Whole Numbers - Columbia Southern University · CHAPTER 1: Whole Numbers ... Copyright © 2008 by Pearson Education, Inc. EXAMPLE 7 American Red Cross. In 2003, ... and Canada Source:

11Whole Numbers

1.1 Standard Notation

1.2 Addition

1.3 Subtraction

1.4 Rounding and Estimating; Order

1.5 Multiplication and Area

1.6 Division

1.7 Solving Equations

1.8 Applications and Problem Solving

1.9 Exponential Notation and Order of Operations

Real-World ApplicationRaces in which runners climb the steps inside abuilding are called “run-up” races. There are 2058 steps in the International Towerthon, KualaLumpur, Malaysia. Write expanded notation for thenumber of steps.

This problem appearsas Exercise 13 inSection 1.1.

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Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.

2

CHAPTER 1: Whole Numbers

We study mathematics in order to be able to solve problems. In this section, we study how numbers are named. We begin with the concept ofplace value.

Place Value

Consider the numbers in the following table.

A digit is a number 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 that names a place-value lo-cation. For large numbers, digits are separated by commas into groups ofthree, called periods. Each period has a name: ones, thousands, millions, bil-lions, trillions, and so on. To understand the population of China in the tableabove, we can use a place-value chart, as shown below.

EXAMPLES What does the digit 8 mean in each number?

1. 278,342 8 thousands

2. 872,342 8 hundred thousands

3. 28,343,399,223 8 billions

4. 1,023,850 8 hundreds

5. 98,413,099 8 millions

6. 6328 8 ones

Do Margin Exercises 1–6.

Periods

PLACE-VALUE CHART

Hu

nd

red

s

Ten

s

On

es

Hu

nd

red

s

Ten

s

On

es

Hu

nd

red

s

Ten

s

On

es

Hu

nd

red

s

Ten

s

On

es

Hu

nd

red

s

Ten

s

On

es

Trillions Billions Millions Thousands Ones

1 3 0 6 3 1 3 8 1 2

1 billion 306 millions 313 thousands 812 ones

Three Most Populous Countries in the World

Source: The World Factbook, July 2005 estimates

COUNTRY

China

POPULATION 2005

IndiaUnited States

1,306,313,8121,080,264,388

295,734,134

1.11.1 STANDARD NOTATION

What does the digit 2 mean in eachnumber?

1. 526,555

2. 265,789

3. 42,789,654

4. 24,789,654

5. 8924

6. 5,643,201

Answers on page A-1

ObjectivesGive the meaning of digits instandard notation.

Convert from standardnotation to expandednotation.

Convert between standardnotation and word names.

To the student:

In the Preface, at the front ofthe text, you will find a StudentOrganizer card. This pulloutcard will help you keep track ofimportant dates and usefulcontact information. You canalso use it to plan time for class,study, work, and relaxation. Bymanaging your time wisely, youwill provide yourself the bestpossible opportunity to besuccessful in this course.

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EXAMPLE 7 American Red Cross. In 2003, private donations to the Ameri-can Red Cross totaled about $587,492,000. What does each digit name?Source: The Chronicle of Philanthropy

Do Exercise 7.

Converting from Standard Notation to Expanded Notation

To answer questions such as “How many?”, “How much?”, and “How tall?”, weuse whole numbers. The set, or collection, of whole numbers is

The set goes on indefinitely. There is no largest whole number, and the small-est whole number is 0. Each whole number can be named using various no-tations. The set without 0, is called the set of natural numbers.

Let’s look at the data from the bar graph shown here.

The number of computer majors in 2003 was 17,706. Standard notationfor the number of computer majors is 17,706. We write expanded notationfor 17,706 as follows:

� 7 hundreds � 0 tens � 6 ones.

17,706 � 1 ten thousand � 7 thousands

Fewer Computer Majors

Year

23,416

23,090

23,033

17,706

2000

2001

2002

2003

2004

The number of computerscience and computer

engineering majors in thefall in the United States

and Canada

Source: Computing Research Association Taulbee Survey

15,950

1, 2, 3, 4, 5, . . . ,

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, . . . .

onestenshundredsthousandsten thousandshundred thousandsmillionsten millionshundred millions

5 8 7, 4 9 2, 0 0 0

7. Presidential Library. In thefirst year of operation, 280,219 people visited theRonald Reagan PresidentialLibrary in Simi Valley, California.What does each digit name?Sources: USA Today research by BruceRosenstein; National Archives & RecordsAdministration; Associated Press

Write expanded notation.

8. 1895

9. 23,416, the number of computermajors in 2000

10. 4218 mi (miles), the diameter of Mars

Answers on page A-1

�

� 5�

3


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EXAMPLE 8 Write expanded notation for 3031 mi (miles), the diameter ofMercury.

3031 3 thousands 0 hundreds 3 tens 1 one, or

3 thousands 3 tens 1 one

EXAMPLE 9 Write expanded notation for 563,384, the population ofWashington, D.C.

Do Exercises 8–12 (8–10 are on the preceding page).

Converting Between Standard Notationand Word Names

We often use word names for numbers. When we pronounce a number, weare speaking its word name. Russia won 92 medals in the 2004 SummerOlympics in Athens, Greece. A word name for 92 is “ninety-two.” Word namesfor some two-digit numbers like 27, 39, and 92 use hyphens. Others like 17 useonly one word, “seventeen.”

EXAMPLES Write a word name.

10. 35, the total number of gold medals won by the United States

Thirty-five

11. 17, the number of silver medals won by the People’s Republic of China

Seventeen

Do Exercises 13–15.

TOP COUNTRIESIN SUMMER

OLYMPICS 2004 TOTAL

35 39 29

27 27 38

32 17 14

17 16 16

14 16 18

103

92

63

49

48

Australia

People’s Republic of China

Germany

Russia

United States of America

BRONZEBRONZE

MEDAL COUNT

GOLDGOLD SILVERSILVER

Source: 2004 Olympics, Athens, Greece

� 3 thousands � 3 hundreds � 8 tens � 4 ones

563,384 � 5 hundred thousands � 6 ten thousands

��

��

11. 4180 mi, the length of the NileRiver, the longest river in theworld

12. 146,692, the number ofLabrador retrievers registered in 2004Source: The American Kennel Club

Write a word name. (Refer to thefigure below right.)

13. 49, the total number of medalswon by Australia

14. 16, the number of silver medalswon by Germany

15. 38, the number of bronzemedals won by Russia

Answers on page A-1

�

4


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For word names for larger numbers, we begin at the left with the largestperiod. The number named in the period is followed by the name of the period; then a comma is written and the next period is named.

EXAMPLE 12 Write a word name for 46,605,314,732.

Forty-six billion,

six hundred five million,

three hundred fourteen thousand,

seven hundred thirty-two

The word “and” should not appear in word names for whole numbers. Although we commonly hear such expressions as “two hundred and one,” the use of “and” is not, strictly speaking, correct in word names for wholenumbers. For decimal notation, it is appropriate to use “and” for the deci-mal point. For example, 317.4 is read as “three hundred seventeen andfour tenths.”


EXAMPLE 13 Write standard notation.

Five hundred six million,

three hundred forty-five thousand,

two hundred twelve

Standard notation is 506,345,212.

Do Exercise 20.

Write a word name.

16. 204

17. $51,206, the average salary in2004 for those who have abachelor’s degree or moreSource: U.S. Bureau of the Census

18. 1,879,204

19. 6,449,000,000, the worldpopulation in 2005Source: U.S. Bureau of the Census

20. Write standard notation.

Two hundred thirteen million, one hundred five thousand, three hundred twenty-nine

Answers on page A-1

Study Tips USING THIS TEXTBOOK

Throughout thistextbook, you will finda feature called “StudyTips.” One of the most

important ways inwhich to improve yourmath study skills is to

learn the proper use ofthe textbook. Here we

highlight a few pointsthat we consider

most helpful.

� Be sure to note the symbols , , , and so on, that correspond to the objectivesyou are to master in each section. The first time you see them is in the margin at thebeginning of the section; the second time is in the subheadings of each section; andthe third time is in the exercise set for the section. You will also find symbols like [1.1a]or [1.2c] next to the skill maintenance exercises in each exercise set and the reviewexercises at the end of the chapter, as well as in the answers to the chapter tests andthe cumulative reviews. These objective symbols allow you to refer to the appropriateplace in the text when you need to review a topic.

