrant (--) To graph a point on a coordinate plane, we draw a dot point indicated by the coordinates. at the — 550 Math 87 Example 1 Graph the following points on (a) Solution (3,4) (2,-3) (b) a coordinate plane. (-1,2) (c) (d) (0,-4) do these exercises. We darken a horizontal and vertical line for the x- and y-axes. For now, we will let the distance between adjacent lines represent a distance of 1 unit. To graph each point, we begin at the origin. To graph (3,4), we move to the right (positive) 3 units along the x-axis. From there we turn and move up (positive) 4 units and make a dot. We label the location (3, 4). It is best to use quad-ruled graph paper to We follow a similar procedure for each point. y (3,4) • — H.2) +-21 -6 -5 -4 -3 -2 12 -1 (0,- Example 2 Find the coordinates 4 3 5 6 (2,-3) for points A, B, and C on this coordinate plane. J f r-fi- ^4 B p 4- d ,A o. c J -( 5 -i ) -* » -: 3 -; l - ,G \ > ; i -1 D [F E t 1 1 <t !> i Rectangular Coordinates Solution We first find the point on the x-axis that is directly 551 above or below the designated point. That is the first coordinate. Then we determine how many units above or below the x-axis the point That is. the second coordinate. is Point A (4, 3) Point B (-3,4) Point C (-5, 0) Practice Refer to the coordinate plane in Example 2 to find the coordinates of the following points. a. Point D b. Point c. Point F d. Point/ E Sketch a coordinate plane and graph the following points. Problem set e. (5,0) g. (-1,-3) 1. What is the quotient divided by the 119 2. sum f. (3,-2) h. (-2,4) when the product of 0.2 and 0.05 of 0.2 is and 0.05? The table shows how many students earned certain scores on the last test. Find the (a) mode and (b) range of these scores. Class Test Scores Score 100 95 90 85 80 75 70 Number of Students iiii m WW m i III ii in i i 3. Melissa finished a 3-mile race in 20 minutes. her average speed in miles per hour? 4. Use two unit multipliers to convert 1 What was km 2 to square meters. 552 Math 87 5. Tim has $5 in quarters and $5 in dimes. What is the ratio of the number of quarters to the number of dimes? Use a ratio box to solve Problems 6-8. 3000 meters in 9 minutes. At that how long will it take Jaime to run 5000 meters? 6. Jaime ran the 7. Sixty 8. To 9. Write an equation to solve this problem. Sixty percent of what number? 10. is first rate, 20 percent more than what number? customers, the merchant reduced all prices by 25 percent. What was the reduced price of an item that cost $36 before the price reduction? attract Draw a diagram of this statement. is 150 Then answer the questions that follow. Diane kept - of her baseball cards and gave the remaining 234 cards to her brother. (a) How many some (b) cards did Diane have before she gave to her brother? How many baseball 11. Compare: a - b 12. If a card is cards did Diane keep? Qb-a drawn from spades, 13 hearts, 13. a king? (b) a heart? a > b normal deck of 52 cards (13 13 diamonds, 13 clubs), what is the a probability that the card (a) if is Find the area of this trapezoid. Dimensions are centimeters. 5 13 in 553 Rectangular Coordinates Find the volume of each solid. Dimensions are in inches. 6 15. 14. H in. h- 3 16. 17. The skateboard cost $36. The tax rate is 6.5 percent. (a) What is the tax on the skateboard? (b) What is the total price, including tax? Complete the table. Fraction Decimal Percent 0.15 (a) 66- percent in. (b) 18. What number 19. Multiply and write the product in scientific notation. is (6 20. of 48? x 10- 8 )(8 x 10 4 ) Find the missing numbers in this function. y = ,x X y 6 -3 21. Use a ratio box to solve this problem. The ratio of the measures of the two acute angles of the right triangle is 7 to 8. What is the measure of the smallest angle of the triangle? 22. Graph the following points on a coordinate plane: (a) (3,-2) (b) (-5,0) (c) (-4,-3) 554 Math 87 23. Find the coordinates for points M, N, and P on this coordinate plane. y M N -6 12 -5 -4 -3 -2 -1 - x 3 4 5 6 1 ¥-2 3 4 Solve: 24. 1-p 2^ + Add, 26. 6 = 36 25. V144 1 f HIV4 27. = 5 4 (1.5) -s- 60 min 1 hr mi min 2 29. 9.5 - 4- v (-18) + (-12) \ ( 2 , 9y LESSON 120 1.3 subtract, multiply, or divide, as indicated: 28. 30. 7y - 5 3.4 y - (-6X3) -3 Estimating Angle Measures We have practiced reading the measure of an angle from a protractor scale. The ability to measure an angle with a protractor is an important skill. The ability to estimate the measure of an angle is also a valuable skill. In this lesson we will learn a technique to help us estimate the measure of an angle. We will also practice using a protractor as we check our estimates. 555 Estimating Angle Measures To estimate a measurement, we need a mental image of the units to be used in the measurement. To estimate angle a mental measure, we need a mental image of a degree scale protractor. We can "build" a mental image of a protractor the face of a clock. from a mental image we already have — — This clock face is a full circle. A full circle is 360°. A clock face is divided into 12 numbered divisions. From one numbered division to the next is — of a full circle. One twelfth of 360° is 30°. Thus, the measure of the angle formed by the hands of the clock at 1 o'clock is 30°, at 2 o'clock is 60°, and at 3 o'clock is 90°. A clock face is further divided into 60 smaller 1 divisions. From one small division to the next is -en- of a circle. One sixtieth of a circle is 6°. the next on the Here Thus from one minute mark face of a clock is 6°. we have drawn an angle on the vertex of the angle is at face of a clock. the center of the clock. the angle is set at 12. The other side of the angle is at to One The side of "8 minutes after." Since each minute of separation represents 6°, the measure of this angle is 8 x 6°, which is 48°. With some practice we can usually estimate the measure of an angle within 5° of its actual measure. 556 Math 87 Example (a) Record your estimate of the measure of ZBOC. (b) Use a protractor to find the measure of (c) Z#OGL By how many degrees did your estimate miss your measurement? Solution (a) We use a mental image of a clock face on ZBOC with one side of the angle set at 12. Mentally other side minutes is after. mZBOC is (b) We more than 10 minutes we is after. 72°, see that the Perhaps we it is 12 estimate that 72°. trace angle so that Since 12 x 6° we may BOC on our paper and extend the sides can use a protractor. We find that mZBOC is 75°. (c) Our estimate, 72°, misses our measurement, 75°, by 3 C 557 Estimating Angle Measures Practice Find the measure of each angle shown on the clock face. a. ZAOB b. ZAOC c. ZAOD In practice problems d-g, estimate the measure of each angle. and measure each angle. By how many degrees did your estimate miss your measurement? Then use d. Problem set 120 a protractor ZBOC In ZDOC e. f. ZFOE ZFOB g. May the merchant bought 3 tons of beans at an average price of $280 per ton. In June the merchant bought 5 tons of beans at an average price of $240 per ton. What was the average price of all the beans bought by the merchant in 2. What is May and the quotient June? when 9 squared is divided by the square root of 9? 3. The Adams' car has a 16-gallon gas tank. How many tanks of gas will the car use on a 2000-mile trip averages 25 miles per gallon? 4. In a triangular prism, vertices to the what number is the ratio of the of edges? if the car number of 558 Math 87 Use a ratio box to solve Problems 5-7. 58 dollars equals 100 marks, what is the cost in dollars of an item that sells for 250 marks? 5. If 6. Sixty 7. The average number is 20 percent less than what number? during the sale. before the sale number If of customers increased 25 percent the average was 120 per day, number of customers what was the average of customers per day during the sale? 8. Write an equation to solve this problem. Sixty percent of 50? 9. Use two unit multipliers to convert 1.2 m 2 is what to square centimeters. 10. Draw a diagram of this statement. Then answer the questions that follow. Twenty-four of the eggs were cracked. This was - of the total number (a) How many (b) What percent of eggs in the crate. eggs were in the crate? of the eggs in the crate were not cracked? 11. Compare: x + y 12. Evaluate: 13. The perimeter Q Q if x - y if a = -4, b = -3, and c = -2 of a certain square of the square in square inches. 14. What is the y > probability of this spinner (a) stopping on 3? (b) not stopping on 3? is 1 yard. Find the area Estimating Angle Measures 15. Find the volume of each solid. 559 Dimensions are in centimeters. (b) (a) Find the 16. including 7 percent tax, of 20 square yards of carpeting priced at $14.50 per square yard. 17. Complete the total price, table. Fraction Percent 3;% (b) (a) 33- percent Decimal 18. What 19. Multiply and write the product in scientific notation. is (3 20. (a) of $24? x 10 3 )(8 x 10~ 8 ) Find the circumference of this circle. (b) Find the area of this circle. Leave n as n 21. Use the clock face to estimate the measure of each angle. (a) (b) (c) 22. ZBOC ZCOA ZDOA Graph each point on a coordinate plane. (a) (-3,2) (b) (0,-5) (c) (3,-4) — 560 Math 87 23. Find the coordinates for points R, S, and T on coordinate plane. y ff c. -4<» R -3 -2-1- s, -( 5 -! ) —* % -: 3 -; ! - ; > ; i <l -1 ! i f -3 -4 r p -6 ' Solve: 24. 0.8m + Add, 26. 1 1.5 25. A [.7 2—X - 7 2 = 13 subtract, multiply, or divide, as indicated: 33 - V49 + 3 • 24 27. 3 yd 2 ft + ( 28. 12.5 \ 3- + 2.7 5 30. (-4) - (-8K-3X-2) -2 29. 2.7 + f 3 V -- 7^ in. 5^ in. ,2] 1^J this Similar Triangles LESSON 561 Similar Triangles 121 We often use "tick marks" to indicate that the measures of angles are equal. Y marks indicate that angles A and Z have equal measures. The double tick marks indicate that angles B and Y have equal measures. In these figures the single tick If three angles in one triangle have the same measures as three angles in another triangle, the triangles are called similar triangles. Similar triangles look alike. (a) (b) These triangles are not the same size, but they have the same shape because the angles of the same measures are in both triangles. These triangles are similar triangles. If an angle in one triangle has the same measure as an angle in a similar corresponding angles. The sides opposite corresponding angles are called corresponding sides. triangle, the angles are called Example 1 List the pairs of corresponding angles and the pairs of corresponding sides for these triangles. 562 Math 87 Solution The left marks tell us that the angles in the triangle on the have the same measures as the angles in the triangle on tick the right. This means that the triangles are similar triangles. Corresponding Angles Corresponding Sides ZN and ZT LM ZL and ZS NM and ZM and ZR NL and TS and SR TR The lengths of corresponding sides in similar triangles are proportional. This means that the ratios formed by corresponding sides are equal. (b) (a) Every side in triangle side in triangle sides (a) is (b). If we and put the sides Shortest Sides twice as long as the corresponding write the ratios of corresponding for triangle (a) on top, we Longest Sides Middle Sides 6 8 10 3 4 5 Each one of these ratios equals triangle (b) on top, we get Shortest Sides 2. If Middle Sides we put the sides of Longest Sides 3 4 _5_ 6 8 10 Each one of these ratios equals one get half. 563 Similar Triangles Example 2 Solution Write the equal ratios of corresponding sides for these triangles. The angles are equal, so the triangles are similar. We to write the lengths of the sides of the left triangle decide on top. a_ z y Example 3 Find the length of side Solution a. We will write a proportion and solve for a. We decide to write the ratios so that the sides from the left 6_ 3 10 a triangle are on top. Since these ratios are equal, we can connect them with an equals sign, cross multiply, and solve for a. A 10 = 1 6a = 30 a = 5 Practice a. equal ratios a cross multiplied solved Identify each pair of corresponding angles of corresponding sides in these two and each pair triangles. 564 Math 87 b. Find the length of side x. 12 Problem set 1. 121 2. 3. Ginger gave the clerk $10 for a tape that cost $8.95 plus 6 percent tax. How much money should she get back? Three hundred billion is how much less than two Write the answer in scientific notation. During the second semester Joe's test scores trillion? were 95, 90, 80, 85, 90, 100, 85, 90, 95, 80 Find the (a) median, (b) mode, and (c) range of these scores. Use a 4. ratio box to solve Coming down Problems 4 and the long per hour. If it took him 5 how long was the hill? 5. Nelson averaged 24 miles minutes to come down the hill, hill, how Nelson traveled 3520 yards in 5 minutes, could he travel in 8 minutes at the same rate? 5. If 6. The points (3, 2), (3,-2), (-1,-2), and (-1,2) are far y the vertices of a square. The area is of the square how many H.2) (3,2) square units? 12 -5 -4 -3 -2 -1 -•3 4 x 5 1 (-1 , -2) (3, -2) 3 7. Three fourths of a yard is how many inches? .< Similar Triangles 8. 565 box to solve this problem. The ratio of leeks to radishes growing in the garden was 5 to 7. If 420 radishes were growing in the garden, how many leeks were there? Use a ratio Write equations to solve Problems 9-11. 9. Forty is 250 percent of what number? 10. Forty is what percent 11. What decimal number 12. Use a ratio box to solve this problem. The tuition increased 10 percent this year. what was the 13. What of 60? 40 percent of 6? is the tuition this year If tuition last year? the average of the two is $2310, is numbers marked by arrows on this number line? 1.7 I I 1.8 I I I I I I » | 14. Complete the I | 2.0 1.9 I II I I I I I I I I I I I | I Fraction table. Compare: x + y Q x - y I I + I I I Decimal ifxis I I | I Percent 3.25 (a) 15. 2.1 I (b) positive and y is negative 16. Multiply and write the product in scientific notation. (5.4 17. x 10 8 )(6 x 10" 4 ) Find (a) the circumference and with a radius of 10 millimeters. (b) the area of a circle 12 18. Find the area of this trapezoid. Dimensions are in feet. n. 566 Math 87 19. Find the volume of each of these solids. Dimensions are in meters. 20. (a) (b) Refer to the figure shown. What are the measures of the following angles? 21. (a) ZX The triangles are similar. Z.Y (b) Find (c) x. ZA Dimensions are in centimeters. 10 15 22. Use two unit multipliers 23. Estimate by first to convert 10 yards to inches. rounding each number to one nonzero 2 = 12 digit: (38,470)(607) 79 Solve: 24. 1.2m + 0.12 = 12 Add, 26. 27 25. 1-y - subtract, multiply, or divide, as indicated: V225 - VT2T 15 1 mi hr 1 hr 60 min 1760 yd 1 mi 567 Scale and Scale Factor / 28. f A 6- - 3.45 5- + 3- - 29. 3 2 2 (-2) - 30. (-2) \ 4.5 - 18y V + (-4X-3) + (+3) - (+4) (+3) Scale and Scale Factor LESSON 122 Scale In the preceding lesson we discussed similar triangles. Scale models and scale drawings are other examples of similar shapes. Scale models and scale drawings are reduced (or enlarged) renderings of actual objects. triangles, the lengths of and the if a As is true of similar corresponding parts of scale models objects they represent are proportional. The scale of the model is stated as a ratio. For instance, model airplane is — the size of the actual airplane, the scale is stated as — We may or 1:24. , use the given scale to write a proportion to find a measurement either on the model or on the actual object. A ratio box helps us put the numbers in the proper places. Example 1 A model of the is Solution airplane model is how many We is built on a scale of 1:24. the wingspan If 18 inches, the wingspan of the actual airplane feet? will construct a ratio problems. In one column we are the scale numbers. In number The second number refers measures. The first entries in the ratio box box as we do with other numbers which the other column we write the write the ratio of the scale refers to the model. to the object. We can use the to write a proportion. Ratio scale ratio Measure Model 1 18 Object 24 w 1 24 w — 18 w = 432 568 Math 87 model was given in inches. Solving the proportion, we find that the full-size wingspan is 432 inches. We are asked for the wingspan in feet, so we convert units The wingspan from inches of the to feet. lft 432in: We Example 2 find that the Sarah model is = 36 ft 12 in: wingspan of the airplane molding a model of a car from is clay. 36 The feet. scale of the 1:36. If the height of the car is 4 feet 6 inches, is what should be the height of the model in inches? Solution First we convert 4 feet 6 inches to inches. 