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Arthur F. Coxford James T. Fey Christian R. Hirsch Harold L. Schoen Gail Burrill Eric W. Hart Ann E. Watkins with the assistance of Emma Ames, Robin Marcus, Mary Jo Messenger, Jaruwan Sangtong, Rebecca Walker, Edward Wall, and Marcia Weinhold Contemporary Mathematics in Context A Unified Approach CORE-PLUS MATHEMATICS PROJECT Course 1 Reference and Practice Book
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Page 1: course i

Arthur F. CoxfordJames T. FeyChristian R. HirschHarold L. SchoenGail BurrillEric W. HartAnn E. Watkinswith the assistance ofEmma Ames, Robin Marcus,Mary Jo Messenger, Jaruwan Sangtong,Rebecca Walker, Edward Wall, and Marcia Weinhold

Contemporary Mathematics in ContextA Unified Approach

C O R E - P L U S M A T H E M A T I C S P R O J E C T

Course1

Reference and Practice Book

Page 2: course i

Project DirectorsArthur F. Coxford, University of MichiganJames T. Fey, University of MarylandChristian R. Hirsch, Western Michigan UniversityHarold L. Schoen, University of Iowa

Senior Curriculum DevelopersGail Burrill, University of Wisconsin-MadisonEric W. Hart, Western Michigan UniversityAnn E. Watkins, California State University, Northridge

Professional Development CoordinatorBeth Ritsema, Western Michigan University

Evaluation CoordinatorSteven W. Ziebarth, Western Michigan University

Project CollaboratorsEmma Ames, Oakland Mills High School, MarylandRobin Marcus, University of MarylandMary Jo Messenger, Howard County Public Schools, MarylandJaruwan Sangtong, University of MarylandRebecca Walker, Western Michigan UniversityEdward Wall, University of MichiganMarcia Weinhold, Western Michigan University

Editorial and Production AssistantsJames Laser, Western Michigan UniversityKelly MacLean, Western Michigan UniversityWendy Weaver, Western Michigan University

Photo AcknowledgmentsCover images: Images © 1997 Photodisc, Inc.Cover Design: Oversat Paredes Design

This project was supported, in part,by the

National Science FoundationOpinions expressed are those of the authorsand not necessarily those of the Foundation

Copyright © by the McGraw-Hill Companies, Inc. All rights reserved. Except as permitted underthe United States Copyright Act, no part of this publication may be reproduced or distributed inany form or by any means, or stored in a database or retrieval system, without prior permissionof the publisher.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027

ISBN 1-57039-440-7

Printed in the United States of America.

2 3 4 5 6 7 8 9 10 079 07 06 05 04 03 02 01

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�Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

�Summary and Review of Middle School Mathematics . . . . . . . 5Numbers and Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Patterns and Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Geometry and Spatial Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Data Analysis, Statistics, and Probability . . . . . . . . . . . . . . . . . . 44

�Maintaining Concepts and Skills . . . . . . . . . . . . . . . . . . . . . . . . . . 51Exercise Set 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Exercise Set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Exercise Set 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Exercise Set 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Exercise Set 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Exercise Set 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Exercise Set 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Exercise Set 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Exercise Set 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Exercise Set 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Exercise Set 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Exercise Set 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Exercise Set 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Exercise Set 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Exercise Set 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Exercise Set 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Exercise Set 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Exercise Set 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Exercise Set 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Exercise Set 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Contents

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�Practicing for Standardized Tests . . . . . . . . . . . . . . . . . . . . . . . . . 97Practice Set 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Practice Set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Practice Set 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Practice Set 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Practice Set 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Practice Set 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Practice Set 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Practice Set 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Practice Set 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Practice Set 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

�Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Solutions to Check Your Understanding . . . . . . . . . . . . . . . . . . . 126

Solutions to Exercise Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Solutions to Practice Sets for Standardized Tests . . . . . . . . . . . . . 156

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I N T R O D U C T I O N 3

Course 1 of the Contemporary Mathematics in Context (CMIC) series introducesimportant ideas and problem-solving skills in algebra and functions, statistics andprobability, geometry, and discrete mathematics. Many of those concepts and skillswill extend mathematical knowledge you’ve acquired in earlier study. However, inorder to make use of that prior knowledge, you may need periodic reminders of keyideas and practice with the skills that put those ideas to work. This Reference andPractice (RAP) book includes information and exercises that should be very helpfulin reviewing and polishing the mathematics that you will need in Course 1.

This book has three main sections: Summary and Review of Middle SchoolMathematics, Maintaining Concepts and Skills, and Practicing forStandardized Tests.

The first section, Summary and Review of Middle School Mathematics, con-tains summaries of key ideas in five strands:

■ numbers and operations

■ patterns and algebra

■ geometry and spatial sense

■ measurement

■ data analysis, statistics, and probability

Introduction

Page 6: course i

The examples in this section illustrate application of the above strands to specif-ic problems. Summaries of each topic are followed by a short problem review setto Check Your Understanding. It is a good idea to solve the Check YourUnderstanding problems, then check your solutions against the answer key at theback of the book.

The second section, Maintaining Concepts and Skills, contains twenty sets ofreview exercises from the various content strands mixed together as they might bein a cumulative examination or in a real-life problem situation. These mainte-nance exercise sets draw material from middle grades mathematics and should beused as periodic reviews to keep ideas from all strands fresh in your mind.Exercise Sets 1–10 review middle grades mathematics and can be used at anytime during the course. Since Exercise Sets 11–20 include some material from thefirst part of Course 1, they can be used any time during the second half of thecourse. Additional exercise sets for maintaining Course 1 concepts and skills areincluded in the Course 1 Teaching Resources book.

The third section, Practicing for Standardized Tests, presents ten sets of ques-tions that draw on all content strands. The questions are presented in the form oftest items similar to how they often appear in standardized tests such as stateassessment tests or the Preliminary Scholastic Aptitude Test (PSAT). Use thesetest sets any time during the school year to become familiar with the formats ofstandardized tests and to develop effective test-taking strategies for performingwell on such tests.

Because you will probably use this book when studying outside of regularmathematics class sessions, answers to the problems are given at the end of thebook. It is often possible to learn a lot by studying worked examples and work-ing back from given answers to the required solution process. However, it’s bet-ter to try to solve the problems on your own first, and then look at the answer key.

4 I N T R O D U C T I O N

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S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S 5

This section consists of brief summaries and illustrative examples for key topicsthat you studied in middle school mathematics. The summaries are organized bystrand: numbers and operations; patterns and algebra; geometry and spatial sense;measurement; and data analysis, statistics, and probability. As you progressthrough Course 1, there may be activities or problems for which you need to usepreviously-learned ideas that you don’t completely remember. You can use thissection to refresh your understanding of those ideas.

Within each topic summary, you will find brief explanations of related concepts and methods together with worked examples and exercises that areintended as a reference. These don’t need to be studied from beginning to end.However, you should scan through this section so that you have an idea of whatmathematics is reviewed and where various topics and subtopics are located. You can then refer to this section for specific information to help you when youneed it.

Summary and Review ofMiddle School Mathematics

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�1 Numbers and Operations

We use numbers and operations in almost all aspects of everyday life, in busi-ness, and in science. In elementary school, your study of numbers focused onwhole numbers, 0, 1, 2, 3, …, and the basic operations of addition, subtraction,multiplication, and division. In the middle grades, you probably spent a greatdeal of time developing skills for using integers, fractions, decimals, percents,and proportions to order, count, measure, and compare all kinds of objects andcollections of objects.

The following sections review the key concepts and skills required to usenumber system operations and properties. The Check Your Understanding prob-lems provide an opportunity for you to practice those skills.

1.1 IntegersThe set of integers includes the set of whole numbers and their opposites{…, �3, �2, �1, 0, 1, 2, 3, …}. The integers can be graphed on a number line.

Integers greater than 0 are called positive, and integers less than zero are callednegative. The absolute value of a number n, written | n |, is its distance from zeroon the number line.

EXAMPLE 1 � Absolute Value—Both | 5 | and | �5 | equal 5 because 5 and �5are each 5 units from zero on the number line.

In mathematical problems, positive integers are often used to measureincreases in a quantity. Negative integers are used to measure decreases. Thereare several basic rules for operations with integers—operations that combinegains and losses.

■ To add two integers with the same sign, add their absolute values and use thecommon sign. For example, 5 � 7 � 12 and �3 � (�9) � �12.

■ To add two integers with opposite signs, subtract the smaller absolute valuefrom the larger. Use the sign of the number with the larger absolute value.For example, �12 � 8 � �4 and 5 � (�1) � 4.

6 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

–4 –2 0–7 –6 –5 –3 –1 1 4 62 3 5 7

0 1 4 62 3 5 7–4 –2–7 –6 –5 –3 –1

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1 • N U M B E R S A N D O P E R A T I O N S 7

■ To find the difference of two integers, define the problem in terms of addition. Subtraction is the same as adding the opposite. In other words, a � b � a � (�b). For example, 10 � (�3) � 10 � 3 � 13 and �8 � 4 � �8 � (�4) � �12.

■ To multiply two integers with the same sign, multiply their absolute values.The answer will be positive. For example, (�4)(�6) � 24.

■ To multiply two integers with opposite signs, multiply their absolute valuesand make the product negative. For example, (�9)(2) � �18.

■ To divide two integers, define the problem in terms of multiplication,a � b � c provided a � b � c. For example, �8 � 2 � �4 because �8 � (2)(�4).

It is helpful to think about integer addition and subtraction in terms of motionon a number line.

EXAMPLE 2 � Adding Integers—If the Central High football team loses4 yards on one play and gains 6 yards on the next play, the netgain is 2 yards because �4 � 6 � 2.

EXAMPLE 3 � Subtracting Integers—If the temperature outside is �2˚ andit drops another 5˚, the new temperature is �7˚ because �2 � 5 � �7.

EXAMPLE 4 � Multiplying Integers—In many large cities, workers use publictransportation for their commute to work. If a commuterbuys a $50 fare card and has a $6 commute everyday, thevalue of her fare card decreases by $30 each week, because5(�6) � �30.

–4 –2 0–7 –6 –5 –3 –1 1 4 62 3 5 7

–4

+6

–4 –2 0–7 –6 –5 –3 –1 1 4 62 3 5 7

–5 –2

Page 10: course i

EXAMPLE 5 � Dividing Integers—When large groups of people go out fordinner together, restaurants like to put all orders on one check.

a. If the total bill for a group of 15 people is $165 and everyoneagrees to split the bill equally, then each person owes $11because �165 � 15 = �11.

b. If the total bill for another party is $144 and each person’sshare is $9, then there must be 16 people in the partybecause �144 � �9 = 16.

The rules for order of operations involving integers are the same as those forwhole numbers: Perform operations within parentheses first. Next, workingfrom left to right, perform indicated multiplications and divisions. Finally, per-form indicated additions and subtractions.

EXAMPLE 6 � Calculate: �4[6 � (�3)] � (�12).

�4[6 � (�3)] � (�12) � �4(9) � (�12)

� �36 � (�12)

� 3

Check Your Understanding 1.1

Solve these problems to check your understanding of integers andoperations on them.

1. Write the following sets of integers in order from least to greatest.

a. 15, �2, 11, �7

b. 10, �23, �3, 0

2. Evaluate:

a. | 16 | b. | �9 | c. �| 23 |

d. (�6) � (�31) e. (�6)(�10) f. �(�5)

g. (�28) � (�4) h. 28(�3) � (�6) � (�2)

3. Jim has $220 in his bank account. If he writes five checks for $20 eachand then deposits $75, how much will he have in his account?

8 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

Page 11: course i

1 • N U M B E R S A N D O P E R A T I O N S 9

1.2 Common FractionsFractions are used to show the relationship of a part to a whole or the divisionof a whole into several parts. Two fractions are equivalent if they represent thesame portion of a whole.

EXAMPLE 1 � Meanings of Fractions—If a class of 30 students has 20 girlsand 10 boys, then �

23

00� is the fraction of the whole class that

is girls. If five of the boys order two large pizzas, sharingequally would give �

25� of a pizza to each boy.

EXAMPLE 2 � Pictures of Fractions—The fraction �23

00� can also be written as �

23�

because both represent the same part of a whole.

�2300� �

23

EXAMPLE 3 � Equivalent Fractions—The fractions �16

64�, �1

2050�, and �

14� are all

equivalent because they represent the same part of a wholequantity.

�1664� �

12050

� �14

Fractions can be compared in several useful ways. You can write the fractionswith a common denominator, compare the fractions to benchmarks like quar-ters and halves, or change the fractions to decimals.

Page 12: course i

EXAMPLE 4 � Comparing Fractions—In Mrs. Green’s class 9 out of 16 students ride the bus; in Mr. Brown’s class 11 out of 24 students ride the bus. Mrs. Green’s class has a greater frac-tion of students riding the bus because �1

96� � �

24

78� and �

12

14� � �

24

28�.

You can reach the same conclusion by comparing both frac-tions to the benchmark �

12� . Since �1

96� is a little greater than �

12� and

�12

14� is a bit less than �

12� , it is easy to see that �1

96� > �

12

14�.

Changing the fractions to decimals shows that �196� � 0.5625 and

�12

14� � 0.4583�, so �1

96� > �

12

14�.

To add and subtract fractions, you must write the fractions in terms of a com-mon denominator. Once you have the common denominator, you add or sub-tract the numerators.

EXAMPLE 5 � Adding and Subtracting Fractions

a. �23

� � �12

� � �76

� because �23

� � �46

� and �12

� � �36

b. �78

� � �14

� � �58

� because �14

� � �28

To multiply two fractions, multiply the numerators and multiply the denominators.

�ab

� � �dc

� � �badc�

To divide two fractions, you define the problem in terms of multiplication.

�ab

� � �dc

� � �ab

� � �dc

EXAMPLE 6 � Multiplying Fractions

a. �23

� • �58

� � �1204� which is equivalent to �

152�

b. �95

�(15) � �95

� • �115� � �

1355

� or 27

EXAMPLE 7 � Dividing Fractions

a. �23

� � �58

� � �23

� • �85

� � �1165�

b. 5 � �45

� � �51

� • �54

� � �245�

10 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

Page 13: course i

1 • N U M B E R S A N D O P E R A T I O N S 11

To use fractions well, you need to recognize which fraction operations arecalled for in a problem. The rules for order of operations involving fractions arethe same as those for whole numbers and integers. It helps to check calculationswith fractions by making estimates with numbers that are rounded to nearbysimple fractions or whole numbers.

EXAMPLE 8 � Fraction Problems—These examples show how operationswith fractions might be used.

a. Assembly and sealing of a mailing box requires

�78� yard of tape for the bottom

�12� yard of tape for the sides

�34� yard of tape for the top

To find the total amount of tape required for each box youneed to add these fractions. The answer is �

187� because �

12� is

equivalent to �48� and �

34� is equivalent to �

68�.

b. Similar boxes with dimensions �23� as long will require:

�23� • �

78�� �

12

44� yard of tape for bottom

�23� • �

12� � �

26� yard of tape for the sides

�23� • �

34� � �1

62� yard of tape for the top

c. If there are 25 yards of tape in a roll, this is enough for 11 boxes because:

25 � �187� � 25 • �1

87� � �

21070

� � 11�11

37�

As you see in Example 8 Part c, fraction problems can involve mixed numbers—the sum of a whole number and a fraction between 0 and 1. To perform opera-tions with mixed numbers, you can convert them to standard fractions and use therules for fraction operations. With addition and subtraction, you can often treat thewhole number and fraction parts separately, and then combine the results.

EXAMPLE 9 � Mixed Numbers

a. 2�34� � �

141� because 2 is equivalent to �

84�.

b. �430� � 13�

13� because 39 � 13(3).

c. 1�34� � �

12� � �

144� because 1�

34� is equivalent to �

74� and

�74� • �

21� � �

144�. The fraction �

144� can be rewritten as �

72� or 3�

12�.

Page 14: course i

Check Your Understanding 1.2

Solve these problems to check your understanding and skill withcommon fractions.

1. Evaluate and express your answers as equivalent fractions in simplestform.

a. �25� � �1

30� b. �

58� � �

34�

c. �45� • �1

72� d. �

95� � �1

121�

2. Write three equivalent fractions for each of the following numbers.

a. �35� b. �1

82�

c. �182� d. 3�

45�

3. Solve these fraction problems.

a. Pauline found that the amount of lemon juice added to a recipe was halfthe amount of sugar, and the amount of sugar was 6 times the amount ofsalt. If she added 1�

12� teaspoons of salt, how much lemon juice did she add?

b. A school pledged $2,000 toward cancer research. After one month,the students collected �

14� of the amount needed to reach half of their

goal. How much money had they collected?

4. In a 10 km race Samantha ran for 8�170� km and walked the rest.

a. How far did she walk?

b. What fractional part of the race did she run?

5. Bill won 7 out of 9 tennis matches in April and 15 out of 18 matchesin June.

a. In which month did he have the better record?

b. What number of wins in June would have given him the same recordfor both months?

6. In a magic square, rows, columns, and diagonals all have the same sum.Complete the following table to make a magic square.

12 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

5 ? ?

? 6�14� 8�

34�

? ? 7�12�

Page 15: course i

1 • N U M B E R S A N D O P E R A T I O N S 13

1.3 Decimals The decimal system of writing numbers expresses each whole number as a sumof ones, tens, hundreds, thousands, ten-thousands, and so on. For example,4,375 � 4(1,000) � 3(100) � 7(10) � 5(1). In a similar way, decimal fractionsexpress numbers between 0 and 1 as sums of tenths, hundredths, thousandths,and so on.

EXAMPLE 1 � Expressing Decimals as Fractions and Fractions as Decimals

a. 0.254 � 2��110�� � 5���1

100�� � 4��1,0

100��

b. 3.1415 � 3(1) � 1��110�� � 4 ���1

100�� � 1 ��1,0

100�� � 5 ��10,

1000��

c. �12� � 0.5 because �

12� is equivalent to �1

50�.

d. �34� � 0.75 because �

34� is equivalent to �1

7050� which is

equivalent to �17000� � �1

500� or �1

70� � �1

500�.

It is relatively easy to convert a fraction to its equivalent decimal form by usinga calculator. To find the equivalent decimal form of a fraction such as �

ab

�, simplydivide a by b. In many cases, this division will give an answer requiring only afew decimal places. In some quite important cases, an infinite repeating deci-mal will be required.

EXAMPLE 2 � Finite and Repeating Decimals

a. �58� � 0.625, �

95� � 1.8, and �2

101� � 0.55

b. �13� � 0.333…, �

23� � 0.666…, and �1

52� � 0.41666…

Operations with decimal fractions are now most often done with calculators orcomputers. To check your work with those tools, it helps to make mental esti-mates of the results. To make these estimates, use simple rounded values of thenumbers involved in a calculation.

EXAMPLE 3 � Estimating Decimal Calculations

a. 2.43 � 4.78 should be a bit more than 7.

b. 23.45 � 14.9 should be a bit less than 25 � 15 � 375.

c. 4.85 � 1.6 should be about 4.5 � 1.5 � 3.

Very large and very small numbers are often written in a special decimal formcalled scientific notation: a � 10b, where 0 ≤ a < 10. This form not only makessuch numbers easier to write, but it also makes calculations easier. Some impor-tant numbers are given in scientific notation form in the following chart.

Page 16: course i

EXAMPLE 4 � Scientific Notation

Check Your Understanding 1.3

Solve the following problems to check your understanding of decimals.

1. Write each decimal as a sum of common fractions and as a single fraction.

a. 0.128 b. 0.0205 c. 3.142

2. Write the decimal equivalents to these fractions and mixed numbers.

a. �35

� b. �192� c. 4�

25

� d. �95

3. For every 40 hours Simon works, he earns 2.5 hours of leave. Simonwants to save 60 hours of leave to travel to Mexico. If he has worked142.5, 170, and 163.5 hours in September, October, and November,how many more hours does he need to work until he has enough leaveto make his trip?

4. Calculate the number of seconds in a year and express your answer inscientific notation. Use 365.25 as the number of days per year as youmake your calculations.

5. Write each number in decimal form and identify which are finite andwhich are infinite repeating decimals.

a. �56

� b. �78

� c. �27

14 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

Significance Number Scientific Notation

Approximate population ofthe USA in 1998 270,000,000 2.7 � 108

Speed of light 300,000,000 m/sec 3.0 � 108 m/sec

1 AU: The mean distancebetween the Sun and Earth 150,000,000,000 m 1.5 � 1011 m

1 Angstrom: Unit used tomeasure wavelengths 0.0000000001 m 1 � 10�10 m

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1 • N U M B E R S A N D O P E R A T I O N S 15

1.4 Ratios and ProportionsWhenever it makes sense to compare two numbers, there are two calculationsto consider—finding the difference of the two numbers or finding the ratio ofthe two numbers by division. Ratios are often expressed as common fractionsor as decimal unit rates. Two ratios are equivalent when the corresponding frac-tions are equivalent or when they give the same unit rates.

EXAMPLE 1 � Ratio Comparison

a. In both swimming pools pictured below, the length andwidth differ by 10 feet. However, in one pool the ratio oflength to width is 20 to 10 or 2 to 1; in the other pool theratio of length to width is 40 to 30 or 4 to 3.

b. In one school homeroom there are 18 girls and 12 boys; inanother there are 12 girls and 8 boys. The differencebetween the number of girls and boys in the first homeroomis greater than the difference in the second homeroom.But the ratio of girls to boys is the same in both rooms—a ratio of 18 to 12 is equivalent to a ratio of 12 to 8because both are equivalent to a ratio of 3 to 2 or threegirls for every two boys.

c. If a van travels 400 miles on a 25-gallon tank of gasolineand a car travels 360 miles on a 12-gallon tank, the vantravels farther on a full tank of gas. But the best com-parison of fuel economy is to find the rates of miles pergallon of gasoline. For the van it is 16 miles per gallon;for the car it is 30 miles per gallon.

30 10

20

40

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EXAMPLE 2 � Comparing Ratios and Rates—Crackers are sold in medium orlarge boxes. The net weight of the medium box is 10.5 ounces,and it costs $2.49. The large box has a net weight of 16 ounces,and it costs $3.99.

To determine which box is the better buy, it is helpful to findthe unit rate for each size box: �

21

.04.95� � 0.237 dollars per ounce

for the medium box and �31.969

� � 0.249 dollars per ounce for thelarge box. Therefore, since the unit price for the medium boxis less than the unit price for the large box, the medium box isthe better buy.