� Read and study each step of each example. The examples include important sidecomments that explain each step. These examples and annotations have beencarefully chosen so that you will be fully prepared to do the exercises.

� Stop and do the margin exercises as you study a section. This gives you immediatereinforcement of each concept as it is introduced and is one of the most effective waysto master the mathematical skills in this text. Don’t deprive yourself of this benefit!

� Note the icons listed at the top of each exercise set. These refer to the manydistinctive multimedia study aids that accompany the book.

cba

5


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6


What does the digit 5 mean in each case?

1.11.1 Student’sSolutionsManual

VideoLectures on CD Disc 1

Math TutorCenter

InterActMath

MyMathLabMathXLEXERCISE SET For Extra Help

1. 235,888

5 thousands

2. 253,777

5 ten thousands

3. 1,488,526

5 hundreds

4. 500,736

5 hundred thousands

Used Cars. 1,582,370 certified used cars were sold in 2004in the United States. Source: Motor Trend, April 2005, p. 26

In the number 1,582,370, what digit names the number of:

5. thousands? 2 6. ones? 0

7. millions? 1 8. hundred thousands?

5


13. 2058 steps in the International Towerthon, KualaLumpur, Malaysia

2 thousands 0 hundreds 5 tens 8 ones, or 2 thousands 5 tens 8 ones

14. 1776 steps in the CN Tower Run-Up, Toronto, Ontario,Canada

1 thousand 7 hundreds 7 tens 6 ones

15. 1268 steps in the World Financial Center, New York City,New York

1 thousand 2 hundreds 6 tens 8 ones

16. 1081 steps in the Skytower Run-Up, Auckland, New Zealand

1 thousand 0 hundreds 8 tens 1 one, or 1 thousand 8 tens 1 one��

��

��

��

��

InternationalTowerthon,

Kuala Lumpur,Malaysia

CN TowerRun-Up,Toronto

WorldFinancial

Center,New York

SkytowerRun-Up,

Auckland,New Zealand

2058

1776

12681081

Source: New York Road Runners Club

Step-Climbing Races

Step-Climbing Races. Races in which runners climb the steps inside a building are called “run-up” races. The graph belowshows the number of steps in four buildings. In Exercises 13–16, write expanded notation for the number of steps in each race.

9. 5702 10. 3097 11. 93,986 12. 38,453

9. 5 thousands 7 hundreds 0 tens 2 ones, or 5 thousands 7 hundreds 2 ones��

�� 11. 9 ten thousands 3 thousands 9 hundreds 8 tens6 ones

��

12. 3 ten thousands 8 thousands 4 hundreds 5 tens3 ones

��

10. 3 thousands 0 hundreds 9 tens 7 ones, or 3 thousands 9 tens 7 ones��

��

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Overseas Travelers. The chart below shows the residence and number of overseas travelers to the United States in 2004. InExercises 17–22, write expanded notation for the number of travelers from each country.

7

Exercise Set 1.1

17. 519,955 from Australia

18. 3,747,620 from Japan

19. 308,845 from India

20. 333,432 from Spain

21. 4,302,737 from the United Kingdom

22. 384,734 from Brazil

OVERSEAS TRAVELERS TO U.S., 2004

Source: U.S. Department of Commerce, ITA,Office of Travel and Tourism Industries

Australia 519,955

Brazil 384,734

United Kingdom 4,302,737

India 308,845

Japan 3,747,620

Spain 333,432

Write a word name.

23. 85 Eighty-five 24. 48 Forty-eight 25. 88,000

Eighty-eight thousand

26. 45,987

Forty-five thousand, nine hundred eighty-seven

Write standard notation.

31. Two million, two hundred thirty-three thousand, eight hundred twelve 2,233,812

32. Three hundred fifty-four thousand, seven hundred two

354,702

33. Eight billion 8,000,000,000 34. Seven hundred million 700,000,000

Write a word name for the number in each sentence.

35. Great Pyramid. The area of the base of the GreatPyramid in Egypt is 566,280 square feet.

Five hundred sixty-six thousand, two hundred eighty

36. Population of the United States. The population of the United States in April 2006 was estimated to be298,509,533.Source: U.S. Bureau of the Census

Two hundred ninety-eight million, five hundred ninethousand, five hundred thirty-three

17. 5 hundred thousands 1 tenthousand 9 thousands9 hundreds 5 tens 5 ones18. 3 millions 7 hundredthousands 4 ten thousands7 thousands 6 hundreds2 tens 0 ones, or 3 millions7 hundred thousands 4 tenthousands 7 thousands6 hundreds 2 tens�

��

��

��

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19. 3 hundred thousands 0 ten thousands 8 thousands 8 hundreds 4 tens 5 ones, or 3 hundred thousands 8 thousands8 hundreds 4 tens 5 ones 20. 3 hundred thousands 3 ten thousands 3 thousands 4 hundreds 3 tens 2 ones21. 4 millions 3 hundred thousands 0 ten thousands 2 thousands 7 hundreds 3 tens 7 ones, or 4 millions 3 hundred

thousands 2 thousands 7 hundreds 3 tens 7 ones 22. 3 hundred thousands8 ten thousands 4 thousands 7 hundreds 3 tens 4 ones��

��

��

27. 123,765 28. 111,013 29. 7,754,211,577 30. 43,550,651,808

28. One hundred eleven thousand, thirteen

27. One hundred twenty-three thousand, seven hundred sixty-five

30. Forty-three billion, five hundred fifty million, six hundredfifty-one thousand, eight hundred eight

29. Seven billion, seven hundred fifty-four million, two hundredeleven thousand, five hundred seventy-seven

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37. Busiest Airport. In 2004, the world’s busiest airport,Hartsfield in Atlanta, had 83,578,906 passengers.Source: Airports Council International World Headquarters, Geneva, Switzerland

Eighty-three million, five hundred seventy-eight thousand,nine hundred six

38. Prisoners. There were 2,131,180 total prisoners,federal, state, and local, in the United States in 2004.Source: Prison and Jail Inmates at Midyear 2004, U.S. Bureau ofJustice Statistics

Two million, one hundred thirty-one thousand, one hundred eighty

Write standard notation for the number in each sentence.

39. Light travels nine trillion, four hundred sixty billionkilometers in one year. 9,460,000,000,000

40. The distance from the sun to Pluto is three billion, six hundred sixty-four million miles. 3,664,000,000

41. Pacific Ocean. The area of the Pacific Ocean is sixty-four million, one hundred eighty-six thousand square miles.

64,186,000

42. Internet Users. In a recent year, there were fifty-fourmillion, five hundred thousand Internet users in China.Source: Computer Industry Almanac, Inc.

54,500,000

To the student and the instructor : The Discussion and Writing exercises, denoted by the symbol , are meant to be answeredwith one or more sentences. They can be discussed and answered collaboratively by the entire class or by small groups.Because of their open-ended nature, the answers to these exercises do not appear at the back of the book.

DW

43. Explain why we use commas when writing largenumbers.

44. Write an English sentence in which the number370,000,000 is used.DWDW

To the student and the instructor : The Synthesis exercises found at the end of every exercise set challenge students to combineconcepts or skills studied in that section or in preceding parts of the text. Exercises marked with a symbol are meant to besolved using a calculator.

45. How many whole numbers between 100 and 400contain the digit 2 in their standard notation?

138

46. What is the largest number that you can name onyour calculator? How many digits does that numberhave? How many periods?

All 9’s as digits. Answers may vary. For an 8-digit readout, itwould be 99,999,999. This number has three periods.

SYNTHESIS

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Write an addition sentence thatcorresponds to each situation.

1. John has 8 music CD-ROMs inhis backpack. Then he buys 2 educational CD-ROMs at thebookstore. How many CD-ROMsdoes John have in all?

2. Sue earns $20 in overtime payon Thursday and $13 on Friday.How much overtime pay doesshe earn altogether on the two days?

Write an addition sentence thatcorresponds to each situation.

3. A car is driven 100 mi fromAustin to Waco. It is then driven93 mi from Waco to Dallas. Howfar is it from Austin to Dallasalong the same route?