4 feet 6 inches = 54 inches Then we construct a ratio box using 1 and 36 as the ratio numbers, write the proportion, and solve. Ratio scale Measure Model 1 m 1 m Object 36 54 36 54 36m == 54 m - 54 36 m - 2 -I The height of the model car should be 1— inches. Scale We have solved proportions by using cross products. Sometimes factor more quickly by noting the scale factor. The scale factor is the number of times larger (or smaller) the terms of one ratio are when compared with the terms of the other ratio. The scale factor in the proportion a proportion can be solved Scale and Scale Factor below is 569 6 because the terms of the second ratio are 6 times the terms of the first ratio. /x 6 X 18 24 \ x 6 Jf Example 3 Solution Solve: — %7 = n Instead of finding cross products, we note that 3 times 5 equals 15. Thus the scale factor is 5. We use this scale factor to find n. 15 3 35 \x We Practice a. find that n is 5 35. The blueprints were drawn to a scale of 1:24. If a length of a wall on the blueprint was 6 in., what was the length in feet of the wall in the house? b. carving a model ship from balsa wood on a scale of 1:36. If the ship is 54 feet long, the model ship should Bret be is how many inches long? Solve by using the scale 5 c. Problem set 122 factor. 15 d. w ~3 42 21 box to solve this problem. The regular price of the item was $45, but the item was on sale for 20 percent off. What was the sale price? 1. Use a 2. With one ratio toss of a die, what is the probability of rolling (a) an odd number greater than 1? (b) an even number less than 2? 570 Math 87 3. Ann averaged 10 points per game. In her next 9 games, Ann averaged 15 points per game. How many points per game did Ann average during her In her 6 games, 15 games? first 4. first Ingrid started her trip at 8:30 a.m. with a full tank of gas and an odometer reading of 43,764 miles. When she stopped for gas at 1:30 p.m., the odometer read 44,010 took 12 gallons to fill the tank, her car averaged many miles per gallon? miles. how 5. In Problem many 6. If it 4, Ingrid traveled at an average speed of how miles per hour? box to solve this problem. If 5 dollars equals 30 kronas, what is the cost in dollars of an item priced at Use a ratio 75 kronas? 7. Write an equation to solve this problem. Three Tom's favorite number is 60. What is fifths of Tom's favorite number? 8. On a coordinate plane graph the points (-3, and (-3,-2). If these of a rectangle, 2), (3, 2), points designate three of the vertices what are the coordinates of the fourth vertex of the rectangle? 9. What is set of numbers? the ratio of counting numbers to integers in this {-3,-2,-1,0, 1,2} 10. Find a 2 if {a = 3. 1 Write equations to solve Problems 11-13. 11. Forty is what percent of 250? 12. What is 60 percent of $40? 13. Forty percent of what number is 60? ' 571 Scale and Scale Factor 14. problem. The number of students in chorus increased 25 percent this year. If there are 20 more students in chorus this year than there were last year, how many students are in chorus this Use a ratio box to solve this year? 15. BC is how much Segment AB? longer than segment B • > 1 1 1 1 1 ' 1 ' 1 inches • 1 Graph on a number 17. Complete the ' table. 1 ' line: 1 1 ' 1 4 3 x < 3 Fraction Decimal Percent 1.4% (b) (a) Multiply and write the product in scientific notation. (1.4 19. 1 1 1 2 1 16. 18. ' " 1 x 10" 6 )(5 x 10 4 ) y = -2x X y Find the missing numbers in this function. 3 20. Find (a) the circumference and (b) the area of a circle -2 that has a diameter of 2 feet. 21. Estimate the measure of /.ABC. Then trace the angle, extend the sides, and measure the angle with a protractor. C 572 Math 87 22. Find x. Then find the area of the Dimensions are in inches. triangle on the left. Solve: 23. -277 + 8 = 20 24. 0.3x - 2.7 = 9 5 oc 25. 12 = — 53 Add, 26. 120 17 subtract, multiply, or divide, as indicated: V5 3 52 - 1 gal 27. 1 qt 1 qt 28. ( (0.25) I ™ ) 1 1- - 1.2 4 29. 7l3 ) ( I 1 pt 3 1 1-3.5 4 J (-2)(3) -,(3)(-4) (-2)( -3) LESSON 123 - (4) Pythagorean Theorem The longest side of a right triangle is called the hypotenuse. Every right triangle has a property that makes the right triangle a very important triangle in mathematics. The area of the square drawn on the hypotenuse of a right triangle equals the sum of the areas of the squares drawn on the other two sides. Pythagorean Theorem 573 32 = 9 5 2 = 25 "I \ 4 2 = 16 The triangle on the left is a right triangle. On the right we have drawn squares on the sides of the triangle. The areas of the squares drawn on the sides are 9, 16, and 25. Notice that the area of the largest square equals the sum of the areas of the other two squares. 25 = 16 + 9 This property of right triangles was known to the Egyptians as early as 2000 b.c, but it is named for a Greek who lived about 650 b.c. The Greek's name was Pythagoras, and the property is called the Pythagorean theorem. The Greeks are so proud of this mathematician that they have issued a postage stamp that illustrates the theorem. Here we show a reproduction of this stamp. EAAAI-APX.3.50 U\J\S\J\/\S\S\f\ To solve problems that require the use of the Pythagorean theorem, we will sketch the right triangle and draw the squares on each side. Example 1 Copy this triangle. Draw a square on each side. Find the area of each square. Then find c. 6 cm 8 cm 574 Math 87 Solution We copy the triangle and sketch a square on each side of the triangle, for using a side of the triangle one side of each square. 6 cm We were given the lengths of two shorter sides. The areas of the squares on these sides are 36 cm 2 and 64 cm 2 the 8 . cm 36 + 64 = 100 The sum and 64 is 100, so the area of the largest square is 100. This means that a side of the largest square must be 10 because 10 2 = 100. Thus J Ocm of 36 c = 10 cm 6 cm 8 Example 2 In this triangle, find a. cm Dimensions are in inches. Solution We copy the triangle and draw a square on each side. The area of 13 x 13, or 169 in. The area of one of the smaller squares is 144 in. 2 So a 2 plus 144 must equal 169. the largest square 13 is 2 169 . . ^^ a 2 + 144 = 169 we subtract 144 from both sides, we find that a 2 equals 25. 144 If 12 a 2 + 144 = 169 -144 -144 a z = 25 This means that a equals 5 because 5 squared is 25. a = 5 in. a2 r Pythagorean Theorem Example 3 Solution 575 Find the perimeter of this triangle. Dimensions are in centimeters. We can use the Pythagorean theorem to find side c. The areas of the two smaller squares are 16 and 9. The sum of these areas is 16 25, so the area of the largest square \ ~i Thus the length of side c is Now we add the lengths of the is 25. 5. 9 sides to find the perimeter. Perimeter = 4 Practice a. cm + 3 cm + 5 cm = 12 cm Use the Pythagorean theorem to find side a. b. Use the Pythagorean theorem to find side b. Problem set 1. 123 The meal cost $15. Christie of the cost of the meal. left a tip that was 15 percent How much money did Christie leave for a tip? 2. 3. Twenty-five ten-thousandths is how much greater than twenty millionths? Write the answer in scientific notation. Find the the 4. (a) number mean, (b) mode, and the months of a leap median, of days in (c) (d) range of year. The 2-pound box cost $2.72. The 48-ounce box cost $3.60. The smaller box cost how much more per ounce than the larger box? (There are 16 ounces in a pound.) 576 Math 87 5. Use a box to solve this problem. If 80 pounds of $96, what would be the cost of 300 pounds of ratio seed cost seed? pound how many ounces? 6. Five eighths of a 7. Use a ratio box to solve this problem. The ratio of stalactites is cavern was 9 to to stalagmites in the 5. If the total and stalagmites was 1260, how many stalagmites were in the cavern? number of stalactites Write equations to solve Problems 8 and 8. Ten percent of what number 9. Twenty 10. is what percent The ordered pairs 11. The 20? of 60? (1, 0), points that lie on the coordinate plane and is 9. and (3, 2) designate Graph the points on a -1), (0, same line. draw the line. by 20 percent. If the cost before the decrease was $3.40, what was the cost after the decrease? Use a ratio box to solve cost of a 10-minute call to Boise decreased the problem. 12. What is the area of the shaded region of this rectangle? y —4- (6,3), o_ — -1 1 3 -2 - 1 2 3 4 5 f -2 L 1 ' 13. Use a ratio box to solve this problem. On a 1:60 scale model airplane, the wingspan is 8 inches. The wingspan of the actual airplane feet is this? is how many inches? How many 577 Pythagorean Theorem 14. Complete the Decimal Fraction table. i| Percent (b) (a) 15. Divide 365 by 12 and write the quotient as a mixed number. 16. Multiply and write the product in scientific notation. (8.1 x* - x 10- 6 )(9 x 10 10 ) xy x— Evaluate: 18. Use the Pythagorean theorem to find 19. x if y = 8 and 17. c. Find the volume of this solid. Dimensions are in centimeters. 10 20 h- 20. H Refer to the figure shown. Find the measures of the following angles. (a) 21. 7 Z.X (b) Z.Y (c) The triangles are similar. Find x. Dimensions are in inches. 8 22. Estimate: < 41 39 2 » 395 > fl 81 ' Solve: 23. ZZ 4n + 1.64 = 2 24. 3-x 3 1 = 49 578 Math 87 25. ¥- = 25 Add, 26. £ 75 subtract, multiply, or divide, as indicated: 33 + 4 2 - A/225 3- + 27. 6.3 + 22 5 ( 28. ^ (0.75)(40) 3- V 3 7 7 days 1 week 30. 60 min 1 hr 24 hr 1 day Estimating Square Roots • Special Angles LESSON 124 Estimating square roots -12 - (6)(-3) (-12) - (-6) + (3) 29. We review square roots by remembering that because 2 times 2 equals 4, we say that the square root of 4 2x2 = 4 Because 3 times 3 equals 3x3 9, ^4 = so we say that the = 9 Because 5 times 5 equals 25, is 2. so a/9 2 square root of 9 = is 3. 3 we say that the square root of 25 is 5. 5 x 5 = 25 so \/25 = 5 The number 30 does not have a whole number square root. We can estimate the square root of 30 by guessing and we know how to estimate square roots, then we will know if the answer we get for a square root on a calculator checking. is If reasonable. Let's estimate the square root of 30. 4 x 4 = 16 5 x 5 = 25 6 x 6 = 36 4 is too small 5 is too small 6 is too large Estimating Square Roots • Special Angles 579 Since 30 is between 25 and 36, we see that the square root of 30 is a number between 5 and 6. If we use the square root key on a calculator, we get 30 = 5.4772256 This result Example Solution is reasonable because the square root of 30 number between 5 The square root whole numbers? of 90 We and is a 6. is between which two consecutive begin by guessing. 6 x 6 = 36 7 x 7 = 49 6 is too small 7 is too small 8 x 8 = 64 8 is too small 9 x 9 = 81 9 is too small 10 is too large 10 x 10 = 100 Since 90 is between 81 and 100, we see that the square root of 90 is a number between 9 and 10. Special angles If the sum of the measures of two angles the angles are complementary angles. of two angles is 180°, If the is 90°, we say that sum of the measures we say that the angles are supplementary angles. Angles A and B are complementary angles because their combined measures total 90°. Angles C and Dare supplementary angles because their combined measures total 180°. Some people confuse the meanings of these words. We can avoid confusion by remembering that the "c" in complementary resembles a right angle. <5 omplementary looks like complementary 580 Math 87 The "s" in supplementary resembles total 180°. fupplementary looks like two right angles, which supplementary J* I When two lines cross, four angles are formed. of opposite angles are called vertical angles. The pairs The measures of vertical angles are equal. A and B are a pair of vertical angles, and the measure of angle A equals the measure of angle B. Angles C and D are Angles a pair of vertical angles, and the measure of angle C equals the measure of angle D. easy to see why vertical angles are equal. Let's angle Cbe a 150° angle. It is let A make a straight angle (the angles are supplementary), so angle A is a 30° angle. The 150° angle The 150° angle and angle and angle B make a straight angle (the angles are supplementary), so angle B is a 30° angle. Now either 30° angle angle D is a 150° angle. From this we can see and angle why D form a straight angle, so vertical angles are equal. . Estimating Square Roots Practice Each square root below whole numbers? i. is b. a/7 • Special Angles 581 between which two consecutive \/70 I. a[700 Describe each pair of angles in d and e as complementary or supplementary. d. e. C f. Problem set 1. 124 /D Find the measure of Za. Gabriel paid $20 for 2^ pounds of cheese that cost $2.60 per pound and 2 boxes of crackers that cost $1.49 each. How much money should he get back? 2 What is the probability that the spinner will not stop on a prime number? 3. What is the average of the first 10 counting numbers? 4. At an average speed of 50 miles per hour, would it take to complete a 375-mile trip? 5. Use a ratio box to solve this how long problem. The Johnsons traveled 300 kilometers in 4 hours. At that rate, how long will it take them to travel 500 kilometers? Write the answer in hours and minutes. 6. Three fourths of Bill's favorite number is is one half of Bill's favorite number? 7. Use a ratio box to solve this to losers in the contest contestants, 36. What number problem. The ratio of winners was 1 to 15. If how many winners were there were 800 there? 582 Math 87 Write equations to solve Problems 8 - 10. 8. Three hundred 9. Twenty is 6 percent of is what number? what percent of 10? 10. What 11. The ordered is 6.5 percent of $40? pairs and (2, 4) designate Graph the points on a (-2, -4), (0, 0), on the same line. coordinate plane and draw the line. points that 12. lie Arrange in order of size from 2, least to greatest: 2 2 a/2,-2 , 13. problem. The population of the colony decreased by 30 percent after the first winter. If the population after the first winter was 350, what was the population before the first winter? 14. Nathan used Use a ratio mold box to solve this this graph 10 model car from clay. The car was 4 feet high, and he to a scale 9 8 CD 7 6 used the graph to see that the model should be 2 inches high. If the CD N 4 o < length of the car's bumper is 5 feet, use the graph 5 3 2 1 12 proper length of the model's to find the Model 4 3 (in inches) bumper. 15. Compare: xy Q 16. Complete the table. — if x is positive Fraction (a) and y Decimal (b) is negative Percent 72% Estimating Square Roots 17. Multiply and write the product in scientific notation. x 10 6 )(6 x 10 3 ) (4.5 18. Each square root whole numbers? is between which two consecutive V40 (a) 19. Find (a) (b) Find (b) the area of a circle Use — for n. a. Find the volume of each solid. ^-^\ 21. 23. \'20 the circumference and that has a radius of 7 inches. 20. 583 Special Angles • Dimensions are in centimeters. 22. In this figure, find the measures of angles a, b, and c. Solve: 24. 4-x + 7 = 70 25. 2 Add, 15 45 w 3.3 subtract, multiply, or divide, as indicated: ,\ 2 + 82 26. V6 29. 8- 6 2.5 27. - 13 1 ton - 350 lb 30. 