EXAMPLE 3 � Solving Proportions

a. For a typical television screen, the ratio of width to heightis about 3 to 2. To determine the height of a large screenthat has a width of 24 inches, write and solve the propor-tion �

32� � �

2x4�. Since 24 is 8 times 3, x must be 8 times 2. So

the height will be about 16 inches.

b. On a map of Washington, D.C. with a scale of 1 inch �3 miles, the distance from the Lincoln Memorial to theU.S. Capitol Building measures 1.25 inches. This meansthat the actual distance would be (1.25)(3) � 3.75 miles.

Check Your Understanding 1.4

Check your understanding and skill with ratios and proportions bysolving the following problems.

1. Solve these problems using proportional reasoning if it is appropriate.If proportional reasoning is not appropriate, explain why.

a. It took Jan 6 hours to fly from Baltimore to San Francisco, a distanceof 2,464 miles. How long will it take her to fly from Chicago to San Francisco, a distance of 1,863 miles?

b. If Tony pays $1.98 for 2 pounds of apples, how much will he haveto pay for 5 pounds of apples?

c. The price of a pizza depends on the area of the pizza. If a 6-inch-diameter pizza costs $4.50, then a 9-inch-diameter pizzashould cost how much?

16 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

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1 • N U M B E R S A N D O P E R A T I O N S 17

2. A company that does market research for radio stations found that 50 people out of a sample of 400 people listened to radio stationWXKR. Of the 50 people, 30 were 25 years old or younger. Based onthis information,

a. If 600 people had been contacted, how many would you expect tolisten to WXKR?

b. How many of those listeners would you expect to be 25 years old oryounger?

3. A rectangular tank contains 250 mL of water when filled to a heightof 12 cm. If the tank is filled to a height of 18 cm, how much water willit contain?

4. Which is the better buy on Pix cereal—the super-size box containing14 ounces for $3.85 or the large-size box containing 9.5 ounces for $2.59?

1.5 Percents The most common way to describe the relation of a part to a whole is using thelanguage of percents. To describe a part as some percent of a whole, imaginethe whole divided into 100 equal size pieces and ask how many of those pieceswould be in the part that you are interested in.

EXAMPLE 1 � Meaning of Percent

a. A quarter is 25% of a dollar because there are 100 centsin a dollar and 25 cents in a quarter.

b. A centimeter is 1% of a meter because there are 100 cen-timeters in a meter.

c. If a school year is 9 months long, it takes up 75% of thecalendar year. The fraction �1

92� is equivalent to �1

7050�.

Since percents are special kinds of fractions, they can be expressed inseveral equivalent forms. It is often helpful to write percent information usingcommon fraction and decimal equivalents.

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EXAMPLE 2 � Representing Percents—The fraction 35% is equivalent to �13050�

and to 0.35. This percent is also shown in the following diagram.

The most common percents are those that represent the mostfrequently occurring fractions like:

�12

� � 50% �13

� � 33 �13

�% �14

� � 25% �15

� � 20%

�16

� � 16 �23

�% �18

� � 12.5% �110� � 10%

When problems involve reasoning with percents, the questions usually requirefinding a missing number or percent in a statement like “A is B% of C.”

EXAMPLE 3 � Percent Calculations

a. To find 45% of 300, you can convert the percent to a frac-tion or decimal and multiply: ��1

4050��(300) or 0.45(300).

Both give a result of 135.

b. To find the percent of 80 represented by 32, you can writethe common fraction �

38

20� and find an equivalent fraction

with denominator 100. It is generally easier to do thedivision and convert the resulting decimal to a percent (32 � 80 � 0.40 or 40%).

c. If 90 is 30% of some number, to find the unknown numberyou need to solve the equation 90 � 0.30x. That means x � 90 � 0.30 or x � 300.

Problems involving percents often ask questions about the percent increase ordecrease in some measurement.

18 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

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1 • N U M B E R S A N D O P E R A T I O N S 19

EXAMPLE 4 � Percent of Increase and Decrease

a. If the $48 price of a sweater is reduced by 35%, the new price will be $31.20 because 0.35(48) � 16.80 and48 � 16.80 � 31.20. You could arrive at the same resultby observing that the new price is 65% of the original and.65(48) � 31.20.

b. If the rent on an apartment is now $600 per month andnext month will increase by 7%, the new rent will be$642. You can figure that amount by calculating0.07(600) � 42 and 600 � 42 � 642 or 1.07(600) � 642.

c. In 1992, the longest bungee jump was made in Franceusing an 820-foot-long bungee cord. The cord stretched to a length of 2,000 feet during the jump. To determine the percent of increase, you compare the increase in length to the original length. In this case, that gives 1,180 � 820 � 1.44, so the cord length increased byapproximately 144%.

Check Your Understanding 1.5

Solve the following problems to check your understanding and skillwith percents.

1. Complete this chart by filling in the missing fractions, decimals, andpercents.

a.

b.

c.

d.

e.

2. Find the unknown numbers or percents.

a. What is 38% of 200?

b. 25 is what percent of 40?

c. 20 is 65% of what number?

Fraction Decimal Percent�34�

0.1

0.875�23�

�1270�

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3. A local department store advertised a special 4th of July sale with dis-counts of 25% on clearance merchandise that was already 50% off theregular price. If a clearance item regularly sold for $50, how muchwould it cost during this sale? This price represents a discount of whatpercent of the original price?

4. The following chart shows unit shipments (in thousands) of U.S. con-sumer telecommunications products for 1987 and 1996. Complete thecolumn of percent increase for shipments of each product.

Source: 1999 New York Times Almanac. New York, NY: Penguin Reference, 1998.

5. Future population relates directly to the total fertility rate (TFR). Theworld population will remain constant when the TFR is 2.1. Completethe chart to show the percent decrease in TFR as world populationmoves toward becoming constant.

Source: www.overpopulation.com/tfr.html

20 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

Product 1987 1996 Percent IncreaseCorded phone 13,335 21,700

Cordless phone 9,900 22,800

Answering machine 14,716 20,050

Fax and/or fax modem 1,907 4,700

Region 1990 1998 Percent DecreaseWorld 3.4 2.9

Less-Developed Countries 4.7 3.2

More-Developed Countries 1.9 1.6

Fertility Rates 1990�1998

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2 • P A T T E R N S A N D A L G E B R A 21

�2 Patterns and Algebra

Number patterns often occur in mathematics and its applications. Tables, graphs,and symbolic expressions can be used to represent and study those patterns.

2.1 Expressing Patterns To discover number patterns in problems, it helps to organize data in a table ordiagram and then look for a rule that connects the elements.

EXAMPLE 1 � Numerical Patterns—Jamie is saving money to buy a bike. Shehas $10 at the start of the summer and plans to save $20 of herbaby-sitting money each week. This table shows the growth inJamie’s savings:

Jamie begins with $10 and adds $20 each week, so after w weeks, she will have a total of 10 � 20w dollars.

EXAMPLE 2 � Geometric Patterns—Three stages of a geometric pattern are shownbelow. This pattern is made of square tiles and triangular tiles.

You could continue the pattern and generate a table showingthe number of triangles, the number of squares, and the totalnumber of tiles at each stage.

Week 1 2 3

Savings 10 + 20 10 + 20 + 20 10 + 20 + 20 + 20

Stage 3Stage 2Stage 1

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At each stage after the first, two triangles and one square areadded. At Stage 4, there will be 5 squares and 8 triangles, for atotal of 13 tiles.

The sketches and the table of tile numbers could be extended to account for anystage of the pattern. But that strategy would not be helpful for finding the num-ber of tiles needed to build Stage 100 of the pattern! Symbolic expressions for thenumber patterns would be more efficient.

EXAMPLE 3 � Symbolic Expressions—As a pattern emerges in the table forExample 2, you can probably see ways to express the numbersof tiles in terms of the pattern stage number n.

The total number of tiles at Stage n could be expressed anoth-er way by examining the pattern of the numbers in the last row,rather than as the sum of the number of squares and the num-ber of triangles.

EXAMPLE 4 � Order of Operations—The symbolic expressions of algebragive rules for arithmetic calculations. Expressions for multipli-cation, like “a � b” are often written without the multiplicationsign as “ab.” To write and use symbolic rules as intended, youneed to follow the mathematical standards for order of opera-tions. The key order of operations conventions for linearexpressions are

22 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

Stage Number 1 2 3 4

Number of Squares 2 3 4 ?

Number of Triangles 2 4 6 ?

Total Number of Tiles 4 7 10 ?

Stage Number 1 2 3 n

Number of Squares 1 � 1 2 � 1 3 � 1 n � 1

Number of Triangles 2(1) 2(2) 2(3) 2n

Total Number of Tiles 2 � 2 3 � 4 4 � 6 (n � 1) � 2n

Stage Number 1 2 3 n

Total Number of Tiles 4 7 10 3n � 1

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2 • P A T T E R N S A N D A L G E B R A 23

■ Do calculations within parentheses first.

■ Working from left to right, do multiplication and divisionbefore addition and subtraction.

As you learn about new kinds of algebraic expressions, you’lllearn more order of operations guidelines.

If x � �3, then:

a. 5 � 7x � 5 � (�21) � �16.

b. �4(5 � 8x) � �4(5 � (�24)) � �4(29) � �116.

Check Your Understanding 2.1

Check your understanding and skill in expressing patterns by com-pleting the following problems.

1. Louis weighs 280 pounds and has just joined a weight loss programthat guarantees he will lose 10 pounds per month until he reaches histarget weight. Write an expression for Louis’ weight after m months.

2. The first three stages of a geometric pattern are shown below.

a. Sketch the next stage of the pattern.

b. Complete the following table:

3. Use order of operations rules to evaluate the following expressions.

a. �7x � 12 when x � 4 b. 3(4x � 13) when x � �2

c. 13 � 5(8 � 2x) when x � �5 d. 2L � 2W when L � 10 and W � 6

Stage 3Stage 2Stage 1

Stage Number 1 2 3 4 5 n

Number of Squares

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2.2 GraphsNumerical information is very often presented in graphs. Solving problems oftenrequires reading the stories told by those graphs.

EXAMPLE 1 � Reading Points on Graphs—Jamie started saving for a new bikewith $10 and plans to save $20 each week from summer baby-sitting earnings. You could show the pattern of growth inJamie’s savings toward a bike using a graph like the one below.

The point (3, 70) indicates that after 3 weeks, Jamie will havesaved $70.

EXAMPLE 2 � Rates of Change in Graphs—The next graph shows thescheduled trip of an express train from Washington, D.C., toNew York City. The train makes only two short stops—inBaltimore and Philadelphia.

24 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

0 6543210

120

100

80

60

40

20

Saving Time (in weeks)

Sav

ing

s B

alan

ce (

in $

)

7:000

10:009:008:00

300

250

200

150

100

50

Time of Day

Dis

tan

ce T

rave

led

(in

mile

s)

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2 • P A T T E R N S A N D A L G E B R A 25

Trip Leg Distance Time SpeedWashington to Baltimore 50 miles 0.5 hours 100 mph

Baltimore to Philadelphia 100 miles 1 hour 100 mph

Philadelphia to New York 100 miles 1.5 hours 67 mph

From the graph you can deduce the following facts.

EXAMPLE 3 � Trends in Graphs—The next graph shows the operating cost foran airplane on trips of different lengths. From the graph you cansee that operating cost increases as trip length increases.However, as trips get longer the cost per mile actually decreases.

The point (200, 2,000) shows that for a trip of 200 miles, thecost is $2,000 to operate the plane. The point (600, 2,500)shows that for a trip of 600 miles, the cost is $2,500 to operatethe plane. The cost per mile for a trip of 200 miles is $10; thecost per mile for a trip of 600 miles is $4.17.

500

0 6004002000

3,000

2,500

2,000

1,500

1,000

Trip Length (in miles)

Op

erat

ing

Co

st (

in $

)

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Check Your Understanding 2.2

Complete the following problems to check your understanding oftables and graphs.

1. The chart below shows the cost of parking in a garage.

a. If you park your car in the garage at 8:00 A.M., how much will park-ing cost if you leave at 8:45 A.M.? At 11:48 A.M.? At 3:20 P.M.?

b. Sketch a graph showing the cost of parking in this garage for timesbetween 0 and 8 hours.

2. Rachel and her younger brother,Micah, ran a 100-meter race. Their distances covered at various times in the race are shown in the following graph.

Use the graph to answer the following questions.

a. Who won the race? By how much time and by how much distance?

b. At what distance in the race (if any) were Rachel and Micah tied?

c. How many seconds from the start (if ever) were Rachel and Micah tied?

d. Who was ahead after 20 seconds? By how much?

e. What was each runner’s average speed for the race?

26 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

Time CostUp to 1 hour $2.00

Up to 2 hours $3.50

Up to 3 hours $4.50

Up to 4 hours $5.00

Up to 5 hours $5.50

Maximum $6.00

0 302520151050

100

80

60

40

20

Time Running(in seconds)

Dis

tan

ce C

ove

red

(in

met

ers)

RachelMicah

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2 • P A T T E R N S A N D A L G E B R A 27

2.3 Equations and Inequalities Equations and inequalities compare values and/or expressions. When equationsand inequalities are used to represent conditions of a problem, their solutionsusually give helpful information.

EXAMPLE 1 � Solving Linear Equations—Jamie started a saving plan for anew bike with $10 and plans to save $20 each week. If Jamiewants to purchase a mountain bike that costs $170, she’ll cer-tainly wonder how long she’ll have to work and save in orderto have enough money to buy the bike.

To answer this question, you can write and solve the equation10 � 20w � 170.

You might reason that since she already has $10, Jamie needsto save $160 more. At $20 per week, it would take her 8 weeksto save $160. Therefore, Jamie could afford the bike in8 weeks.

You can also reason algebraically to solve such equations. To solve a linear equation algebraically, undo the operations of an expression in the reverse orderthat you would perform them to calculate the value of the expression.

EXAMPLE 2 � Algebraic Reasoning

a. To solve Jamie’s equation with algebraic reasoning youcan think:If 10 � 20w � 170then 20w � 160 (subtract 10 from both sides)then w � 8 (divide both sides by 20)

The order of operations indicates that in 10 � 20w you wouldfirst multiply w by 20, then add 10. Therefore, in solving, first“undo” the addition, then “undo” the multiplication.

b. To solve 2(a � 5) � 8, reason like this:If 2(a � 5) � 8then a � 5 � 4 (divide both sides by 2)so a � 9 (add 5 to both sides)

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You can check any proposed solution for an equation by substituting it in the original equation. In Example 2 Part b, since 2(9 � 5) � 8, you know that a � 9is a solution.

Some problems lead to a different type of comparison called an inequality.Linear inequalities are solved by the same process as equations, with one caution.When you multiply or divide both sides of an inequality by a negative number,you must reverse the direction of the inequality.

EXAMPLE 3 � Solving Inequalities

a. To solve 5b ≥ �20, divide both sides by 5 to get b ≥ �4.

b. To solve 1 � 3y < 13, subtract 1 from both sides to get�3y < 12. Then divide both sides by �3 and reverse thedirection of the inequality sign to get y > �4.

You can make an informal check of the solution to an inequality by testingvarious numbers from the proposed solution set in the original inequality.However, in general there will be infinitely many solutions to inequalities.

Check Your Understanding 2.3

Check your understanding of equations and inequalities by solvingthe following problems.

1. Solve each of the following equations for x and check your solution.

a. 4x � 3 � 17 b. 3(x � 1) � 15

2. Solve each of the following inequalities for n.

a. �2n > 18

b. 5n � 2 < �8

3. Renting a video at a local store costs $3.50 for the first day and $2.50for each additional day.

a. Write an equation for the cost C of renting a video for d days.

b. Write and solve an equation with a solution that answers the ques-tion, “How long have you kept a video if the bill is $16?”

28 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

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3 • G E O M E T R Y A N D S P A T I A L S E N S E 29

�3 Geometry and Spatial Sense

From flower blossoms and spider webs to cereal boxes and kitchen floors, geo-metric figures surround us in our daily lives. The mathematical properties and rela-tionships of geometric figures have applications in art, science, and everyday life.

3.1 Shapes and PropertiesDifferent shapes have unique properties that may make them practical choices forparticular applications. Circles appear as wheels and jar lids, while rectanglesappear as windows, and triangles appear in the trusses of bridges.

Various polygon shapes appear in street signs such as those shown below.

Polygons are classified by properties such as the number of sides. Triangles(polygons with three sides) may be further classified by angle type or by the num-ber of congruent sides. A polygon having all sides the same length and all anglesthe same measure is called a regular polygon.

ONE WAY STOP

MENAT

WORK

YIELD

SCHOOLZONE

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EXAMPLE 1 � Types of Triangles—This is a right triangle because it has aright angle:

This is an isosceles triangle because exactly two sides (and twoangles) are of equal measure:

This is an equilateral triangle because all of the sides (and allof the angles) are of equal measure:

Another property that distinguishes various polygons is the number of pairs of parallel sides. Parallel lines are straight lines that never meet, no matter howfar they are extended. Quadrilaterals, polygons with four sides, may be furtherclassified according to the number of parallel sides.

30 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

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3 • G E O M E T R Y A N D S P A T I A L S E N S E 31

EXAMPLE 2 � Special Quadrilaterals—This figure is a trapezoid because ithas exactly 1 pair of parallel sides. You should recognize thenotation for the pair of parallel sides in this figure, A�B� and C�D�,as A�B� || C�D�.

This figure is a parallelogram because both pairs of oppositesides are parallel. A�B� || C�D� and A�D� || B�C�

Special angle relationships hold when parallel lines are crossed by another line(which is then called a transversal). Many pairs of angles formed by the paralleland crossing lines have equal measure.

EXAMPLE 3 � Angles Formed by Parallel Lines and a Transversal—In the dia-gram below, line l is parallel to line m, (l || m), and m∠ 5 � 65˚.These facts imply the measures of all remaining angles.

m ∠ 5 � m ∠ 8 � 65˚

m ∠ 5 � m ∠ 4 � m ∠ 1 � 65˚

m ∠ 2 � 180˚ � 65˚ � 115˚

m ∠ 2 � m ∠ 3 � m ∠ 6 � m ∠ 7 � 115˚

These angle relationships imply other properties of polygons. One you have proba-bly learned is that the sum of the measures of the angles in any triangle is 180˚.

A B

D C

A B

D C

l

m

n

12

3 4

56

78

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EXAMPLE 4 � Angle Measures in Any Triangle—In the triangle below, you canuse the given facts to find m∠ B.

m∠ A � m∠ B � m∠ C � 180˚

45˚ � m∠ B � 40˚ � 180˚

So, m∠ B � 95˚.

Another very important geometric principle, the Pythagorean Theorem, relatesthe side lengths of any right triangle:

a2 � b2 � c2

In this equation, a and b are the lengths of thelegs, and c is the length of the hypotenuse.

EXAMPLE 5 � Scott let out all 50 yards of string on his kite. His sister, Katie,stood directly under the kite, 30 yards from Scott. Assumingthat the kite string was a straight line, about how high above theground was the kite?

Using the Pythagorean Theorem you can write the equation:

302 � h2 � 502

So, 900 � h2 � 2,500

h2 � 1,600

h � �1,600�

or h � 40

If Scott was holding the kite string about 1 yard above theground, then the kite was about 41 yards above the ground.

32 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

A

B

C45˚ 40˚

a b

c

S K30

50 h

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3 • G E O M E T R Y A N D S P A T I A L S E N S E 33

Check Your Understanding 3.1

Complete the following problems to check your understanding ofshapes and their properties.

1. Can two sides of a triangle be parallel to each other? Explain why orwhy not.

2. Name the polygons that describe the shapes of the street signs shownon page 29. Identify those that are regular polygons.

3. Use the given information to find the measure of each labeled angle inthe following parallelogram with one diagonal drawn from vertex A tovertex C.

Hint: Line AC is atransversal betweenparallel lines AB andDC and also a trans-versal between parallellines AD and BC.

4. Find the length of each diagonal in this rectangle.

3.2 Symmetry and Transformations Symmetry is a very important property of shapes and designs. Symmetry isusually explained by describing motions of a figure that will leave its imageunchanged.

A figure that has line symmetry can be folded along a line so that one half ofthe figure exactly matches the other half.

45˚30˚a

b c d

A B

CD

5 cm

12 cm

A B

CD

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EXAMPLE 1 � Line Symmetry—The figures below have line symmetry, andthe dotted lines indicate the lines or axes of symmetry.

A figure that has rotational symmetry can be turned less than a full turn about itscenterpoint and appear unchanged.

EXAMPLE 2 � Rotational Symmetry—The figures below have rotational orturn symmetry.

The first figure can be turned through 180˚ (a half-turn) andappear unchanged. The second figure can be turned through90˚ (a quarter-turn), 180˚, or 270˚ and appear unchanged.

There are three basic motions or transformations that change the position of a figure without changing its size or shape. The informal names for those motionsare flips, turns, and slides.

EXAMPLE 3 � Rigid Motions are special transformations thatmove a shape in a plane without changing itsshape or size. Consider this simple shape andits images under various rigid motions.

34 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

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3 • G E O M E T R Y A N D S P A T I A L S E N S E 35

a. When a shape is transformed by a flip (or reflection) overa line, the shape and its image are symmetric with respectto the line.

b. A turn (or rotation) turns a shape about a point (called thecenter of the turn) through a specified angle. Turns areusually measured in a counterclockwise direction.

90˚ counterclockwise 180˚ turn about 270˚ counter-turn about point A point B clockwise turn

about point C

c. A slide (or translation) moves a figure a specified dis-tance in a specified direction without turning it. As inyour work with integer addition, a slide can be describedby an arrow.

A B C

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Check Your Understanding 3.2

Complete these problems to check your understanding of symmetryand transformations.

1. Describe the symmetries of a regular octagon.

2. Describe the symmetries of each figure below. Indicate any lines ofsymmetry and angles of turn symmetry.

a. b. c.

3. For each part, identify the figure on the right that can be obtained fromthe figure on the left by the indicated transformation.

a. A flip I II III IV

b. A turn

c. A slide

d. A flip

e. A turn

f. A slide

36 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

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4 • M E A S U R E M E N T 37

�4 Measurement

Measurement uses numbers to describe the size of things. Rulers measure thelength of line segments, clocks measure periods of time, scales and balancesmeasure weights and masses, and protractors measure angles. In each case,measurement compares the size of a given object or time period to some standardunit like a meter, a second, a gram, or a degree. In many figures, it is possible to usebasic measurement data and arithmetic calculations to derive other size information.