4. A coaxial cable 5 ft (feet) long isconnected to a cable 7 ft long.How long is the resulting cable?

Answers on page A-1

9

1.2 Addition

Addition and Related Sentences

Addition of whole numbers corresponds to combining or putting things together.

The addition that corresponds to the figure above is

3 4 7. This is read “3 plus 4 equals 7.”

Addend Addend Sum

We say that the sum of 3 and 4 is 7. The numbers added are called addends.

EXAMPLE 1 Write an addition sentence that corresponds to this situation.

Kelly has $3 and earns $10 more. How much money does she have?

An addition that corresponds is . This sentence is alsocalled an equation.

Do Exercises 1 and 2.

Addition also corresponds to combining distances or lengths.

EXAMPLE 2 Write an addition sentence that corresponds to this situation.

A car is driven 44 mi from San Francisco to San Jose. It is then driven 42 mi from San Jose to Oakland. How far is it from San Francisco to Oakland along the same route?


Berkeley

Pa c i f i cO c e a n San

FranciscoBay

San PabloBay

Oakland

San Jose

580

580

680

280

San Francisco

0 5 10

Miles

It is 44miles from

San Franciscoto San Jose.

It is 42 milesfrom San Jose

to Oakland.

PleasantonDaly City

San Gregorio

44 mi � 42 mi � 86 mi

$3 � $10 � $13

��

We combine two sets. This is the resulting set.

A set of 3 iPods A set of 4 iPods A set of 7 iPods

1.21.2 ADDITIONObjectivesWrite an addition sentencethat corresponds to asituation.

Add whole numbers.

Use addition in findingperimeter.

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Addition of Whole Numbers

To add whole numbers, we add the ones digits first, then the tens, then thehundreds, then the thousands, and so on.

EXAMPLE 3 Add:

Place values are lined up in columns.

Add ones. We get 13 ones, or 1 ten 3 ones. Write 3 in the ones column and 1 above the tens. This is called carrying, or regrouping.

Add tens. We get 17 tens, or 1 hundred 7 tens.Write 7 in the tens column and 1 above thehundreds.

Add hundreds. We get 18 hundreds, or 1 thousand 8 hundreds. Write 8 in the hundreds column and 1 above thethousands.

Add thousands. We get 11 thousands.

Addends

Sum

EXAMPLE 4 Add:

Add ones. We get 24, so we have 2 tens 4 ones. Write 4 in the ones column and 2 above the tens.

Add tens. We get 35 tens, so we have 30 tens5 tens. This is also 3 hundreds 5 tens. Write 5 in the tens column and 3 above the hundreds.

Add hundreds. We get 19 hundreds.

1 9 5 4

� 4 9 8 7 8 9 2 7 6 33

92

1

5 4

� 4 9 8 7 8 9 2 7 6 �

� 33

92

1

4

� 4 9 8 7 8 9 2 7 6

� 3 92

1

391 � 276 � 789 � 498.

1 1 8 7 3

� 4 9 9 5 61

81

71

8

1 1 8 7 3

� 4 9 9 5 61

81

71

8

8 7 3

� 4 9 9 5 � 61

81

71

8

7 3

� 4 9 9 5� 6 8

1

71

8

3

� 4 9 9 5� 6 8 7

1

8

6878 � 4995.

Add.

5.

6.

7.

8.

Answers on page A-1

� 8 9 4 1 9 8 7 8 6 7 2 3 1 9 3 2

� 6 3 7 8 9 8 0 4

6203 � 3542

� 5 4 9 7 7 9 6 8

10


We show you these stepsfor explanation. You needwrite only this.

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Do Exercises 5–8 on the preceding page.

How do we do an addition of three numbers, like ? We do so byadding 3 and 6, and then 2. We can show this with parentheses:

. Parentheses tell what to do first.

We could also add 2 and 3, and then 6:

.

Either way we get 11. It does not matter how we group the numbers. Thisillustrates the associative law of addition, .

EXAMPLE 5 Insert parentheses to illustrate the associative law of addition:

.

We group as follows:

.


We can also add whole numbers in any order. That is, . Thisillustrates the commutative law of addition, .

EXAMPLE 6 Complete the following to illustrate the commutative law ofaddition:

We reverse the appearance of the two addends:

.


THE ASSOCIATIVE LAW OF ADDITION

For any numbers a, b, and c,

.

THE COMMUTATIVE LAW OF ADDITION

For any numbers a and b,

.

Together the commutative and associative laws tell us that to add morethan two numbers, we can use any order and grouping we wish.

a � b � b � a

�a � b� � c � a � �b � c�

5 � 4 � 4 � 5

�5 � 4 �

a � b � b � a2 � 3 � 3 � 2

5 � �1 � 7� � �5 � 1� � 7

5 � �1 � 7� � 5 � 1 � 7

a � �b � c� � �a � b� � c

�2 � 3� � 6 � 5 � 6 � 11

2 � �3 � 6� � 2 � 9 � 11

2 � 3 � 6

Insert parentheses to illustrate theassociative law of addition.

9.

10.

Complete the following to illustratethe commutative law of addition.

11.

12.

Answers on page A-1

�7 � 1 �

�2 � 6 �

�5 � 1� � 4 � 5 � 1 � 4

2 � �6 � 3� � 2 � 6 � 3

11

1.2 Addition

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EXAMPLE 7 Add from the top.

We first add 8 and 9, getting 17; then 17 and 7, getting 24; then 24 and 6,getting 30.

177 246 6 30

EXAMPLE 8 Add from the bottom.

8 8 309 22

13


Finding Perimeter

Addition can be used when finding perimeter.

PERIMETER

The distance around an object is its perimeter.

EXAMPLE 9 Find the perimeter of the octagonal (eight-sided) resortswimming pool.

Perimeter 13 yd 6 yd 6 yd 12 yd 8 yd

12 yd 13 yd 16 yd

The perimeter of the pool is 86 yd.

��

��

6 yd

6 yd13 yd

16 yd

13 yd

12 yd

12 yd

8 yd

30

� 6 7 9 8

30

� 6 7 9 8

� 6 7 9 8

Add from the top.

13.

14.

15. Add from the bottom.

Answers on page A-1

� 5 4 9 9

� 4 7 9 6 8

� 5 4 9 9

Try to writeonly this.

You still write the answer here.

Study Tips

USE OF COLOR

As you study the step-by-stepsolutions of the examples,note that color is used toindicate substitutions and tocall attention to the new stepsin multistep examples.Recognizing what has beendone in each step will helpyou to understand thesolutions.

12


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EXAMPLE 10 Find the perimeter of the soccer field shown.

The letter m denotes meters (a meter is slightly more than 3 ft). Note that forany rectangle, the opposite sides are the same length. Thus

Re-ordering theaddition

100 m 180 m 280 m

The perimeter is 280 m.


��

50 m � 50 m � 90 m � 90 m � Perimeter

50 m � 90 m � 50 m � 90 m � Perimeter

90 m

50 m

Find the perimeter of each figure.

16.

17.

Solve.

18. Index Cards. Two standardsizes for index cards are 3 in. (inches) by 5 in. and 5 in.by 8 in. Find the perimeter ofeach card.

Answers on page A-1

3 in.

5 in.

5 in.

8 in.

16 ft

16 ft

15 ft15 ft

9 in.

5 in.

4 in.

5 in.

6 in.

13

1.2 Addition

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

CALCULATOR CORNER

Adding Whole Numbers To the student and the instructor: This is thefirst of a series of optional discussions on using a calculator. A calculatoris not a requirement for this textbook. There are many kinds of calculatorsand different instructions for their usage. We have included instructionshere for a minimum-cost calculator. Be sure to consult your user’s manualas well. Also, check with your instructor about whether you are allowed touse a calculator in the course.

To add whole numbers on a calculator, we use the and keys.For example, to add 57 and 34, we press . Thecalculator displays , so To find we press . The display reads

, so

Exercises: Use a calculator to find each sum.

314 � 259 � 478 � 1051.1051

�874�952�413

314 � 259 � 478,57 � 34 � 91.91

�43�75

��

1. 55 2. 12173 � 4819 � 36

3. 1602 4. 734276 � 458925 � 677

5.

1932

6.