28. (18) (-3) + - 7.2 3- -*- (2)(-3) (-2) - (-4) 584 Math 87 Multiplying Three or LESSON 125 Numbers More Signed Powers of Negative • Numbers Multiplying three or more signed numbers One way to multiply three or more signed numbers is to multiply the factors in order from left to right, keeping track of the signs with each step, as we show here. (-3)(-4)(+5)(-2)(+3) (+12X+5H-2H+3) (+60)(-2)(+3) (-120)(+3) -360 problem multiplied (-3)(-4) multiplied (5)(12) multiplied (60)(-2) multiplied (-120)(3) Another way to keep track of the signs when multiplying signed numbers is to count the number of negative factors. Notice the pattern in the multiplications below. -1 = -1 (-1X-1) = +1 (-D(- 1)(-1) = -1 (-i)(-i)(-i)(-i) = +i (-D(- lX-iH-iX-i) = -i odd even odd even odd When there is an even number of negative factors, the product is positive. When there is an odd number of negative factors, the product Example 1 Solution is negative. Find the product: (+3)(+4)(-5)(-2)(-3) There are three negative factors (an odd number), so the product will be a negative number. We multiply and get (+3)(+4)(-5)(-2)(-3) = -360 We did not consider the signs of the positive factors because positive factors do not affect the sign of the product. I Powers Powers of negative We remember that many times the base numbers ( Example 2 Solution Simplify: (a) the exponent of a (a) used as a is _ 3)4 means (-2) 4 (-2) indicates how 3 )(_3)(_ 3 ) 5 The expression (- 2) 4 means (- 2)(- 2)(- 2)(- 2). Since there is an even number of negative factors, the product is a positive number. Since 2 4 (b) power 585 Numbers factor. r_3)(_ (b) of Negative 16, is we find that (-2) 4 +16. is (-2) 5 means (-2)(-2)(-2)(-2)(-2). This an odd number of negative factors, so the The expression time there is product is a negative number. Since 2 5 equals 32, find that (-2) 5 equals -32. Practice Problem set we Simplify: +5 )(-4X+3)(-2)(+l) a. (-5)(-4)(-3)(-2)(-l) c (-2) 3 1. The dinner bill totaled $25. Mike left a 15 percent tip. much money did Mike leave for a tip? 125 2. d. When the square root of 9 the difference 3. The on a (-3) 4 table is ( e. (-9) 2 is how much shows a class test. b. (a) the mode and (b) Class Test Scores 100 95 90 85 80 70 How earned by students the 29 scores in the class. Score (-D 5 subtracted from 9 squared, greater than 9? tally of the scores Find f. Number of Students III m m m llll 1 i i mi the median of 586 Math 87 4. The plane completed the 5. Use a ratio for 4 hours of work. 2^ hours. If the flight covered 1280 kilometers, what was the plane's average speed in kilometers per hour? box hours of work flight in to solve this at the How same problem. Jeremy earned $25 much would he earn for 7 rate? how many 6. Eight fifths of a kilometer 7. Use a ratio box to solve this problem. If 40 percent of the lights were on, what was the ratio of lights on to lights is meters? off? 8. Use the Pythagorean theorem longest side of a triangle to find the length of the whose vertices are (3, 1), (3, -2), and (-1,-2). 9. Use a box ratio buying the was the problem. Sam saved $25 a sale that offered 20 percent off. What to solve this suit at regular price of the suit? Write equations to solve Problems 10 and 11. 10. What decimal number 11. What percent 12. Use problem. The merchant bought the item for $30 and sold it for 60 percent more. How much profit did the merchant make on the item? 13. Compare: a ratio of 25 box is percent of 150? 20? to solve this (-D (-D (-D (-D 14. is 1 o (-U (-D (-D (-D (-D Use a ratio box to solve this problem. The — scale model of the rocket stood 54 inches high. What was the height of the actual rocket? 15. Graph on a number line: x > Powers 16. Complete the Decimal Fraction table. (b) Multiply and write the product in scientific notation. (4.8 18. Quadrilateral is 34°. (a) x 104)(8 x 10-8) ABCD is a rectangle. The measure of ZACB Find the measure of each of these angles. ZCAB (b) ZCAD 19. These two triangles are similar. Find x. 20. Find (a) the circumference and with a diameter of 2 feet. 21. Estimate the measure of ZAOB. (b) the area of a circle Then use a protractor to measure the angle. 22. 587 Percent 1.75 (a) 17. Numbers of Negative Which a. of these Vl3 numbers b. is between 12 and 14? a/130 c. a/150 588 Math 87 23. Find the volume of this 24. right prism. Find the volume of this right circular cylinder. A V 6 U /« Solve: 25. 6-/ J -5 = 26. 5 3 Add, 27. m 2.5 25 subtract, multiply, or divide, as indicated: 102 29. (a) 30. (a) • 1- - 25 - 8- + 1- 28. 12.5 (b) (-12)(+3)(-2)(-l) 7 (-5H+4H-3H+2) I-™-™- 1 (-6)(-3)(-2) ) (b) (-4)(-l) (-3)(+2) LESSON 126 12 Semicircles A semicircle is half of a circle. Thus, the length of a semicircle is half the circumference of a whole by a semicircle and a diameter is circle. The area enclosed half the area of the full circle. We will practice finding the lengths of semicircles the areas they enclose by calculating the perimeters and and areas of figures that contain semicircles. Unless directed otherwise, we will use 3.14 as the approximation for /rwhen we perform the calculations. A calculator will be helpful. 589 Semicircles Example 1 Solution Find the perimeter of this figure. Dimensions are in meters. The perimeter of the semicircle half the perimeter of a circle is whose diameter is 10. 15.7m = Perimeter of semicircle = Now we can write all the dimensions. 15.7 =10m + 4m « 33.7 m Perimeter + 4m Example 2 Find the area of this figure. Dimensions are in meters. Solution We divide the figure into two parts, + 15.7 m ~q and then we find the area of each part. The area of the figure equals A 1 plus A 2 10 . _d A = A2 = nr 2 3.14(25 LxW=4x m 2 ) - 39.25 = 40 10 Find (a) and (b) the Dimensions the perimeter area of this figure. are in centimeters. £ ~i Problem set 126 1. 2 m 2 2 « 79.25 Total area Practice m m -y, \ 4 L , r The merchant sold the item for $12.50. If 40 percent of the selling price was profit, how much money did the merchant earn in profit? 590 Math 87 2. With one toss of a a prime number? 3. Bill's what die, is the probability of rolling average score for 10 tests was 88. If his lowest score, 70, is not counted, what was his average for the remaining tests? 4. The 36-ounce container cost $3.42. The 3-pound container cost $3.84. The smaller container cost how much more per ounce than the larger container? 5. Sean read 18 pages in 30 minutes. If he had finished page 128, how many hours will it take him to finish his 308-page book if he reads at the same rate? 6. Matthew was thinking of a certain number. If 3 number was 75, what was - of the number? - of the i 7. Use a ratio box to solve this problem. The ratio of crawfish to tadpoles in the creek was 2 to 21. If there were 1932 tadpoles in the creek, how many crawfish were there? Write equations to solve Problems 8 and 8. What percent of $60 9. What number is 10. The ordered is $45? 45 percent of 60? pairs (-1, 0), and (0,-2) designate Graph the points on a (-3, 4), on the same line. coordinate system and draw the points that 9. lie line. Write equations to solve Problems 11 and 12. 11. What percent of 60 is 75? 12. What percent of 75 is 60? 13. Complete the table. Fraction (a) Decimal (b) Percent 2.2% Semicircles 14. 591 Michelle used this graph to find how long it would take her to drive a certain distance. According to the graph, how long would it take her to drive 75 miles? i CD | 150 cB 100 a) 50 > o c b 2 1 Driving time Qa—b 3 (in hours) is positive and b is negative 15. Compare: ab 16. Multiply and write the product in scientific notation. if a 4 (3.6 x 10" )(9 x 17. Find the area of this figure. Dimensions are in centimeters. 18. Find the perimeter of the figure in 19. Problem 10 8 ) 17. Find the volume of this solid in cubic inches. Dimensions ^^^ ^^ are in feet. ^1 20. What 21. Find mZx. angle is formed by the hands of a clock at 5:00? 22. The triangles are similar. Find x. ^t% 23. Use the Pythagorean theorem Problem 22. to find y in the triangle in 592 Math 87 Solve: 24. 2-w + 4 = 48 - 2.4n 25. 0.12 = 4.8 4 Add, 26. subtract, multiply, or divide, as indicated: 27. ^ffl[l&) 5 lb 1^ 28. 12.5 1 - 8+ 1- 30. oz 2 lb 8 oz 4- 29. 6 3 7 3- * -s- 2.5 4 6 (-3) (4) -2 Surface Area LESSON 127 The total area of the outside surfaces of a geometric solid is called the surface area of the solid. This cube has six faces (surfaces). Each face has an area of 9 square inches. surface area 3 Thus the total is 3 3 9 Example in. 2 + 9 in. in. 2 + 9 in. 2 + 9 Find the surface area of this block. Dimensions are in in. 2 + 9 in. in. in. 2 + 9 ^^^^ Top in. 2 = 54 in. 2 ^^ Side centimeters. Front Solution The block has six rectangular faces. The areas of the top and bottom are equal. The areas of the front and back are equal, and the areas of the left and right sides are equal. We add the areas of these six faces to find the total surface area. Surface Area Area of top = 5 cm X 6 cm Area of bottom = 5 cm X 6 cm = Area of front = 3 cm X 6 Area of back = 3 cm Area of side = 3 Area of side = 3 = 30 Practice cm 2 30 cm 2 cm = 18 cm 2 X 6 cm = 18 cm 2 cm X 5 cm = 15 cm 2 cm X 5 cm = 15 cm 2 = 126 Total surface area 593 cm 2 Find the surface area of this rectangular solid. Dimensions are in meters. Problem set 127 box to solve this problem. The regular price of the dress was $30. The dress was on sale for 25 percent off. What was the sale price? 1. Use 2. Twenty a ratio billion is how much greater than nine hundred million? Write the answer in scientific notation. 3. The mean of the following numbers than the median? Use a calculator. is how much less 3.2, 4.28, 1.2, 3.1, 1.17 \a 2 - b2 a = 10 and b = 8 4. Evaluate: 5. paid at a rate of $8.50 per hour, will she earn if she works 6^ hours? 6. Use If Glenda if is a ratio box to solve this flour costs $2.48, what is problem. If how much 6 kilograms of the cost of 45 kilograms of flour? 7. Five eighths of a mile is how many yards? (1 mi = 1 760 yd) 594 Math 87 8. problem. The ratio of Whigs to Tories at the assembly was 7 to 3. If 210 party members had assembled, how many were Tories? Use a ratio box to solve this Write equations to solve Problems 9—11. 9. What percent of $60 is $3? 10. Sixty percent of what number 11. What 10 percent of 4? 12. Use a ratio box to solve fraction the item at If is is 6? problem. The merchant sold a 30 percent discount from the regular price. the regular price this was $60, what was the sale price? 13. The coordinates 14. Compare: a 3 15. Use a ratio box to solve this problem. Begin by converting 60 feet to inches. Brandon is making a model plane at a 16. and -2) are the coordinates of the vertices of a right triangle. Find the length of the hypotenuse of this triangle. (-2, -2), (-2, 2), Qa 2 if a is (1, negative the length of the actual plane 1:36 scale. If how many inches long should he Complete the table. Fraction (a) make his Decimal 1.7 is 60 feet, model? Percent (b) 17. Divide 1000 by 33 and write the quotient as a decimal with a bar over the repetend. 18. Multiply and write the product in scientific notation. (8 19. x 10~ 4 )(3.2 x 10- 10 ) Find the perimeter of this figure. Dimensions are in meters. Surface Area 20. y = -2x - Find the missing numbers in this function. X 1 y 3 -2 21. Find the surface area of this cube. Dimensions are in millimeters. 5 22. Find the volume of circular right -10- this cylinder. Dimensions are in 10 centimeters. 23. Find mZb. 24. The triangles are similar. Find x. Solve: 25. 5-x + 3.9 4 = 48 26. 75 2 Add, subtract, multiply, or divide, as indicated: ' 27. 3.2 + V 28. 25 42 — 12 1 5^ 2 8y A f 8.5 V + 1- 595 596 Math 87 29. 1 yd 2 ft 3 in. (_10)(-4) 30. (-4) LESSON 128 + 2 yd 1 ft 10 in. - (3K-2H-1) - (-2) Solving Literal Equations • Transforming Formulas Solving literal A literal equation is an equation that contains letters instead of numbers. We can rearrange (transform) literal equations equations by using the rules we have learned. Example Solve for 1 Solution x: x + a = b We solve for x by getting x alone on one side of the equation. We do this by adding -a to both sides of the equation. x + a = b -a add -a x = b - a added -a Example 2 Solution Solve for To solve x: equation for x, Formulas we divide both sides of the equation by ax = b equation — = -a divided by y a Example 3 a formula is written. Solve for W: A = LW a. simplified it is we can use to solve necessary to change the are literal equations that certain kinds of problems. Often way both sides ax = b x = — Transforming formulas to Solving Literal Equations Solution This is that W is to the right of the equals sign To undo the multiplication by of the equation by L. A = a = L, we m divided by L simplified x - a = b a. Solve for b. Solve for w: wx = y c. The formula for the area of a parallelogram is x: Solve this equation for 1. see is equation W We 597 multiplied by can divide both sides and LW A= Problem set 128 Transforming Formulas a formula for finding the area of a rectangle. L. Practice • bh b. Max paid $20 for 3 pairs of socks priced at $1.85 per pair and a T-shirt priced at $8.95. The sales tax was 6 percent. How much money should he get back? 2. If the spinner is spun once, what is the probability that the spinner will end up pointing to a one-digit prime number? how many 3. At $2.80 per pound, the cheddar cheese costs cents per ounce? 4. Brenda's average score after 6 tests was 90. If her lowest score, 75, is not counted, what was her average score on the remaining 5 tests? 598 Math 87 5. The ordered pairs on the same line. coordinate system and draw the points that 6. 7. Use a lie box and (0, 1) designate Graph the points on a (-2, -1), (2, 3), line. problem. Justin finished problems in 4 minutes. At that rate, how long will take him to finish the remaining 27 problems? The ratio to solve this it was $46. The tax was 6 percent. much money was needed to pay for the glove? price of the glove How 8. Write an equation to solve this problem. is 225 percent of 40? 9. Arrange in order of size from 2, 10. 3 f What number least to greatest: 2 V2, 2 , < Use a ratio box to solve this problem. The ratio of residents to visitors in the community pool was were 60 people in the pool, 2 to 3. If there how many were visitors? Write equations to solve Problems 11 and 12. 66- percent 11. Sixty-six 12. Seventy-five percent of 13. Use 14. Complete the is of what number is 2.4? problem. The number of students enrolled in chemistry increased 25 percent this year. If there are 80 students enrolled in chemistry this year, how many were enrolled in chemistry last year? a ratio box to solve this table. Fraction (a) 15. what number? Graph the positive number line. Decimal [b] Percent 105% integers that are less than 4 on Solving Literal Equations • Transforming Formulas 599 16. Divide 6.75 by 81 and write the quotient rounded to three decimal places. 17. Multiply and write the product in scientific notation. x 10- 10 )(6 x 10- 6 ) (4.8 x2 + bx + c x = 18. Evaluate: 19. Find the area of this figure. Dimensions are in millimeters. 20. Find the surface area of this right triangular prism. Dimen- if 3, b = -5, and c = 6 sions are in centimeters. 21 . Find the volume of this cylinder. Dimensions are in inches. -10 h- 22. Find mZB. 23. Solve for x: x + 24. Solve for x: ax = b c = H d Solve: 25. 12x + 8 = 14 Add, 27. 26. 15 m 8 32 subtract, multiply, or divide, as indicated: 25 - [3 2 + 2(5 - 3)] 600 Math 87 ton - 100 pounds 28. 1 29. 3- 4 30. ( {\( o\ 5 2 6 1V Jv 3 y (_3)(-2)(+4)(-l) + (-3) + (-4) - (-2) Graphing Functions LESSON 129 In the table below we have recorded the values of y that paired with x values of 2,0, and- 1 y - x + X 2 for the function y = x + are 2. 2 y 4 2 -1 1 The word "linear" comes from the word "line." The function y = x + 2 is called a linear function because the graphs of all pairs of x and y for this function lie on the same straight line. In the figure below, we graph the pairs of x and y from the table above and draw the line through these points. The graph of every pair of x and y that y = x + 2 will be on this line. satisfies the equation Graphing Functions Example Make shows three pairs of x and y values for the function y = 2x - 1. Graph these number pairs on a coordinate plane and draw a line through the points to show other a table that number Solution 601 pairs of the function. we think of three numbers we would use in place of x. We decide to use 2, 0, and -2. We use small numbers so that the pairs of points can be graphed on a small First like to graph. y = 2x X 1 y 2 -2 Next we use the equation to find the values of y, and we record these numbers in the table. y = 2x X 2 1 V 3 -1 -2 -5 we have recorded three x, y pairs. They are (2,3), and (-2, -5). Now we graph the points and draw a In the table (0, -1), line through the points. 