The following sections review the key concepts and skills required to answermeasurement questions. They also provide some further problems to check yourunderstanding and practice your skills.

4.1 Measuring Angles

The common unit for measuring angles and rotations is the degree. There are 90˚in a right angle or quarter-turn, 180˚ in a half-turn, and 360˚ in a full turn. Anglescan be measured by a protractor.

EXAMPLE 1 � Using a Protractor—In the diagram below, the measure of∠ AOB = 45°, the measure of ∠ AOC � 120°, and the measure of ∠ BOC � 75° (since 120° � 45° � 75°). The measure of ∠ DOC � 60°

C

B

ADO

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Check Your Understanding 4.1

Check your understanding of basic ideas of angle measurement bycompleting the following problems.

1. In the following diagram, what degree measure is indicated for eachangle.

a. ∠ PQR b. ∠ PQS

c. ∠ SQT d. ∠ RQT

2. Without using a protractor, draw angles with the following measures.Then check your estimates using a protractor.

a. 20˚ b. 150˚c. 60˚ d. 110˚

4.2 Measuring Perimeter The distance around a geometric figure is called its perimeter. For irregularfigures, the perimeter can only be estimated by measuring polygonal paths thatapproximate the true boundary. For many standard figures, it’s possible tomeasure a few key dimensions and then calculate the total boundary length.

38 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

RS

P

T

Q

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4 • M E A S U R E M E N T 39

EXAMPLE 1 � Estimating Perimeter—A trip around the lake shown in thisscale drawing is about 15 kilometers because the perimeter ofthe drawing is about 15 centimeters and the drawing scale is1 cm � 1 km.

The perimeter of a rectangle can be calculated from its length (l) and width (w)by the formula P � 2l � 2w.

If dimensions of the rectangle are called base (b) and height (h), the formulabecomes P � 2b � 2h.

The perimeter of a circle is called its circum-ference. The circumference of any circle can becalculated from the diameter (d) using the formulaC � �d. The constant � (“pi”), is approximately3.14. Since the diameter of a circle is two timesthe radius (r), the circumference can also be cal-culated using the formula C � 2�r.

EXAMPLE 2 � Perimeter of Rectangles and Circles—The perimeter of the rectangle shown here is 77.4 meters because 2(23.5) � 2(15.2)� 77.4. The circumference of the circle is about 47.1 metersbecause 15� � 47.1.

1 cm = 1 km

l

w

P = 2l + 2w

r

d

C = 2πr = 2d

15.2 m

23.5 m

15 m

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EXAMPLE 3 � Perimeter Problem—The following diagram shows a soccerfield enclosed by a running track. The perimeter of the soccerfield is 2(120) � 2(55) � 350 meters. The perimeter of the run-ning track is (55� � 240) � 413 meters because it consists ofthe circumference of a circle with diameter 55 meters and twosides of the soccer field.

Check Your Understanding 4.2

Check your understanding of perimeter and circumference by solvingthese problems.

1. The following diagram shows pools at a recreation center—a squarediving pool, a rectangular racing pool, and a circular pool for youngchildren. Staff at the center have to scrub the tiles around each poolonce a week to meet health department regulations.

a. What are the perimeters of the various pools?

b. If scrubbing the tiles can be done at the rate of 1.5 meters per minute,how long will the job take each week?

40 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

120 m

55 m

50 m

20 m

15 m

d = 8 m

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4 • M E A S U R E M E N T 41

2. What is the perimeter of the swimming pool pictured below?

3. Earth’s equator is a circle with a circumference of approximately 40,000 kilometers. What are the diameter and radius of that circle?

4.3 Measuring Area The area of a two-dimensional shape describes the size of the region enclosed bythe shape. The area of any figure can be estimated by imagining a grid of unitsquares covering it. With some figures it is possible to use linear measurementsand arithmetic operations to calculate area.

EXAMPLE 1 � Estimating Area—The centimeter grid covering the scale draw-ing of a lake requires about 12 squares, each matching a squarekilometer on the real lake. So the area of the lake is approxi-mately 12 square kilometers.

10 m

4 m6 m8 m

4 m

1 cm = 1 km

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The area of any rectangle or parallelogram is given by the formula A � bh, whereb and h are the base and height of the figure. The area of any triangle is given byA � �

12� bh, where b and h are the base and height of the figure. The area of any

circle with radius r is given by the formula A � �r2.

EXAMPLE 2 � Areas of Rectangles and Circles—The total area enclosed by arunning track shown at the top of page 40 is 6,600 � 756.25�

square meters. The soccer field itself has area (120)(55) � 6,600square meters. The semicircular ends have total area equal to thatof a single circle with radius 27.5 meters or �(27.5)2 ≈ 2,376square meters.

EXAMPLE 3 � Areas of Triangles and Parallelograms—In using formulas tocalculate areas of triangles and non-rectangular parallelo-grams, it’s important to identify the base and height.

a. The area of the triangle shown on the left below is 6 square centimeters because (0.5)(6)(2) � 6.

b. The area of the triangle in the middle is 3 square cen-timeters because (0.5)(3)(2) � 3.

c. The area of the parallelogram is 9 square centimetersbecause (3)(3) � 9.

42 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

A = bhb

h

A = bh12

b

h

A = πr 2

r

120 m

55 m

II

3 cm

2 cm4.5 cm

I

4.4 cm

6 cm

2 cm2.8 cm

III

3 cm

3 cm

3.6 cm

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4 • M E A S U R E M E N T 43

EXAMPLE 4 � Surface Area of Boxes—The box shown below has a total surfacearea of 62 m2 because the different visible faces have areas(3)(2) � 6 m2, (3)(5) � 15 m2, and (5)(2) � 10 m2, and thereare two identical faces of each type.

Check Your Understanding 4.3

Solve these problems to check your understanding of area measure-ment and calculations.

1. Look again at the top view of aswimming pool with attached diving well.

a. What is the surface area of thewater in the pool?

b. If there are 150 people in thepool on a hot day, how muchspace is there for each of them?

2. The rectangular flag design shown at right has three regions, each ofwhich is a different color.

a. What is the area of theparallelogram region?

b. What is the area of each triangular region?

3. On many large farms, irrigation sprinklersrotate around a central pump to spray waterin circular patterns on the fields. What is thearea watered by such a system if the sprinklerarm has a length of 100 meters?

3 m

2 m5 m

50 cm 50 cm

60 cm

10 m

4 m6 m8 m

4 m

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44 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

�5 Data Analysis, Statistics, and Probability

To solve problems and make important decisions, we often need to collect, dis-play, and analyze data about key variables. The best tools for these tasks comefrom the mathematics of statistics and probability.

5.1 Drawing and Reading Graphs It is often hard to make sense of numerical data until the numbers have beenorganized and displayed in a useful graphic form. The choice of graphic displayusually depends on the type of data available.

EXAMPLE 1 � Bar and Circle Graphs—The following table shows data aboutthe distribution of world population in the years 1950 and 2000and a projection to the year 2050.

Source: Population: A Lively Introduction, Third Edition,Volume 53, No. 3, September, 1998.

This sort of categorical data is commonly displayed with bargraphs and circle graphs like those below for the 1950 data.The heights of the various bars are proportional to the popula-tion numbers they represent. The central angles of wedges in acircle graph are fractions of 360˚ that correspond to the fractionsof total population in each region.

Region 1950 2000 2050North America 7% 5% 4%

Europe, Japan, Australia 25% 15% 8%

Africa 9% 13% 22%

Asia less Japan 52% 59% 57%

Latin America 7% 8% 9%

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5 • D A T A A N A L Y S I S , S T A T I S T I C S , A N D P R O B A B I L I T Y 45

Data that give frequencies of measurements like temperatures, test scores,heights, weights, or fuel economy for cars are often displayed in histograms orstem-and-leaf plots. The following histogram and stem-and-leaf plot display pop-ulations of the 50 United States. (Source: 1999 World Almanac and Book of Facts,Mahwah, NJ: World Almanac, 1998.)

EXAMPLE 2 � Histogram—The histogram below groups states with popula-tions less than 1 million, from 1 million up to 2 million, and soon. For example, the bar of height 10 shows that there are 10states with populations at least 1 million and less than 2 million.

Distribution of World Population in 1950

Asia

Europe,Japan,

Australia

Africa

NALA

52%

25%

7%

7%

9%

NA EuropeJapan

Australia

Africa Asia LA

10

20

30

40

50

Region

Per

cen

t o

f W

orl

d P

op

ula

tio

n

5

10

...

2 4 6 8 10 12 14 18 30...State Population (in millions)

Fre

qu

ency

00

16

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EXAMPLE 3 � Stem-and-Leaf Plot—This stem-and-leaf plot records eachstate’s population to the nearest tenth of a million. The stem indi-cates millions and the leaf indicates tenths of a million.

0 5 6 6 6 7 7 81 0 0 1 1 2 3 5 6 8 82 4 5 6 8 93 2 3 4 6 7 84 1 3 4 95 0 0 0 2 66 0 3 6 77 789 410 911 512 013 3

• •

• •

• •

17 318 1

• •

• •

30 4 2 | 4 represents 2.4 million

Check Your Understanding 5.1

Check your understanding of statistical graph methods by completingthe following problems.

1. Use the data in Example 1 (page 44) to construct each graph.

a. A bar graph showing the distribution of world population in the year2000.

b. A circle graph showing the distribution of world population projectedfor 2050.

46 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

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5 • D A T A A N A L Y S I S , S T A T I S T I C S , A N D P R O B A B I L I T Y 47

2. The following numbers are high school graduation rates (% of agegroup) for the 50 United States in 1995–96.

58, 65, 58, 75, 65, 72, 74, 66, 53, 58, 55, 75, 80, 70, 85, 76, 68, 58,72, 74, 76, 70, 85, 57, 71, 83, 83, 65, 75, 83, 63, 62, 62, 89, 71, 73,67, 76, 71, 54, 87, 63, 58, 78, 90, 76, 72, 76, 80, 78Source: 1999 World Almanac and Book of Facts, Mahwah, NJ: World Almanac, 1998.

a. Display these data in a histogram, grouping the data in intervals of10 percentage points.

b. Display the data in a stem-and-leaf plot.

5.2 Data Summaries When you need to give quick summaries of data distributions, there are severalstandard statistical measures. To describe the middle or center of a data set, youcan give the mean or the median. To find the mean of a data set, you add all datavalues and divide by the number of elements in the data set. The median of a dataset is the middle value of the distribution. In the case of an even number of datapoints, it is the value midway between the two points that determines upper andlower halves of the distribution.

EXAMPLE 1 � Mean and Median—The state populations graphed in Example 2of section 5.1 (page 45) have been ordered (populations rounded to the nearest tenth of a million) in the list below.

0.5, 0.6, 0.6, 0.6, 0.7, 0.7, 0.8, 1.0, 1.0, 1.1, 1.1, 1.2, 1.3, 1.5,1.6, 1.8, 1.8, 2.4, 2.5, 2.6, 2.8, 2.9, 3.2, 3.3, 3.4, 3.6, 3.7, 3.8,4.1, 4.3, 4.4, 4.9, 5.0, 5.0, 5.0, 5.2, 5.6, 6.0, 6.3, 6.6, 6.7, 7.7,9.4, 10.9, 11.5, 12.0, 13.3, 17.3, 18.1, 30.4.

The mean of this data set is 5.036 because the sum of all pop-ulation figures is 251.8 million and there are 50 states.

The median state population is 3.5 million because the stateranking 25th has a population of 3.4 million and the state rank-ing 26th has a population of 3.6 million.

Note that the mean is much higher than the median becausethere are a few very populous states that pull up the mean, butnot the median.

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Means and medians don’t tell everything of interest about a data set. It is also use-ful to describe the range of values, the difference between the largest and thesmallest data elements.

EXAMPLE 2 � Range of Data—In the case of state populations, the range is29.9 million, from a low of 0.5 million to a high of 30.4 mil-lion. Can you guess which states are at the two extremes?

Check Your Understanding 5.2

Complete the following problems to check your understanding ofnumerical summaries of data.

1. Reproduced below are the 1995–96 high school graduation rates of the50 United States.

58, 65, 58, 75, 65, 72, 74, 66, 53, 58, 55, 75, 80, 70, 85, 76, 68, 58, 72,74, 76, 70, 85, 57, 71, 83, 83, 65, 75, 83, 63, 62, 62, 89, 71, 73, 67, 76,71, 54, 87, 63, 58, 78, 90, 76, 72, 76, 80, 78

a. Find the mean. b. Find the median. c. Find the range.

2. Suppose the class mean for 20 students on a 10-point test is 7.5, themedian is 8, and the range is 3 points. What will happen to these sum-mary statistics if the teacher:

a. Adds 1 point to every student’s score?

b. Multiplies each score by 10?

c. Includes one new score of 6 for a student who took a makeup test?

48 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

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5 • D A T A A N A L Y S I S , S T A T I S T I C S , A N D P R O B A B I L I T Y 49

5.3 ProbabilityMany activities in science, business, and everyday life have uncertain outcomes.We can’t predict with certainty the outcome of a coin toss, the gender of acoming baby, the weather at some future date, or the winning numbers in oneday’s state lottery. However, we can make useful predictions about likely out-comes of such events. The probabilities of outcomes from random experimentscan be estimated by collecting data in many trials of the experiment or, in somesituations, by carefully analyzing the experiment.

EXAMPLE 1 � Experimental ProbabilityEstimates—In 100 tosses ofa thumbtack, the point wasfacing upward 65 times andfacing downward 35 times.This experience suggestsestimating the probability of“point up” to be 0.65 and the probability of “point down” to be0.35. Based on these estimates, in 500 tosses one would expectabout 65% or 325 of the tosses to land point up.

EXAMPLE 2 � Probability by Analysis—If one tosses a nickel and a dime andrecords the result as heads up (H) or tails up (T) for each coin,it makes sense to predict that the probability of at least one coinbeing a head would be 0.75.

This prediction follows from analysis of the situation—if thecoins are fair, then each has an equal chance of coming upheads or tails. There are four possible, equally likely outcomes:

HnHd , HnTd , TnHd , and TnTd .

Each outcome will occur about one-fourth of the time, andthree of the four outcomes have at least one H.

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Check Your Understanding 5.3

Solve the following problems to check your understanding of proba-bility.

1. Suppose you flip a penny, a nickel, and a dime and note which come upheads and which come up tails.

a. List the possible outcomes of this experiment.

b. Use analysis of your answer in Part a to predict the probabilities that:

■ At least 2 coins will be heads up.■ No coins will be heads up.■ The value of the heads-up coins will be at least 11 cents.

2. Records over a long period of time show that the number of male andfemale births in a large hospital are just about equal. Based on this pattern:

a. If the first baby of a new year is a boy, what is the probability thatthe next baby will also be a boy?

b. If 100 babies are born in the hospital during one year, which of thefollowing results is most likely to be true:

(i) Between 40 and 60 boys(ii) Between 45 and 55 boys(iii) Between 49 and 51 boys(iv) Exactly 50 boys and 50 girls

50 S U M M A R Y A N D R E V I E W O F M I D D L E S C H O O L M A T H E M A T I C S

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M A I N T A I N I N G C O N C E P T S A N D S K I L L S 51

The following sets of exercises give you an opportunity to review the mathe-matical ideas and skills that you acquired in middle school mathematics and inearly Course 1 units. They include questions that require knowledge of:

■ numbers and operations

■ patterns and algebra

■ geometry and spatial sense

■ measurement

■ data analysis, statistics, and probability

Some problems combine ideas from two or more of those strands of mathematics. Ineach case, you will need to determine the appropriate ideas and techniques to apply.

If you need a refresher on some particular topic, look back at the reference material and examples in the first section of this guide. However, it is best to makea good effort at completing an exercise before looking for help in the referencematerial or in the answers that are given at the back of the book. Since practice ofany skill is most effective when distributed in modest amounts throughout theschool year, the practice exercises have been arranged in sets of ten items, so youcould do about one set every other week throughout the school year as compan-ion work to your study of new topics in Course 1. Exercise sets beginning with Exercise Set 11 include some questions over material you studied in earlyCourse 1 units. You should complete those exercises during the second half ofthe course.

Maintaining Concepts and Skills

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1. Consider the following numbers: �4.2, �1, 2.3, 0, 4, �10�.

a. Which of the numbers are integers?

b. Draw a number line and locate each of the numbers on it.

2. On the first three tests in a marking period Paula has scores of 85, 90, and 75.

a. What scores can Paula get on the fourth and final unit test in order to have a meanof at least 85 for the marking period?

b. What will her median score be for the marking period if she gets the lowest possible score in Part a?

c. What will her range of scores be for the marking period if she gets the lowestpossible score in Part a?

3. According to an article in the December 11, 1989, U.S. News and World Report,human hair grows at a rate of �

18� of an inch in a week, and deer antlers grow at the

incredible rate of 2�34� inches per week during the spring antler season. How many

weeks would your hair have to grow in order to match the growth of the deer antlersduring one week in the spring?

4. Rewrite each of the following numbers using scientific notation.

a. 32,700 b. 42,349.1 c. 927,000,000

d. 0.00340 e. 0.00006275 f. 0.0105

5. In the high school of one city there are 1,012 male students and 1,000 female students. In the high school of a nearby city there are 1,145 female students and1,000 male students.

a. What percent of students in the first city’s high school are female?

b. What percent of the students in the second city’s high school are female?

6. Sketch an example of each of the following. Then describe the ways in which thesefigures are alike and the ways in which they differ.

a. A square

b. A rectangle that is not a square

c. A parallelogram that is not a rectangle

52 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 1

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E X E R C I S E S E T 1 53

7. Maria solved the equation 5x � 8 = 12 for x and obtained the solution x = 4. Withoutsolving, check her solution.

8. The following rectangle has a perimeter of 18 centimeters and an area of 18 squarecentimeters.

a. What other rectangles with sides of whole number length have a perimeter of18 cm?

b. Which of the rectangles in Part a has the greatest area?

c. Which of the rectangles in Part a has the smallest area?

9. Copy each shape. Then draw all lines of symmetry.

a. b. c.

10. There are 197 students going on a trip.

a. The trip costs $12 for every 5 students. Find the cost for this trip.

b. In order to cover the cost, how much should each student be charged?

6 cm

3 cm

Exercise Set 1

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1. The greatest number of two-egg omelets made in any 30-minute time period atKathy’s Kitchen was 427. At that rate, how long did it take to make:

a. 1 omelet

b. 12 omelets

c. 100 omelets

2. Using a coordinate grid like the one shown at the right:

a. Give the coordinates of four points that are the vertices of a rectangle.

b. Give the coordinates of four points that are the vertices of a parallelogram that is not a rectangle.

3. Evaluate each expression.

a. 6(3 � 5) ÷ 2 • 6 + 5 � 8(4)

b. �12

� (7 + 2) � 3 • 6 + 5

c. �4 � 7 + 2 � 3 • 4

4. a. Find the length in centimeters of each line segment below.

(i)

(ii)

(iii)

(iv)

b. Find the length in millimeters of each line segment in Part a.

c. Find the length in meters of each line segment in Part a.

5. The normal male adult pulse rate is approximately 74 beats per minute and anabnormal heart may beat as fast as 300 times per minute. What is the percent ofincrease from normal to abnormal?

6. Write each set of numbers in order from least to greatest.

a. 17, �8, 9, �6 b. 43, �13, �33, 0

c. �12

�, �34

�, �23

�, �58

� d. 2.13, 2.044, 2, 2.305

54 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 2

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E X E R C I S E S E T 2 55

7. For her statistics project, Julia asked 60 people what they used their computer formost. The results are in the table below.

a. Construct and label a bar graph displaying the distribution of this data.

b. Construct and label a circle graph for this data.

8. If you want to paint a wall that measures 12 feet by 8�12� feet and has a door that

measures 3�112� feet by 6�

58� feet, you will need to buy enough paint to cover how large

an area?

9. Imagine arranging nine one-centimeter-square tiles in patterns so that each tile shares at least one full edge with another tile.

a. What is the largest possibleperimeter of the resulting figure?

b. What is the smallest possibleperimeter of the resulting figure?

10. Bill earns $137.70 for 25.5 hours of work.

a. Find Bill’s hourly rate.

b. How much will he make if he works 38.5 hours?

Exercise Set 2

Computer Use Number of PeopleInternet Access 12

Word Processing 30

Games 7

Spreadsheets 2

Other 9

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1. You are purchasing items that cost $45, $13, and $29, and returning an item that was$23. How much will you have to pay, assuming a 4% sales tax?

2. Suppose that a large tree has a circumference of 15 feet at its base and 10 feet at apoint 20 feet above the ground.

a. What is the diameter of the tree at its base and at a point 20 feet above ground?

b. If the tree is cut at its base and a log is cut that is 20 feet long, what will the areaof the log be at each end?

3. Determine the percent increase in each type of organ transplant from 1985 to 1995.

Source: Statistical Abstract of the United States, 1997. Washington D.C.:U.S. Bureau of the Census, 1997.

4. Francisco conducted a survey of people leaving thegrocery store in his town. He and his friends ran-domly asked 800 people what foreign languagesthey spoke. The results are in the table below.

a. How is it that the “Number of People” columnadds up to more than 800 people?

b. Approximately what percentage of the peoplesurveyed spoke each foreign language?

c. If Francisco were to survey another 150 people,approximately how many would he expect tospeak each language?

5. The drug store is selling three note pads for $1.99. If you buy only one, how muchwill you pay?

56 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 3

Type 1985 1995 Percent IncreaseHeart 719 2,361

Liver 602 3,924

Kidney 7,695 11,816

Language Number of People

Spanish 300

French 200

German 100

Japanese 50

Russian 80

None 400

Organ Transplants 1985–1995

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E X E R C I S E S E T 3 57

6. Express each rate as a unit rate.

a. 5 pounds for $7.85 b. 364 miles in 7 hours

c. 8 tickets for $88.00 d. 12 feet in 8 seconds

e. $15.00 for three hours f. 10 pounds in 8 weeks

7. The following diagram shows plans for a basketball court. The floor will be madeof wood costing $3 per square foot and the bold lines shown on the diagram will bepainted at a cost of $0.50 per linear foot.

a. How much will the painting cost?

b. How much will the wood flooring cost?

8. Find the surface area and volume of the rectangular prism shown.

9. In May 1992, General Motors Corporation offered one of the biggest common-stock issues in the United States, consisting of 5.5 � 107 shares with a total valueof 2.15 � 109 dollars. What was the price per share for this stock offering? (Source:Guinness Book of Records. New York: Bantam Books, 1993.)