864 � 1 2 1 4 9 0 2 5 3

� 6 9 1 4 1 5 8 2 6

To the instructor and thestudent: This section presenteda review of addition of wholenumbers. Students who aresuccessful should go on toSection 1.3. Those who havetrouble should studydevelopmental unit A near theback of this text and thenrepeat Section 1.2.

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14


Write an addition sentence that corresponds to each situation.



Math TutorCenter

InterActMath


1. Two trucks haul sand to a construction site to use in adriveway. One carries 6 cu yd (cubic yards) and theother 8 cu yd. Altogether, how many cubic yards of sandare they hauling to the site?

6 cu yd 8 cu yd 14 cu yd

2. At a construction site, there are two gasoline containersto be used by earth-moving vehicles. One contains 400 gal (gallons) and the other 200 gal. How manygallons do both contain together?

400 gal 200 gal 600 gal��

3. A builder buys two parcels of land to build a housingdevelopment. One contains 500 acres and the other 300 acres. What is the total number of acres purchased?

500 acres 300 acres 800 acres

4. During March and April, Deron earns extra moneydoing income taxes part time. In March he earned $220, and in April he earned $340. How much extra did he earn altogether in March and April?

$220 � $340 � $560��

Add.

5.

387

6.

1869

7.

5198

8.

10,186

� 2 6 8 3 7 5 0 3

� 3 4 8 2 1 7 1 6

� 3 4 8 1 5 2 1

� 2 3 3 6 4

9.

164

10.

142

11.

100

12.

1010

� 1 1 9 9 9

� 1 9 9

� 6 9 7 3

� 7 8 8 6

13. 8503 14. 3609 15. 5266 16. 34,982280 � 34,702356 � 4910271 � 33388113 � 390

17. 4466 18. 28,84710,120 � 12,989 � 57383870 � 92 � 7 � 497

19.

6608

20.

6354

21.

34,432

22.

67,665

� 2 1, 7 8 6 4 5, 8 7 9

� 1 0, 9 8 9 2 3, 4 4 3

� 2 7 0 0 3 6 5 4

� 1 7 8 3 4 8 2 5

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15

Exercise Set 1.2

23.

101,310

24.

100,111

25.

230

26.

187

� 7 6 1 4 3 2 2 7 3 8

� 8 0 4 4 3 6 2 5 4 5

� 1 1 2 9 9, 9 9 9

� 2 3, 7 6 7 7 7, 5 4 3

27.

18,424

28.

162,340

29.

31,685

30.

4012

� 7 0 3 9 2 0 8 3 4 5 6 6 9 8 9

� 7 9 3 1 8 9 8 6 9 2 0 4

7 2 9 4 8 3 5

� 3 6, 7 1 2 8 3, 1 4 1 4 2, 4 8 7

� 3, 4 0 0 2 , 9 5 4

1 2, 0 7 0

Insert parentheses to illustrate the associative law of addition.

31. 32.

�7 � 1� � 5 � 7 � �1 � 5�

�7 � 1� � 5 � 7 � 1 � 5

�2 � 5� � 4 � 2 � �5 � 4�

�2 � 5� � 4 � 2 � 5 � 4

33. 34.

5 � �1 � 4� � �5 � 1� � 4

5 � �1 � 4� � 5 � 1 � 4

6 � �3 � 2� � �6 � 3� � 2

6 � �3 � 2� � 6 � 3 � 2

35. 36. 37.

6 � 1 � 1 � 6

�6 � 1 �

5 � 2 � 2 � 5

�5 � 2 �

2 � 7 � 7 � 2

�2 � 7 �

38. 39. 40.

7 � 5 � 5 � 7

�7 � 5 �

2 � 9 � 9 � 2

�2 � 9 �

1 � 9 � 9 � 1

�1 � 9 �

Add from the top. Then check by adding from the bottom.

41.

28

42.

25

43.

26

44.

35

� 7 8 7 4 9

� 7 3 2 6 8

� 8 1 9 3 4

� 8 4 9 7

Complete each equation to illustrate the commutative law of addition.

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Find the perimeter of each figure.

16


45. 114 mi 46. 229 yd

39 yd

54 yd

62 yd

28 yd

46 yd

14 mi13 mi

8 mi

10 mi

47 mi

22 mi

49. Explain in your own words what the associativelaw of addition means.

50. Describe a situation that corresponds to thismathematical expression:

80 mi 245 mi 336 mi.��

DWDW

The exercises that follow begin an important feature called Skill Maintenance exercises. These exercises provide an ongoingreview of topics previously covered in the book. You will see them in virtually every exercise set. It has been found that this kindof continuing review can significantly improve your performance on a final examination.

51. What does the digit 8 mean in 486,205? [1.1a]

8 ten thousands

52. Write a word name for the number in the followingsentence: [1.1c]

In fiscal year 2004, Starbucks Corporation had total netrevenues of $5,294,247,000.

Source: Starbucks Corporation

Five billion, two hundred ninety-four million, two hundredforty-seven thousand

57. A fast way to add all the numbers from 1 to 10 inclusive is to pair 1 with 9, 2 with 8, and so on. Use a similar approach toadd all numbers from 1 to 100 inclusive.

Then and 4900 � 50 � 100 � 5050.49 � 100 � 490049 � 51 � 100.2 � 98 � 100, . . . ,1 � 99 � 100,

47. Find the perimeter of a standard hockey rink.

570 ft

48. In Major League Baseball, how far does a batter travel incircling the bases when a home run has been hit?

360 ft

90 ft

90 ft

200 ft

85 ft

SKILL MAINTENANCE

SYNTHESIS

53. Is it possible for a narrower soccer field to have agreater perimeter than the one in Example 10? Why orwhy not?

54. Is it possible for a rectangle to have a perimeter of12 cu yd? Why or why not?DWDW

Add.

55. 56,055,667 56. 49,187,23239,487,981 � 8,709,486 � 989,7655,987,943 � 328,959 � 49,738,765

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Subtraction and Related Sentences

TAKE AWAYSubtraction of whole numbers applies to two kinds of situations. The first iscalled “take away.” Consider the following example.

A bowler starts with 10 pins and knocks down 8 of them.

From 10 pins, the bowler “takes away” 8 pins. There are 2 pins left. The sub-traction is

We use the following terminology with subtraction:

10 8 2 .

Minuend Subtrahend Difference

The minuend is the number from which another number is being subtracted.The subtrahend is the number being subtracted. The difference is the resultof subtracting the subtrahend from the minuend.

EXAMPLES Write a subtraction sentence that corresponds to each situation.

1. Juan goes to a music store and chooses 10 CDs to take to the listeningstation. He rejects 7 of them, but buys the rest. How many CDs doesJuan buy?

There are 10 CDsto begin with.

He rejects 7 of them. He buys the remaining 3.

10 7 3� �

CLASSIC

MASTERS

GUITARGREATS

GUITARGREATS

MELLOWTUNES

��

8

10 10 � 8 � 2

10 � 8 � 2.

1.31.3 SUBTRACTIONObjectivesWrite a subtraction sentence that corresponds to a situation involving “take away.”

Given a subtraction sentence, write a relatedaddition sentence; and givenan addition sentence, writetwo related subtractionsentences.

Write a subtraction sentencethat corresponds to asituation of “How much do I need?”

Subtract whole numbers.

Study Tips

HIGHLIGHTING

� Highlight importantpoints. You are probablyused to highlighting keypoints as you study. If thatworks for you, continue todo so. But you will noticemany design featuresthroughout this book thatalready highlight impor-tant points. Thus you maynot need to highlight asmuch as you generally do.

� Highlight points that youdo not understand. Usea unique mark to indicatetrouble spots that canlead to questions to beasked during class, in atutoring session, or whencalling or contacting theAW Math Tutor Center.

17

1.3 Subtraction

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Write a subtraction sentence thatcorresponds to each situation.

1. A contractor removes 5 cu yd ofsand from a pile containing67 cu yd. How many cubic yardsof sand are left in the pile?

2. Sparks Electronics owns a fieldnext door that has an area of20,000 sq ft (square feet).Deciding they need more roomfor parking, the owners have12,000 sq ft paved. How manysquare feet of field are leftunpaved?