602 Math 87 We note that, had we decided to use other numbers for the graph of x, the table would contain different x, y pairs. However, these pairs would also be on the line we have drawn. Practice Make a table for each of these linear functions. Find three pairs of x and y draw the graphs Problem set 1 29 for each function. Then plot the pairs and of the functions. a. y = x + b. y = -2x 1. The shirt regularly priced was the sale price? 2 at $21 was on sale for - off. What 2. Nine hundred seventy-five billion is how much than one trillion? Use words to write the answer. 3. What is the (a) range and (b) mode less of this set of numbers? 16, 6, 8, 17, 14, 16, 12 4. Use a ratio box to solve this problem. Riding her bike from home to the lake, Sonia averaged 18 miles per hour (per 60 minutes). If it took her 40 minutes to reach the lake, 5. how far The points (3, did she ride? -2), (-3, -2), and (-3, 6) are the vertices Use find the length of the hypo- of a triangle. Find the perimeter of the triangle. [Hint the Pythagorean theorem to : tenuse.) how many 6. Five sixths of a yard 7. Use a ratio box to solve this problem. The ratio of earthworms to cutworms in the garden was 5 to 2. If there were 140 worms in the garden, how many earthworms were there? is inches? Graphing Functions 603 Write equations to solve Problems 8-10. 8. Sixty is 125 percent of what number? 9. Sixty is what percent of 25? 10. What number 11. Use a is 60 percent of 125? problem. The average cost of a new car increased 8 percent in one year. Before the increase, the average cost of a new car was $16,550. ratio box to solve this What was the average cost of a new car after the increase? 12. 40 green marbles, and 50 blue marbles. If a marble is drawn from the can, what is the probability that the marble will not be red? 13. Complete the In a can there are 30 red marbles, table. Decimal Fraction Percent 5 14. Compare: x + y Qx-y if x is whole number an integer Multiply and write the product in scientific notation. (1.8 x 16. a is and y 15. (b) (a) 6 10 10 ) (9 x 10" 6 ) Betw een which two consecutive whole numbers does a/200 lie? 17. Find three pairs of x and y for the function y = x + 1. Graph these number pairs on a coordinate plane and draw a line through these points. 18. Find the area of meters. this figure. Dimensions are in centi- 604 Math 87 19. Find the surface area of this rectangular solid. Dimensions are in inches. 20. Find the volume of this right circular cylinder. Dimensions ^^ are in centimeters. 10 21. The polygon ABCD is a rectangle. Find D 22. mZx. A Use two unit multipliers to convert 2 m 2 to square centimeters. 23. Solve for x in each (a) x - y = literal z equation. (b) w = xy Solve: 24. 25. 21 Add, 6x + 5 = 7 7 subtract, multiply, or divide, as indicated: 26. 62 + 5(20 - 27. 2 yd - 4 ft [4 5 in. 2 + 3(2 - 1)]| 28. 5- + 3.5 - - 605 Formulas and Substitution ( 29. (0.2) K 3Q A (5.5) 2 J (5)(-3)(2)(-4) + (-2)(-3) -6 Formulas and Substitution LESSON 130 A formula is a literal equation that describes a relationship between two or more variables. Formulas are used in mathematics, science, economics, the construction industry, food preparation, and wherever measurement is used. To use a formula, we replace the letters in the formula with measures that are known. Then we solve the equation for the measure we wish to find. Example 1 Solution Use the formula d = rt to find t when d is 36 and r is 9. This formula describes the relationship between distance (c/), rate (r), and time [i). We replace d with 36 and r with 9 and then solve the equation for t. d = formula rt 36 = 9t t Another way to find d d = 4 = _ divided by 9 to first solve the is t -- = substituted rt t formula for t. formula divided by r r Then replace d and r with 36 and 36 9, respectively, = t substituted 4 = t divided 9 and simplify. 606 Math 87 Example 2 Solution Use the formula F = 1.8C + 32 F when Cis to find 37. This formula is used to convert measurements of temperature from degrees Celsius to degrees Fahrenheit. We replace C with 37 and simplify. F = 1.8C + 32 formula F = 1.8(37) + 32 substituted F = 66.6 + 32 multiplied F = 98.6 added Thus 37 degrees Celsius equals 98.6 degrees Fahrenheit. Practice a. Use the formula A = bh A = -bh to find b when A is 20 and h when A is 20 and h is 4. b. Use the formula to find b 2 is 4. Problem set c. Use the formula F = 1.8C + 32 1. The main course 130 to find F when Cis -40. The beverage cost $1.25. Dessert cost $2.40. Jason left a tip that was 15 percent of the total price of the meal. How much money did Jason cost $8.35. leave for a tip? 2. Twelve hundred-thousandths is how much greater than twenty millionths? Write the answer in scientific notation. 3. Arrange the following numbers in order. Then find the median and the mode of the set of numbers. 8, 12, 9, 15, 8, 10, 9, 8, 7, 4 4. One card is to be drawn from a normal deck What is the probability of drawing a 5? 5. Use a ratio box to solve this problem. Milton can exchange $200 for 300 Swiss francs. At that rate, how many dollars would a 240-franc Swiss watch cost? of 52 cards. Formulas and Substitution 607 how many pounds? 6. Three eighths of a ton 7. problem. The jar was filled with red beans and brown beans in the ratio of 5 to 7. If there were 175 red beans in the jar, what was the total number of beans in the jar? Use box a ratio is to solve this Write equations to solve Problems 8-10. 8. What number 9. Ten percent of what number 10. Fifty-six is 11. Use season, the If percent of 800? is 2500? what percent of 700? a ratio percent. is 2.5 box to solve this room problem. During the off- were reduced by 35 were $90 per day, what would rates at the resort the usual rates be the cost of a 2-day stay during the off-season? 12. What is the area of the shaded region of this rectangle? »- X 13. Use a ratio box to solve this floor plan of her house. feet. What is On problem. Liz is drawing a the plan, 1 inch equals 2 the floor area of a room that measures 6 inches by 7^ inches on the plan? 14. Complete the table. Fraction ij Decimal (a) Percent (b) 608 Math 87 15. Multiply and write the product in scientific notation. (2.8 x 10 5 )(8 x lCr 8 ) 16. Use the formula c = 2.54n 17. Make 18. Find the perimeter of this figure. Dimensions are in to find c is 12. shows three pairs of numbers for the function y = 2x. Then graph the number pairs on a coordinate plane and draw a line through the points. a table that 4 inches. 19. when n Find the surface area of this cube. Dimensions are in inches. 20. Find the volume of this right circular cylinder. Dimensions are in centimeters. 21. FindmZx. 150 22. The triangles are similiar. Find y Dimensions c are in centimeters. xy<\Q y 23. 10 Use the Pythagorean theorem to find x in the triangle above. Simple Interest 609 Solve: 24. —=— 35 Add, 26. 1-x - 25. 5 28 - {21 - 3[2 + 6 lb 10 oz 27. 2(2)]} 1 2.75 - 1- V (-6) - 3- 29. 0.5 4 3 (7)(-4) 8 oz lb \ ( 30. = 32 subtract, multiply, or divide, as indicated: + 2 28. 3 3 + 5 + • 2- + 10 3 (~8)(-9) (-3) (-2) Simple Interest LESSON 131 When you money bank uses your money to make more money. The bank pays you to let them use your money. The amount of money you deposit is called the principal. The amount of money they pay you is called interest. The interest is a percentage of the money deposited. If you deposit $100 at 6 percent simple interest,* the bank will pay you $6 a year to use your money. If you take your money out after 3 years, the bank will pay you a total of put your in a bank, the $118. $100.00 $6.00 $6.00 $6.00 third year interest $118.00 total principal first year interest second year interest *With simple interest, interest is paid on the principal only and not on previous interest earnings. Most bank accounts pay compound interest, which pays interest on previous interest earnings. 610 Math 87 Example Roger deposits $700 in the bank for 3 years. end of Solution First How much money percent simple interest will he have in the bank at the at 8 3 years? we calculate the interest for 1 year. $700.00 x 0.08 $ 56.00 The interest for 3 years will 3 At the end of Practice a. $3000 Problem set at x $56.00 = $168.00 3 years, How much be he will have $700.00 $168.00 principal $868.00 total interest for 3 years earned in 3 years on a deposit of 7 percent simple interest? interest is b. Jena deposited $5000 at 7 percent simple interest. Six months later she withdrew the deposit and the interest she had earned. How much money did she withdraw? {Hint: Consider six months as — of a year.) 1. bought 3 paperback books for $5.95 each. The tax rate was 6 percent. If he pays for the purchase with a $20 bill, how much money should he get back? 131 2. Bill When the sum and -, what is of | and | is divided the quotient? by the product of - 3. What 4. George burned 100 calories running 1 mile. How many miles would he need to run to burn 350 calories? 5. If a is the sum of the first five positive dozen roses cost $4.90, what roses? is odd numbers? the cost of 30 611 Simple Interest how many 6. Three fourths of a day 7. The average of four numbers was 8. Three of the numbers were 2,4, and 6. What was the fourth number? is hours? Write equations to solve Problems 8-10. 8. One hundred 9. What number fifty is is 10. Sixty percent of 11. The points (3, 1), a right triangle. what percent of 60? 60 percent of 60? what number (-1, 1), What is is 150? and (-1,-2) are the vertices of the length of the hypotenuse of the right triangle? Use a 12. ratio The box The 13. was reduced by 40 what was the regular price of the dress sale price 13. Problems 12 and to solve car the car was $48, model was is built 180 inches, on a 1:36 percent. If the price? scale. If the length of how many inches Jong is the model? 14. The square root of 80 whole numbers? 15. Complete the table. is between which two consecutive Fraction Decimal 1.25 (a) Percent (b) 16. Graph x > -2 on a number 17. Multiply and write the product in scientific notation. (6.3 18. line. x 10 7 )(9 x 10- 3 ) The probability that it will rain is that it will not rain? -. What is the probability 612 Math 87 19. How much $4000 at earned in 5 years on a deposit of 9 percent simple interest? interest is mc 20. Find the value of 21. These triangles are 22. Find the volume of triangular prism. if x m similar. = c = 10, and x - 2, Find x. this Dimen- sions are in inches. 23. Find mZx. Solve: 44 11 24. 25. 42 Add, 4 = 36 3 subtract, multiply, or divide, as indicated: 26. 16 - {27 - 3 [8 - 27. 60 mi 1 hr ( 29. 3-w - 20 + 1 hr 60 min 1 ^ 3- * 1- 3 5 (3 2 - 2 3 )]} 28. 3- + 1.5 + 4- 3 30. (-3) 2 + (-2) 3 6 5. 3 . Compound Compound Probability LESSON 132 61 Probability We know that the probability of getting heads on one toss of a coin is -. We can state this fact with the following equation which P{H) stands in for "the probability of heads." P{H) = 2 The probability of getting two heads in a row We can see this if we draw a diagram. time, we can get heads or tails. If the up If first either the either first we comes up heads, the second toss heads or toss If times - , or - toss a coin one toss could come tails. came up heads or is - tails, the second toss could come up tails. — HH HT TH TT We see that there are 4 possible outcomes. HH The HT TH They are TT probability of each of these 4 outcomes is one fourth. 61 4 Math 87 Thus, the probability of getting HH is P (H, H) = \ . \ The probability of independent events occurring in a specified order is the product of the probabilities of each event. P{H,H,T) Thus ' P so P and - is 1 2 ' ' [H, T, T, • H) - • 2 - 1 2 8 is 16 f^ [T, H, T, H, T) is 2 K J Example 1 What the Solution is the probability of getting a 2 on first spin and a 1 on the second spin? The a 1 probability of getting a 2 is -. The \ . The probability of getting probability of independent events ocCTjrring in a specified order is the product of the individual probabilities. P{2, Example 2 is 32 1) = - - = — 4 16 • 4 Jim tossed a coin once and it turned up heads. What is the probability that he will get heads on the next toss of the coin? Solution Past events do not affect the probability of future events. There are only 2 possible outcomes. The next toss of the coin will turn turn Example 3 Solution up either heads or tails. up heads What The probability that it will is \. the probability of rolling a 4 above 3 in two rolls of a single die? is P(4) = 6 and then P{>3) = a number . Compound P(4,>3) = So 12 6 Practice 615 Probability Use this information to answer each question: The probability of a bird is \ The probability of a dog is . a. What is the probability of getting bird, bird, dog in that order? b. What is the probability of getting bird, dog, bird, dog in that order? Problem set 132 Sherman deposited $3000 1. an account paying 8 percent simple interest yearly. He withdrew his money and interest 3 years later. How much money did he withdraw? 2. What is in the square root of the sum of 3 squared and 4 squared? 3. Find (a) the median and (b) the mode of the following quiz scores. Class Quiz Scores 4. Score Number of Students 100 2 95 7 90 6 85 6 80 3 70 3 The trucker completed the 840-km haul in 10 hours 30 minutes. What was the trucker's average speed in kilometers per hour? 5. Use $28 a ratio box to solve this problem. Barbara earned hours of work. At that she earn for 9 hours of work? 6. A for 6 rate, how much would mile is about eight fifths of a kilometer. About many meters is eight fifths of a kilometer? how 61 6 Math 87 7. 60 percent of the students were boys, what was the ratio of boys to girls? If (-2,-1), and (-2, 11) are the vertices of a right triangle. Use the Pythagorean theorem to find the length of the hypotenuse of this triangle. 8. The points 9. Use a ratio a jacket at (3, 11), Mike paid $48 for 25 percent off of the regular price. What was box to solve this problem. the regular price of the jacket? Write equations to solve Problems 10 and 11. 10. What decimal number is 11. What percent 2? 12. Use a of 2.5 box is 2 percent of 360? Troy bought a baseball card for $6 and sold it for 25 percent more than he paid for it. How much profit did he make on the ratio to solve this problem. sale? 13. What is the probability of having a coin turn up tails on 4 consecutive tosses of a coin? 14. How much deposited 15. at earned in 6 months on $4000 9 percent simple interest? interest is Complete the table. Decimal Fraction Percent 5 (a) 8 16. (b) Multiply and write the product in scientific notation. (3 x 10- 4 )(4 x 10 8 ) 17. Convert 300 kg 18. Solve the formula d = rt for to grams. t. Compound 19. Make 20. Find the perimeter of this figure. Dimensions are in Probability shows three pairs of numbers for the function y = -x. Then graph the number pairs on a coordinate plane and draw a line through the points. a table that centimeters. 21. Find the surface area of this right triangular prism. Dimensions are in feet. 22. What fraction of 42 23. These Find triangles are similar. x. rotate one triangle so that like is 7? Mentally [Hint: looks it 8 x other the triangle.) 10 Solve: 16 48 2.5 / 24. Add, 25. 52 27. 1 3 = 21 3 - - 2(10 + [40 3 yd 2 1 2- 4 30. 2-x - subtract, multiply, or divide, as indicated: 26. 28. 617 (4) ( 1.H V 3ft ft yd - 3 2 )] 1 1] yd 29. 