10. How many different 3-digit numbers can be made using the digits 2, 5, and 8 if thedigits are not repeated? How many of those numbers are divisible by 2? By 3? By 5?

3 cm

6 cm12

8 cm14

15’

19’

5’

5’

45’

12’ 12’

Exercise Set 3

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1. Round each number to the indicated decimal place.

a. 8.932 to tenths b. 350.068 to hundredths c. 6,740.4958 to hundredths

2. In the following diagram, lines k and m are parallel, and quadrilaterals ABCD,DBCE, and EBCF are all parallelograms.

a. Which parallelogram has the greatest area?

b. Which parallelogram has the greatest perimeter?

3. In 1996, Svetlana Masterkova of Russia set the current women’s record for running1 mile with a time of 4:12.56. The men’s record was set in 1993 by NoureddineMorceli of Algeria with a time of 3:44.39. (Source: Guinness Book of Records. NewYork: Bantam Books, 1999.)

a. Write these times out in words.

b. Determine how much faster Noureddine was than Svetlana.

4. Consider the figure at the right.

a. Find the area of the figure.

b. Find the perimeter of the figure.

5. Apply what you know about grouping and order of operations to complete the following.

a. Evaluate 5 + 2 � 9 � 1 + 12 ÷ 3. b. Use three 8s and three 15s to make 13.

c. Use two 4s and two 2s to make 8.

6. Charlene withdrew $45.00 from her savings account. The new balance was $358.71.

a. Let S represent the balance in Charlene’s savings account before the with-drawal. Write an equation to represent the situation described above.

b. Solve your equation from Part a and check your solution in the original problem.

4

4

28

6

5

k

m

A

B C

D E F

58 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 4

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E X E R C I S E S E T 4 59

7. What percent of each row in the figure below is shaded?

8. Suppose you have recordedchecks you have written anddeposits that you have madein your checkbook for thelast week as shown below,but you have not calculatedthe new balance after eachentry. Use your estimationskills to determine whetheryou can write a $78.00check to the school to payfor your class trip.

9. Find the range, mean, and median of the following set of numbers:

6.72, 5.803, 3.5, 7, 8.07.

10. Compare the ratios of salaries of various types of physicians to determine whichmedical specialty had the greatest comparative growth between 1988 and 1994.

Source: Statistical Abstract of the United States: 1997. Washington, D.C.: U.S. Bureau of the Census, 1997.

Row 1

Row 2

Row 3

Row 4

Exercise Set 4

Item Debit Credit Balance$372.56

Check # 101 $129.35

Deposit $58.42

Check # 102 $119.65

Check # 103 $74.48

Check # 104 $54.12

Deposit $140.91

Check # 105 $57.23

Check # 106 $51.08

Medical Specialty Mean Net Mean Net Ratio of Mean Income, 1988 Income, 1994 Net Income,

1994 to 1988General/Family Practice $77,900 $121,200

Internal Medicine $102,000 $174,900

Surgery $155,000 $255,200

Pediatrics $76,200 $126,200

Obstetrics/Gynecology $124,300 $200,400

Mean Net Income for Physicians

Page 62: course i

1. The Willard estate is being divided equally among three sons, Adam, Buckley, andCharles. Each son keeps half of his share of the estate and divides the other half amonghis daughters. Adam has two daughters, Diana and Erin. Buckley has three daughters,Frances, Gail, and Heather. Charles has four daughters, Ingrid, Julia, Katherine, andLillian. What fractional part of the Willard estate does each daughter receive?

2. The following diagram shows a picture and aframe around it. The frame on the picture is1.5 centimeters wide. Find the

a. area of the picture.

b. area of the picture and the frame.

c. total length of wood needed to make theframe.

3. Perform the indicated operations.

a. �12

� � �34

� • �16

� b. 3 ��13

� + �34

�� � �34

c. �53

� ÷ ��18

� � �12

�� d. 3�12

� + 6 �14

� � 5�38

4. In a game for two players, each player spins one of the spinners pictured below. If theresults on the two spinners match, Player 1 is the winner. Otherwise, Player 2 wins.

a. Does each player have an equal chance of winning?

b. If the game is not fair, what unequal payoffs to each player would make it fair?

1 1

22 33

Player 1 Spinner Player 2 Spinner

60 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 5

4 cm

6 cm

Page 63: course i

E X E R C I S E S E T 5 61

5. While shopping at the grocery store you remember that you only brought $15.00and some coupons with you. Use your estimation skills to determine which of thefollowing items you can buy. Assume that the items are listed in priority order.

6. The table below shows the growth of the average annual cost of cable TV C from1990 to 1997.

Source: Nielsen Media Research.

a. Graph the data from the table on a coordinate grid with years on the horizontalaxis and the average annual cost of cable TV on the vertical axis.

b. Describe the trend you see.

c. Predict the average annual cost of cable TV for the year 2000.

7. Express each fraction as a terminating or a repeating decimal.

a. �17

� b. �59

� c. �1135� d. �

58

Year 1990 1991 1992 1993 1994 1995 1996 1997

Cost, C 87.90 94.41 101.18 108.23 108.33 121.82 133.25 142.42

Exercise Set 5

Item Price CouponChocolate Silk Ice Cream $4.49 60¢ coupon

Chicken $2.79

Broccoli $0.99

Beans $0.53

Butter $2.89 30¢ coupon

Milk $1.39

Ground Sirloin $2.29

Rolls $1.49 50¢ coupon

Laundry Detergent $2.39 50¢ coupon

Page 64: course i

8. Use >, <, or = to indicate the relationship between the pairs of items in Columns Aand B.

9. What percent of the figure below is shaded?

10. Consider the following list of high temperatures (˚F) for the first two weeks ofOctober in a city in Maryland.

65, 72, 60, 64, 75, 59, 71, 63, 60, 67, 72, 85, 86, 63

a. Make a stem-and-leaf plot of the data.

b. Find the mean high temperature for this time period.

c. Find the median high temperature for this time period.

62 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 5

A B Relationship

The largest prime The largest prime factor of 56 factor of 38

The least common The least common multiple of 3, 2, and 7 multiple of 3, 5, and 13

�112� 2�28�The number of prime The number of prime factors of 302 factors of 462

Page 65: course i

E X E R C I S E S E T 6 63

1. A store bought 40 gallons of milk at $1.10 per gallon and sold the milk at a20% profit. How much per gallon did the store charge for the milk?

2. To water a rectangular patch of grass, a landscape company can install several different kinds of sprinkler systems.

What percent of the grass will be watered by each of the following plans?

a. Two sprinkler heads with a watering radius of 8 meters, placed at points Aand B

b. Sprinkler heads with a watering radius of 15 meters, placed at points C and D

3. Diet-right cookies contain 4 grams of fat for a serving of 13 cookies. This is 6% ofthe recommended daily amount of fat.

a. What is the number of grams of fat recommended daily?

b. How many cookies could you eat before exceeding the recommended dailyamount of fat?

4. The quality of service by different airlines is often compared using records of “on-time” arrivals for their flights. The following data show the number of minutes thatflights between New York and Los Angeles for two airlines were early (negativenumbers) or late (positive numbers) on a sample of dates.

Airline 1: �5, 10, 0, 5, �5, 50, �4, 8, �15, 12

Airline 2: 0, 5, 8, 4, 0, 10, 6, �5, 15, 7

a. Which statistic (mean, median, range) would make Airline 1 look best?

b. Which statistic would make Airline 2 look best?

c. What single flight distorts the statistics comparing the two airlines and why?

35 meters

25 meters

C

D

A B

8 meters 8 meters

Exercise Set 6

Page 66: course i

5. Replace each __ with >, <, or = to make a true statement.

a. �1102� __ �

191� b. �

1382� __ �

2478�

c. �35

� __ �47

� d. �89

� __ �1291�

6. The highest road in the world is in China. It reaches an altitude of 18,480 feet abovesea level. The lowest road in the world is along the Red Sea in Israel. It is 1,290 feetbelow sea level. What is the difference in altitude between the highest and lowestroads in the world?

7. The first four stages of a pattern representing triangular numbers are shown below.

a. Draw Stages 5 and 6 of the pattern.

b. Make a table showing the number of circles in each stage of the pattern.

c. Use the table to predict how many circles will be in Stage 10 of the pattern.Explain how you arrived at your prediction.

8. You want to fence a garden plot that you need to divide into three separate sections as shown. The two end sections are each 100 feet by 100 feet, and the center section is 40 feet by 100 feet. How far apart could you place the fence posts so that they will be equally spaced on every side? What is the greatest distance that you could have between the fence posts? Remember that you must place a post at the corners of each of the three sections.

9. Fill in each blank with the appropriate number.

a. 35 cm = __________ m = __________ mm

b. 0.5 ft = __________ in. = __________ yd

c. 1500 minutes = __________ hours = __________ days

10. A satellite travels around the Earth once every 90 minutes. How many times does itcircle the Earth in one day?

Stage 1 Stage 2 Stage 3 Stage 4

64 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 6

Page 67: course i

E X E R C I S E S E T 7 65

1. You can represent multiplication of simple fractions by arrays. This array represents �34� • �

27� . Draw an array to represent �

23� • �

35�

and give the product.

2. Solve each of the following equations for x and check your solutions.

a. 2x � 3 = 7 b. �x +

32

� = 5 c. 3(x + 1) = 21

3. If the temperature drops 15˚F between 6 P.M. and midnight and the temperature atmidnight is 62˚F, what was the temperature at 6 P.M.?

4. Order each set of numbers from smallest to largest.

a. 4.037, 4.04, 4.13, 4.007, 4.105

b. �34

�, �58

�, �12

�, �13

�, �25

c. �3, 4, 0, �5, �4.2, �3.7

5. Suppose you want to tile the top of a table that is 30 inches by 18 inches. What isthe largest square tile you could use if you want to use only whole tiles?

6. Suppose you are told that a bag contains 100 marbles, some red and some blue. Youare to pick a marble from the bag, observe its color, and replace it in the bag. Youthen repeat this experiment for a total of 60 draws.

a. If your experimentation produces 15 red and 45 blue marbles, what estimatesabout the number of each color marble in the whole bag would make most sense?

b. If the bag actually contains 40 blue and 60 red marbles, how many red and bluemarbles would you expect in your 60 draws?

7. Sketch and give dimensions of figures with these properties:

a. A rectangle with an area of 12 square meters and a perimeter of 16 meters

b. A right triangle with an area of 12 square meters

c. A circle with a circumference of 10π meters

8. In constructing a circle graph, you know that a certain sector should be 6�12�% of a

circle. How large an angle should you measure for this sector?

Exercise Set 7

Page 68: course i

9. The following table gives rankings for National League baseball teams in team batting and team pitching for the 1998 season.

Source: The World Almanac and Book of Facts, 1999. Copyright ©1998World Almanac Education Group. All rights reserved.

a. Construct a scatterplot of these data pairs and explain what the resulting patternof points says about the relation between the team’s batting and pitching.

b. Draw the line y = x on the scatterplot and explain the relationship between theteam’s batting and pitching shown by points above, on, and below that line.

10. A stock had the following changes in one week: �1�34� , ��1

16� , ��

11

36� , �2, ��

12�. If the

price of the stock was $23�18� at the beginning of the week, then what was the price

at the end of the week?

66 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 7

Team Bat Rank Pitch RankColorado 1 15

Houston 2 2

San Francisco 3 7

Atlanta 4 1

Chicago 5 11

Philadelphia 6 14

Cincinnati 7 10

Milwaukee 8 12

New York 9 4

St. Louis 10 8

Pittsburgh 11 6

San Diego 12 3

Los Angeles 13 5

Montreal 14 8

Florida 15 16

Arizona 16 13

Page 69: course i

E X E R C I S E S E T 8 67

1. A submarine, starting out at sea level, dove 42 feet, then came up 15 feet, and thendove 23 feet. Where is it relative to sea level?

2. The following table shows winning speeds in the Daytona 500 stock car race at five-year intervals from 1960 to 1995.

Source: The World Almanac and Book of Facts, 1999. Copyright©1998 World Almanac Education Group. All rights reserved.

a. Construct a plot-over-time for these data.

b. Write a short statement describing the trend in winning speeds and explain howthe graph supports your statement.

3. Suppose you are preparing a newsletter for printing that is to have 1�

12�-inch margins at the top and bottom, �

34�-inch

margins at the right and left sides, and 3 columns with �14� inch between the columns. How wide will each of your columns be if the paper used for printing measures 8�

12� by 11 inches? How long will each column be?

4. Solve each proportion.

a. �8x

� = �1488�

b. �140� = �

2a5�

c. �176� = �

4y.8�

Exercise Set 8

Year Speed1960 124.7

1965 141.5

1970 149.6

1975 153.6

1980 177.6

1985 172.2

1990 165.8

1995 141.7

Page 70: course i

5. Choose the correct angle measure for each angle below.

a. b.

30˚ 60˚ 90˚ 50˚ 80˚ 100˚

c. d.

90˚ 120˚ 150˚ 5˚ 20˚ 45˚

6. Suppose you have $80.00 and wish to buy movie tickets that cost $6.50 each. Howmany tickets can you buy?

7. Evaluate each expression.

a. 3 � 2 � (7 � 10) b. (�6)(7) � 2(4 � 9 � 12)

c. �3 � 3(4 � 6) d. (�2)2 � 22

8. Mr. Smith gets paid every 6 days, and Mrs. Smith gets paid every 15 days. If theyboth get paid today, how long will it be before they get paid again on the same day?

9. Solve each of the following for x and graph the solution(s) on a number line.

a. 3x � 5 = 2x � 9

b. 10 � 3x ≤ 28

c. �56

�x � 13 > 3

10. If a man made 39 putts in his last 18 holes of golf, what is his average number ofputts per hole? If he used 117 strokes for those 18 holes, what percent of his aver-age number of strokes per hole were putts?

68 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 8

Page 71: course i

E X E R C I S E S E T 9 69

1. Mrs. Ames had the following transactions in her checking account one week. Shedeposited $239, and wrote checks for $62, $78, and $250. If she started with $725,how much money did she have in her account at the end of the week?

2. Time zones are determined by lines of longitude. The table below gives time zonesfor certain cities. Traveling east from 0 you set your clock forward; traveling west,you set your clock back.

a. If it is 6 A.M. in Rome, what time is it in Montreal?

b. If you are in Montreal and it is 2 P.M., what time is it in Hong Kong?

3. If you bought twelve 2-liter bottles of soda for a party and your glasses hold �18� of a

liter, how many glasses of soda will you be able to serve at your party?

4. Consider the two spinners pictured below.

Suppose you spin both spinners and find the sum of the two numbers.

a. Find the probability that the sum is 2.

b. Find the probability that the sum is 4.

c. If you repeated this process 60 times, approximately how many times would youexpect to get a sum of 5?

1 2

34

Spinner 1

1

2

Spinner 2

Exercise Set 9

City Time DifferenceMontreal �5

London 0

Rome +1

Hong Kong +8

Page 72: course i

5. The exchange rate from francs to American dollars is 6.3371 francs for eachAmerican dollar. You want to buy a souvenir that costs 8.5 francs. How much wouldit cost in American dollars?

6. Earth’s orbit around the sun is very close to a circle with a radius of 93-million miles.

a. What is the total distance traveled by the Earth in a year?

b. What is the average speed of the Earth in miles per day and in miles per houralong its orbit around the sun?

7. Write each decimal as a fraction.

a. 0.14

b. 2.6

c. 0.3�

d. 0.2�3�

8. In figuring nine-week grades, a social studies teacher wants homework to count for20%, quizzes to count for 20%, class participation to count for 20%, and exams tocount for 40% of the grade.

a. What average would a student get if her scores in those categories were 80, 70,90, and 75 respectively?

b. If a student had a homework score of 90, a quiz score of 80, and an exam scoreof 80, what class participation score will give a nine-week average of at least 80?

9. Draw a figure that meets each description.

a. Has line symmetry but not turn symmetry

b. Has turn symmetry but not line symmetry

10. Bill ran into a friend when visiting his mother in a nursing home. Bill only comesto the home every 45 days. His friend Charlie said he only comes every 50 days.They wondered why they had never met before. How many days will it be beforethey meet again?

70 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 9

Page 73: course i

E X E R C I S E S E T 1 0 71

1. Rewrite the following numbers as percents.

a. 0.25 b. 0.004 c. 2.8 d. 1�12

� e. �35

2. There are 5 blue marbles, 3 yellow marbles, and 2 green marbles in a bag. Supposeyou picked a marble, noted the color, and returned the marble to the bag 60 times.Estimate the number of times you would expect to pick:

a. A blue marble

b. A yellow marble

c. A blue or green marble

3. It takes the planet Saturn about 30 years to rotate around the sun. The planetNeptune requires 165 years. Once they line up, how many years will go by beforethey line up again?

4. A person on a diet lost the following weights in pounds during the first six weeksof his diet program: 2�

12�, 3, 4�

14�, 1�

12�, 2�

12�, and 3�

12� . If he weighed 183 pounds at the start

of the diet, how much did he weigh after six weeks?

5. Find the perimeter and area of the right triangle shown below.

6. Order each of the following sets of numbers from least to greatest.

a. 0.63, 0.6�, 0.067, 0.6�3�, 0.066�

b. �2.3, ��52

� , �1, ��152� , �2

c. 1.035, 1.47, 0.99, 1.047, 1.009

7. A weather forecaster kept track of the temperature over a six-hour period of time.At the beginning of the period, the temperature was �9.5˚F. It then rose 4.1˚F,dropped 1.8˚F, dropped 2.3˚F, rose 2.2˚F, rose 7.3˚F, and then dropped 0.8˚F. Whatwas the temperature at the end of the six hours?

8. Suppose a 2-liter bottle of cola that costs $0.95 in America sells for $23.09 inRussia. At this rate, what would an item that costs $1.00 in America sell forin Russia?

2.5 cm6.5 cm

Exercise Set 10

Page 74: course i

9. The following data show results from national mathematics tests of eighth gradestudents in 1996 and money spent per student in the same year in a sample of states.

Source: The World Almanac and Book of Facts, 1999. Copyright ©1998World Almanac Education Group. All rights reserved.

a. Which of the following data plots might provide the most useful picture of therelation between test scores and expenditures for schools?

(i) histogram (ii) box plot

(iii) scatterplot (iv) stem-and-leaf plot

(v) circle graph (vi) bar graph

b. Construct the graphic display of your choice in Part a and write a brief sum-mary explaining what the display shows about the relation of the two variables.

10. Consider all numbers x such that 0 ≤ x ≤ 10.

a. List all integers in this range.

b. For what numbers is �1x� ≥ 1?

c. For what integers in this range is x2 < 10?

d. List all prime numbers in this range.

72 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 10

State Percent Math Cost per Student Proficient in $100

AL 45 47

AZ 57 49

DE 55 73

IN 68 60

ME 77 65

MI 67 72

NM 51 46

SC 48 51

WA 67 60

WI 75 71

Page 75: course i

E X E R C I S E S E T 1 1 73

1. The first 4 stages of a pattern are shown below:

a. Make a table showing the number of square tiles used for each of the first eight stages of the pattern.

b. Write an equation using NOW and NEXT that describes the number of tiles in onestage given the number of tiles in the previous stage.

c. Use your NOW-NEXT equation to predict the number of square tiles needed atStage 10.

d. Write an expression for the number of square tiles used at the nth stage ofthe pattern.

e. Use your expression in Part d to check the number of square tiles that will beused in Stage 10. Compare your answer to that obtained in Part c.

2. Simplify:

a. �| 3 � 5 |

b. 60 � | �68 |

c. | �18 + 3 |

d. | �8 | � 2 | 3 |

3. The greatest common factor (GCF) of two numbers is 14 and one of the numbers is56. What is the smallest possible number the other number could be if it is greaterthan 20?

4. In a golf tournament, the top ten finishers had scores of5, �4, �1, 0, 3, �3, 1, 2, �1, 3.

a. Order the scores from lowest to highest.

b. What was the difference between the lowest and highest scores?

Stage 1 Stage 2 Stage 3 Stage 4

Exercise Set 11

Page 76: course i

5. Write the following numbers using scientific notation.

a. 6,800,000,000

b. 248,000

c. 0.045

d. 0.000389

6. If you have a roll of wrapping paper that contains 40 square feet of paper and it is2�

12� feet wide, how long is the roll of paper?

7. Draw a figure that has exactly two lines of symmetry.

a. What relationship do you notice between the lines of symmetry?

b. What else must be true about the figure?

8. A rug normally costs $1,850 but is now on sale for $1,099. What is the percentmarkdown for the rug?

9. If a punch is made by mixing 4 liters of orange juice, 3 liters of grapefruit juice, and2 liters of ginger ale, what percent of the punch is not juice?

10. A bag contains 4 red marbles and 2 white marbles. Without looking into the bag,one marble is pulled out. The color is noted, and then another marble is pulled outin the same way. Determine the probabilities that:

a. A red marble is drawn first.

b. A white marble is drawn first.

c. A white marble will be followed by a red marble.

d. Both marbles will be the same color.

e. The two marbles will be of different color.

74 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 11

Page 77: course i

E X E R C I S E S E T 1 2 75

1. Find the perimeter and area of each figure below.

a. b.

c. d.

2. Simplify:

a. �1,000 ÷ (�100) b. 192 � (�258)

c. (�2)(�22)(�5) d. (�79)(�1)(�2)(�3)

3. In July 1969, the first moonwalk was made by Apollo 11 astronauts. Approximately600 million people watched this event on TV. If this was about �

15� of the world’s

population at the time, determine the world population in 1969. (Source: Guinness,1999.)

4. Solve each of the following for n.

a. 3n + 14 = 2 � n b. 2(5n � 1) = 7(n + 1)

c. 3n + 4 < 1 d. �4 �

2n

� + 5 ≥ 9

5. For each calculation, first estimate the answer by rounding the numbers to thenearest integer; second, determine the actual answer (to four decimal places); then,determine the difference between your two answers as a percent of the answer.What observations can you make about estimating?

a.

b.

c.

d.

8 ft

8 ft 8 ft

10 cm

6 cm

8 in.

12 in.

3 in.