Answers on page A-1

18


2. Kaitlin has $300 and spends $85 for office supplies. How much moneyis left?


Related Sentences

Subtraction is defined in terms of addition. For example, is that numberwhich when added to 4 gives 7. Thus for the subtraction sentence

Taking away 4 from 7 gives 3.

there is a related addition sentence

Putting back the 4 gives 7 again.

This can be illustrated using a number line. Both addition and subtrac-tion correspond to moving distances on a number line. The number linesbelow are marked with tick marks at equal distances of 1 unit. On the left, thesum is shown. We start at 3 and move 4 units to the right, to end up at7. The addition that corresponds to the situation is

On the right above, the difference is shown. We start at 7 and move4 units to the left, to end up at 3. The subtraction that corresponds to the situ-ation is

This leads us to the following definition of subtraction.

SUBTRACTION

The difference is that unique whole number c for which

For example, is the number 9 since We know that answers we find to subtractions are correct only because of

the related addition, which provides a handy way to check a subtraction.

13 � 9 � 4.13 � 4

a � c � b.a � b

7 � 4 � 3.

7 � 4

0 1 2 3 4 5 6 7 8 9

Move 4 unitsto the left. Start at 7.

0 1 2 3 4 5 6 7 8 9

Move 4 unitsto the right.Start at 3.

3 � 4 � 7.3 � 4

7 � 3 � 4.

7 � 4 � 3

7 � 4

Amount to begin with

$300

Amount spentfor office supplies

$85

Amount left

$215� �

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Write a related addition sentence.

3.

4.

Write two related subtractionsentences.

5.

6.

Answers on page A-1

11 � 3 � 14

5 � 8 � 13

17 � 8 � 9

7 � 5 � 2

19

1.3 Subtraction

EXAMPLE 3 Write a related addition sentence:

This number gets added (to 3).

The related addition sentence is


EXAMPLE 4 Write two related subtraction sentences:

This addend gets subtracted from the sum.

(7 take away 3 is 4.)

4 � 7 � 3

4 � 3 � 7

4 � 3 � 7.

8 � 3 � 5.

8 � 3 � 5

8 � 5 � 3

8 � 5 � 3.

This addend gets subtracted from the sum.

(7 take away 4 is 3.)

3 � 7 � 4

4 � 3 � 7

The related subtraction sentences are and


How Much Do I Need?

The second kind of situation to which subtraction can apply is called “howmany more.” You have 2 notebooks, but you need 7. You can think of this as“how many do I need to add to 2 to get 7?” Finding the answer can be thoughtof as finding a missing addend, and can be found by subtracting 2 from 7.

What must be added to 2 to get 7? The answer is 5.

Missing addend Difference

EXAMPLES Write a subtraction sentence that corresponds to each situation.

5. Jillian wants to buy the roll-on luggage shown in this ad. She has $30. Sheneeds $79. How much more does she need in order to buy the luggage?

7 � 2 �� 72 �

Need 7 notebooks

Have 2notebooks

5 notebooks

3 � 7 � 4.4 � 7 � 3

By the commutative law ofaddition, there is also anotheraddition sentence:

8 � 5 � 3.

⎫⎪⎪⎬⎪⎪⎭ ⎫⎪⎬⎪⎭

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Write an addition sentence and arelated subtraction sentencecorresponding to each situation.

7. It is 348 mi from Miami toJacksonville. Alice hascompleted 200 miles of thedrive. How much farther doesshe need to travel?

8. Cedric estimates that it will take1200 bricks to complete the sideof a building, but he has only800 bricks. How many morebricks will be needed?

9. Subtract.

Answers on pages A-1–A-2

� 4 0 9 2 7 8 9 3

400

20


CALCULATORCORNER

Subtracting WholeNumbers To subtract whole numbers on a calculator, we use the and keys. For example, to find we press . The calculator displays ,so We can check this result by adding thesubtrahend, 47, and the difference, 16. To do this, we press . The sum is the minuend, 63, so the subtraction is correct.

Exercises: Use a calculatorto perform each subtraction.Check by adding.

1. 28

2. 47

3. 67

4. 119

5.2128

6.2593 � 9 8 1 3

1 2 , 4 0 6

� 2 8 4 8 4 9 7 6

612 � 493

145 � 78

81 � 34

57 � 29

�74�61

63 � 47 � 16.16

�74�36

63 � 47,��

Thinking of this situation in terms of a missing addend, we have:

To find the answer, we think of the related subtraction sentence:

or

6. Cathy is reading Ishmael by Daniel Quinn as part of her philosophy class.It contains 263 pages, of which she has read 250. How many more pagesdoes she have left?

250 � � 263

Now we write a related subtraction sentence:

. 250 gets subtracted.


Subtraction of Whole Numbers

To subtract numbers, we subtract the ones digits first, then the tens digits,then the hundreds, then the thousands, and so on.

EXAMPLE 7 Subtract:

Subtract ones.

Subtract tens.

Subtract hundreds.

Subtract thousands.

Do Exercise 9.

5 4 4 8

� 4 3 2 0 9 7 6 8

5 4 4 8

� 4 3 2 0 9 7 6 8

4 4 8

� 4 3 2 0 9 7 6 8

4 8

� 4 3 2 0 9 7 6 8

8

� 4 3 2 0 9 7 6 8

9768 � 4320.

263 � 250 � 13

Total numberof pagesis

Pages to bereadplus

Pages alreadyread

$79 � $30 � $49.� 79 � 30,

� 7930 �

Plus

$30 $7930 � � 79

This is for explanation.

You should write only this.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

⎫⎪⎬⎪⎭

⎫⎪⎪⎬⎪⎪⎭ ⎫⎪⎬⎪⎭ ⎫⎪⎪⎬⎪⎪⎭

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Sometimes we need to borrow.

EXAMPLE 8 Subtract:

We cannot subtract 9 ones from 6 ones, but we cansubtract 9 ones from 16 ones. We borrow 1 ten to get 16 ones.

13

We cannot subtract 7 tens from 3 tens, but we cansubtract 7 tens from 13 tens. We borrow 1 hundred to get 13 tens.

11 13

We cannot subtract 8 hundreds from 1 hundred, but wecan subtract 8 hundreds from 11 hundreds. We borrow 1 thousand to get 11 hundreds.

We can always check the answer by adding it to the number being subtracted.

11 13

4 3 6 7

� 1 8 7 9 65

21

43

616

4 3 6 7

� 1 8 7 9 65

21

43

616

6 7

� 1 8 7 9 6 2

1 4

3 6

16

7

� 1 8 7 9 6 2 4

3

616

6246 � 1879.

21

1.3 Subtraction

Check:

6 2 4 6

� 1 8 7 9 41

31

61

7This iswhat youshouldwrite.

The answerchecks becausethis is the topnumber in thesubtraction.


EXAMPLE 9 Subtract:

We cannot subtract 7 ones from 2 ones. We have 9 hundreds, or 90 tens. We borrow 1 ten to get 12 ones. We then have 89 tens.


EXAMPLE 10 Subtract:

We have 8 thousands, or 800 tens. We borrow 1 ten to get 13 ones. We then have 799 tens.

EXAMPLES

4 3 3 6

� 3 6 6 7 8 0 07 9 9

313

8003 � 3667.

4 2 5

� 4 7 7 9 08 9

212

902 � 477.

11. Subtract: 12. Subtract:11

3 0 5 6

� 2 9 6 8 6 05 9

21

414

6024 � 2968.

2 2 3 8

� 3 7 6 2 6 0 05 9 9

010

6000 � 3762.


Subtract. Check by adding.

10. 11. � 2 3 9 8

7 1 4 5 � 2 3 5 8

8 6 8 6

Subtract.

12. 13. � 2 9 8

5 0 3 � 1 4

7 0

16. � 7 4 8 9

9 0 3 5

Answers on page A-2

Subtract.

14. 15. � 3 1 4 9

6 0 0 0 � 6 3 4 9

7 0 0 7

To the instructor and thestudent: This section presenteda review of subtraction of wholenumbers. Students who aresuccessful should go on toSection 1.4. Those who havetrouble should studydevelopmental unit S near theback of this text and thenrepeat Section 1.3.

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22


Write a subtraction sentence that corresponds to each situation.