3.5 *) - (-3X-2H-D + (-5M4X-3K2) -1 618 Math 87 Volume of a Pyramid and a Cone LESSON 133 Pyramids A pyramid is a geometric solid that has three or more triangular faces and a base that a polygon. is Each of these figures is a pyramid. The volume of a pyramid is - the volume of a prism that has the same base and height. To find the volume of a pyramid, we will first find the volume of a prism that has the same base and height. Then we will divide the result by 3. Example 1 The cube just contains the pyramid. Each edge of the cube Solution is 6 centimeters. (a) Find the volume of the cube. (b) Find the volume of the pyramid. (a) The volume of the cube equals the area of the base times the height. Area of base: 6 cm x 6 Volume = base (b) = 36 cm2 height = (36 cm 2 )(6 cm) = 216 cm3 The volume of the pyramid is | the volume of the cube. Dividing by 3 (or multiplying by |), we find that the volume of the pyramid is 216 Cones The volume the • cm of a cone same base and cm 3 is ^ height. = 72 the cm 3 volume of the cylinder with 9 Volume Example 2 Solution of a Pyramid and a Cone 61 Find the volume of this circular cone. Dimensions are in centimeters. We volume of a cylinder with the same base and height as the first find the cone. Volume of cylinder = area of circle = (3.14) (10 = 9420 Then we • cm) 2 height • 30 cm cm 3 find - of this volume. Volume of cone = - (volume 1 of cylinder) cm 3 9420 3 = 3140 Practice cm3 Find the volume of each figure. Dimensions are in centimeters. b. a. Problem set 133 1. box to solve of the item was $24, but Use a What 2. Ten ratio is problem. The regular price was on sale for 25 percent off. this it the item's sale price? billion is how much greater than nine hundred eighty million? Write the answer in scientific notation. 3. The median of the following numbers than the mean? 1.4, 0.5, 0.6, 0.75, 5.2 is how much less 620 Math 87 4. 5. 6. 7. Nelda worked for 5 hours and earned $24. How much did Nelda earn per hour? Christy worked for 6 hours and earned $33. How much did Christy earn per hour? Christy earned how much more per hour than Nelda? 24 kilograms of seed costs $31, what is the cost of 42 kilograms of seed at the same rate? Use a ratio box to solve the problem. If kilometer is about - of a mile. A mile kilometer is about how many yards? A A card was drawn from is 1760 yards. a deck of 52 playing cards. card was then replaced. Another card was drawn. was the probability that both cards were hearts? Write equations to solve Problems 8 and 8. What percent 9. Fifty percent of 10. Trinh left of $30 is The What 9. $1.50? what number $5000 in an account interest annually. A 1 is 2-? that paid 8 percent simple How much interest was earned in 3 years? 11. Use a ratio box to solve this problem. A merchant sold an item at a 20 percent discount from the regular price. If the regular price was $12, what was the sale price of the item? 12. The points triangle. (0, 4), (-3, 2), and Find the area of the 13. Use two unit multipliers 14. Use a (3, 2) are the vertices of a triangle. to convert 6 ft 2 to square inches. box to solve this problem. Jessica sculptured a figurine from clay at ~ of the actual size of the model. If the model was 6 feet tall, how many inches tall was the ratio figurine? Volume 15. Complete the of a Fraction table. Pyramid and a Cone 16. 0.5 (b) Multiply and write the product in scientific notation. (6.3 17. Percent Decimal (a) 621 x 10 6 )(7 x 10" 3 ) Tim can get from point A to point B by staying on the sidewalk and turning left at the corner C, or he can take the shortcut and walk straight from point A to point B. Shortcut 40 yd How many 30 yd yards does he save by taking the shortcut instead of staying on the sidewalk? Begin by using the Pythagorean theorem to find the length of the shortcut. 18. (a) Solve the formula A = -bh for h. 2 \ (b) Use the formula A = - bh 2 b = 8. to find 19. Make 20. Find the volume of the pyramid. Dimensions are in meters. 21. Find the volume of the cone. Dimensions are in centimeters. h when A = 16 and shows three pairs of numbers for the function y = — 2x + 1. Then graph the number pairs on a coordinate plane and draw a line through the points to show other number pairs of the function. a table that 622 Math 87 22. Refer to the figure to find the measure of each angle. 23. (a) Z.B (b) Z£ (c) AA Problem 8 cm. Find CD. In the figure in EA is 22, EB is 4 cm, BD is 6 cm, and Solve: 24. 25 7 -4- 25. 1-w + 16 Add, 26. 27. subtract, multiply, or divide, as indicated: - 52 {4 - [3 2 - lmin min 60 sec 2 3 (2 2 - 3 1 l 2 )]) ft yd (-7) LESSON 134 3 - (-3) ^ «v V 4 y v 3y *i 1- + 2- - 34 30. 2 440 yd 1 28. 17 = 49 5 6 - o\ 29. (2)(-3) 6 + (-4) - (-3K-2K-D Probability, Chance, and Odds and odds are different ways of expressing the likelihood that an event will occur. Recall that probability is the ratio of the number of ways a particular event can happen to the total number of equally likely possible outcomes. Probability, chance, Probability = number of favorable outcomes number of possible outcomes Thus, the probability that this spinner will end up in region AisT. . Probability, Chance, and Odds 623 We can use the word chance and a percent to describe a probability. Since the fraction - is equivalent to 25 percent, we can say that the chance of the spinner stopping in region A is 25 percent. While probability is the ratio of the number of favorable outcomes to the number of possible outcomes, odds is the ratio of the number of favorable outcomes to the number of unfavorable outcomes. Using the spinner example, one outcome is A, and three outcomes are not A. Thus, the odds of the spinner ending up in region A are 1 to 3 Note that odds are usually expressed by using the word "to" and not by using a division line. However, odds are reduced as are other ratios. Example 1 Solution A 20 percent chance of rain was forecast. (a) What is (b) What are the (a) the probability that odds that it it will rain? will rain? To express chance as probability, we simply write the percent as a fraction and reduce. 200/0 The probability that it = -25- = I 100 5 will rain is — -1 (b) Since the probability of rain is -, the probability that it will not rain is -. Thus, for every favorable outcome there are 4 unfavorable outcomes. Therefore, the odds that it will rain are 1 to 4. 624 Math 87 Example 2 Solution The odds drawn from that a marble a bag will be red is 3 to 2. (a) What is the probability that a red marble will be drawn? (b) What is the chance of drawing a red marble? If the odds are 3 to (a) (b) The 2, we mean 3 favorable outcomes 2 unfavorable outcomes 5 possible outcomes probability of drawing a red marble favorable 3^ possible 5 The chance of drawing a red marble is is - = 60% Practice Use this information to answer questions a-d. In a bag there are 4 blue marbles, 3 red marbles, 2 green marbles, and Problem set 1 yellow marble. a. What is b. What are the c. What is d. What are the 1. The regular price off. What was the total sale price including Use box 134 tax? the chance of drawing a green marble? the chance of drawing a blue or yellow marble? sales tax 2. On odds of drawing a blue marble? odds of drawing a red or green marble? a ratio and was $72.50, but it was on to find the sale price. sale for 7% Then 20% sales find the total price. average score was 87. What score does he need to average on his next 2 tests to have a 6-test average of 90? his first 4 tests, Eric's Probability, 3. In a bag there are 6 red marbles, 9 green marbles, One marble blue marbles. What (a) is to 625 Chance, and Odds and 12 be drawn from the bag. the probability that the marble will be is blue? 4. (b) What is (c) What are the a If the chance that the marble will be green? odds that the marble will not be red? box of 12 dozen pencils costs $10.80, then what is the cost per pencil? 5. How much interest is earned on $5000 at 8 percent simple interest in 6 months? 6. One fourth of the students in the class earned an A. third of the students earned a B. A, Use a 7. how many ratio The box 6 students earned an students earned a grade lower than a B? to solve Problems 7 and 8. by the checkpoint was and trucks passed by the checkpoint, ratio of cars to trucks passing 3500 cars many were cars? 5 to 2. If how 8. If One There were 20 percent more rainy days in April than there were in March. If there were 10 rainy days in March, then how many rainy days were there in April? [Hint: Let the number of rainy days in March equal 100 percent.) Write equations to solve Problems 9 and 10. 9. What is 120 percent of $240? 10. Sixty is what percent 11. The points (3, 2), (6, of 150? -2), (-2, -2), and vertices of a trapezoid. (a) Find the area of the trapezoid. (b) Find the perimeter of the trapezoid. (-2, 2) are the 626 Math 87 12. Arrange these numbers in order from least to greatest. 2 V6, 6 -6, 0.6 , 13. Complete the table. Fraction Multiply and write the product in scientific notation. (2 15. Percent (b) (a) if 14. Decimal x 10- 6 )(5 x 10- 9 ) Divide 0.02 by 1.1 and write the answer with a bar over the repetend. 16. Convert 12 inches to centimeters. 17. (a) (b) 18. Solve the formula C (1 in. = nd for Use the formula C = nd and ;ris 3.14. = 2.54 cm) d. to find d when C is Find three pairs of numbers for the function y = 2x + 1. Then graph the number pairs on a coordinate plane and draw a line through the points to show other number pairs of the function. 19. Find the perimeter of this figure. Dimensions are in centimeters. 20. Find the surface area of this cube. Dimensions are in feet. 3 21. 62.8 Find the volume of this cylinder. Dimensions are in meters. h- -10- -H Volume, Capacity, and Weight 22. in the Metric System 627 Refer to the figure to find (a)-(c). mZACB mZCAB mZCDE (a) (b) (c) 23. 20 cm deep is filled with water. Find the volume of the water in the aquarium. An aquarium that is 40 cm long, 10 cm wide, and Solve: 24. 0.8m - = 6 1.2 25. — — = 3.2 24 Simplify: 26. 92 - V5T + V9 28. (1.2) ( 1 29. 2- - 3 1 -*• V 30. i\ 4- 4y 2 kilogram - 50 grams 27. 1.5 1 - - 4 6 (-3X- Volume, Capacity, and Weight in the Metric System LESSON I 135 Metric units of volume, capacity, and weight are related. The relationship describes the volume and the weights of a quantity of water under certain standard conditions. There are two commonly used references. One milliliter of One cubic centimeter can contain 1 milliliter of water has a volume of 1 cubic centimeter and a weight of 1 gram. water, which has a weight of 1 gram. f-7\ 628 Math 87 water has a volume of 1000 cubic centimeters and a weight of 1 kilogram. One liter of ^^z '/./.// / / One thousand cubic cen- z777 7777 /Zx ////// ' y / / / / /// Z* / / / / / / // 2 liter of water, which has / X , timeters can contain 1 , ////////// 7-Z / / 71 ' / a weight of 1 kilogram. v'/ // v / // V < A V < v'< /,* ' Example 1 Ray has a fish v V aquarium 50 cm long and 20 cm wide. If the aquarium is filled with water to a depth of 30 cm, (a) how many liters of water would be in the aquarium, and (b) what is the weight of the water in that /A V vV V is 30 cm 20 cm 50 cm the aquarium? Solution we First find the volume of the water (50 cm)(20 cm)(30 cm) = 30,000 (a) Each cubic centimeter of water thousand milliliters is 30 liters. (b) Each liter is 30 kg. wanted to find the volume of a vase. She Jan beaker with water and then used all but 240 milliliters to filled fill a 1 -liter the vase. is cm 3 1 milliliter. Thirty of water has a weight of 1 kg, so the weight of the water in the aquarium Example 2 in the aquarium. 1 L Volume, Capacity, and Weight (a) What was (b) If volume the Solution (a) of the vase? the weight of the vase weight of the vase 629 the Metric System in filled was 640 grams, what was the with water? beaker contained 1,000 mL of water. Since Jan used 760 mL (1,000 mL - 240 mL), the volume of the inside of the vase was 760 cm 3 The 1-liter . (b) The weight vase (640 Practice g) is 1400 g. a. What is the weight of 2 liters of water? b. What is the c. When the bottle was filled with water, the weight increased by Problem set of the water (760 g) plus the weight of the 1 kilogram. d. A 1. How much 135 3 liters of water? How many milliliters of water were added? tank that is 25 cm long, 10 cm wide, and 8 can hold how many liters of water? of 2. volume of $7000 at With two cm earned in 9 months on a deposit 8 percent? [Hint 9 months is ^ of a year.) interest is tosses of a coin, (a) What is the probability of getting two heads? (b) What is the chance of getting (c) What are the On deep two tails? odds of getting heads, then tails? 3. 4 days of their trip the Schmidts averaged 410 miles per day. On the fifth day they traveled 600 miles. How many miles per day did they average for the first 5 days of their trip? 4. The 18-ounce container cost $2.16. The 1 -quart container cost $3.36. The smaller container cost how much more the first per ounce than the larger container? 630 Math 87 Use a 5. ratio that rate, word The Problems and 5 6. 160 words in 5 minutes on his typing test. how long would it take him to type an 800- essay? guinea pigs to rats running the maze was 7 Of the 120 guinea pigs and rats running the maze, many were guinea pigs? ratio of to 5. how 7. to solve Adam typed At 6. box was thinking of a certain number. If was 48, then what was -o of the number? Kelly Write an equation to solve Problems 8 and 8. What percent of $60 9. What is 10. fraction The points of the number 9. $20? is 50 percent of -? (-3, 4), (5, -2), and (-3, -2) are the vertices of a triangle. 11. (a) Find the area of the (b) Find the perimeter of the triangle. triangle. A glass aquarium with dimensions as 5 kg shown has when a mass of What empty. 20 cm is the mass of the aquarium when it is half full of water? 20 cm 25 cm 12. Complete the Fraction table. 0.875 (a) Q - b Decimal Percent (b) 13. Compare: a + b 14. Multiply and write the product in scientific notation. (6.4 a a is positive and b negative if x 10 6 )(8 x 10- 8 ) is . Volume, Capacity, and Weight 15. Convert 36 inches to centimeters. 16. (a) Solve the formula (b) Use the formula and h is 6. 17. A A = -bh = \bh in 631 the Metric System (1 in. = 2.54 cm) for b. to find b when A is 24 Find three pairs of numbers for the function y = -2x. Then graph the number pairs on a coordinate plane and draw a line through the points to show other number pairs of the function. 18 . Find the area of this figure Dimensions are in millimeters. 19. Find the surface area of the cube. Dimensions are 100 in centimeters. 100 100 20. Find the volume of the cylinder. Dimensions are H— 40- in inches. 21. Refer to the figure to find (a)-(c). (a) (b) (c) 22. mZYXZ mZWXV mZWVX Problem 21, ZX is 21 cm, YX is 12 cm, 14 cm. Write a proportion to find WV. In the figure in and XV is 632 Math 87 23. A pyramid is cut out of a cube of plastic with dimensions as shown. the What 6 in. is volume of the pyramid? 6 6 in. in. Solve: 24. + 0.4/7 5.2 = 12 25. 18 y Add, subtract, multiply, or divide, as indicated: 2 26. V5 28. 3.5 - + V5 2 - 4 2 32 ( + 2 1- V {3 30. 36 28 ^ ^) 3 3 29. 12.75 + 2 ft + (_8) + (_4K+5) " (2K_3) 1 in. 3- + 2 5 ] 6 -s- yd - 27. 8 Appendix Supplemental Practice Problems for Selected Lessons This appendix contains additional practice problems for concepts presented in selected lessons. It is very important that no problems in the regular problem sets be omitted to make room for these problems. This book is designed to produce long-term retention of concepts, and long-term practice of all the concepts is The practice problems in the problem sets provide enough initial exposure to concepts for most students. If a student continues to have difficulty with certain concepts, some of these problems can be necessary. assigned as remedial exercises. Supplemental Practice Problems Supplemental Practice for List the whole numbers from 1-10 for Selected Lessons that are factors of: 1. 36 2. 3600 3. 350 4. 1326 5. 4320 6. 950 7. 12,000 8. 35,420 9. 36,270 123,450 1. 1,000.000 Lesson 5 10. Supplemental Reduce each Practice for Lesson 16 2. 20 24 24 24 16 24 18 28 8. 3±5 10. 6 36 18 11. 35 8 24 18 Supplemental 4. 3. 32 30 9. 12 _8_ 6. 5. 2520 fraction to lowest terms. 15 i- 12. 12. 4 ii 32 15 Write the prime factorization of each of these numbers. Practice for 1. 81 2. 300 3. 2000 4. 625 5. 450 6. 1200 7. 440 8. 750 9. 10,000 10. 128 11. 