13 m

12 m

Exercise Set 12

Problem Estimated Answer % ErrorAnswer

7.9 + 2.1 + 8.7 � 9.3

�7.9 + 2�.1 + 8.�7 – 9.3�(7.9 + 2.1 + 8.7 � 9.3)3

4.2(7.9 + 2.1 + 8.7 � 9.3)

Page 78: course i

6. On the first five quizzes of a marking period, a student has a mean of 6 on a scaleof 0 to 10. Find the student’s new mean quiz score if, on the next 10-point quiz, thestudent earned

a. A score of 10

b. A score of 6

c. A score of 4

7. In 1990, the United States population over 65 years of age was 31.2 million, whichwas 12.6% of the total U.S. population. What was the total U.S. population in 1990?

8. Use proportions to answer each of the following questions.

a. How many grams of protein are contained in 320 g of tuna if 85 g of tuna con-tain 24 g of protein?

b. Joe exchanges 50 U.S. dollars for 70 Canadian dollars. How many Canadian dol-lars should he receive for 135 U.S. dollars?

c. One yard is equivalent to 36 inches. How many yards are there in 207 inches?

d. Jasmine used 2�14� cups of rice to serve 8 people. How much rice will she need to

serve 30 people?

e. If 8 ounces is equal to 240 milliliters, how many ounces are in one liter?

9. Think about the meaning of percent as you answer the following questions.

a. What is 25% of 40?

b. 50% of what number is 80?

c. 60 is what percent of 180?

d. What is 10% of 20% of 500?

e. 150 is what percent of 75?

f. What is 0.5% of 50?

10. Every twentieth person in line for concert tickets was to be given a poster, and everyfiftieth person was to be given a CD. What person in line would be the first to getboth a poster and a CD?

76 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 12

Page 79: course i

E X E R C I S E S E T 1 3 77

1. A legend on a map indicates that 2 inches is equal to 12 miles. How many miles isit from Baltimore, Maryland, to Washington, D.C., if the two cities are 6�

13� inches

apart on the map?

2. The number of visitors V to a swimming pool varies with the day’s high tempera-ture in degrees Fahrenheit T according to the following equation:

V = 150 + 25(T � 80).

If 500 people visit the pool one day, what would you predict was that day’s hightemperature?

3. Order each set of numbers from least to greatest.

a. 8, �3, �11, �1, 15, 0

b. �3.26, 3.2, 3.26, �3.2

c. �18

�, �196�, �

34

�, �12

�, �58

4. The graph below shows the distance from home as time passed during a walk to anearby store and back home.

a. Write the segments of the trip so the walking rates are in ascending order.

b. Indicate the time of arrival at the store on a sketch of the graph.

5. For the numbers 680 and 1,000, find the

a. prime factorization.

b. greatest common factor (GCF).

c. least common multiple (LCM).

Time

I

II

III

IV

Dis

tan

cefr

om

Ho

me

Exercise Set 13

Page 80: course i

6. The following data show average class sizes in a sample of 10 states.

Source: The World Almanac and Book of Facts, 1999. Copyright ©1998 World Almanac Education Group. All rights reserved.

a. Calculate the mean and median class size of the sample.

b. Calculate the mean absolute deviation (MAD) of class size in the sample andexplain what that statistic tells about average class size that the measures of cen-ter don’t reveal.

7. Nancy is in charge of feeding 75 volunteers who are working on a Habitat forHumanity project in Georgia. The following recipe serves 50. How much of eachingredient will she need so that she can make enough spaghetti for all of the workers?

78 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 13

State Average Class SizeAL 16.6

AZ 19.7

DE 16.6

IN 17.3

ME 13.7

MI 19.1

NM 16.7

SC 15.7

WA 20.2

WI 16.1

Spaghetti for Fifty

10 pounds ground beef �12� cup shortening

�12� cup flour 2 gallons water

10 medium onions, chopped 1�14� cups chopped green pepper

1�14� cups chopped celery 40 bay leaves

�12� cup chili powder �

13� cup salt

3 tablespoons pepper 7 pounds spaghetti, cooked

32 ounces each tomato paste, tomato sauce

3 tablespoons each oregano, basil, and thyme

Page 81: course i

E X E R C I S E S E T 1 3 79

8. Solve each of the following problems.

a. What is 20% of 135? b. What percent of 8 is 12?

c. 5 is 30% of what number?

9. The following diagram shows a soccer field enclosed by a running track. The trackis 6 meters wide with several lanes for runners.

a. If a runner keeps close to the inside of the track (averaging 0.5 meters off theinside edge), how far will she run in 1 lap?

b. If a different runner is in the outside lane (averaging 5.5 meters off the insideedge), how far will she run in 1 lap?

c. How much will it cost to lay sod over the entire infield region if sod costs$1.25 per square meter?

10. The carton pictured below has dimensions as shown.

a. What length of tape would be required to tape along each edge of the carton?

b. How much wrapping paper, in square centimeters, would be needed to wrapthe carton?

c. If the carton is to hold smaller cubical boxes that are each 3 centimeters on anedge, how many of those smaller boxes will the carton hold?

9 cm6 cm

6 cm

112 m

56 m

Exercise Set 13

Page 82: course i

1. The figure at the right shows a pentagon with two diagonals drawn. Find the sum of the fiveinterior angles of the pentagon, and explainhow you know your answer is correct.

2. The figure below shows the outline of a school that is in the shape of an “H” andhas both horizontal and vertical line symmetry.

a. What is the area of the floor of the school?

b. What is the perimeter of the school’s exterior wall?

3. Where should parentheses be placed so that when you evaluate the expression 6 • 3 � 5 ÷ 2 • 6 + 5, the answer is �9.5?

4. Rank the following creatures from smallest to largest in terms of weight.

50 feet

50 feet

35 feet

100 feet

A B

C

D

E

80 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 14

Creature Weight RankLargest cockroach: Macropanesthia rhinoceros 1�

14� ounces

Smallest mammal: Bumblebee bat 0.65 ounces

Smallest bird: Male bee hummingbird 0.056 ounces

Smallest non-flying mammal:Savi’s white tooth pygmy shrew

0.52 ounces

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E X E R C I S E S E T 1 4 81

5. The golden ratio is approximately 1.618. This ratio is found in many places and isused by many artists and architects. If an artist draws a person correctly, the ratio ofthe distance from the top of the head to the navel compared to the distance from thenavel to the feet should be equal to the golden ratio. How tall would a statue be ifthe artist made the distance from the navel up to the top of the head 6.8 feet?

6. The following table shows program formats of U.S. radio stations in 1998.

Source: The World Almanac and Book of Facts, 1999. Copyright ©1998 World Almanac Education Group. All rights reserved.

a. Calculate the percent of stations in each format category.

b. Construct a bar graph showing the share held by each category.

c. On a circle graph, what would the degree measure be for the sector correspond-ing to Country music? For the sector corresponding to Rock?

d. Explain why the share of radio stations with each format might not match theshare of the listening audience by each format.

Exercise Set 14

Station Format NumberCountry 2,393

Adult Contemporary 1,562

News, Talk, Sports 1,356

Religion 1,075

Rock 782

Oldies and Classic Hits 975

Spanish and Ethnic 565

Adult Standards 563

Urban, Black 347

Top 40 379

Other 383

Page 84: course i

7. Draw shapes to match each description below. Indicate lengths when necessary.

a. A triangle with two acute angles

b. A parallelogram with equal length diagonals

c. A trapezoid with a height of 2 inches

d. A circle with a circumference of 16π cm

e. A rectangle with an area of 14 square units

f. A square with an area of 81 square units

8. What fractional part of this design is shaded?

9. The sketch below shows a cylindrical oil storage tank with a diameter of 50 feet anda height of 20 feet.

a. How long is each of the reinforcing bands around the tank?

b. What is the surface area of the tank—the vertical wall and the circular bases?

c. What is the volume of the tank?

d. One cubic foot is equivalent to about 7.5 gallons. What is the capacity of the oiltank in gallons?

10. Suppose during a 25% off sale you paid $84 for a jacket before tax. What was theoriginal price of the jacket?

50 ft

20 ft

82 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 14

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E X E R C I S E S E T 1 5 83

1. The ratio of a man’s height to the length of his foot is about 7. A short teenage boy,whose feet have stopped growing, has a foot that is 10�

12� inches long. How tall can

he hope to grow by the time he reaches his adult height?

2. Find the area of the shaded part of each diagram below.

a. b.

3. Frances Redmond has the longest fingernails in the United States. In the last 12years, they have grown to be 17.25 inches long. The longest fingernails in the worldbelong to an Indian, Shridhar Chillal. His nails have not been cut since 1952, andin 1997 the longest one was 48 inches. Determine the difference in fingernailgrowth rate between the American and the Indian. (Source: Guinness Book ofRecords. New York: Bantam Books, 1999.)

4. The following back-to-back stem-and-leaf plot gives ratings by two different judgesin an ice skating contest. Possible scores range from 0 to 6.0.

a. Calculate the summary statistics needed to construct box plots of the same dataand draw those plots.

b. Calculate the mean and the mean absolute deviation (MAD) for each judge’s scores.

c. Based on the various summary statistics you’ve calculated and the box plotsyou’ve drawn, what conclusion would you reach on the question of whether thetwo judges give similar or different ratings overall to a group of skaters?

d. What data and plot would be helpful in determining whether the two judges givesimilar ratings to individual skaters?

5. Suppose during a 20% off sale you buy a pair of slacks that usually costs $79. Ifsales tax is 5%, how much should you pay for the slacks?

6

5

10

515

5

10

5 15

Exercise Set 15

Judge 1 Judge 23 4

6 6 7 8 95 5 4 4 2 3

9 8 6 6 7 7 8 85 4 3 3 2 5 3 4 4 5

9 9 8 7 6 7 90 6 0 0

Page 86: course i

6. Complete the table for regular polygons.

7. Simplify each expression.

a. 9 � 2(5 � 7) + 3 • 52 b. �3 + (2 � 7) � 3 + 2 c. �12

� ��34

� + �58

�� � 2

8. The school Booster Club is planning to sell state championship T-shirts. Theyexpect the following expenses and income.

Expenses: $50 art-screen fee, $5.75 per shirt

Income: $10 per shirt

a. Write an expression for the cost of n shirts.

b. Write an expression for the income earned from the sale of n shirts.

c. Write two equivalent expressions for the profit earned from the sale of n shirts.

d. What is the minimum number of shirts that must be sold in order not to lose money?

9. The standard dimensions of high school basketball courts are 84 feet long by50 feet wide. For a college court, the standard dimensions are 94 feet long by50 feet wide.

a. What is the difference in area of standard college and high school courts?

b. What is the difference in perimeter of standard college and high school courts?

10. Sketch a graph of each line.

a. y = �2x + 5 b. y = �34

� x � 2 c. y = x + 1

d. y = �x � 3 e. y = ��23

� x f. y = 3

84 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 15

Regular # of Lines Smallest Angle ofPolygon # of Sides of Symmetry Turn SymmetryTriangle 3

Quadrilateral 4

Pentagon 5

Hexagon 6

Octagon 8

n-gon n

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E X E R C I S E S E T 1 6 85

1. Suppose a frog is down in a well that is 100 feet deep. If the frog is able to leap up9 feet but slips back 3 feet on every jump, how many jumps will it take the frog toget out of the well?

2. What are the perimeter and area of thisparallelogram? Find at least two ways tomake each calculation.

3. Evaluate each expression.

a. 43 � 2 • 52 + 9 b. �3(4 � 12) ÷ 3 + 1

c. | �6 | � 2 | 5 | d. ��12

� (6 � 5) � 4��35

� � 4�25

��e. �368 � 24 � 2(�48)

4. Consider the following vertex-edge graphs.

(i) (ii) (iii)

a. Which of the graphs contain an Euler circuit? Explain your answer.

b. Of the graphs that do not contain an Euler circuit, which contain an Euler path?Explain your reasoning.

5. Solve each linear equation.

a. 3x � 12 = 24 b. 6x � 17 = 20 � 9x c. 10 � (5 � 2x) = 7

d. �4(2x � 8) = 3(4 � x) e. x = 2x � 9 f. �12

� x � 2 = �52

� x � 10

6. Determine whether each of the following is equivalent to �35�. Explain your reasoning.

a. 3 divided by 5 b. 3 times the reciprocal of 5

c. 5 times the reciprocal of 3 d. �15

� times 3

7. Suppose the scores on a 100-point test for a class of 20 students have mean 75%,median 80%, and range 40 (from 55% to 95%). How will the mean, median, andrange change (if at all) if the teacher:

a. Increases each student score by 5 points? b. Divides each score by 10?

Exercise Set 16

4 cm

3.6 cm3 cm

2 cm

Page 88: course i

8. Very large and very small numbers are often written in scientific notation to makecalculations easier. Give the scientific notation form for the numbers in the table.

a.

b.

c.

d.

e.

f.

Source: 1999 New York Times Almanac. New York: Penguin Reference, 1998.

9. A standard basketball hoop has an inside diameter of 18 inches. A standard basket-ball has a circumference of 30 inches.

a. What is the circumference of a standard basketball hoop?

b. If a standard basketball drops straight down through the center of the basket, howmuch clearance will there be between the ball and the basket hoop?

10. Complete the chart by filling in the missing fractions, decimals, and percents.

a.

b.

c.

d.

e.

86 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 16

Significance Number Scientific Notation

Population of the world 5,930,000,000,000in 1998

The U.S. national debt $5,543,600,000,000in 1998

Net worth of richest $39,800,000,000American, Bill Gates

Number of page views per 95,000,000day to Yahoo!, the most popular web site

1 cm expressed in miles 0.000006214

1 foot expressed in miles 0.0001894

Fraction Decimal Percent�58�

0.142

0.375�16�

�1230�

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E X E R C I S E S E T 1 7 87

1. Write each fraction as an equivalent fraction in simplified form.

a. �1664� b. �

2450� c. �

5226� d. �

172200

2. A trapezoid is used to create a tile pattern as shown at the right.

Describe the transformations that will map the shaded figure onto each of the positions 1�4.

3. Write each set of numbers in order from least to greatest.

a. 1�17

�, 25%, �45

�, 0.63, 1.2 b. 82.5%, �79

�, 0.88, 1.80

4. Andrea’s parents have rented a suite at a Phoenix Mercury basketball game for herbirthday party. The suite cost $180 to rent. In addition they must buy $12 tickets tothe game for each person. Write and solve equations or inequalities to answer thefollowing questions.

a. How much will it cost for 8 people to be at the party?

b. If Andrea’s parents paid $240, how many people were at the party?

c. How many people can be at the party if the cost must be less than $320?

5. In a store giftwrap department, one of the standard boxesis 12 inches by 15 inches by 8 inches high.

a. Ribbon to wrap such a box must goaround the box aspictured in the sketch with 20 inches extra to make afancy bow. How many inches of ribbon will berequired?

b. Approximately how much wrapping paper is neededto wrap the box?

6. Two families rented a boat for $275 a week. One family used it for 3 days, and theother family used it the remaining 4 days. How much should each family pay?

7. Perform the indicated calculations.

a. (�3) � (�21) b. 14 � 11 � 32 c. (�6)(�12)

d. (�4)(�7)(5) e. (�56) ÷ 4 f. 18(�3) + (�4) ÷ (�2)

1

2

3 4

Exercise Set 17

Page 90: course i

8. The following table shows the distribution of beach debris, collected in an annualU.S. coastal cleanup.

Source: USA Today. 1992.

a. What type of graph would best display these data: line plot, stem-and-leaf plot,bar graph, histogram, circle graph, or box plot? Explain your reasoning.

b. Display the data by making a graph of the type you recommended in Part a.

9. Solve each of the following problems.

a. Jim has saved $620. How much money will he have after he buys a basketballfor $25 and receives his two-week paycheck for $195?

b. River Hill High School had a dance to raise money for band uniforms. They sold300 tickets at $5 each. They hired three security guards at a cost of $65 each andtwo custodians at $75 each. How much money did they raise?

c. Last year Jack saved 15% of his salary. If his salary was $29,900, how muchmoney did he save?

10. The tiling pattern at the right consists of large and small squares. The large tiles inthe arrangement are 30 cm by 30 cm, and the small tiles are 18 cm by 18 cm. Usingthe arrangement shown, what is the shortest length of a strip so that the right-handend of the tiles match up?

88 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 17

Paper and Paperboard 11%

Rubber 2%

Plastic 59%

Metal 12%

Glass 12%

Wood 3%

Cloth 1%

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E X E R C I S E S E T 1 8 89

1. Evaluate each expression.

a. | 41 | b. | �6 |

c. �| 12 | d. 6 � 3 • 2 + 7(1 � 8)

e. (�4)2 + 42 � 2(�3) f. �6(3) � 2(9 � 20)

2. For each table below, determine whether the pattern of change is linear, exponen-tial, or neither. If the pattern is linear or exponential, write a NOW-NEXT equationdescribing the pattern.

a. x 0 1 2 3 4

y 7 4 1 �2 �5

b. x 0 1 2 3 4

y 1 5 25 125 625

c. x 0 1 2 3 4

y 4 5 7 10 14

d. x 0 1 2 3 4

y 6 12 24 48 96

3. Temperature can be roughly determined by the number of times a particular speciesof cricket chirps per minute. At 72˚ Fahrenheit, one species of cricket chirps about140 times per minute and at 80˚ about 172 times per minute. If this is a linearrelationship, estimate the temperature when a cricket chirps 180 times a minute?

4. International Falls, Minnesota, is often the coldest spot in the lower 48 states duringthe winter season. During one week in January, the low temperatures in Fahrenheitwere: �5°, �15°, �20°, �35°, �4°, 6°, and 10°.

a. What was the mean low temperature for that week?

b. What was the range of low temperatures for that week?

c. What was the median of the low temperatures for that week?

Exercise Set 18

Page 92: course i

5. Indicate whether each inequality is true or false.

a. �7 > �9 b. �8 < 10 c. �12 > �4

d. �35

� < �170� e. �

94

� > �172� f. �2.16 < �2.75

6. One serving of Special G cereal contains 9% of the recommended daily allowanceof sodium. If one serving contains 220 milligrams of sodium, what is the recom-mended daily allowance of sodium?

7. Box kites like the one shown below utilize “spreaders”(rods that form Xs inside the kite) to keep the kite open.

a. If each edge along the base of the kite measures 18 inches, how long should each spreader rod be?

b. The fabric on the kite shown is 6 inches wide. Howlong a strip of fabric is needed to make this kite?

8. An 18-ounce box of breakfast cereal has dimensions: 3 inches by 8 inches by 12 inches.

a. What are the dimensions of the various faces of this box?

b. What is the total surface area of the box?

c. What is the volume of the box?

9. Examine the digraph for a project shown below.

a. Find the length of all paths from S to F.

b. Find the critical path and the earliest finish time for the project.

10. A store ran an advertisement during a sale that read “Buy one CD and get a secondCD for half price.” If you paid $21.75 for two CDs during the sale, what was the regular price of a CD?

Start

B5

S0

Finish

A1

C3

D4

E3

G2

F0

90 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 18

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E X E R C I S E S E T 1 9 91

1. Which is the better buy, a box of cookies that weighs 125 grams for $2.39 or a boxthat weighs 200 grams for $3.79?

2. Cindy is investigating how fast a particular bee population will grow under con-trolled conditions. She began her experiment with 2 bees. The next month shecounted 10 bees.

a. If the bee population is growing linearly, how many bees can Cindy expect tofind after 2, 3, and 4 months?

b. If the bee population is growing exponentially, how many bees can Cindy expectto find after 2, 3, and 4 months?

c. Write an equation for the number of bees B after m months assuming lineargrowth. Write another equation assuming exponential growth.

d. How many months will it take for the number of bees to reach 200 assuminglinear growth? Assuming exponential growth?

3. Estimate each of the following to the closest integer.

a. �102� b. �87� c. �139�

d. ��15� e. �4� + �10�

4. Solve each linear equation.

a. 6(2x � 5) = 9x b. 3 � (x � 9) = 14

c. 257 + 2x � 97 = x + 140 d. 3(15x + 2) = 25x + 16

5. If it takes 5 seconds for the sound of thunder to travel one mile, how long does ittake the sound of thunder to travel �

14� of a mile?

6. The thinnest glass has a minimum thickness of 0.000984 inches and a maximumthickness of 0.00137 inches. It is made in Germany and used in electronic andmedical equipment. (Source: Guinness Book of Records. New York: Bantam Books,1993.)

a. Rewrite these values in scientific notation.

b. The thicker measurement is what percent of the thinner?

7. If a team won 6 out of 9 games in December, then won 4 games and lost 7 games during the rest of the season, what percent of their games did they win during theseason?

Exercise Set 19

Page 94: course i

8. Write a NOW-NEXT equation and a y = … equation to match each graph below.

a. b.

c. d.

9. To compare gasoline prices in two neighboring states, students in a CMIC class col-lected data from 10 service stations in each state. The costs per gallon (in dollars)for regular unleaded gasoline were:

State 1: 1.179, 1.139, 1.089, 1.219, 1.149, 1.039, 1.129, 1.169, 1.099, 1.159

State 2: 1.089, 1.069, 1.099, 1.229, 1.159, 1.129, 1.119, 1.089, 1.389, 1.299

a. What are the mean and median prices in the sample of stations from each state?

b. What is the range of prices in each state?

c. One group of students claimed that it would be easier and just as accurate to dealonly with data in the form 17, 13, 8, 21, 14, and so on. Could you take the mean,median, and range from their calculations and easily find the exact values for theactual data? If so how? If not, why not?

d. Which state seems to have the lower gasoline prices, and what data summariesbest support your conclusion?

10. Evaluate each expression

a. 3�49

� • �34

� b. 3�49

� � �49

� c. �35

� � �54

d. 2�14

� � �12

� � �78

� e. �23

� • �14

� � �56

� • �12

� f. �34

� � �12

� � 4�14

y

x4 8–8 –4

–8

8

4

–4

y

x4 8–8 –4

–8

8

4

–4

y

x4 8–8 –4

–8

8

4

–4

4 8–8 –4

–8

8

y

x

4

–4

92 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 19

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E X E R C I S E S E T 2 0 93

1. Write an inequality to represent each of the following situations.

a. The number of people who attended the 1999 Women’s World Cup Soccer finalin Pasadena, California. (More than 80,000 people attended, breaking a recordfor attendance at any women’s sporting event anywhere in the world.)

b. The age of voting United States citizens. (United States citizens who are at least18 years of age may vote.)

c. The weight of riders on a waterslide at Rainbow Falls at Six Flags America hasa maximum weight limit of 250 pounds.