Math TutorCenter

InterActMath


1. Lauren arrives at the Evergreen State Fair with 20 ridetickets and uses 4 to ride the Upside Down Ride. Howmany tickets remain?

2. A host pours 5 oz (ounces) of salsa from a jar containing16 oz. How many ounces are left? 11 oz16 oz � 5 oz �

20 � 4 � 16

3. Frozen Yogurt. A dispenser at a frozen yogurt storecontains 126 oz of strawberry yogurt. A 13-oz cup is soldto a customer. How much is left in the dispenser?

4. Chocolate Cake. One slice of chocolate cake with fudgefrosting contains 564 cal (calories). One cup of hotcocoa made with skim milk contains 188 calories. Howmany more calories are in the cake than in the cocoa?

376 cal564 cal � 188 cal �126 oz � 13 oz � 113 oz

Write a related addition sentence.

5.

or

6.

or

7.

or

8.

or 9 � 9 � 09 � 0 � 9,

9 � 9 � 0

13 � 8 � 513 � 5 � 8,

13 � 8 � 5

12 � 5 � 712 � 7 � 5,

12 � 5 � 7

7 � 4 � 37 � 3 � 4,

7 � 4 � 3

9.

or

10.

or

11.

or

12.

or 51 � 18 � 3351 � 33 � 18,

51 � 18 � 33

43 � 16 � 2743 � 27 � 16,

43 � 16 � 27

20 � 8 � 1220 � 12 � 8,

20 � 8 � 12

23 � 9 � 1423 � 14 � 9,

23 � 9 � 14

Write two related subtraction sentences.

13. 14. 15. 16.

0 � 8 � 88 � 8 � 0;

8 � 0 � 8

7 � 15 � 88 � 15 � 7;

8 � 7 � 15

9 � 16 � 77 � 16 � 9;

7 � 9 � 16

9 � 15 � 66 � 15 � 9;

6 � 9 � 15

17. 18. 19. 20.

10 � 52 � 4242 � 52 � 10;

42 � 10 � 52

9 � 32 � 2323 � 32 � 9;

23 � 9 � 32

8 � 19 � 1111 � 19 � 8;

11 � 8 � 19

6 � 23 � 1717 � 23 � 6;

17 � 6 � 23

Write an addition sentence and a related subtraction sentence corresponding to each situation.

21. Kangaroos. There are 32 million kangaroos in Australiaand 17 million people. How many more kangaroos arethere than people?

22. Interstate Speeds. Speed limits on interstate highwaysin many western states were raised from 65 mph (milesper hour) to 75 mph. By how many miles per hour werethey raised?

75 � 65 � 10� 75;65 �

32 � 17 � 15� 32;17 � ISB

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23

Exercise Set 1.3

41.

56

42.

12

43.

454

44.

385

� 4 1 8 8 0 3

� 2 3 6 6 9 0

� 7 8 9 0

� 2 4 8 0

45.

3749

46.

6150

47.

2191

48.

4418

� 1 5 8 9 6 0 0 7

� 1 0 9 2 3 0 0

� 2 5 8 6 4 0 8

� 3 0 5 9 6 8 0 8

Subtract.

25.

44

26.

53

27.

533

28.

203

� 3 2 3 5 2 6

� 3 3 3 8 6 6

� 3 4 8 7

� 2 1 6 5

29. 39 30. 45 31. 234 32. 189887 � 698981 � 74773 � 2886 � 47

33.

5382

34.

3535

35.

3831

36.

7404

� 5 9 9 8 0 0 3

� 3 8 0 9 7 6 4 0

� 2 8 9 6 6 4 3 1

� 2 3 8 7 7 7 6 9

37.

7748

38.

10,334

39.

43,028

40.

84,405

84,703 � 29890,237 � 47,209 � 5 , 8 8 8

1 6 , 2 2 2 � 4 , 8 9 9

1 2 , 6 4 7

23. A set of drapes requires 23 yd of material. The decoratorhas 10 yd of material in stock. How much more must beordered?

24. Marv needs to bowl a score of 223 in order to win histournament. His score with one frame to go is 195. Howmany pins does Marv need in the last frame to win thetournament? 223 � 195 � 28� 223;195 �

23 � 10 � 13� 23;10 �

49.

86

50.

282

51.

4813

52.

7019

� 1 0 7 3 8 0 9 2

� 3 0 2 7 7 8 4 0

� 1 8 8 4 7 0

� 7 4 1 6 0

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24


53.

5745

54.

9604

55.

95,974

56.

7208

15,017 � 7809101,734 � 576010,002 � 3985843 � 98

57.

9975

58.

12,134

59.

83,818

60.

292,174

311,568 � 19,39483,907 � 8921,043 � 890910,004 � 29

61.

4206

62.

1458

63.

10,305

64.

5445

� 1 1 , 5 9 8 1 7 , 0 4 3

� 3 7 , 6 9 5 4 8 , 0 0 0

� 6 5 4 3 8 0 0 1

� 2 7 9 4 7 0 0 0

65. Describe two situations that correspond to thesubtraction one “take away” and one“missing addend.”

66. Is subtraction commutative (is there acommutative law of subtraction)? Why or why not?DW

$20 � $17,DW

SKILL MAINTENANCE

SYNTHESIS

Add. [1.2b]

67.

1024

68.

12,732

69.

90,283

70.

29,364

� 6 8 5 1 7 7 2 0 6 7 8 9 8 0 0 4

� 3 2 , 4 0 6 5 7 , 8 7 7

� 3 6 5 4 9 0 7 8

� 7 8 9 4 6

71. 1345 72. 924 73. 22,692 74. 10,9209909 � 101112,885 � 9807901 � 23567 � 778

75. Write a word name for 6,375,602. [1.1c]Six million, three hundred seventy-five thousand, six hundred two

76. What does the digit 7 mean in 6,375,602? [1.1a]

7 ten thousands

77. Describe the one situation in which subtraction iscommutative (see Exercise 66).

78. Explain what it means to “borrow” in subtraction.DWDW

Subtract.

79. 2,829,177 80. 11,706,36921,431,206 � 9,724,8373,928,124 � 1,098,947

81. Fill in the missing digits to make the subtraction true:

9,

3; 4

81 � 7,251,140.48,621 � 2,097,

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Round to the nearest ten.

1. 37

2. 52

3. 73

4. 98


5. 35

6. 75

7. 85

Answers on page A-2

4037

??

30

25


Rounding

We round numbers in various situations when we do not need an exact an-swer. For example, we might round to see if we are being charged the correctamount in a store. We might also round to check if an answer to a problem isreasonable or to check a calculation done by hand or on a calculator.

To understand how to round, we first look at some examples using num-ber lines, even though this is not the way we generally do rounding.

EXAMPLE 1 Round 47 to the nearest ten.

Here is a part of a number line; 47 is between 40 and 50. Since 47 is closerto 50, we round up to 50.


42 is between 40 and 50. Since 42 is closer to 40, we round down to 40.

Do Exercises 1–4.


45 is halfway between 40 and 50. We could round 45 down to 40 or up to50. We agree to round up to 50.

When a number is halfway between rounding numbers, round up.

Do Exercises 5–7.

Here is a rule for rounding.

ROUNDING WHOLE NUMBERS

To round to a certain place:

a) Locate the digit in that place.b) Consider the next digit to the right.c) If the digit to the right is 5 or higher, round up. If the digit to the

right is 4 or lower, round down.d) Change all digits to the right of the rounding location to zeros.

504540

42 504540

47 504540

1.41.4 ROUNDING AND ESTIMATING; ORDER

ObjectivesRound to the nearest ten,hundred, or thousand.

Estimate sums anddifferences by rounding.

Use � or � for to write atrue sentence in a situationlike 6 10.

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a) Locate the digit in the tens place, 8.

6 4 8 5

b) Consider the next digit to the right, 5.

6 4 8 5

c) Since that digit, 5, is 5 or higher, round 8 tens up to 9 tens.

d) Change all digits to the right of the tens digit to zeros.

6 4 9 0 This is the answer.

EXAMPLE 5 Round 6485 to the nearest hundred.

a) Locate the digit in the hundreds place, 4.

6 4 8 5


6 4 8 5

c) Since that digit, 8, is 5 or higher, round 4 hundreds up to 5 hundreds.

d) Change all digits to the right of hundreds to zeros.