780 12. Lesson 22 Supplemental Simplify: Practice for Lesson 23 1540 l. — 28 2. 3. 48 40 6. 7. 15 15 9. 3 10. 4 16 12 36 4. 10 12 8 6 30 50 36 8. 12 ii. 9 15 6 8 12. 8 24 10 635 636 Math 87 Supplemental Add or subtract, as indicated: Practice for Lesson L.e*3>uii 26 ô l. 4. 7. 5- + 25 5 *} S 5- + 34 4 7 1 6- - 2- 5- - 13 Supplemental 2. 8 5. o o 7- + 1- 3. 5 7 6- + 5- 7 6. - 3- 5 9. 6 - 11. 5 4- - 15 3 12. 6- - 26 5 Multiply or divide, as indicated: 41 | 2^ x 3 3. i| x 3^ 2- 5. - x 3- 6. 2- x 1- 8 5 3| x 4. 7 x 3 1 3- * 3 8. 4 3- 1- -s- Supplemental Add 11. 9. 5 12 1- 4 2- 4- 2 6-5-3- 4 3 4 3 3 2- + - 2 10. 6 2. 1. 7. S 2- 4 Practice for Lesson 30 9 9 1 8. 7 5 7 8- + 2- 8 8 4 2- + 3- 8 8 8 10. ~ . q 12. 3 - - 38 5 2 or subtract, as indicated: Practice for Lesson35 1. f + £ 3 4. 6 4 5 1 8 2 2 . 2 5. Q 3 l | + a 3 1- 5 7. 8. 8 2 12 3 11. 3 + | + | 5 9. 1 12 8 4- - 14 12. 7 3 1 6 2 6- - 35 _ f 6. 3 3- - 16 10. 2 9 2— 3. 3 5- - l| 4 6 1 Supplemental Practice Problems for Selected Lessons Supplemental Practice for __ Lesson 37 Name 637 these decimal numbers. -, 1. 1K10 16.125c 9 2. no 5.03 3. 105.105 4. 0.001 5. 160.166 6. 4000.321 r Write each of these as decimal numerals. Supplemental practice for 7. One hundred 8. One hundred and 9. One hundred twenty and twenty-three thousandths twenty-three thousandths three thousandths 10. Five hundredths 11. Twenty and nine hundredths 12. Twenty-nine and 13. One thousand and two hundred twelve thousandths 14. One thousand two hundred and twelve thousandths Round ± to the nearest 23.459 five tenths whole number. 2. 164.089 3. 86.6427 6. 0.1084 9. 3.14159 Lesson 38 Round 4. two decimal 12.83333 Round 7. to places. 5. 6.0166 to the nearest thousandth. 0.08333 8. 0.45454 10. Round 283.567 to the nearest hundred. 11. Round 283.567 to the nearest hundredth. 12. Round 126.59 to the nearest ten. 638 Math 87 Supplemental Practice for Add or subtract, as indicated: 1. 45.3 + 2.64 + 3 3. 3.6 2. 0.4 + 0.5 + 0.6 + 0.7 4. 12.8 + 6.32 + 15 6. 10 + 1.0 + 0.1 + 0.01 Lesson 40 '5. 7. 10. Supplemental Practice for Lesson 42 + 2.75 + 0.194 + 3 278.4 + 3.26 + 1.475 14.327 - 6.5 8. 10 - 4.76 11. - 9.67 9. - 0.019 12. 10.8 0.1 - 4.321 6.5 - 4.937 5 Find the perimeter of each of these polygons. Dimensions are in centimeters. 8 1 2 8 12 6 10 20 15 3. 4. - 5 5 12 9 I 10 20 14 16 12 18 Supplemental Practice for Lesson 48 Change each of these numbers to a reduced fraction or mixed number. 1. 0.48 2. 3.75 3. 0.125 4. 12.6 5. 0.025 6. 1.08 Change each of these numbers 8. 7. 8 I 3 to a decimal number. 9. 25 Supplemental Practice Problems 10. 6 11. 6 Supplemental Practice for for Selected — 20 12. 639 Lessons 59 Complete each division. * 1. 0.15 4. 5 7. 4.3 - 0.01 10. 4.5 h- 0.5 2. 14.4 * 0.06 3. 18 * 0.4 5. 12.5 * 0.04 6. 288 -h 8. 1.5 9. 9 1.8 Lesson 50 Supplemental Practice for 0.8 -5- 11. 2.5 8 - 0.12 - 0.04 12. -s- 1.2 12.5 * 0.5 Simplify: 1. 82 5. 32 f 2. 2 6. 52 3. 3 : 4. 10 5 7. 4 : 8. 15 2 Lesson 52 + 2 : 10 4 10 Supplemental Practice for 3 4< 82 10 9. - - ¥ 11. 25 2 12. 54 - 53 Change: 1. 40 inches 2. 200 seconds to feet and inches Lesson 58 to minutes and seconds Simplify: 3. 3 21 ft Add and 3 5. + Supplemental 1 4. in. 2 hr 90 min simplify: yd yd 2 ft 7 in. 1 ft 8 in. Find the area of each Practice for 6. + 2 figure. min 23 hr 45 min 48 5 hr 18 Lesson 60 9 10 sec Dimensions are in centimeters. 2. l. sec 8 8 640 Math 87 4. 3. J 5 8 4 10 15 12 6 5 18 Supplemental Practice for Subtract: 1. 5 2. 10 3. 4 yd 6 4. 1 ft - 7 in. 3 ft 10 in. Lesson 66 - 13 sec in. hr 10 8 5. Supplemental min 1 yd yd - 2 - 3 min 28 ft 8 in. min - 25 min 40 2 4 ft sec sec 6. in. - 2 ft 9 in. Copy and complete the min 30 hr 48 min 43 3 hr 17 sec 2 sec table Practice for Fraction Lesson 71 Decimal Percent (1) (2) 5 6 1.2 (3) (5) 8% (6) If (7) (8) 0.075 (9) (11) Supplemental (4) (10) 125% (12) Evaluate Practice for 1. ab - be + abc 2. xy + Lesson 75 x y 5 if if a - x = 8 5, b = 4, and y = 4 and c = 2 Supplemental Practice Problems for Selected Lessons 3. abc - ab a = if b = 6, and c = 4, 641 3 c 4. Supplemental m - mn m if 5. wx + xz - 6. ab - ac z = —3 and n = -1 4 = 1.2, a = 4, if iv ab if 2 x = b = and z = 0.5, and c = 3, 0.1 2 Find each sum. Practice for l. (-36) + (+54) 2. (-15) + (-26) 3. (-6) + (-12) + (+15) 4. (+4) 5. (_ 6 ) 6. (-9) + (-15) + (+50) 7. (+42) + (-23) + (-19) 8. (-54) + (+76) + (-17) Lesson 78 ( 9. V 11. Supplemental + -A -32 / (_ 8 ) + (_ 7 ) + + (-12) + (+21) (_ 2 ) ( ( -2- + v 4; V (-4.3) + (+2.63) Find the area of each + 10. 12. triangle. 3 v / 5^ +2- 6y + (-3.2) + (-1.8) (-1.7) Dimensions are in centimeters. Practice for 3. Lesson 79 6. 642 Math 87 Supplemental Practice for Lesson 80 Supplemental Practice for Lesson 85 Change each percent 1# what is 50 o/o of 250 ? 2. What is 5% 3. What is 25% 4. What is 90% 5. What is 100% 6. What is 1% 7. What is 8% 8. What is 75% 9. What is 80% of 21? 10. What is 2% 11. What is 70% of 80? 12. What is 80% of 48? of 65? of 48? Find the circumference of each of 40? of 30? of 5000? of 64? of 600? of 70? Dimensions are in circle. centimeters. l. 3. 2. for n Use 4. 22 -j for n Leave n as n 6. 5. Use 3.14 Wessons decimal number before multiplying. Use 3.14 Supplemental to a fraction or for Use n 22 -j for n Leave n as n Rewrite each subtraction or addition. Then find the sum. L M) " M) 2 " - (-15) 3. (+8) 5. (-3) + (-4) - 7. (-2) - - (-3) M2) + (+20) 4. (+6) - (18) (-5) 6. (+3) - (-4) (-4) 8. (+2) - (3) - - (+5) (-4) Supplemental Practice Problems for Selected Lessons Supplemental Practice for 9. (-6) - (-7) + (8) 11. (-3) - (-1) - (-8) - (2) 10. (+8) - (+9) - (-12) 12. (-9) - (10) - (-11) Express in the customary form of scientific notation. 1. 0.15 x 10' 2. 48 x 10- 8 3. 20 x 10 5 4. 0.72 x 10- 4 5. 0.125 x 10 12 6. 22.5 x 10~ 6 7. 17.5 x 10 10 8. 0.375 x 10" 8 Lesson 89 Supplemental Practice for Find each area. Dimensions are in centimeters. 2. l. Lesson 96 10 12 10 15 18 12 3. 8 10 Ll 24 20 Supplemental Simplify: Practice for Lesson 97 62 2. 1. « 100 6-% 6. 10 3. 100 3 Change each percent 5. 16 to a fraction. 87-% 7. 83-% 3 8. 3-% o 643 644 Math 87 Find "what percent." Supplemental Practice for what percent of 40? Sixteen is 10. Eight what percent of 12? 11. What percent of 30 is 6? 12. What percent of 30 is 5? 9. is Find the area of each circle. Dimensions are in centimeters. 2. l. 3. Lesson 103 Use 3.14 for 4. Practice for 22 -j for k Leave n as n 5. Use 3.14 Supplemental Use n for 6. n Use 22 -j for Leave n as n n Write each product in scientific notation. ± (1 2 x 1Q5 ^ 3 x 1Q6) 2 (3 3. (2.5 x 10 5 )(4 x 10 7 ) 4. (4.2 5. (4 x 10" 3 )(2 x 10- 8 ) 6. (6 7. (2 x 8. (1.6 x 9. (6 x 10- 4 )(4 x 10 8 ) Lesson 104 11. 10- 4 )(6.5 x 10" 8 ) (1.4 x 10 7 )(8 x 10- 5 ) x 1Q6 ^ 6 x ^ x 10 8 )(2.5 x 10 12 ) x 10" 7 )(4 x 10" 5 ) lQ- 5 )(7 x lO" 7 ) x 10- 9 )(3 x 10 5 ) 10. (7 12. (7.5 x 10" 8 )(4 x 10 6 ) 1 Supplemental Practice Problems Supplemental 8 x 10 8 l. 2. 3 : X 2.4 x 10 12 10 4 8 x 10 7 10 9 2 x x 10 12 5 x 10 7 Lesson 119 9. 10 12 6.3 x 10 8 7.5 10. 11. x 10 7 4 x 10 5 1.8 x 10 7 X 10 8 1 8. 10 12 3 6. 4.2 x 10 6 x 2 x 5. 7. 7 3. x 10 6 3 x 10 8 1.2 Practice for 645 3.6 x 10 6 6 x 10 3 4 x 10 4 4. Supplemental Lessons Write each quotient in scientific notation. Practice for Lesson 110 for Selected 12. 9 x 10 4 6 x 10 11 4 x 10 6 5 x 10 10 Find the coordinates of each of these points of this coordinate plane. 1. Points 2. Points 3. Point C 4. Point D 5. Point E 6. Point F J 1 I p c -1 B o A E -< 5 -! > -' i -. 3 -; > - — ; ^1 C ,F ) ; i *l !i ( D, 9 c. c ' On a piece of graph paper draw an x-axis and a y-axis. Then graph and label the following points. 7. 10. (4,-3) (3, 4) 8. 11. (-3,4) (3, 0) 9. 12. (-3,-4) (0, 3) Glossary The absolute value number that equals the number from the origin. Since quality of a distance of the graph of the the graphs of -3 and +3 are both 3 units from the origin, the absolute value of both numbers An acute angle and angle is 3. whose degree measure is between 0° 90°. A triangle in which all three angles are acute acute triangle angles. addend One to find a sum. of two or more numbers adjacent angles Two common common The angles vertex. angles that have a lie added common side and a on opposite sides of their side. algebraic addition numbers that are to be to The combining of positive and/or negative form a sum. algorithm A particular process for solving a certain type of problem. Often the process is repetitive, as in the long division algorithm. altitude of a triangle The perpendicular distance from the base of a triangle to the opposite vertex; also called the height of the triangle. angle a In geometry, the figure common formed by two rays that have endpoint. area The number of square units of a certain size needed to cover the surface of a figure. 647 648 Math 87 The sum of a group of numbers divided by number of numbers in the group. average base (2) (1) A designated side The lower number (or a face) of a geometric figure. in an exponential expression. In the exponential expression 2 5 the number 5 is the exponent. , One hundredth centimeter the number 2 is the base and the of a meter. A way of expressing the likelihood of an event; the chance probability of an event expressed as a percent. The perimeter circumference common factors of a circle. Identical factors of two or more indicated products. complementary angles Two angles whose sum is 90°. composite number A counting number that is the product of two counting numbers, neither of which is the number 1. congruent polygons Two polygons in which the corresponding sides have equal lengths and the corresponding angles have equal measures. coordinate(s) number line; The number associated with a point on a also the ordered pair of numbers associated with a point in the Cartesian plane. corresponding sides Sides of similar polygons that occupy corresponding positions. Corresponding sides are always opposite angles whose measures are equal. counting numbers these numbers decagon A are Sometimes called the natural numbers, 1, 2, 3, 4, 5, ... . 10-sided polygon. decimal fraction A decimal number. 649 Glossary A base 10 numeral that contains a decimal decimal number point. A decimal point dot placed in a decimal a place value reference point. decimal point is The place number to use as to the left of the always the units' (ones') place. denominate number A combination of a number and a descriptor that designates units. Examples are 4 ft, 16 tons, 42 miles per hour. The bottom number denominator The distance diameter The difference digit across a circle through directed or its center. result of subtraction. In the base 10 system, 5, 6, 7, 8, in a fraction. any of the symbols 0, 1, 2, 3, 4, 9. numbers Another name for signed numbers. The number to be divided. In the expression 10 + the dividend is 10 and the divisor is 2. dividend 2, one whole number is divided by another whole number and the quotient is a whole number (the remainder is zero), we say that the first whole number is divisible by the second whole number: 10 is divisible by 2. divisible If The number by which another number is divided. In the expression 10 2, the divisor is 2 and the dividend is 10. Also, a factor of a number. Both 2 and 5 are divisors of 10. divisor -5- edge A line segment of a polyhedron where two faces intersect. equation A statement that two quantities are equal. equilateral triangle same length. A triangle whose sides all have the 650 Math 87 equivalent fractions Fractions that have the same value. To determine an approximate value. estimate A way expanded form number of writing a the products of the digits as the sum and the place values of the of digits. The upper number in an exponential expression. A number that tells how many times another number is to be used as a factor. In the expression 2 5 5 is the exponent and exponent , 2 is the base. exponential expression An expression that indicates that one number is to be used as a factor a given number of times. The expression 4 3 tells us that 4 is to be used as a factor 3 times. The value of 4 3 is 64. A face flat surface of a geometric solid. Noun. One of two or more numbers that are to be multiplied. In the expression 4xy, the factors are 4, x, and y. (2) Verb. To write as a product of factors. We can factor the number 6 by writing it as 2 x 3. factor (1) A part of a whole or the indicated fraction numbers, such as division of two -. 5 fraction line The line segment that separates the numerator and the denominator of a function that for there is A fraction. number pairs related by a certain rule so every number to which the rule may be applied, set of exactly one resulting number. geometric solid A three-dimensional geometric figure. Spheres, cones, and prisms are examples of geometric solids. gram A basic unit of mass in the metric system. One number is said to be greater than a second number if the graph of the number is to the right of the graph greater than of the second number. Glossary See altitude of a height A heptagon hexagon A hypotenuse 651 triangle. seven-sided polygon. six-sided polygon. The side of a right triangle that is opposite the right angle. improper fraction A whose numerator equal to or greater than the denominator; thus, a fraction equal to or greater than 1. fraction is independent events Two events are said to be independent if the outcome of one event does not affect the probability that the other event will happen. If a dime is tossed twice, the outcome (heads or tails) of the first toss does not affect the probability of getting heads or tails on the second toss. The whole numbers and the opposites of the positive -2, -1, 0, whole numbers. The members of the set integers . 1 . . , 2 To share a common point or points. Lines that intersect meet at a common point. intersect inverse operation Two operations are inverse operations if one operation will "undo" the other operation. If we begin with 3 and multiply by 2, the product is 6. If we divide 6 by 2, we will undo the multiplication by 2, and the answer will be 3, the original number. invert When said of a fraction, to interchange the numerator and denominator. isosceles triangle A triangle with at least two sides of equal length. kilogram kilometer One thousand grams. One thousand meters. , 652 Math 87 common denominator (LCD) Of two or more fractions denominator that is the least common multiple of the least a denominators of the least fractions. common multiple (LCM) The smallest whole number member that every of a set of whole numbers will divide evenly. than One number is less than a second number if the graph of the number on a number line is to the left of the graph of the second number. less A straight collection of points line line segment liter The A part of a line. basic unit of capacity in the metric system. lowest terms In reference to a fraction, and denominator contain no mean extending without end. Of a common when the numerator factors. set of numbers, the average of the set of numbers. The middle number when a median set of numbers is arranged in order from the least to the greatest. meter The basic unit of length in the metric system. One thousandth milliliter of a One thousandth millimeter liter. of a meter. mixed number A numerical expression composed of a whole number and a fraction, such as 2-. 2 mode most The number in a set of numbers that appears the often. multiple A product of a selected counting number and any other counting number. Multiples of 3 include 3, 6, 9, and 12. Glossary One multiplier of two numbers that are to 653 be multiplied; a factor. Numbers to the left of zero on the number negative numbers line. nonagon A nine-sided numeral Symbol polygon. or groups of symbols used to represent a number. The top number numerator of a fraction. obtuse angle An angle whose measure and less than 180°. A triangle obtuse triangle An octagon an obtuse angle. of describing the likelihood of an event; the ratio of favorable outcomes to unfavorable Two numbers whose sum opposites greater than 90° eight-sided polygon. A way odds that contains is is 0. outcomes. Thus, a positive number and a negative number whose absolute values are equal. The numbers - 3 and +3 are a pair of opposite numbers. The point on origin zero is a number line with which the number associated. Lines in the same plane that do not intersect. parallel lines parallelogram A quadrilateral that has two pairs of parallel sides. A pentagon Per hundred. Forty percent is 40 per hundred. Hundredth. Forty percent is forty hundredths. percent (2) five-sided polygon. (1) perimeter the figure. Of a plane geometric figure, the distance around . 