2. Determine the percent decrease in drug use for teens from 1985 to 1995.

Source: Statistical Abstract of the United States, 1997. Washington, D.C.:U.S. Bureau of the Census, 1997.

3. Judy and George own a peanut company. They ship cans of peanuts all over thecountry. If the cans are 3 inches high and have a diameter of 3 inches, give thedimensions of boxes that could hold exactly 48 cans of peanuts. Explain which boxyou think would be best to use and why.

4. Dr. Messenger records hisodometer reading and theamount of gas he puts in thecar every time he fills up atthe service station. He doesthis so that he can check forchanges in gas mileage thatmight indicate that his car hasa problem. Some entries areshown below. Estimate thegas mileage for these fill-ups.

Exercise Set 20

Type 1985 1995 Percent DecreaseAlcohol 56.1% 40.6%

Cigarettes 50.7% 38.1%

Date Odometer Fuel Added Gas MileageReading

4/21/99 74,780 12.782

5/4/99 75,044 13.326

5/24/99 75,317 13.016

6/6/99 75,482 9.116

6/14/99 75,709 10.812

Drug Use Decline:Percent of 12-to 17-Year-Olds Who Have Used Drugs

Page 96: course i

5. Suppose that a spinner with three equal-size sectors is spun and a fair coin isflipped in a game of chance.

What are the probabilities of these outcomes?

a. 1 on the spinner and heads on the coin

b. Odd number on the spinner and tails on the coin

c. Even number on the spinner

d. Even number on the spinner and tails on the coin

6. Solve each proportion.

a. �1x2� = �

130� b. �

1y5� = �

192�

c. �21..34� = �

4x.2� d. �

x �

22

� = �4x

7. The longest space flight lasted 437 days, 17 hours, 58 minutes, and 16 seconds. Itwas made by a Russian doctor, Valeriy Poliyakov, on the Mir 1 space station during1994 and 1995. Express the length of the space flight in days as a decimal roundedto 4 decimal places.

8. The standard box for 3 golf balls is like that pictured below. Each ball has a diameterof 1.75 inches. Find the dimensions of the box.

Side View of Box

9. Suppose you want to use �14�-inch grid paper to make a scale drawing of a room that

is 14 feet by 16 feet. You decide that each �14� inch should represent �

12� foot. What are

the dimensions of your drawing?

94 M A I N T A I N I N G C O N C E P T S A N D S K I L L S

Exercise Set 20

1

23

Page 97: course i

E X E R C I S E S E T 2 0 95

10. The following table gives the percent of eighth grade students judged proficient in1996 national tests of mathematics and science. Data are from a random sample of19 states.

Source: The World Almanac and Book of Facts, 1999. Copyright ©1998 World Almanac Education Group. All rights reserved.

a. Construct back-to-back stem-and-leaf plots of the mathematics and science scores.

b. Draw box plots for these data.

c. Write a brief argument supporting your conclusion about the subject whichUnited States students seem to know better—mathematics or science.

Exercise Set 20

State Percent Math Percent Science Proficient Proficient

AL 45 18

AK 68 31

AZ 57 23

CT 70 36

DE 55 21

HI 51 15

IN 68 30

LA 38 13

ME 77 41

MD 57 25

MI 67 32

MO 64 28

NM 51 19

NY 61 27

SC 48 17

UT 70 32

WA 67 27

WV 54 21

WI 75 39

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96

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P R A C T I C I N G F O R S T A N D A R I Z E D T E S T S 97

Because you work closely with your classmates and teachers on a daily basis,they have a good idea of what you know and are able to do with respect to themathematics that you are studying this year. However, your school district or statedepartment of education may ask you to take tests that they use to measure theachievement of all students, classes, or schools in the district or state.

External standardized tests usually contain questions in formats that caneasily be scored to produce simple percent-correct ratings of your knowledge. Ifyou want to perform well on such tests, it helps to have some practice with testitems in a multiple-choice format. The following 10 sets of multiple-choice taskshave been designed to give you that kind of practice and to offer some strategicadvice in working on such items. You will find helpful Test Taking Tips at the endof each of the practice sets.

Practicing for Standardized Tests

Page 100: course i

1. If N is an odd integer, which of the following numbers is also an odd integer?

(a) N � N (b) N + N (c) 3N � 1

(d) N � 1 (e) N + 5

2. A T-shirt sells for $18 in a retail store. If this price is 120% of the wholesale price,what is the wholesale price?

(a) $14.40 (b) $15.00 (c) $16.00

(d) $16.20 (e) $21.60

3. Which of the following figures is the result of a half-turn about point T of the figure below?

(a) (b) (c)

(d) (e)

4. Jenny needs to pack 50 bagels in bags that hold 6 bagels each. What is the smallestnumber of bags Jenny will need to pack all the bagels?

(a) 7 (b) 8 (c) 9

(d) 10 (e) 11

TT

TT

T

T

98 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 1

Page 101: course i

5. In a quadrilateral, two of the angles have a measure of 90˚ each. The measure of athird angle is 100˚. What is the measure of the remaining angle?

(a) 70˚ (b) 80˚ (c) 170˚

(d) 190˚ (e) 280˚

6. Each of six faces of a cube is painted either red or white. When the cube is tossed,the probability of the cube landing with a white face up is �

13�. How many faces are

white?

(a) 1 (b) 2 (c) 3

(d) 4 (e) 5

7. If a > 0 and b < 0, which of the following must be negative?

I. ab II. �ba

� III. a � b

(a) I only (b) II only

(c) III only (d) I and II

(e) All of them

8. Jane bought some peppermint patties. She gave half of them to her brother and thena third of those left to her sister. Now she has 6. How many peppermint patties didshe buy?

(a) 18 (b) 24 (c) 30

(d) 36 (e) 42

9. A�B�, C�E�, and D�E� intersect at point E and the measure of �AEC is 90˚. The measureof �BED is twice as much as the measure of �CED. What is the measure of �CED?

(a) 15˚

(b) 22.5˚

(c) 30˚

(d) 45˚

(e) 60˚

Practice Set 1

P R A C T I C E S E T 1 99

A

C

D

B

E

Page 102: course i

10. Which fraction has the greatest value?

(a) �157� (b) �

155� (c) �

153�

(d) �151� (e) �

59

100 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 1

Test Taking Tip

Test general properties of numbers by using specific numbers.

Example Look back at Item 1 on page 98. To use this strategy, choose a specificodd number such as 3 to substitute for N in each of the listed expressions.

For choice (a): 3 � 3 = 9. 9 is an odd integer.

For choice (b): 3 + 3 = 6. 6 is not an odd integer.

For choice (c): 3(3) � 1 = 8. 8 is not an odd integer.

Explain why choices (d) and (e) are not correct choices. So, the answeris (a).

■ Find, if possible, another test item in the practice set for which this strategymight be helpful. Try it.

■ Keep this strategy in mind as you work on future practice sets.

Page 103: course i

1. What is the product of the values of the digit 2 in 13.265 and in 0.312?

(a) 4 (b) �140� (c) �

1400�

(d) �1,0

400� (e) �

10,4000�

2. 2 � 39 is not equal to

(a) 2 � (30 + 9)

(b) 2 � (40 � 1)

(c) (2 � 30) + 9

(d) (2 � 9) + (2 � 30)

(e) (2 � 50) � [(2 � 10) + (2 � 1)]

3. A librarian recorded that 10 students checked out books on Monday. Her recordsshow the number of books each student checked out as follows:

3, 4, 5, 2, 2, 4, 3, 3, 2, 2

What is the average number of books these students checked out?

(a) 2 (b) 3 (c) 4

(d) 5 (e) 10

4. Jay had 36 inches of wire. He bent the wire to form a square without any wire over-lapping. What is the area in square inches of the square he formed?

(a) 3 in.2 (b) 9 in.2 (c) 18 in.2

(d) 24 in.2 (e) 81 in.2

5. Which of the following is the smallest?

(a) 0.5 ÷ 10

(b) 0.0052

(c) 0.05 � 10

(d) 5.0 � 10�4

(e) 5.0 � 103

Practice Set 2

P R A C T I C E S E T 2 101

Page 104: course i

6. If 24 out of 30 students are wearing white shirts on a given day, what percent ofthe students are wearing other color shirts?

(a) 6% (b) 10% (c) 20%

(d) 40% (e) 80%

7.Suppose s = 2 and t = �3. Find the value of 2st + (s2t).

(a) �144 (b) �24 (c) 0

(d) 24 (e) 144

8. �110� ��

230� + 10� =

(a) 1 (b) 1�23

� (c) 7�23

(d) 10�23

� (e) 16�23

9. In the figure below, what is the value of x + y + z? Note: The figure is not drawn toscale.

(a) 120˚

(b) 150˚

(c) 240˚

(d) 270˚

(e) 330˚

102 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 2

90˚

30˚ z ˚y ˚x ˚

Page 105: course i

10. Jim uses 0.8% of a 24-hour day for lunch. Approximately how many minutes doeshe take for his lunch?

(a) 2 (b) 12 (c) 19

(d) 29 (e) 48

Practice Set 2

P R A C T I C E S E T 2 103

Test Taking Tip

Know and be able to use the distributive property.

The distributive property is useful in simplifying algebraic expressions. You musttake care, however, to recognize when to use it and to use it correctly. The distrib-utive property for multiplication over addition states that a(b + c) = ab + ac. Inparticular, �(b + c) = �b + (�c).

Example Look back at Item 2 on page 101. Equivalent expressions may be sub-stituted for 39, as in choices (a) and (b). Applying the distributive prop-erty to the expression in choice (a) yields the expression in choice (d),not that in choice (c). Note that the expression in choice (e) can berewritten as follows:

2 � [50 � (10 + 1)] = 2 � (50 � 10 � 1) = 2 � 39

So, the answer is (c).

■ Find, if possible, another test item in the practice set which can be simplifiedusing the distributive property. Try it.

■ Keep this caution in mind as you work on future practice sets.

Page 106: course i

1. ��56

� � ��130� � �

12

�� =

(a) ��34

� (b) ��14

� (c) ��16

(d) �16

� (e) �152�

2. Two hundred pounds of corn will feed 60 pigs for 1 day. How much corn will beneeded to feed 90 pigs for 2 days?

(a) 300 (b) 400 (c) 600

(d) 1,200 (e) 2,400

3. What is 10% of 25% of 600?

(a) 15 (b) 60 (c) 90

(d) 150 (e) 210

4. Triangles ABC and DEF are similar. What is the length of E�F�?

(a) 1.25

(b) 3.2

(c) 5

(d) 11

(e) 20

5. �3x

� < 8 is equivalent to

(a) x < 24 (b) x < 5

(c) x < �83

� (d) x > 5

(e) x > 24

6. Which list contains three equivalent fractions?

(a) �23

�, �46

�, �68

� (b) �35

�, �57

�, �160�

(c) �34

�, �192�, �

1158� (d) �

39

�, �168�, �

1326�

(e) �46

�, �184�, �

1221�

104 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 3

8

5

A

B C

2

D

EF

Page 107: course i

7. What is the difference between the largest and smallest three-digit numbers that aredivisible by 5?

(a) 90 (b) 95 (c) 100

(d) 890 (e) 895

8. If the mean of a set of five numbers is 7 and one of the numbers in that set is 3, whatis the average of the other four numbers?

(a) 4 (b) 5 (c) 6

(d) 7 (e) 8

9. Evaluate each of the following expressions. Which has the largest value?

(a) 3 � 2(�10) (b) 3 + 4(6 � 4) (c) (2 � 5)2 + 10

(d) 3 • 6 + 2 � 10 (e) 3 + 4 • 5 � 20

10. A fish tank has the shape shown below. The width is three-fourths of the length, andthe height is half of the length. What is the volume of the tank in cubic feet?

(a) 9

(b) 18

(c) 24

(d) 34

(e) 68length = 4 feet

Practice Set 3

P R A C T I C E S E T 3 105

Test Taking Tip

Be sure to apply arithmetic operations in the correct order: Operations with-in parentheses first, next exponentiation, next multiplication and division inorder from left to right, and then addition and subtraction in order from leftto right.

Example Look back at Item 9. To evaluate the expression in choice (a), first multi-ply 2 by �10 giving �20, then subtract �20 from 3 to get 23.

■ Evaluate the expressions in choices (b) through (e) using the correct order ofoperations to confirm that the expression with the largest value is choice (a).

■ Keep the correct order of operations in mind as you evaluate expressions in yourfuture work.

Page 108: course i

1. One pound of cherries costs $1.99. If Nina buys 2 pounds of cherries and pays witha $10 bill, how much change will she get back?

(a) $3.98 (b) $6.02 (c) $7.12

(d) $8.01 (e) $9.11

2. A and B are numbers shown on the number line below. What is B � A?

(a) 0

(b) �112�

(c) 1�112�

(d) 1�152�

(e) 2�112�

3. To obtain a certain color of purple paint, Mike combines 4 liters of red paint, 3 litersof blue paint, and 5 liters of white paint. What portion of the purple paintwas white?

(a) �75

� (b) �57

� (c) �152� (d) �

152� (e) �

172�

4. In the figure below, x =

(a) 23

(b) 33

(c) 67

(d) 113

(e) 157

5. Which of the following lists three fractions in ascending order?

(a) �56

�, �140�, �

162� (b) �

140�, �

56

�, �162� (c) �

56

�, �162�, �

140�

(d) �162�, �

140�, �

56

� (e) �140�, �

162�, �

56

6. The price of an item last year was $100. This year the price was increased by 20%then later decreased by 10%. What is the price of that item now?

(a) $101 (b) $102 (c) $108 (d) $110 (e) $118

x ˚

67˚

0 2 31

A B

106 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 4

Page 109: course i

7. A jar contains 54 balls: some blue, some white, some red, and some green. If theprobability of selecting a green ball is �

29�, how many green balls are in the jar?

(a) 2 (b) 6 (c) 8 (d) 10 (e) 12

8. Which of the following is a false statement about whole numbers?

(a) Every even number has 2 as a factor.

(b) Every number has 1 as a factor.

(c) Every odd number has 3 as a factor.

(d) Every composite number has at least 3 factors.

(e) Every prime number has exactly 2 factors.

9. Which of the following points can be joined to the point (�3, 5) by a line segmentthat crosses neither the x-axis nor the y-axis?

(a) (5, �3) (b) (�2, �3) (c) (2, �3) (d) (2, 3) (e) (�2, 3)

10. �1320� is equivalent to

(a) 0.004% (b) 0.04% (c) 0.4% (d) 4% (e) 40%

Practice Set 4

P R A C T I C E S E T 4 107

Test Taking Tip

When feasible, use familiar benchmarks to compare fractions.

Fractions are easily compared to benchmarks such as �12� or 1. These comparisons

may save you the time of finding a common denominator.

Example Look back at Item 5 on page 106. To use this strategy, compare eachfraction to �

12� as shown below.

�140� < �

12

�, �162� = �

12

�, and �56

� > �12

�.

From these comparisons, it is easy to see how to list the given threefractions in ascending order without rewriting the fractions with a com-mon denominator. The correct answer is (e).

■ Find, if possible, another test item in the practice set for which this strategymight be helpful. Try it.

■ Keep this strategy in mind as you work on future practice sets.

Page 110: course i

1. The price of a pack of gum drops from 75 cents to 60 cents. What is the percentdecrease of the price of the gum?

(a) 15% (b) 20% (c) 25%

(d) 30% (e) 35%

2. Which number cannot be written as the quotient of two integers?

(a) 0 (b) �4� (c) �12�

(d) 2.35 (e) 0.555…

3. In a school, there is one teacher for every 25 students. Which of the following state-ments is not true about the relationship of the number of teachers T and number ofstudents S at that school?

(a) T = 25S

(b) 25T = S

(c) �TS

� = �215�

(d) S � 25T = 0

(e) There will be 7 teachers if there are 175 students.

4. A parallelogram must be a square if:

(a) It has two pairs of congruent angles.

(b) Each pair of the parallel sides are congruent.

(c) All angles are right angles.

(d) All sides are congruent.

(e) It has one right angle and all sides are congruent.

5. Pete and Andre each decided to start saving their money. Each month, Pete can save$3 and Andre can save $5. At this rate, after how many months will Andre haveexactly $10 more than Pete?

(a) 2 (b) 3 (c) 4

(d) 5 (e) 8

108 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 5

Page 111: course i

6. Ester started her trip with 14 gallons of gas in the tank of her car. Her car consumes4.5 gallons of gas for every 100 miles driven. How many gallons of gas remainedin the tank after she drove 250 miles?

(a) 2.75 (b) 3.25 (c) 3.75

(d) 11.25 (e) None of the above

7. If a is a positive integer and a2 = 4, what is a3?

(a) 6 (b) 8 (c) 10

(d) 12 (e) 16

8. Lines AB, CD, and EF intersect at point G. What is the value of x?

(a) 35˚

(b) 45˚

(c) 55˚

(d) 65˚

(e) 145˚

9. The circle graph below shows the distribution of grades for a mathematics test. If250 students took the test, how many more students received a grade of C thanreceived a grade of A?

(a) 10

(b) 20

(c) 25

(d) 55

(e) 80

Practice Set 5

P R A C T I C E S E T 5 109

55˚E

CA

F

DB

x ˚

G

B24%

C32%

D18%A

22% F4%

Page 112: course i

10. Samut drove his car to work for a week. He found that the total distance he traveledin that week was 320 miles, and he used 11.5 gallons of gas. Approximately howmany miles can his car travel on one gallon of gas?

(a) 24 miles per gallon

(b) 25 miles per gallon

(c) 26 miles per gallon

(d) 28 miles per gallon

(e) 30 miles per gallon

110 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 5

Test Taking Tip

Create a table to compare quantities in a problem situation.

You may find it easier to solve a problem by constructing a table of values whenyou are unsure of how to tackle the problem algebraically.

Example Look back at Item 5 on page 108. To use this strategy, create a table like theone below showing Pete’s and Andre’s savings at the end of each month.

From the table, you can see that Andre has exactly $10 more than Peteafter 5 months of saving. So the answer is (d).

■ Find, if possible, another test item from Practice Sets 1�5 for which this strategy might be helpful. Try it.

■ Keep this strategy in mind as you work on the next practice set and in yourfuture work.

Month Pete’s Savings Andre’s Savings1 $3 $5

2 $6 $10

3 $9 $15

4 $12 $20

5 $15 $25

Page 113: course i

1. What is the ratio of the length of a side of a square to its perimeter?

(a) 1:1 (b) 1:2 (c) 1:4 (d) 2:1 (e) 4:1

2. �23

� ÷ ���17

� � �67

�� =

(a) ��499� (b) �4 (c) ��

1145� (d) ��

79

� (e) ��449�

3. What is the area in square centimeters of the shaded portion of the figure below?

(a) 100 � 100π

(b) 100 � 50π

(c) 100 � 25π

(d) 40 � 50π

(e) 40 � 25π

4. Which of the following numbers is not a prime number?

(a) 41 (b) 53 (c) 79 (d) 83 (e) 93

5. The bar graph below shows the amount of time that 30 ninth-grade students spendon homework nightly. What percent of students spends less than two hours on theirhomework nightly?

(a) 13.3%

(b) 20.0%

(c) 26.7%

(d) 40.0%

(e) 60.0%

6. Which of the following lists is ordered from largest to smallest?

(a) 0.19, 0.345, �23

�, 0.7 (b) 0.7, 0.19, 0.345, �23

� (c) �23

�, 0.345, 0.19, 0.7

(d) 0.7, �23

�, 0.345, 0.19 (e) 0.19, �23

�, 0.345, 0.7

Practice Set 6

P R A C T I C E S E T 6 111

10 cm

10 cm

0

2

4

6

8

10

12

14

Nu

mb

er o

f S

tud

ents

Less thanOne Hour

OneHour

TwoHours

More thanTwo Hours

Page 114: course i

7. Through what angle does the minute hand of a clock turn as it moves from 12 to 4?

(a) 20˚ (b) 40˚ (c) 110˚ (d) 120˚ (e) 150˚

8. Which of the following graphs shows the relationship between the perimeter of asquare and the length of a side?

(a) (b)

(c) (d)

(e)

0

4

8

12

16

20

2 4 6 8 10Side Length

Per

imet

er

0

24

0

4

8

12

16

20

2 4 6 8 10Side Length

Per

imet

er

0

24

0

4

8

12

16

20

2 4 6 8 10Side Length

Per

imet

er

0

24

0

4

8

12

16

20

2 4 6 8 10Side Length

Per

imet

er

0

24

0

4

8

12

16

20

2 4 6 8 10Side Length

Per

imet

er

0

24

112 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 6

Page 115: course i

9. The table below shows the values of two variables, P and Q. If P is proportional toQ, what are the values of m and n?

P 3 9 m

Q 5 n 30

(a) m = 15, n =11 (b) m = 28, n = 11 (c) m = 15, n = 24

(d) m = 18, n = 15 (e) m = 15, n = 18

10. What percent of the total area is shaded in the figure below?

(a) 12.5%

(b) 20%

(c) 80%

(d) 87.5%

(e) 140%

Practice Set 6

P R A C T I C E S E T 6 113

Test Taking Tip

When finding the area of a region formed by overlapping figures, subtract thesmaller area from the larger area.

Example Look back at Item 3 on page 111. To use this strategy, first determinethe area of the square, then subtract the area of the quarter-circle.

Area of the Square: 10(10) = 100 cm2.

Area of the Quarter-Circle: �14�(π)(102) = 25π cm2.

The area of the shaded region is determined by subtracting these areasto obtain the expression

100 � 25π. So, the answer is (c).

■ Find, if possible, another item in the practice set for which this strategy might behelpful. Try it.

■ Keep this strategy in mind as you work on future problems of this type.

8

20

5

4

Page 116: course i

1. If a is a positive integer and a2 = 4, what is (�3)a?

(a) �9 (b) �6 (c) 6

(d) 8 (e) 9

2. Which fraction represents the largest value?

(a) �160� (b) �

79

� (c) �57

(d) �45

� (e) �34

3. The perimeter of a rectangle is 28. The ratio of the width and length of the rec-tangle is 3:4. What is the length of a diagonal of the rectangle?

(a) 5 (b) 10 (c) 20

(d) 22.1 (e) 22.8

4. One pound is approximately 0.454 kilograms. Sue weighs the equivalent of 55 kilo-grams. What is her approximate weight in pounds?

(a) 25 lb (b) 108 lb (c) 110 lb

(d) 121 lb (e) 150 lb

5. The table below shows the number of books that a sample of 25 students carry toschool. What is the median number of books?