EXAMPLE 6 Round 6485 to the nearest thousand.

a) Locate the digit in the thousands place, 6.

6 4 8 5


6 4 8 5

c) Since that digit, 4, is 4 or lower, round down, meaning that 6 thousandsstays as 6 thousands.

d) Change all digits to the right of thousands to zeros.



Caution!

7000 is not a correct answer to Example 6. It is incorrect to round from theones digit over, as follows:

6485, 6490, 6500, 7000.

Note that 6485 is closer to 6000 than it is to 7000.

We can use the symbol �, read “is approximately equal to,” to indicate thatwe have rounded 6485 to 6490. Thus, in Example 4, we can write

6485 � 6490.


8. 137

9. 473

10. 235

11. 285

Round to the nearest hundred.

12. 641

13. 759

14. 750

15. 9325

Round to the nearest thousand.

16. 7896

17. 8459

18. 19,343

19. 68,500

Answers on page A-2

Study Tips

USE ABBREVIATIONS

If you take notes and havetrouble keeping up with yourinstructor, use abbreviations tospeed up your work. Considerstandard abbreviations like“Ex” for “Example,” “�” for “isapproximately equal to,” “�”for “therefore,” and “⇒” for“implies.” Feel free to createyour own abbreviations as well.

26


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Sometimes rounding involves changing more than one digit in a number.

EXAMPLE 7 Round 78,595 to the nearest ten.

a) Locate the digit in the tens place, 9.


c) Since that digit, 5, is 5 or higher, round 9 tens to 10 tens. To carry this out,we think of 10 tens as 1 hundred 0 tens, and increase the hundreds digitby 1, to get 6 hundreds 0 tens. We then write 6 in the hundreds place and0 in the tens place.

d) Change the digit to the right of the tens digit to zero.

This is the answer.

Note that if we round this number to the nearest hundred, we get thesame answer.


Estimating

Estimating can be done in many ways. In general, an estimate made byrounding to the nearest ten is more accurate than one rounded to the nearesthundred, and an estimate rounded to the nearest hundred is more accuratethan one rounded to the nearest thousand, and so on.

In the following example, we see how estimation can be used in makinga purchase.

EXAMPLE 8 Estimating the Cost of an Automobile Purchase. Ethan andOlivia Benson are shopping for a new car. They are considering a Saturn ION.There are three basic models of this car, and each has options beyond the basicprice, as shown in the chart on the following page. Ethan and Olivia have al-lowed themselves a budget of $16,500. They look at the list of options and wantto make a quick estimate of the cost of model ION�2 with all the options.

Estimate by rounding to the nearest hundred the cost of the ION�2 with allthe options and decide whether it will fit into their budget.

7 8 , 6 0 0

�

�

7 8 , 5 9 5

7 8 , 5 9 5

20. Round 48,968 to the nearest ten,hundred, and thousand.

21. Round 269,582 to the nearestten, hundred, and thousand.

Refer to the chart on the next pageto answer Margin Exercises 22 and 23.

22. By eliminating options, find away that Ethan and Olivia canbuy the ION�2 and stay withintheir $16,500 budget.

23. Tara and Alex are shopping for anew car. They are considering aSaturn ION�3 and have alloweda budget of $19,000.

a) Estimate by first rounding tothe nearest hundred the costof an ION�3 with all theoptions.

b) Can they afford this car with abudget of $19,000?

Answers on page A-2

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24. Estimate the sum by firstrounding to the nearest ten.Show your work.

25. Estimate the sum by firstrounding to the nearesthundred. Show your work.

Answers on page A-2

� 1 6 8 2 3 8 6 8 5 6 5 0

� 6 6 3 5 2 3 7 4

28


First, we list the base price of the ION�2 and then the cost of each of theoptions. We then round each number to the nearest hundred and add.

� 8 2 5 2 2 0 2 5 0 7 2 5 3 9 5 4 0 0

1 4 , 9 4 5

Estimated answer 1 7 , 7 0 0

� 8 0 0 2 0 0 3 0 0 7 0 0 4 0 0 4 0 0

1 4 , 9 0 0

Air conditioning is standard on the ION�2, so we do not include that cost.The estimated cost is $17,700. Since Ethan and Olivia have allowed them-selves a budget of $16,500 for the car, they will need to forgo some options.

Do Exercises 22 and 23 on the preceding page.

EXAMPLE 9 Estimate this sum by first rounding to the nearest ten:

We round each number to the nearest ten. Then we add.

� 8 5 3 1 4 9 7 8

78 � 49 � 31 � 85.

Estimated answer 2 5 0

� 9 0 3 0 5 0 8 0


Base Price: $12,975 Base Price: $14,945 Base Price: $16,470

Each of these vehicles comes with several options. Note that some of the options are standard on certain models. Others are not available for all models.

Antilock Braking System with Traction Control: $400

Head Curtain Side Air Bags: $395

Power Sunroof (Not available for ION�1): $725

Rear Spoiler (Not available for ION�1): $250

Air Conditioning with Dust and Pollen Filtration (Standard on ION�2 and ION�3):

$960

CD/MP3 Player with AM/FM Stereo and 4 Coaxial Speakers (Standard on ION�3):

ION�1—$510ION�2—$220

Power Package: Power Windows, Power Exterior Mirrors, Remote Keyless Entry, and Cruise Control (Not available for ION�1 and Standard for ION�3):

$825

MODEL ION�1 SEDAN (4 DOOR) 2.2-LITER ENGINE, 4-SPEED AUTOMATIC TRANSMISSION

MODEL ION�2 SEDAN (4 DOOR) 2.2-LITER ENGINE, 5-SPEED

MANUAL TRANSMISSION

MODEL ION�3 SEDAN (4 DOOR) 2.2-LITER ENGINE, 5-SPEED

MANUAL TRANSMISSION

Source: Saturn

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26. Estimate the difference by firstrounding to the nearesthundred. Show your work.

27. Estimate the difference by firstrounding to the nearestthousand. Show your work.

Use � or � for to write a truesentence. Draw a number line ifnecessary.

28. 8 12

29. 12 8

30. 76 64

31. 64 76

32. 217 345

33. 345 217

Answers on page A-2

� 1 1 , 6 9 8 2 3 , 2 7 8

� 6 7 3 9 9 2 8 5

29


EXAMPLE 10 Estimate the difference by first rounding to the nearest thou-sand:

We have

� 2 8 4 9 9 3 2 4

9324 � 2849.

Estimated answer 6 0 0 0

� 3 0 0 0 9 0 0 0


Order

We know that 2 is not the same as 5, that is, 2 is not equal to 5. We express thisby the sentence We also know that 2 is less than 5. We symbolize thisby the expression We can see this order on the number line: 2 is to theleft of 5. The number 0 is the smallest whole number, so for any wholenumber a.

ORDER OF WHOLE NUMBERS

For any whole numbers a and b:

1. (read “a is less than b”) is true when a is to the left of b onthe number line.

2. (read “a is greater than b”) is true when a is to the right of bon the number line.

We call � and � inequality symbols.

EXAMPLE 11 Use � or � for to write a true sentence: 7 11.

Since 7 is to the left of 11 on the number line,

EXAMPLE 12 Use � or � for to write a true sentence: 92 87.

Since 92 is to the right of 87 on the number line, 92 � 87.

A sentence like is called an equation. It is a true equation. Theequation is a false equation. A sentence like is called aninequality. The sentence is a true inequality. The sentence is afalse inequality.


23 � 697 � 117 � 114 � 8 � 11

8 � 5 � 13

87 88 89 90 91 9286 93

7 � 11.

7 8 9 10 11 126 13

a � b

a � b

1 2 3 4 5 60 7

2 � 5

0 � a2 � 5.

2 � 5.

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Math TutorCenter

InterActMath


1. 48 50 2. 532 530 3. 467 470 4. 8945 8950

5. 731 730 6. 17 20 7. 895 900 8. 798 800

9. 146 100 10. 874 900 11. 957 1000 12. 650 700

13. 9079 9100 14. 4645 4600 15. 32,850 32,900 16. 198,402 198,400

Round to the nearest hundred.