654 Math 87 perpendicular lines Two lines that intersect and form right angles. The number of diameters equal to the circumference circle. Approximate values of pi are 3.14 and — pi [n) of a plane In mathematics, a flat surface that has point A location on a line, no boundaries. on a plane, or in space with no size. A polygon closed, plane geometric figure whose sides are line segments. polyhedron positive A geometric numbers solid whose faces are polygons. Numbers to the right of zero on the number line. power The value of an exponential expression. The expression 2 is read as 2 to the fourth power and has a value of 16. Thus, 16 is the fourth power of 2. The word power is also used 4 to describe the exponent. prime factorization The expression of a composite number as a product of its prime factors. prime factors The factors of a number that are prime numbers. prime number A whole number greater than 1 whose only whole number divisors are 1 and the number itself. prism A polyhedron with two congruent parallel bases. probability A way of describing the likelihood of an event; the ratio of favorable outcomes to product all possible outcomes. The result obtained when numbers proper fraction the denominator. A fraction are multiplied. whose numerator is less than . Glossary Two proportion 655 equivalent ratios. A Pythagorean theorem description of a property of right triangles that states that the area of a square constructed the longest side of a right triangle is on equal to the areas of the squares constructed on the other two sides of the triangle. Any one quadrant coordinate system, number of the four sectors of a rectangular which is lines that intersect formed by two perpendicular at the origins of both number lines. A four-sided quadrilateral The quotient polygon. result of division. An expression that contains radical signs, radical expression 3 V* Vl6 and vxy which indicate roots such as , > , of a number. The distance from the center of a circle to a point on radius the circle. The difference between the numbers in a set of numbers. range rate A ratio of of a to b ray A is and smallest two measures. A comparison ratio largest of two numbers by division. The ratio written | part of a line that begins at a point called the origin and continues without end. Two numbers whose product is since - x - = 1. reciprocals of - is 3 4 3 rectangle A and The reciprocal 4 parallelogram that has four right angles. all A polygon in which all sides have equal angles have equal measures. regular polygon lengths 1. 656 Math 87 The repeating digits of a decimal number often indicated by a bar. In the number 0.083 the repetend is 3. repetend A parallelogram that has four sides whose lengths rhombus are equal. One right angle of the angles two perpendicular A right triangle root lines. formed at the intersection of A right angle has a measure of 90°. triangle that contains a right angle. The solution to an equation; also, the value of a radical expression. The number that relates corresponding sides of scale factor similar geometric figures. scalene triangle A triangle whose three sides are of different lengths. scientific notation of a decimal semicircle sequence A method of writing a number as a product number and a power of 10. A half circle. An ordered list of numbers arranged according to a certain rule. signed numbers Numbers that are either positive numbers or negative numbers. Two have the same shape but may not be the same size. The corresponding angles of similar triangles are equal in measure and the lengths of the corresponding sides are proportional. similar triangles square A triangles that rectangle with sides of equal length. square root A number which, when multiplied by itself, equals the given number. A square root of 49 is 7 because 7 7 = 49. • Glossary An straight angle sum The angle is 180°. result of addition. supplementary angles is whose measure 657 Two angles the sum of whose measures 180°. The surface area total area of the surface of a geometric solid. A trapezoid quadrilateral with exactly one pair of parallel sides. A three-sided triangle polygon. The process unit conversion of changing a denominate number to an equivalent denominate number that has different units. unit multiplier value A fraction of denominate numbers whose is 1. The unit price A variable price of one unit of measure of a product. letter used to represent a number that has not been designated. vertex A point of an angle, polygon, or polyhedron where two or more volume lines, rays, or The number segments meet. of cubic units of a certain size that equals the space occupied by a geometric solid. whole number The numbers 0, 1, 2, 3, 4, ... . * Index Angles Absolute value in addition of signed numbers, 347-349 definition of, 88 554-556 naming, 88 sides of (see Rays) sum of, in triangles, of decimal numbers, 184 symbol for, 88 Symbols) of exponents to multiply types 184 for, 8, 10, 473-474 finding averages with, 151-152 of fractions, 41, 162-163 of integers on number lines, 324-326 missing numbers in, 11-12 of mixed measures, 263 of mixed numbers, 121-123 on number lines, 20-21, 325-326 279-280 504-505 (see also of, acute, 88, 456 complementary, 579 obtuse, 88, 456 right, 88, 456 powers of for, of, measuring, 456-458, 504-505, 384-385 order of operations 456 554-556 of signed numbers, 325 Acute angles, 88, 456 Acute triangles, 339 Addends, definition of, 1 Addition aligning decimals of, estimating measures definition of, 325 algebraic, charts of degrees 505-506 579-580 supplementary, straight, 88, 456, vertical, vertex of, 580 88 Area conversion of units of, 499-500 formulas for, 596-597 of geometric figures: circles, rule for equations, 467-469 complex shapes, 271-272, 432-433 sequence of of signed numbers, 325-326, rectangles, part-part-whole pattern of, 63-66 393-396 numbers in, 320 347-349 some and some more 44-47 of whole numbers, 3 parallelograms, 312-313, 467 244-246, 467 semicircles, pattern of, trapezoids, triangles, 588-589 433-434 352-353, 467 surface, of geometric solids, 592-593 Algebra addition, 384-385 Arithmetic, operations division, 334 minus sign 334-335 symbols in, 334 See Operations of arithmetic Arrowheads on number 384 multiplication, 334 in, operations of arithmetic of. in, lines, 15, 20-21 with rounding numbers, 156-157, 176-177 as symbol for lines, 32 659 660 Math 87 Asterisk, as symbol Common for See also Least multiplication, 2 Averages, 151-152, 292-293 symbol Common denominators; Least common multiples; for repeating decimals, 215-217 Multiples Comparison of decimals, in areas of parallelograms, 313 and exponents, 236-237 volume of, of solids, 410-411 Binary operations, definition of, Borrowing, in subtraction of mixed measures, 296-297 Braces, 343. See also Symbols; Symbols of inclusion Brackets, 343. See also Symbols; Symbols of inclusion 172-173 in insufficient information 312-313, 354 in multiples, 143-144. See also Bar graphs, 198 Bases, 237 definition common multiples Common Bar, as denominators, 160-161. 1 Canceling, 131-132, 283. See also Reducing fractions Capacity, 299-301, 627-628 Cent sign, 6, 7. See also Money Center, of circles, 375 problems, 452 symbols of [see Symbols of Comparison; Symbols) Complementary angles, 579 Complex shapes, area of, 271-272, 432-433 Composite numbers definition of, 105 of, 105-106 factoring, 106-108 divisors Compound probability, 613-615 Cones, 366, 619 Conversion of decimals to fractions, 220 of equivalent measures, 300-301 of fractions to decimals, 220-222 Centimeters, 83, 409 69-70 Chance. See Probability of fractions to numbers, Charts numbers to fractions, 114-115 of numbers to percents, 257 of of characteristics of quadrilaterals, 308 of degrees in angles, 456 of percents to numbers, 258 of fraction-percent equivalents, with unit multipliers, 284-285, 371-372, 498-499 495 of operations of arithmetic in algebra, 334 of place values, 23, 167, of weights, 240 Circle graphs, 199 Circles area of, 467-469 circumference of, 375 definition of, 375 parts of, 375 as symbols, 156-157, 176-177, 448 Circumference, 375 definition of, 375 finding with pi, 380-381 Commas, in large 499-500 371-372 Coordinates, rectangular, 548-551 Corresponding angles, 561-563 Corresponding sides, 561-563 of units of area, numbers, 23-24 Counting numbers, 3, 488. See also Composite numbers; Prime numbers Cubes, 366, 367, 409-410 Cubic measurements, 409-410 Cylinders, 366, 367 Decagons, 92 Decimal-part-of-a-number problems, 330-331 Index Decimal numbers adding, 184 charts of, 167 comparing, 172-173 Denominators [Cont.) numbers ordinal in, 131-132 228-230 dividing of, by whole numbers, 211-213 fraction-percent equivalents of, 316-317 Diagrams of fraction-of-a-group problems, of 117-119 number families, 490 Diameter, definition of, as fractions, 166-168, 220-222 Difference, definition multiplying, 207-208 Digits 10, 241 on number lines, 180-181 and operations with fractions, 388-389 place value of, in, 172-173, 211-212 repeating, 215-217 rounding of, 176-177 Directed numbers, 325. See also Negative numbers; Positive numbers; Signed numbers Dividend, definition definition tests for, 172-173, 177 in words, 171-172 Decimal points of, 1 of, 28 29 229-230 207-208, in converting improper 69 by decimal numbers, 228-230 of decimal numbers, 211-212 fractions, definition 241 for subtraction, 8, placing with money, for, 9, 211-212, 229-230 211-212, for multiplication, 334 algebraic, aligning decimals aligning, for addition, 8, 184 185 6-7 placing with powers of 10, 241, in 167 of, 23, 215-217 repeating, Division zeros, for division, 9, 275-276, 304-305 scientific notation, 275-276, of, 1 151-152 135-137, 437-438 finding averages with, of fractions, symbol of inclusion, line, as of 343-344 mixed numbers, 140 by 100 304-305 of, 456 measuring, in angles, 456-458 505-506 504-505 Denominators common [see Common 225-226 repeated method of, in factoring, 108 rules for, 404-406, 428 of signed numbers, definition of, 36, symbols 135-137, 437-439 least common common multiple) for, 427-429 bars, 2 boxes, 2 36 [see also Symbols) in unit price problems, 233 of whole numbers, 4 lines, [see Least [see also Rules) denominators) in decimal fractions, 166-167 in dividing fractions, 279-280 77-79 for, in reducing fractions, remainders in, 211-212, in straight angles, in triangles, 258 to find percents, order of operations Degrees definition 375 of, 1 Divisibility subtracting, 185 and terminal place value 167-168 placing of zeros 36 77-79, in reducing fractions, dividing by, by powers of 661 662 Math 87 Expanded notations, Powers of 10 Divisors definition of, 1 as factors, 28-29 inverting, 136-137, 437-439 of numbers, Dollar signs, Dots, as 105-106 6, 7. symbol 241. See also Exponential expressions, 236-237. See also Exponents Exponents See also Money for multiplication, 2 adding, in multiplication of powers of 10, 473-474 definition of, 236 expressions with, 237 to show repeated Edges, of polyhedrons, 366, 367 multiplication, 236-237 Ellipsis, definition of, 3 Endpoint, 32 Equal fractions. See Equivalent Equal groups problems, 59-61 Equalities/inequalities. See Greater than/less than; Inequalities; Symbols; Symbols of comparison) Equals sign, 16, 447. See also Symbols; Symbols of comparison Equations addition rule definition of, division rule 393-396 for, 100 lor, to find percents, literal, Factors definition of, 1 as divisors, Feet, 82, 28-29 409 Formulas 595-597 for areas, definition 596, 605 of, 605-606 596-597 transforming, Fraction lines, 2 Fractional-part-of-a-number with mixed numbers, 510-511 multiplication rule for, 403-406 100-102 Equivalent fractions definition of, 73 multiplying to find, 73-74 Equivalent measures. See also Linear measures conversion of, 300-301 tables of, 82, 83, problems, 330-331, 422-424 Fraction-of-a-group problems, 117-119, 414 Equilateral triangles, 339 300 Equivalents, fraction-decimalpercent, with repeated division method, 108 with tree method, 107 substituting letters in, 404-406 442-443 596 solving, Faces, of polyhedrons, 366, 367 Factoring fractions 316-317 Fractions adding, 41, 162-163 canceling, 131-132, 283 common denominators conversion of to numbers, 69-70 decimal-percent equivalents, 316-317 as decimals, Estimating definition 554-556 measures of angles, with rounded numbers, 157-158 square roots, 578-579 Evaluation of expressions with positive numbers, 334-335 with signed numbers, 514-515 in, 160-161 of, 166-168, 220-222 36 denominators [see Denominators) dividing, 135-137, 437-439 equivalent, 69, 73-74 improper (see Improper fractions) lowest terms of, 78 663 Index Hyphens, Fractions (Cont.) mixed numbers multiplication as, 114-115 numbers, 24 Hypotenuse 41 of. for two-digit on number lines, 37-38 numerators (see Numerators) and operations with decimals, definition of, 572 and Pythagorean theorem, 572-575 388-389 percent equivalents proper. of. 494-495 110-111 reciprocals of, 135-137 Improper 127-128. [see also definition Reciprocals) fractions) subtracting, 41, 482-483 162-163 terms of, 69-70 69 69 of. of, mixed numbers as, 114-115 simplifying, 110-111 whole numbers as, 115 reducing [see Reducing simplifying. 110-111, fractions, conversion Inches, 82. 410 Inequalities 78 definition whole numbers as. 115 Functions. 321. 483-485. See also graphing of. symbols for. Linear functions 448 448 447 (see also of, Greater than/less than; Symbols of comparison) Insufficient information problems. See Grams Gal. See Gallons g. Gallons, 82. 451-452 Integers 299-301 adding on number Geometric solids. 366-367. 409-410. 592-593 Grams, 370-372 Graphs, 197-199 of inequalities, 448 600-602 447-448 324-326 definition Interest, lines, of rectangular coordinates, of, 489 609-610 International system (SI. or Systeme International). See Metric system of linear functions, on number lines, Intersecting lines, definition of, Isosceles triangle, 339, 340 548-551 types 198 of, bar, 199 199 circle. line, picture, kg. 197-198 Greater than/less than, 16, 447-448. See also Inequalities: Symbols: Symbols of comparison See Kilograms Kilograms. 82, 370-372 Kilometers, 82-83 km. See Kilometers L. See Liters Larger-smaller-difference Height in parallelograms, in triangles, in volume 313 353 of solids. 410 Heptagons, 92 Hexagons. 