(a) 3 (b) 4 (c) 5 (d) 6 (e) 7

114 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 7

Number of Books Number of Students3 3

4 5

5 7

6 5

7 5

Page 117: course i

6. Which of the following numbers is the smallest?

(a) 3.2 � 10�3

(b) 3.2 � 103

(c) 3.82 � 10�4

(d) 3.82 � 104

(e) 3.82 � 10�3

7. The enrollment at Cedar Creek High School this year is 1,250 students. Last yearthe enrollment was 1,000. By what percent did the enrollment change between lastyear and this year?

(a) 20% (b) 25% (c) 80%

(d) 125% (e) 250%

8. A number x is multiplied by itself and the result is added to 3 times the originalnumber. This can be expressed algebraically as:

(a) x + 3 (b) x2 + 3 (c) 2x + 3

(d) x2 + 3x (e) 2x + 3x

9. A shop announces a clearance sale. The price of each item is 60% off. The originalprice of a watch is $65. By how many dollars will the price of the watch be reduced?

(a) $26 (b) $39 (c) $46

(d) $49 (e) $56

Practice Set 7

P R A C T I C E S E T 7 115

Page 118: course i

10. Which figure has the largest area?

(a) (b)

(c) (d)

(e)

116 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 7

Test Taking Tip

Ratios and proportions can be helpful in solving problems involving percents.

Example Look back at Item 7 on page 115. The amount of change was 250 stu-dents. To find the percent change, use the ratio of change amount to thebeginning value. So the percent change is equivalent to �1

2,05000� = �1

2050� or 25%.

■ Find, if possible, another test item in this practice set for which ratios and/or pro-portions can be used to help find a percentage.

■ Keep this tip in mind as you work on future practice sets.

4 cm

3.5 cm 3 cm

2.5 cm 5 cm

5 cm

2.5 cm

4 cm

6 cm

4 cm

2 cm

Page 119: course i

1. If 19 + m = 28 + n, what is m � n?

(a) �47 (b) �19 (c) 9

(d) 19 (e) 47

2. What is the volume of a cube with a surface area of 24 square centimeters?

(a) 4 cm3 (b) 8 cm3 (c) 16 cm3

(d) 32 cm3 (e) 64 cm3

3. The area of a circle is 64πsquare inches. How many inches is the circumference ofthat circle?

(a) 4π (b) 8π (c) 16π (d) 32π (e) 64π

4. Suppose the ratio of girls to boys in a class of 36 students is 5:4. How many boysare in the class?

(a) 9 (b) 16 (c) 17

(d) 20 (e) None of the above

5. Manie has a 60 cm � 18 cm piece of poster board. She wants to cover the posterboard using several sheets of colored paper. If she cuts each piece of colored paperinto squares of the same size, what is the largest size square that she can use to coverthe poster board without overlapping the colored paper?

(a) 2 cm � 2 cm (b) 3 cm � 3 cm (c) 4 cm � 4 cm

(d) 6 cm � 6 cm (e) 10 cm � 10 cm

6. In which of the following lists are the numbers ordered from smallest to largest?

(a) 0, �1, �7, �9, 2 (b) 0, �9, �7, �1, 2 (c) �1, �7, �9, 0, 2

(d) 0, 2, �1, �7, �9 (e) �9, �7, �1, 0, 2

7. 15% is equivalent to

(a) 0.15 (b) 1.5 (c) 10.5 (d) 15 (e) 150

8. In a bag of chips, �16� are green, �

14� are yellow, �1

12� are blue, �

13� are white, and �

16� are red.

If someone takes a chip from the bag without looking, which color is it most likelyto be?

(a) Green (b) Yellow (c) Blue (d) White (e) Red

Practice Set 8

P R A C T I C E S E T 8 117

Page 120: course i

9. The length of a rectangle is 8 cm, and its perimeter is 20 cm. What is the area of therectangle in square centimeters?

(a) 16 cm2 (b) 28 cm2

(c) 40 cm2 (d) 96 cm2

(e) 160 cm2

10. �ABE can be rotated onto �DBC. What point is the center of rotation?

(a) Point A

(b) Point B

(c) Point C

(d) Point D

(e) Point E

118 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 8

Test Taking Tip

Be careful not to confuse perimeter and area.

Perimeter is a measure of the distance around the border of a figure, while area isa measure of the region enclosed by a figure. Perimeter is measured in units oflength, while area is measured in square units. The perimeter of a circle is calledits circumference.

Example Look back at Item 3 on page 117. The area of a circle is given as 64π square inches. If you are frequently confused as to which formulaapplies to the area, 2πr or πr 2, remember that square units are obtainedby squaring the radius. Thus, in this problem πr2 = 64π in .2.

Since π(8 in.)2 = 64π in.2, the radius must be 8 inches. Substitute 8 inches into the formula for circumference:

2π(8 in.) = 16π in.

So, the answer is (c).

■ Look back at Practice Sets 1�8 and identify the items which require you todistinguish between perimeter and area.

■ Keep this caution in mind as you work on future practice sets.

B

A

C

DE

Page 121: course i

1. ABCD is a rectangle. AC = 10, BC = 6. What is the perimeter of ABCD?

(a) 8

(b) 14

(c) 28

(d) 38

(e) 48

2. Which digit of 0.2�3�1�6� has a place value of �1,0

100�?

(a) 0 (b) 1 (c) 2

(d) 3 (e) 6

3. �35

� + ���46

� + �56

�� =

(a) �310� (b) �

340� (c) �

1330�

(d) �2330� (e) �

6330�

4. Which of the following is not equivalent to �1220�?

(a) �35

� (b) �1255� (c) �

2315�

(d) �2470� (e) �

3660�

5. Supa has 3 ribbons of different colors and lengths. She has 3.5 meters of blue rib-bon, 4.9 meters of red ribbon, and 5.6 meters of white ribbon. She would like to cutthese ribbons so that all pieces of the three different-colored ribbons have the samelength. What is the longest length into which she could cut the ribbons?

(a) 0.3 meters (b) 0.5 meters (c) 0.7 meters

(d) 0.9 meters (e) 1.1 meters

Practice Set 9

P R A C T I C E S E T 9 119

D C

A B

Page 122: course i

6. In the figure shown below, m�A is 75˚, m�D is 25˚, and m�EFC is 170˚. Find thedegree measure of �EBC.

(a) 35˚

(b) 70˚

(c) 85˚

(d) 95˚

(e) 145˚

7. If 6x � 4 = 16, then x =

(a) 2 (b) �73

� (c) �130� (d) �

230� (e) 20

8. The histogram at the right shows test scores for 40 students. What percent of thestudents scored at least 65?

(a) 17%

(b) 30%

(c) 42.5%

(d) 50%

(e) 57.5%

120 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 9

E

A

B C D

F

0

2

4

6

8

10

12

14

Nu

mb

er o

f S

tud

ents

20 35 50 65

16

80 95

Scores

Page 123: course i

9. If x = ��32

� , then what is the value of 5 + 4x?

(a) �13.5 (b) �1 (c) 1.5

(d) 3 (e) 8.5

10. If the sum of three consecutive odd integers is 15, what is the largest of those threeintegers?

(a) 5 (b) 7 (c) 9

(d) 11 (e) 13

Practice Set 9

P R A C T I C E S E T 9 121

Test Taking Tip

Equivalent fractions are generated by multiplying by various forms of 1.

Example Look back at Item 4 on page 119. To use this strategy, test each choiceto determine if the reduced form of �

12

20� or �

35� can be multiplied by a form

of 1 to obtain the fraction in that choice.

Choice (a): �35� • �

11� = �

35�

Choice (b): �35� • �

55� = �

12

55�

Show that choices (c) and (e) are also equivalent to �12

20� by multiplying �

35�

by other forms of 1 to obtain each choice.

Choice (d): �35� • �

99� = �

24

75� ≠ �

24

70�, so �

12

20� ≠ �

24

70�

So, the answer is (d).

■ Find, if possible, another test item in the practice set for which this strategymight be helpful. Try it.

■ Keep this strategy in mind as you work on future practice sets.

Page 124: course i

122 P R A C T I C I N G F O R S T A N D A R D I Z E D T E S T S

Practice Set 10

1. If a = 2 and b = 3, then what is (ab)2?

(a) 23 � 23 (b) 23 + 23 (c) 2 � 2 � 3

(d) 2 � 3 � 2 � 3 (e) 2 � 3 � 3

2. For five days, a student paid an average of $4 per day for lunch. How much moneydid the student pay for lunches for the five days?

(a) $1.25 (b) $4.00 (c) $5.00

(d) $16.00 (e) $20.00

3. 32 is 16% of

(a) 160 (b) 200 (c) 250

(d) 300 (e) 400

4. Which equation describes the relationship in the table shown below?

x �1 0 1 2 3 4

y �7 �5 �3 �1 1 3

(a) y = x � 5

(b) y = x � 6

(c) y = 2x � 5

(d) y = �x � 5

(e) y = �12

�x � 5

5. The figure below consists of 6 congruent squares. The area of the entire figure is54 square centimeters. What is the perimeter of the figure?

(a) 42 cm

(b) 45 cm

(c) 54 cm

(d) 60 cm

(e) 72 cm

Page 125: course i

Practice Set 10

P R A C T I C E S E T 1 0 123

6. Which of the following numbers is 86.0749 rounded to the nearest hundredth?

(a) 86.07 (b) 86.08 (c) 86.10

(d) 90.00 (e) 100.00

7. Which ratio is not equivalent to 21:14?

(a) 33:22 (b) 18:12 (c) 15:10

(d) 8:6 (e) 6:4

8. For positive integers x, y, and z, if �1x

� > �1y

� > �1z

�, then which of the following state-ments is false?

(a) x < y (b) x � z < 0 (c) y < z

(d) x � y < 0 (e) z < x

9. If n = 8, then what is the value of �3n

21�

66

�?

(a) 3 (b) 12 (c) 36 (d) 75 (e) 570

10. Point A has coordinates (�3, 5). When using the y-axis as a reflection line, theimage of point A is point B. What are the coordinates of point B?

(a) (3, �5) (b) (�3, �5) (c) (5, 3)

(d) (5, �3) (e) (3, 5)

Test Taking Tip

Reciprocals and opposites reverse order relations.

If x and y are both positive or both negative and x < y, then �1x

� > �1y

�, and �x > �y.

Example Look back at Item 8. The relationships among x, y, and z are the reverseof the relationships among their reciprocals. Since�1x

� > �1y

� > �1z

�, x < y < z. Choices (a) and (c) are clearly true from this obser-vation. Choices (b) and (d) are true because subtracting a larger positiveinteger from a smaller positive integer will always yield a number lessthan 0. Choice (e) is false since the order relation should be reversed. So,the answer is (e).

■ Look back at Practice Sets 1�10 and identify the items for which this remindermight be helpful.

■ Keep this fact in mind in your future work with inequalities.

Page 126: course i

124

Page 127: course i

S O L U T I O N S 125

Check Your Understanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Exercise Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Practice Sets for Standardized Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Solutions

Page 128: course i

�Solutions to Check Your Understanding

Check Your Understanding 1.1, p. 81. a. –7, –2, 11, 15 b. –23, –3, 0, 10

2. a. 16 b. 9 c. –23 d. 25

e. 60 f. 5 g. 7 h. –81

3. $195

Check Your Understanding 1.2, p. 121. a. �

170� b. ��

18

� c. �175� d. �

15058

2. There are many possible equivalent forms of each given fraction or mixed number.We give only three for each.

a. �160�, �

195�, �

1220� b. �

46

�, �23

�, �1105�

c. �64

�, �32

�, �1150� d. �

159�, �

3180�, �

5175�

3. a. 4�12

� teaspoons of lemon juice b. $250

4. a. 1�130� km b. �

18070

5. a. June b. 14 wins

6. The table entries are:

126 S O L U T I O N S

5 11�14

� 2�12

3�34

� 6�14

� 8�34

10 1�14

� 7�12

Page 129: course i

S O L U T I O N S 127

Check Your Understanding 1.3, p. 141. a. 0.128 = �

110� + �

1200� + �

1,0800� = �

11,02080

b. 0.0205 = �1200� + �

10,5000� = �

102,00500�

c. 3.142 = 3 + �110� + �

1400� + �

1,0200� = �

31,,104020

2. a. 0.6 b. 0.75 c. 4.4 d. 1.8

3. 484 hours

4. 3.15576 � 107 seconds

5. a. 0.833�; infinite repeating decimal

b. 0.875; finite decimal

c. �27

� = 0.285714285714…; infinite repeating decimal

Check Your Understanding 1.4, p. 16–171. a. About 4.54 hours b. $4.95 c. $10.13

2. a. 75 people b. 45 people

3. 375 mL

4. The large box

Check Your Understanding 1.5, p. 19–201.

a.

b.

c.

d.

e.

Fraction Decimal Percent�34� 0.75 75%

�110� 0.1 10%

�78� 0.875 87.5%

�23� 0.666... 66�

23�%

�1270� 0.85 85%

Page 130: course i

2. a. 76 b. 62.5% c. Approximately 30.77

3. The sale price is $18.75. This represents a total discount of 62.5%.

4.

5.

Check Your Understanding 2.1, p. 231. 280 � 10m

2. a.

b.

3. a. �16 b. �63 c. �77 d. 32

128 S O L U T I O N S

Product 1987 1996 Percent Increase

Corded phone 13,335 21,700 62.7%

Cordless phone 9,900 22,800 130.3%

Answering machine 14,716 20,050 36.2%

Fax and/or fax modem 1,907 4,700 146.5%

Region 1990 1998 Percent DecreaseWorld 3.4 2.9 14.7%

Less-Developed Countries 4.7 3.2 31.9%

More-Developed Countries 1.9 1.6 15.8%

Stage Number 1 2 3 4 5 n

Number of Squares 1 3 5 7 9 2n�1

Fertility Rates 1990–1998

Page 131: course i

S O L U T I O N S 129

Check Your Understanding 2.2, p. 261. a. Charges will be $2.00, $5.00, and $6.00 respectively.

b.

2. a. Rachel won by 5 seconds. She was ahead by about 10 meters when she crossedthe finish line.

b. They were tied after Rachel had run 60 meters.

c. They were tied after about 15 seconds.

d. Rachel was ahead by about 5 meters.

e. Rachel averaged 4 meters per second. Micah averaged 2�23

� meters per second.

Check Your Understanding 2.3, p. 281. a. x = 5 b. x = 6

2. a. n < �9 b. n < �2

3. a. C = 3.50 + 2.50(d � 1) for d ≥ 1 b. 6 days

Check Your Understanding 3.1, p. 331. No two sides of a triangle can be parallel. Suppose that sides a and b meet at point

C. Then the other side is determined by the endpoints of those sides and thus,because it intersects both of those sides, it is not parallel to either.

2. The one-way sign is a rectangle. The stop sign is a regular octagon. The men-at-work sign is a square. The yield sign is an equilateral triangle. The school zone signis a pentagon. The stop sign, men-at-work sign, and yield sign are regular polygons.

6

5

4

3

00

642

2

1

Parking Time (in hours)

Par

kin

g C

ost

(in

do

llars

)7

1 753 8

Page 132: course i

3. m�a = 105˚, m�b = 45˚, m�c = 30˚, m�d = 105˚

4. Both diagonals are 13 cm long.

Check Your Understanding 3.2, p. 361. A regular octagon has 8 lines of symmetry (4 through midpoints of opposite sides

and 4 through opposite vertices) and rotational symmetries of 45˚, 90˚, 135˚, 180˚,225˚, 270˚, and 315˚ about the center of the octagon.

2. a. 6 lines of symmetry as shown at the right,and symmetry rotations of 60˚, 120˚, 180˚,240˚, and 300˚ about the center of thesnowflake.

b. There is only one line of symmetry: a vertical line through the center of the figure.

c. This has only half-turn symmetry.

3. a. IV b. IV c. II

d. I e. I f. III

Check Your Understanding 4.1, p. 381. a. 60˚ b. 115˚ c. 50˚ d. 105˚

2. a. b. c. d.

Check Your Understanding 4.2, p. 40–411. a. Racing pool: 140 m; diving pool: 60 m; child’s pool: 8π m

b. About 150 minutes or 2.5 hours per week.

2. 64 m

3. Diameter ≈ 12,732 km; Radius ≈ 6,366 km

130 S O L U T I O N S

Page 133: course i

S O L U T I O N S 131

Check Your Understanding 4.3, p. 431. a. 196 m2 b. About 1.3 m2

2. a. 3,000 cm2 b. 1,500 cm2 each

3. 10,000π ≈ 31,416 m2

Check Your Understanding 5.1, p. 46–471. a. b.

2. a.

Graduation Rate (%)20 40 60 80 100

4

8

12

20

16

00

Fre

qu

ency

Asia

Europe,Japan,

Australia

Africa

LA

NA

57%

22%

8%

4%

9%

2050

Distribution of World PopulationDistribution of World Population

NA EuropeJapan

Australia

Afric Asia LA

10

20

30

40

50

Region

Per

cen

t o

f Wo

rld

Po

pu

lati

on 2000

60

Page 134: course i

b. U.S. High School Graduation Rates

5 3 45 7 8 8 8 8 8

6 2 2 3 3 5 5 5 6 7 8

7 0 0 1 1 1 2 2 2 3 4 4 5 5 5 6 6 6 6 6 8 8

8 0 0 3 3 3 5 5 7 9

9 0 5 | 3 represents 53%

Check Your Understanding 5.2, p. 481. a. Mean = 71.02 b. Median = 72 c. Range = 37

2. a. Mean and median will increase by 1 point to 8.5 and 9 and the range will staythe same.

b. Mean, median, and range will be multiplied by 10 to 75, 80, and 30.

c. Mean will decrease to about 7.43. Median might decrease (depending on the dis-tribution of the original set of scores). Range might or might not change (it couldonly get larger).

Check Your Understanding 5.3, p. 501. a. HpHnHd, HpHnTd, HpTnHd, TpHnHd, HpTnTd, TpTnHd, TpHnTd, TpTnTd

b. ■ P(at least 2 heads) = �48

■ P(no heads) = �18

■ P(heads-up value ≥ 11) = �38

2. a. �12

b. The safest bet is that the number of boys will be between 40 and 60 since all theother options are included in this wide net.

132 S O L U T I O N S

Page 135: course i

S O L U T I O N S 133

�Solutions to Exercise Sets

Exercise Set 1, p. 52–531. a. �1, 0, 4 b.

2. a. 90 or higher b. 87.5 c. 15

3. 22 weeks

4. a. 3.27 � 104 b. 4.23491 � 104 c. 9.27 � 108

d. 3.40 � 10�3 e. 6.275 � 10�5 f. 1.05 � 10�2

5. a. Approximately 49.7% b. Approximately 53.4%

6. a. b.

c.

All three figures are alike because they are all parallelograms, opposite sides par-allel and the same length. The rectangle and the square have 4 right angles andthe parallelogram does not. The square has 4 equal sides and the rectangle doesnot. The parallelogram may or may not have 4 equal sides.

7. 5(4) � 8 = 20 � 8 = 12

8. a. 1 cm � 8 cm, 5 cm � 4 cm, 2 cm � 7 cm

b. 5 cm � 4 cm c. 1 cm � 8 cm

9. a. b. c.

–4 –2 0–5 –3 –1 1 42 3 5

–4.2 10 ≈ 3.162.3

Page 136: course i

10. a. $480.00 (Note: This assumes a minimum charge of $12 for any group of 5 orfewer students.)

b. $2.44

Exercise Set 2, p. 54–551. a. Approximately 4.2 seconds b. Approximately 50.6 seconds

c. Approximately 421.5 seconds or about 7 minutes

2. Answers will vary.

3. a. �63 b. �8.5 c. �21

4. a. (i) 5.2 cm (ii) 7.3 cm (iii) 2.5 cm (iv) 10 cm

b. (i) 52 mm (ii) 73 mm (iii) 25 mm (iv) 100 mm

c. (i) 0.052 m (ii) 0.073 m (iii) 0.025 m (iv) 0.1 m

5. Approximately 305%

6. a. �8, �6, 9, 17 b. �33, �13, 0, 43

c. �12

�, �58

�, �23

�, �34

� d. 2, 2.044, 2.13, 2.305

7. a. Note that the bars can be in any order, if properly labeled.

SpreadsheetsInternetAccess

WordProcessing

Games Other

10

20

30

40

Nu

mb

er o

f P

eop

le

Computer Usage

134 S O L U T I O N S

Page 137: course i

S O L U T I O N S 135

b. Again, the pie segments can be in any order.

8. Approximately 81.6 square feet

9. a. 20 cm b. 12 cm

10. a. $5.40 per hour b. $207.90

Exercise Set 3, p. 56–57 1. $66.56

2. a. D ≈ 4.77 feet at its base; D ≈ 3.18 feet at 20 feet above its base

b. A ≈ 17.9 ft2 at the bottom of the log; A ≈ 7.94 ft2 at the top of the log

3.

WordProcessing

InternetAccess

Games Spreadsheets

Other

Computer Usage

Type 1985 1995 Percent IncreaseHeart 719 2,361 228%

Liver 602 3,924 552%

Kidney 7,695 11,816 54%

Organ Transplants 1985–1995

Page 138: course i

4. a. Some people could speak more than one language.

b�c.

5. $0.67

6. a. $1.57 per pound b. 52 miles per hour c. $11 per ticket

d. 1.5 feet per second e. $5 per hour f. 1.25 pounds per week

7. a. Approximately $244.70 b. $18,000

8. Surface area = 195.75 cm2

Volume = 160.875 cm3

9. Approximately $39.09 per share

10. 6 numbers. Four of them are divisible by 2. All of them are divisible by 3. Two ofthem are divisible by 5.

Exercise Set 4, p. 58–591. a. 8.9 b. 350.07 c. 6,740.50

2. a. All three parallelograms have the same area.

b. Parallelograms ABCD and BCFE have equal perimeters and that perimeter isgreater than the perimeter of parallelogram DBCE.

3. a. Four minutes, twelve and fifty-six hundredths secondsThree minutes, forty-four and thirty-nine hundredths seconds

b. 28.17 seconds

136 S O L U T I O N S

Language Percent of People Number of PeopleSpanish 37.5% 56

French 25% 38

German 12.5% 19

Japanese 6.25% 9

Russian 10% 15

None 50% 75

Page 139: course i

S O L U T I O N S 137

4. a. 86 square units b. 54 units

5. a. 26 b. �15

1

•515

� � �8 +

88

� = 13 c. (4 + 4) ÷ ( 2 ÷ 2) = 8

6. a. S � 45 = 358.71 b. S = $403.71

7. row 1 100%

row 2 66�23

� %

row 3 60%

row 4 57�17

� %

8. Yes, because you have about $86.

9. Range: 4.57; Mean: 6.2186; Median: 6.72

10.

Internal medicine had the greatest comparative growth.

Exercise Set 5, p. 60–621. Diana and Erin each receive �1

12� of the estate. Frances, Gail, and Heather each

receive �118� of the estate. Ingrid, Julia, Katherine, and Lillian each receive �2

14� of the

estate.

2. a. 24 sq cm b. 63 sq cm c. 26 cm

3. a. �38

� b. �52

� c. ��490� d. 4�

38

Medical Specialty Mean Net Mean Net Ratio of Mean Income, 1988 Income, 1994 Net Income,

1994 to 1988General/Family Practice $77,900 $121,200 1.56

Internal Medicine $102,000 $174,900 1.71

Surgery $155,000 $255,200 1.65

Pediatrics $76,200 $126,200 1.66

Obstetrics/Gynecology $124,300 $200,400 1.61

Mean Net Income for Physicians

15

Page 140: course i

4. a. No. Player 1 wins only �13

� of the time and Player 2 wins �23

� of the time.

b. Player 1 should get twice the amount Player 2 gets for winning.

5. You could buy everything except the rolls and the laundry detergent.

6. a.

b. Except for the year 1993�1994, the cost of cable TV has steadily increased.

c. About $162

7. a. 0.1�4�2�8�5�7� b. 0.5� c. 0.86� d. 0.625

8. a. < b. < c. = d. >

9. 57�17

�%

10. a. High Temperatures

5 96 0 0 3 3 4 5 77 1 2 2 58 5 6 5 | 9 represents 59˚F

b. 68.7˚F c. 66˚F

140

130

120

110

9080

969492

100

90

Year

Pri

ce(i

n d

olla

rs)

150

91 979593

Average Annual Cost of Cable1990–1997

138 S O L U T I O N S

Page 141: course i

S O L U T I O N S 139

Exercise Set 6, pp. 63–641. $1.32

2. a. Approximately 46% b. Approximately 40%

3. a. 66.67 grams b. 216 cookies

4.

a. The median would make Airline 1 look best.

b. The range makes Airline 2 look best.

c. The outlier of 50 minutes for Airline 1 distorts the statistics. If you take that out,the mean is 0.6�, the median is 0, and the range is 27 minutes.

5. a. > b. = c. > d. <

6. 19,770 feet

7. a.

b.

c. 55 circles—To get the next number in the table, take the previous number of circlesand add the next stage number to that number. For example, at Stage 6 there are21 circles, so you take 21 plus 7 to get the number of circles at Stage 7. Repeatingthis will give 55 circles at Stage 10.

Stage Number 1 2 3 4 5 6

Number of Circles 1 3 6 10 15 21

Stage 6Stage 5

Airline 1 Airline 2Mean 5.6 5

Median 2.5 5.5

Range 65 20

Page 142: course i

8. Posts could be placed 1, 2, 4, 5, 10, or 20 feet apart. The greatest distance betweenposts is 20 feet.

9. a. 35 cm = 0.35 m = 350 mm b. 0.5 ft = 6 in = 0.16� yd

c. 1,500 minutes = 25 hours = 1�214� days

10. 16 times

Exercise Set 7, pp. 65–661. The product is �

165� or �

25

�.

2. a. 5 b. 13 c. 6

3. 77˚

4. a. 4.007, 4.037, 4.04, 4.105, 4.13 b. �13

�, �25

�, �12

�, �58

�, �34

c. �5, �4.2, �3.7, �3, 0, 4

5. 6-inch-square tiles

6. a. 25 red and 75 blue b. 24 blue and 36 red

7. a.

b. There are many possible triangles. Three are shown here.

3 m

8 m12 m

2 m

6 m

4 m

2 m

6 m

140 S O L U T I O N S

Page 143: course i

S O L U T I O N S 141

c.

8. 23.4˚

9. a.

The points are completely scattered so there seems to be no relation between bat-ting and pitching.

b. Points above the line have a better rank for team pitching than batting. Points onthe line represent teams that have the same rank for pitching and batting. Pointsbelow the line have better ranks for batting than for pitching.

10. The net change is zero, so the price is still $23�18

�.

National League Batting andPitching Rankings for 1998

Pit

chin

g R

ank

Batting Rank

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16

5 m

Page 144: course i

Exercise Set 8, pp. 67–681. 50 feet below sea level

2. a.

b. There does not seem to be any overall trend to these data. The winning speedsincrease from 1960 to 1980 and then decrease from 1980 to 1995.

3. Each column will be 2�16

� inches wide and 8 inches long.

4. a. 3 b. 10 c. 2.1

5. a. 30˚ b. 80˚ c. 150˚ d. 20˚

6. 12 tickets

7. a. 4 b. �40 c. �9 d. 8

8. 30 days

9. a. x = 4

b. x ≥ �6

c. x > �12

10. 2.17 putts per hole

33�13

�%

–4 –2 0–5 –3 –1 1 42 3 5

–8 –4 0–10 –6 –2 2 84 6 10

–12 –6 0–15 –9 –3 3 126 9 15

Winning Speeds in Daytona 500

Sp

eed

(in

mp

h)

Year

120

130

140

150

160

170

180

55 60 65 70 75 80 85 90 95 00

142 S O L U T I O N S

Page 145: course i

S O L U T I O N S 143

Exercise Set 9, pp. 69–701. $574

2. a. Midnight or 12 A.M. b. 3 A.M. the next day

3. 192 glasses

4. a. �18

� or 0.125 b. �14

� or 0.25 c. 15 times

5. $1.34

6. a. Approximately 584.34 million miles

b. 1.6 million miles per day or 66,705 miles per hour

7. a. �11040

� = �570� b. �

2160� = �

153� c. �

13

� d. �2939�

8. a. 78 b. 70 or higher

9. Answers will vary, but one possibility is given below.

a. b.

10. 450 days

Exercise Set 10, pp. 71–721. a. 25% b. 0.4% c. 280%

d. 150% e. 60%

2. a. 30 times b. 18 times c. 42 times

3. 330 years

4. 165�34

�lb

5. Perimeter: 15 cmArea: 7.5 cm2

Page 146: course i

6. a. 0.066�, 0.067, 0.63, 0.6�3�, 0.6� b. ��52

�, ��152�, �2.3, �2, �1

c. 0.99, 1.009, 1.035, 1.047, 1.47

7. �0.8˚F or 0.8˚F below zero

8. $24.31

9. a. A scatterplot

b.

This graph shows that states that spent more money tended to have higher scores.

10. a. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 b. x ≤ 1

c. 0, 1, 2, 3 d. 2, 3, 5, 7

Exercise Set 11, pp. 73–74 1. a.

b. NEXT = NOW + 4 c. 37

d. S = 4n � 3 e. S = 4(10) � 3 = 37

2. a. �2 b. �8 c. 15 d. 2

3. 28

4. a. �4, �3, �1, �1, 0, 1, 2, 3, 3, 5 b. 9

Stage 1 2 3 4 5 6 7 8

Number of Tiles 1 5 9 13 17 21 25 29

144 S O L U T I O N S

Co

st p

er S

tud

ent

in $

100

Percent Math Proficient

40

45

50

55

60

65

70

75

80

40 45 50 55 60 65 70 75 80

Page 147: course i

S O L U T I O N S 145

5. a. 6.8 � 109 b. 2.48 � 105 c. 4.5 � 10�2 d. 3.89 � 10�4

6. 16 feet

7. a. The lines are perpendicular. b. It has 180˚ rotation symmetry.

8. About 40.59%

9. About 22.2%

10. a. �23

� b. �13

� c. �145�

d. �175� e. �

185�

Exercise Set 12, pp. 75–76 1. a. Perimeter: 30 m b. Perimeter: 28 in.

Area: 30 m2 Area: 30 in.2

c. Perimeter: 26 + 3π ≈ 35.42 cm d. Perimeter: 24 ftArea: 60 + 4.5π ≈ 74.14 cm2 Area: 4�48� ≈ 27.7 ft2

2. a. 10 b. 450 c. �220 d. 474

3. 3,000,000,000 or 3 billion

4. a. n = �3 b. n = 3 c. n < �1 d. n ≤ �4

5.

a.

b.

c.

d.

Most estimates seem to have a small percentage of error, but when you cubed theestimate, the error was much greater.

6. a. 6.6� b. 6 c. 5.6�

7. About 247.6 million people

8. a. Approximately 90.35 g of protein b. 189 Canadian dollars

EstimatedProblem Answer Answer % Error7.9 + 2.1 + 8.7 � 9.3 10 9.4 6%

�7.9 + 2�.1 + 8.�7 � 9.3� 3 3.0659 2%

(7.9 + 2.1 + 8.7 � 9.3)3 1,000 830.584 20.40%

4.2(7.9 + 2.1 + 8.7 � 9.3) 40 39.48 1%

Page 148: course i

c. 5.75 yards d. 8�176� cups of rice e. 33�

13

� ounces

9. a. 10 b. 160 c. 33�13

�%

d. 10 e. 200% f. 0.25

10. The 100th person

Exercise Set 13, pp. 77–791. 38 miles

2. 94˚

3. a. �11, �3, �1, 0, 8, 15 b. �3.26, �3.2, 3.2, 3.26 c. �18

�, �12

�, �196�, �

58

�, �34

4. a. III, I, IV, II

b. The point that corresponds to the left end of the horizontal segment III

5. a. 680 = 23 • 5 • 17; 1,000 = 23 • 53 b. 40 c. 17,000

6. a. Mean = 17.17, median = 16.65

b. MAD = 1.524. This statistic tells you how far away the class sizes for those statesare from the mean. Since 1.5 is a small number, it says most of the states haveaverage class sizes that are close to the mean.

7.

146 S O L U T I O N S

Spaghetti for Seventy-Five

15 pounds ground beef �34� cup shortening

�34� cup flour 3 gallons water

15 medium onions, chopped 1�78� cups chopped green pepper

1�78� cups chopped celery 60 bay leaves

�34� cup chili powder �

12� cup salt

4�12� tablespoons pepper 10.5 pounds spaghetti, cooked

48 ounces each tomato paste, tomato sauce

4�12� tablespoons each oregano, basil, and thyme

Page 149: course i

S O L U T I O N S 147

8. a. 27 b. 150% c. 16�23

9. a. About 403 meters b. About 434 meters c. $10,918.76

10. a. 84 cm b. Approximately 288 cm2 c. 12 boxes

Exercise Set 14, pp. 80–82 1. The sum is 540˚ because it is equal to the sum of the measures of the angles of all

three triangles. Since each triangle has 180˚, then the sum must be 540˚.

2. a. 11,500 square feet b. 640 feet

3. 6 • 3 � 5 ÷ 2 • (6 + 5) = �9.5

4. Hummingbird (0.056); pygmy shrew (0.52); bat (0.65); cockroach �1�14��

5. 11 feet tall

6. a.Station Format PercentCountry 23.1

Adult Contemporary 15

News, Talk, Sports 13.1

Religion 10.3

Rock 7.5

Oldies and Classic Hits 9.4

Spanish and Ethnic 5.4

Adult Standards 5.4

Urban, Black 3.3

Top 40 3.7

Other 3.7

Page 150: course i

b.

c. 83˚ for Country, 27˚ for Rock

d. The shares of radio stations with each format might not match the share of the listening audience by each format because there might be many of one format thathave a small listening audience.

7. a. b. c.

d. e. f.

8. �59

9. a. 157.1 feet b. 7,068.6 square feet

c. 39,269.9 cubic feet d. 294,524.3 gallons

10. $112

9

9

7

2

8 cm

2 in.

0

500

1,000

1,500

2,000

2,500

Count

ry

Adult

Conte

mpo

rary

Rock

News,

Tal

k, S

ports

Relig

ion

Old

ies

and

Class

ic Hits

Spani

sh a

nd E

thni

c

Adult

Stand

ards

Urban

, Bla

ckTo

p 40

Oth

er

148 S O L U T I O N S

Page 151: course i

S O L U T I O N S 149

Exercise Set 15, pp. 83–841. 6 feet 1�

12

� inches

2. a. 175 square units b. 30.9 square units

3. Redmond: 1.4375 inches per year; Chillal: 1.07 inches per yearThe difference is 0.3675 inches per year.

4. a.

b. Judge 1: Mean ≈ 5.03, MAD ≈ 0.604

Judge 2: Mean ≈ 4.9, MAD ≈ 0.705

c. The judges give very similar ratings overall to the group of skaters.

d. A scatterplot of data for all skaters, paired (Judge 1, Judge 2), would be helpfulin determining if the judges give similar ratings. The more closely the points areclustered to the y = x line, the more similar the individual ratings are.

5. $66.36

6.

Min Q1 Median Q2 Max

Judge 1 3.6 4.5 5.2 5.7 6.0

Judge 2 3.4 4.2 4.8 5.6 6.0

Regular # of Sides # of Lines Smallest Angle ofPolygon of Symmetry Turn SymmetryTriangle 3 3 120°

Quadrilateral 4 4 90°

Pentagon 5 5 72°

Hexagon 6 6 60°

Octagon 8 8 45°

n-gon n n �36n0°�

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5

Judge 1

Judge 2

Page 152: course i

7. a. 88 b. �9 c. ��2116�

8. a. 50 + 5.75n b. 10n

c. 10n � 50 � 5.75n, 4.25n � 50 d. 12 shirts

9. a. 500 square feet b. 20 feet

10. a. b. c.

d. e. f.

Exercise Set 16, pp. 85–86 1. 17 jumps

2. Perimeter: 19.2 cm, Area: 18 cm2

3. a. 23 b. 9 c. �4

d. 19.5 e. �488

4. a. Only graph (iii) has an Euler circuit because it is the only graph with all even vertices.

b. Both (i) and (ii) contain Euler paths because each has only two odd vertices.

5. a. x = 12 b. x = �3175� c. x = 1

d. x = ��454� e. x = 9 f. x = 6

4

8

–4–8 4 8

–4

–8

x

y

4

8

–4–8 4 8

–4

–8

x

y

4

8

–4–8 4 8

–4

–8

x

y

4

8

y

x–4–8 4 8

–4

–8

4

8

y

x–4–8 4 8

–4

–8

4

8

–4–8 4 8

–4

–8

x

y

150 S O L U T I O N S

Page 153: course i

S O L U T I O N S 151

6. a. Yes; fractions are another way of writing division.

b. Yes; 3 • �15

� = �35

c. No; this would be 5 • �13

� = �53

�.

d. Yes; �15

� • 3 = �35

7. a. The mean and median are both five points higher, and the range stays the same.

b. The mean, median, and range will all be divided by 10.

8. a. 5.93 � 1012 b. 5.5436 � 1012 c. 3.98 � 1010

d. 9.5 � 107 e. 6.214 � 10�6 f. 1.894 � 10�4

9. a. Approximately 56.55 inches b. Approximately 4.23 inches

10.

a.

b.

c.

d.

e.

Exercise Set 17, pp. 87–88 1. a. �

14

� b. �58

� c. �21

� d. �16

2. Position 1 180˚ turn about the center of the shared edgePosition 2 Horizontal flipPosition 3 Horizontal flip followed by 180˚ turn about the center of common edgePosition 4 A slide

3. a. 25%, 0.63, �45

�, 1�17

�, 1.2 b. �79

�, 82.5%, 0.88, 1.80

4. a. $276 b. 5 people c. 11 people

5. a. 106 inches b. 792 square inches

6. $117.86 for 3 days$157.14 for 4 days

Fraction Decimal Percent�58� 0.625 62.5%

�57010� 0.142 14.2%

�38� 0.375 37.5%

�16� 0.16� 16�

23�%

�1230� 0.65 65%

Page 154: course i

7. a. 18 b. �29 c. 72

d. 140 e. �14 f. �52

8. a. Either a bar graph or a circle graph.

b.

9. a. $790 b. $1,155 c. $4,485

10. 90 cm

Exercise Set 18, pp. 89–90 1. a. 41 b. 6 c. �12

d. �49 e. 38 f. 4

2. a. Linear; NEXT = NOW � 3 (start at 7)

b. Exponential; NEXT = 5NOW (start at 1)

c. Neither

d. Exponential; NEXT = 2NOW (start at 6)

3. 82˚

4. a. �9˚ b. 45˚ c. �5˚

P and P11%

R2%

P59%

M12%

G12%

W3%

C1%

0%

10%

20%

30%

40%

50%

Paper and P

aperboard

Rubber

PlasticMetal

GlassW

oodCloth

60%

152 S O L U T I O N S

Page 155: course i

S O L U T I O N S 153

5. a. True b. True c. False

d. True e. True f. False

6. About 2,444 mg of sodium

7. a. 25.46 inches

b. A bit more than 144 inches of fabric

8. a. Front and back: 8 � 12 in.Top and bottom: 8 � 3 in.Right and left sides: 12 � 3 in.

b. SA = 312 square inches c. V = 288 cubic inches

9. a.

b. The critical path is S-B-D-G-F and the earliest finish time is 11.

10. $14.50

Exercise Set 19, pp. 91–921. The 200-gram box is the better buy.

2. a. 18, 26, and 34 respectively b. 50, 250, and 1,250 respectively

c. B = 8m + 2 for linear growth, B = 2(5)m for exponential growth.

d. After 25 months assuming a linear model, after 3 months assuming an exponen-tial model

3. a. 10 b. 9 c. 12

d. �4 e. 5

4. a. x = 10 b. x = �2 c. x = �20 d. x = 0.5

Path LengthS-A-D-G-F 7

S-B-D-G-F 11

S-B-E-G-F 10

S-C-E-G-F 8

Page 156: course i

5. 1.25 seconds

6. a. 9.84 � 10-4 and 1.37 � 10-3 b. 139.2%

7. 50%

8. a. NEXT = NOW + 2 (start at 2) b. NEXT = 2NOWy = 2 + 2x y = 2x

c. NEXT = NOW � �13

� (start at 2) d. NEXT = �12

� NOW

y = ��13

�x + 2 y = ��12

��x

9. a. State 1: mean = 1.137, median = 1.144

State 2: mean = 1.167, median = 1.124

b. State 1: range = 0.180, State 2: range = 0.320

c. Yes, you could find each value by dividing by 100 and then adding 1.009.

d. State 1 has the lower gasoline prices. Both the range and the mean are smallerfor State 1.

10. a. 2�172� b. 7�

34

� c. 1�1270�

d. 2�58

� e. �172� f. 5�

34

Exercise Set 20, pp. 93–951. a. p > 80,000 b. c ≥ 18 c. w ≤ 250

2. Alcohol: 27.6%; Cigarettes: 24.9%

3. Dimensions in inches: 9 � 12 � 12, 6 � 12 � 18, 3 � 12 � 36, 3 � 48 � 9,3 � 24 � 18, 3 � 6 � 72, 6 � 24 � 9, 6 � 6 � 36. The best size box to use wouldbe 9 � 12 � 12 because it would be the easiest to hold and it has a minimum sur-face area. Also, the 6 � 12 � 18 size would be a good choice because it has thesame minimum surface area as the 9 � 12 � 12.

4. 19.81 mpg, 20.97 mpg, 18.10 mpg, 21.00 mpg

5. a. �16

� b. �13

� c. �13

� d. �16

6. a. x = 3.6 b. y = 20 c. x = 6.9 d. x = 4

154 S O L U T I O N S

Page 157: course i

S O L U T I O N S 155

7. 437.7488 days

8. 5.25 � 1.75 � 1.75 inches

9. 7 � 8 inches

10. a. Percent Proficient on TestMath Science

1 3 5 7 8 92 1 1 3 5 7 7 8

8 3 0 1 2 2 6 98 5 4 1

7 7 5 4 1 1 58 8 7 7 4 1 6

7 5 0 0 7 | 1 | 3 represents 13%

b.

c. Looking at the stem-and-leaf plots, it is clearly evident that a much higher percentof students are proficient in mathematics than in science. The mean and median arealso much higher in mathematics. Even though there is a larger range, or spread,in mathematics, it is across higher percents. The statistics shown would allow oneto conclude that students seem to know mathematics better than science.

Math ScienceMean 60.2 26.1

Median 61 27

Range 35 2610 20 30 40 50 60 70 8015 25 35 45 55 65 75

Math

Science

Page 158: course i

�Solutions to Practice Sets for Standardized TestsPractice Set 1, pp. 98–100 Practice Set 2, pp. 101–103

1. (a) 1. (e)

2. (b) 2. (c)

3. (d) 3. (b)

4. (c) 4. (e)

5. (b) 5. (d)

6. (b) 6. (c)

7. (d) 7. (b)

8. (a) 8. (b)

9. (c) 9. (b)

10. (e) 10. (b)

Practice Set 3, pp. 104–105 Practice Set 4, pp. 106–107

1. (d) 1. (b)

2. (c) 2. (c)

3. (a) 3. (d)

4. (b) 4. (a)

5. (a) 5. (e)

6. (d) 6. (c)

7. (e) 7. (e)

8. (e) 8. (c)

9. (a) 9. (e)

10. (c) 10. (e)

156 S O L U T I O N S

Page 159: course i

S O L U T I O N S 157

Practice Set 5, pp. 108–110 Practice Set 6, pp. 111–113

1. (b) 1. (c)

2. (c) 2. (a)

3. (a) 3. (c)

4. (e) 4. (e)

5. (d) 5. (d)

6. (a) 6. (d)

7. (b) 7. (d)

8. (a) 8. (b)

9. (c) 9. (d)

10. (d) 10. (d)

Practice Set 7, pp. 114–116 Practice Set 8, pp. 117–118

1. (e) 1. (c)

2. (d) 2. (b)

3. (b) 3. (c)

4. (d) 4. (b)

5. (c) 5. (d)

6. (c) 6. (e)

7. (b) 7. (a)

8. (d) 8. (d)

9. (b) 9. (a)

10. (d) 10. (b)

Page 160: course i

Practice Set 9, pp. 119–121 Practice Set 10, pp. 122–123

1. (c) 1. (d)

2. (b) 2. (e)

3. (d) 3. (b)

4. (d) 4. (c)

5. (c) 5. (a)

6. (b) 6. (a)

7. (c) 7. (d)

8. (c) 8. (e)

9. (b) 9. (b)

10. (b) 10. (e)

158 S O L U T I O N S