92, 93, 96 word problems, 54-56 lb. See Pounds Least common multiple (LCM), 144. See also Common multiple: Denominators: Multiples 87 664 Math 87 Length measures in metric system, of, 83-84 in U.S. system, 82 table of units of, 82, 83 Metric System (Cont.) length measures in, square units in, 245 tables of, 83, 300 units of capacity in, units of Letters volume Milligrams, in writing equations, 100, 596 Milliliters, graphing of, 299-301 628-629 in, 370 370-372 299-301 Millimeters, 83 Linear functions 600 of, in, weight measures mg. See Milligrams naming angles with, 88 naming lines with, 32-34 naming polygons with, 93 definition 83-84 600-602 Minuend, definition of, 1 Minus sign, in algebraic addition, 384 Missing numbers Linear measures metric, 82, 83 11-12 tables of, 82, 83 in addition, U.S. system, 82 finding in functions, 321, using, 82-84 483-485 Line graphs, 199 Lines intersecting, 87, in 549 naming with letters, 32-34 number (see Number lines) parallel, 12-13 some and some more problems, 44-47 in multiplication, 87 in subtraction, 12 Mixed measures adding, 263 perpendicular, 87 262-263 converting, symbols for, 32, 33 [see also Symbols) Line segments. See Segments Literal equations, 596 Liters, 82, 299-301. See also [see also Conversion; Unit multipliers) 296-297 subtracting, Mixed numbers. See also Fractions adding, 121-124 definition of, 37 Volume m. See Meters dividing, 140 equations with, 510-511 Mass, 371 as Mean, 477-479 Median, 464, 477-479 Measurement. See also Rates; multiplying, 140 Ratios 299-301 (see also Volume) mixed [see Mixed measures) unit conversion problems of, 284-285, 371-372, 499-500 Meters, 82-84, 409 114-115 37-38 110-111 122-124 Milliliters See Millimeters Mode, 464, 477-479 Money forms of writing, 6-7 interest problems, 609-610 and operations of arithmetic with, 8-9 placing decimal points with, 6-7 Metric system of, mL. See fractions, lines, simplifying, mm. measures) definition on number subtracting, equivalent [see Equivalent of liquids, improper 82 in unit price problems, 233-234 665 Index Notations. See Symbols Multiples Number common, 143-144 definition, 143 least common (LCM), 144 Multiplication of decimal numbers, definition 207-208 lines adding on, 20-21, 324-326 [see also Signed numbers) decimal numbers on, 180-181 definition fractions on, of, 1 15 of, 37-38 in division of fractions, 136 graphs on, 447-448 equal groups pattern, 59-61 mixed numbers on, 37-38 as model for addition, 325-326 to find equivalent fractions, 73-74 reading, 16, 38 of fractions, 41 rounding with, 155-156 missing numbers in, 12-13 of mixed numbers, 140 subtracting on, of numbers in scientific notation, 473-474 20-21 zeros as origin on, 15 Number pairs, 321 Numbers by 100 percent, 257 averages of, 151-152, 292-293 operations of, in algebra, 334 order of operations for, 279-280 by powers of 10, 241 commas in, 23-24 comparing of, 16-17 [see also Symbols of comparison) of 472-474 266-268 powers of of rates, 10, repeated, with exponents, 236-237 rules for, 403-406, 428 {see also Rules; specific operations) sequence of numbers in, 321 of signed numbers, 427-429, 584 symbols for, 2, 334 {see also Symbols) with unit multipliers, 284-285 334-335 whole numbers, 4 of variables, of composite, 104-108 {see also Composite numbers) conversion of fractions to, 69-70 conversion of percents to, 258, 357-358 percents, 257 488 {see also Composite numbers; Prime numbers) decimal {see Decimal numbers) decimals in {see Decimal conversion counting, of, to 3, points) directed, 325 {see also Signed numbers) families Negative numbers. See also Signed numbers 488-491 of, missing {see Missing numbers) mixed {see Mixed numbers) of, 325 adding with positive numbers, 325-326, 347-349 definition of, 16 negative {see Negative numbers) in evaluation of expressions, as percents, 257-258, 442-443 514-515 powers of, 585 percents 357-358 absolute value numbers, 324-325 and zeros, 16, 304 Negative sign, 16 Newtons, 371 Nonagons, 92 as signed opposite, 16, 384 ordering of, 16 ordinals, in denominators, 36 of, prime {see Prime numbers) positive {see Positive numbers) rational, 489-490 275-276, 304-305, 399-400, in scientific notation, 473-474 666 Math 87 Numbers Parallel lines, 87. [Cont.) sequences of, 320-321 signed [see Signed numbers) whole {see Whole numbers) Parallelograms, area squares Numerators of, 312-313, 467 as, 307 Parentheses 36 in dividing fractions, 135-137, 437-439 and signed numbers, 514-515 simplifying numbers within, 2-3 reducing fractions, 77-79, in of, 307-308 rectangles as, 307 in words, 24 definition See also Symbols 131-132 symbol of inclusion, 2 also Symbols) as [see using in evaluation of variables, 334-335 Part-part-whole word problems, 63-66 Obtuse angles, 88, 456 Obtuse triangles, 339 Pentagons, 92, 93 Octagons, 92 Percents, common Odds. See Probability One 442-443 fraction equivalents, 494-495 (1) hidden, 437 conversion of, to numbers, 258, 357-358 numbers to, 257, 358 fractions equivalent to, 69, 73 Operations of arithmetic, 1-2 334-335 with fractions and decimals, 388-389 definition with money, 8-9 and probability, 623-624 symbol for, 257 [see also in algebra, 256 fraction-decimal equivalents, 316-317 279-280, 482-483 symbols of, 334 [see also Symbols) within symbols of inclusion, order of, of, 343-344 with whole numbers, 3-4 Symbols) using proportions with problems of. 461-463 zeros with, 257 Perimeters 96 Opposites, of the opposite, 384 definition Order of operations, 279-280, 482-483. See also Operations of arithmetic 193-194 of semicircles, 588-589 word problems with, 97-98 of polygons, 96-97, Origin Perpendicular as zeros on number lines, as point of intersection of x axis Outlines equal groups problems, 59 for part-part-whole pattern, for 64 some and some more pattern, 64 Overbar, as symbol for segments, oz. 33 See Ounces lines, 87. See also Symbols 15 and y axis, 549 Ounces, 299-301, 371-372 for of, ' Pi [n) and area of circles, 468-469 and circumference, 380-381 definition of, 380 symbol for, 380 [see also Symbols) Picture graphs, Pints, 197-198 299-301 Place values charts of, 167 of decimal numbers, 667 Index Product, definition Place values, charts of [Cont.) with powers of 10, 240 of whole numbers, 23 in decimal fractions, 166 of decimals, definition of, Proper fractions, simplifying with percent problems, 461-463 23 Planes, definition trillions, of, of, 87 92 93 92 regular. 93 perimeters of, 96-98, 193-194 vertex of, 93 Polyhedrons, 366-367 Positive numbers. See also Signed of, numbers absolute value of, ratio problems, 288-289 measuring angles with, 457-458, 554-556 pt. See Pints Pyramids, 366, 618-619 Pythagorean theorem, 572-575 Protractors, irregular, naming with 23 Polygons definition of, 110-111 Proportions, 203-204 167-168 through hundred of, 1 See Quarts Quadrants, 549 qt. Quadrilaterals, 92, 93 charts of characteristics 325 adding with negative numbers, 325-326, 347-349 definition of, 16 of, 308 307-309 299-301 classifying, Quarts, Quotient, 1 evaluation of expressions with, 334-335 and scientific notation, as signed and 304 numbers, 324-325 zeros, 16, 304 Pounds, 82, 371-372 Powers, 237-238 of negative numbers, 585 of 10 (see Powers of 10) Powers of 10 chart of place values of, 240 and expanded notations, 241 472-474 multiplying decimal numbers multiplying, 241 placing decimals with, 275-276, 304-305 in scientific notation, 275-276, 399-400, 472-474 use of zeros in writing, 240 Prime factorization, 107, 131 Prime numbers definition of, 104 by, divisors 105-106 106-108 609-610 of, factoring, Principal, Prisms, 365-367, 409 Probability. 538-540, 613-615, 622-624 Radical sign, as symbol for square root, 249 Radius (Radii), 375-376 Range, 464, 477-479 Rates definition problems 253 of, multiplying, of, 266-268 254, 417-419 exchange, 253 mileage, 253 speed, 253 Ratio boxes, use of with implied ratio problems, types of, 418-419 with problems of scale, 567-568 with proportion and percent problems, 461-463 with ratio word problems 288-289, 361-362 Rational numbers, 489-490 Ratios, 188-190. See also Proportions cross products definition implied, of, of, 203-204 188 417-419 as proportions, 203 668 Math 87 567-568 Scale factor, 568-569 Scale, Ratios {Cont.) as rate, 253 word problems, 189-190, in 288-289, 361-362, Scalene triangles, 339 Scientific notation in, 275-276, 304-305 numbers in, 275-276 multiplying numbers in, 473-474 and powers of 10, 275-276, 399-400, 472-474 small numbers in, 304-305 decimals 461-463 large Rays of angles, 88 naming, 32 symbol for, 32 (see also Symbols) Reciprocals, definition 127-128 of, 127 in division of fractions, length 135-137 property of, of, 244-246, 467 complex shapes, 272-273 area perimeter of, 307-308 Rectangular coordinates, graphing, 548-551 Rectangular prisms, 366, 367 Reducing fractions before multiplying, 131-132, 283 (see also Canceling) numerators in, 77-79, 131-132 with prime factorization, 131 Remainders, 211-212, 225-226 Repeated division method, of factoring, 108 Repetend, definition of, 215 Right angles, 88, 456 Right triangles definition of, 339-340 and Pythagorean theorem, 572-575 Rounding of decimal numbers, 176-177 in estimating answers, 157-158 with number lines, 155-156 of whole numbers, 155-158 Rules for equations: addition, 393-396 404-406 multiplication, 403-406 division, for 32-34 of, symbols for, 32, 33 (see also Symbols) 588-589 Semicircles, Sequence 96 as quadrilaterals, 33 of, naming 127 Rectangles in Segments. See also Lines definition of, 32 signed numbers: division, 428 multiplication, 428 missing numbers in, 321 of numbers, 320-321 Signed numbers absolute value of, 325, 347-348 adding, 325-326, 347-349 definition of, as directed dividing, 324-325 numbers, 325 427-429 evaluating expressions with, 514-515 multiplying, 427-429, 584 on number lines, 325-326 order of operations with, 482-483 and parentheses, 514-515 rules for, 428 zeros with, 347-348 Similar triangles, 561-563 Simplifying, 2-3, 110-111, 482-483 Slanted fraction line, 2 Solids. See Geometric solids Some and some more word problems, 44-47 Some went away word problems, 49-52 Spheres, 366, 367 Square roots, 249-250 definition of, 249 estimating, 578-579 symbol for, 249 Symbols) (see also Index Squares perimeter Symbols 97 of, - 505-506 formulas, 605-606 (overbar), 33 II (parallel lines), % Subtraction aligning decimals for, 8. 185 of decimal numbers. 185 of, 1 of fractions, 41, 162-163 larger-smaller-difference pattern 54-56 missing numbers in, 12 of mixed measures, 296-297 of mixed numbers. 122-124 on number lines. 20-21 of. order of operations for, (percent), 87 257 -L (perpendicular lines), 87 K (pi). 380 ^^ (rays), V~~ (square root), 249 32 Symbols of comparison. See Symbols; specific symbols Symbols of inclusion. See also Symbols; specific symbols definition of, 343 operations of arithmetic within, 343-344 279-280 some went away pattern 49-52 of whole numbers, 3 Subtrahend, definition (multiplication), 2 (negative), 16 Straight angles, 88, 456. definition [Cont.) for lines, 32, 33 x, * 308 Square units, in area, 245 as quadrilaterals, Substitution, in 669 of. t. See Tons Tables. See also Metric system; of, 1 U.S. system Sums adding signed numbers, 347 of angles measures in triangles, in of equivalent measures, 300 of fraction-decimal-percent equivalents, 317 504-505 of units of length, 82, 83 definition, 1 Supplementary angles, 579-580 Surface area, of geometric solids, 592-593 Symbols Z 88 arrowheads, 32 (angles), * — also (bars), 596-597 215-217 Trapeziums. 307, 308 Trapezoids. 307-309. 156-157, 176-177, 448 of comparison. = (equals sign), 16. 447 circles, > < (greater than/less than), 16, 447 (division), 2 - ) method of factoring, 107 338-340 352-353. 467 339 measuring angles in. 504-505 180° in. 504 area of, classifying. as polygons, 92, 93 types of inclusion. } — ) [ (braces), ] (brackets). acute, 339 339 340 isosceles. 339. 343 obtuse, 339 right, 339. 340, (division line), 344 (parentheses), of, equilateral. 343 2. 343 433-434 Triangles, (division box). 2 dot (multiplication), 2 ( Tree (division bar), 2 S (dollar sign), 6 { Sums Transformation, of formulas, (asterisk), 6 (cent sign), 6 c / Terms, definition of. 78 Theorem. Pythagorean, 572-575 Time, problems of, 55-56 Tons, 371 Totals, in ratio problems, 362. See 572-575 scalene, 339 similar, 561-563 670 Math 87 Whole numbers Triangular prisms, 366, 367 Two-step word problems, 147-148 [Cont.) operations of arithmetic with, 3-4 place values Undecagons, 92 Unit multipliers, 283-285 with conversion problems, 284-285, 372, multiple, subtracting, 3 with perimeters, 97-98 with probability, 613-615, 498-499 623-624 Unit price problems, 233-234 U.S. Customary system definition of, 82 measuring length in, 82-84 measuring weight in, 371-372 square units in, 245 tables of 82, 300 units of capacity in, 23-24 Word problems 284 of, 23 reading and writing, rounding, 155-158 499-500 definition of, 299-301 253-254, 417-419 ratios, 189-190, 288-289, 361-362, 461-463 with with rates, with with scale, types of, decimal part of a number, 330-331 567-568 477-479 statistics, equal groups, 59-61 fraction-of-a-group, 414 Variables definition 117-119, fractional-part-of-a-number, 334 of, 330-331, 422-424 334-335 with parentheses, 334-335 multiplying, larger-smaller-difference, 54-56 Vertex (Vertices) part-part-whole, of angles, 88 quantitative comparison, of polygons, 93 451-452 some and some more, 44-47 some went away, 49-52 time, 55-56 two-step, 147-148 with unit prices, 233-234 of polyhedrons, 366, 367 Volume, 299-301 of cones, 619 of cubes, 410 definition of, 409 of prisms, 409 of pyramids, 63-66 618-619 of solids, 410 x-axis, 549 y-axis, 549 units of capacity in, 299-300, 627-628 using cubes in measurement of, Zeros 409-410 in adding signed numbers, 347-348 Weight, 370-372, 627-628 Whole numbers adding, 3 averages of, 151-152 definition of, 3, 488-489 dividing, 4 dividing decimal numbers by, 211-212 as fractions, 115 multiplying, 4 with cent sign, 7 with decimal numbers, 172-173, 211-212 with dollar sign, 7 and negative numbers, 16, 304 as origin on number lines, 15 with percents, 257 and positive numbers, 16, 304 terminal, 172-173, 177 in writing powers of 10, 240 1 » « * ' 1 ] Place Value Chart Whole Numbers Places Decimal Places co X X CO X) CO CO s CO d O fl O O -a • H d B d o *« 14 XI 0) d •i-H • d i— CD 4-t 10 o 10 13 3 d CD x 10" 10 10 10 9 12 d H o — X 10 H d T3 d •i-H CD e 10 7 10 co CO -1 6 10 CO d 10 10 1 1 hour 24 hours = 7 days = 1 day 1 week 365 days = 366 days = 1 common 1 leap year 10 years = 1 decade 100 years = 1 century d 9 3 d $ x CO -5 thousand hundre 1 io 10° ten 10"' 10" 2 10" 3 Probability, Chance, minute million tenths d o CD 2 10 d o hundredt CO CD co XI 4— 4 T=> d CO CO thous H d X Time 60 seconds = 60 minutes = CD d o 0) X 9 T3 CO CO d 5 2 pH X X X a co a O 03 1 d CO -a T3 co as -i— 10 X d S r! T> o -t— co CO CD -a fH • co CO x 0^ KT* 10~ 5 1(T 6 Odds favorable Ratio of Probability possible Chance Probability expressed as a percent Odds Ratio of year favorable unfavorable Equivalence Table for Units LENGTH CUSTOMARY U.S. 12 in. = 1 ft 3 ft = 1 yd 5280 ft = 1 mi 1760 yd = lmi METRIC mm 1000 mm = 1 = 1 cm = 1 m m m = 1 km 1 kg 1 L 10 100 1000 WEIGHT U.S. MASS CUSTOMARY 16 oz = 2000 lb = 1 lb 1 METRIC 1000 g = ton LIQUID U.S. MEASURE CUSTOMARY 16 oz = 1 pt = 1 qt 2 pt 4 qt = 1 cm gal METRIC 1000 mL = Classification of Triangles CLASSIFICATION BY SIDES EXAMPLE TYPE CHARACTERISTICS A K Equilateral Isoceles Three sides of equal length At Vx Scalene least two sides of equal length All three sides are of different length CLASSIFICATION BY ANCLES EXAMPLE TYPE Acute A Right t\ CHARACTERISTICS All angles are acute K Obtuse One angle is right One angle is obtuse Abbreviations U.S. CUSTOMARY METRIC ABBREVIATION UNIT UNIT ABBREVIATION inch in. millimeter mm foot ft centimeter cm yard yd meter m mile mi kilometer km ounce oz gram g pound lb kilogram kg degree Fahrenheit °F degree Celsius °C pint Pt liter L quart, gallon qt. gal milliliter mL OTHER ABBREVIATIONS square sq. square mile sq. mi square centimeter sq. cm Statistics Mean: the average of a set of Median: the middle Mode: the Range: the difference between largest number number numbers of a set of that appears the numbers arranged most often in a and smallest in order set of numbers in a set of numbers Classification of Quadrilaterals SHAPE r\ r\ \Z^ NAME CHARACTERISTIC No \ Trapezium sides parallel One pair of parallel sides Trapezoid Two pairs of parallel sides Parallelogram Parallelogram with right angles Rectangle Rectangle with equal sides Square Note that squares and rectangles are types of parallelograms and that a square is a type of rectangle. Regular and Irregular Polygons NAME REGULAR IRREGULAR A Triangle S\ r\ Quadrilateral O o Pentagon Hexagon <r> Pn n o Octagon Geometric Solids TYPE Cube 2 O Q g > O pej Rectangular prism Triangular prism a. Pyramid Cylinder Sphere Cone EXAMPLES m TrJ m m ^> M> A A 8 m © Q A £

Saxon Math 87 An Incremental Development by Hake

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