17. 5876 6000 18. 4500 5000 19. 7500 8000 20. 2001 2000

Round to the nearest thousand.

21. 45,340 45,000 22. 735,562 736,000 23. 373,405 373,000 24. 6,713,855 6,714,000

25.

180

26.

300

27.

5720

28.

640

� 2 8 6 7 3

� 2 3 4 7 8 0 7 4

� 8 8 4 6 9 7 6 2

� 9 7 7 8

Estimate the sum or difference by first rounding to the nearest ten. Show your work.

29.

220; incorrect

30.

180

31.

890; incorrect

32.

2940; incorrect

3 9 4 7

� 9 3 8 7 9 4 3 7 4 8 3 6

9 3 2

� 1 1 1 8 1 7 8

6 2 2

1 7 7

� 6 0 5 5 2 1 4 1

3 4 3

� 5 6 2 5 7 7 4 5

Estimate the sum by first rounding to the nearest ten. Do any of the given sums seem to be incorrect when compared to theestimate? Which ones?

33.

16,500

34.

2400

35.

5200

36.

6600

� 2 7 8 79 4 3 8

� 1 7 4 8 6 8 5 2

� 4 0 2 9 3 8 4 7 2 5 6 8

� 9 2 4 7 7 3 4 8

Estimate the sum or difference by first rounding to the nearest hundred. Show your work.

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31

Exercise Set 1.4

Planning a Kitchen. Perfect Kitchens offers custom kitchen packages with three choices for each of four items: cabinets,countertops, appliances, and flooring. The chart below lists the price for each choice.

Perfect Kitchens lets you customize your kitchen with one choice from each of the following four features:

37. Estimate the cost of remodeling a kitchen with choices(a), (d), (g), and (j) by rounding to the nearest hundreddollars.

$11,200

38. Estimate the cost of a kitchen with choices (c), (f ), (i),and (l) by rounding to the nearest hundred dollars.

$20,500

39. Sara and Ben are planning to remodel their kitchen andhave a budget of $17,700. Estimate by rounding to thenearest hundred dollars the cost of their kitchenremodeling project if they choose options (b), (e), (i),and (k). Can they afford their choices?

$18,900; no

40. The Davidsons must make a final decision on thekitchen choices for their new home. The allottedkitchen budget is $16,000. Estimate by rounding to thenearest hundred dollars the kitchen cost if they chooseoptions (a), (f ), (h), and (l). Does their budget allotmentcover the cost?

$15,800; yes

41. Suppose you are planning a new kitchen and must staywithin a budget of $14,500. Decide on the options youwould like and estimate the cost by rounding to thenearest hundred dollars. Does your budget support your choices?

Answers will vary depending on the options chosen.

42. Suppose you are planning a new kitchen and must staywithin a budget of $18,500. Decide on the options youwould like and estimate the cost by rounding to thenearest hundred dollars. Does your budget support your choices?

Answers will vary depending on the options chosen.

(a) Oak $7450

(b) Cherry 8820

(c) Painted 9630

CABINETS TYPE PRICE

(d) Laminate $1595

(e) Solid surface 2870

(f ) Granite 3528

COUNTERTOPS TYPE PRICE

(g) Low $1540

(h) Medium 3575

(i) High 6245

APPLIANCES PRICE RANGE PRICE

(j) Vinyl $ 625

(k) Ceramic tile 985

(l) Hardwood 1160

FLOORING TYPE PRICE

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Estimate the sum by first rounding to the nearest hundred. Do any of the given sums seem to be incorrect when compared tothe estimate? Which ones?

43.

1600

44.

2800; incorrect

45.

1500

46.

2300

2 3 0 2

� 9 4 3 7 5 8 2 7 5 3 2 6

1 4 4 6

� 2 0 5 6 3

4 2 8 7 5 0

1 8 4 9

� 1 0 4 3 6 2 3 7 0 2 4 8 1

1 6 4 0

� 5 9 5 7 4 5

8 4 2 1 6

Estimate the sum or difference by first rounding to the nearest thousand. Show your work.

47.

31,000

48.

27,000

49.

69,000

50.

74,000

� 1 1 , 1 1 0 8 4 , 8 9 0

� 2 2 , 5 5 5 9 2 , 1 4 9

� 2 2 2 2 7 8 4 2 9 3 4 8 7 6 4 8

� 7 0 0 4 8 9 4 3 4 8 2 1 9 6 4 3

Use � or � for to write a true sentence. Draw a number line if necessary.

51. 0 17 � 52. 32 0 � 53. 34 12 � 54. 28 18 �

55. 1000 1001 � 56. 77 117 � 57. 133 132 � 58. 999 997 �

59. 460 17 � 60. 345 456 � 61. 37 11 � 62. 12 32 �

Daily Newspapers. The top three daily newspapers in the United States are USA Today, the Wall Street Journal, and the NewYork Times. The daily circulation of each as of September 30, 2005, is listed in the table below. Use this table when answeringExercises 63 and 64.

Source: Audit Bureau of Circulations

NEWSPAPER

USA TodayWall Street JournalNew York Times

DAILY CIRCULATION

2,296,3352,083,6601,126,190

63. Use an inequality to compare the daily circulation of theWall Street Journal and USA Today.

or

64. Use an inequality to compare the daily circulation ofUSA Today and the New York Times.

or 1,126,190 � 2,296,3352,296,335 � 1,126,190,2,296,335 � 2,083,6602,083,660 � 2,296,335, ISB

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Exercise Set 1.4

65. Pedestrian Fatalities. The annual number of pedes-trians killed when hit by a motor vehicle has declinedfrom 6482 in 1990 to 4641 in 2004. Use an inequality tocompare these annual totals of pedestrian fatalities.

or

66. Life Expectancy. The life expectancy of a female in2010 is predicted to be about 82 yr (years) and of a maleabout 76 yr. Use an inequality to compare these lifeexpectancies.

or 76 yr � 82 yr82 yr � 76 yr,

66 68 70Years

72 74 76 78 80 82 84

Life Expectancy

Year

2010

2010

Men

Women

2005

2005

Men

Women

2001

2001

Men

Women

74

80

75

81

76

82

Source: U.S. Bureau of the Census

4641 � 64826482 � 4641,

1990Year

2004

Pedestrian Fatalities on the DeclineN

um

ber

of f

atal

itie

s

5000

45000

5500

6000

65006482

4641

Source: National Center for Statistics and Analysis

67. Explain how estimating and rounding can beuseful when shopping for groceries.

68. When rounding 748 to the nearest hundred, astudent rounds to 750 and then to 800. What mistake ishe making?

DWDW


69. 7992 [1.1b] 70. 23,000,000 [1.1b]2 ten millions � 3 millions7 thousands � 9 hundreds � 9 tens � 2 ones

79.–82. Use a calculator to find the sums and differences in each of Exercises 47–50. Then compare your answers with those found using estimation. Even when using a calculator it is possible to make an error if you press the wrong buttons,so it is a good idea to check by estimating.

SKILL MAINTENANCE

SYNTHESIS

Add. [1.2b] Subtract. [1.3d]

73.

86,754

74.

13,589

� 4 5 8 7 9 0 0 2

� 1 8 , 9 6 5 6 7 , 7 8 9 75.

48,824

76.

4415

� 4 5 8 7 9 0 0 2

� 1 8 , 9 6 5 6 7 , 7 8 9

71. Write a word name for 246,605,004,032. [1.1c] 72. Write a word name for 1,005,100. [1.1c]

71. Two hundred forty-six billion, six hundred five million, four thousand, thirty-two72. One million, five thousand, one hundred

77. Consider two numbers a and b, with Is itpossible that when a and b are each rounded down, theresult of rounding a is greater than the result ofrounding b? Why or why not?

78. Why do you think we round up when a 5 is thenext digit to the right of the place in which we arerounding?

DWa � b.DW

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34


Multiplication of Whole Numbers

REPEATED ADDITIONThe multiplication corresponds to this repeated addition:

The numbers that we multiply are called factors. The result of the multi-plication is called a product.

3 � 5 � 15

Factor Factor Product

RECTANGULAR ARRAYSMultiplications can also be thought of as rectangular arrays. Each of the fol-lowing corresponds to the multiplication

When you write a multiplication sentence corresponding to a real-worldsituation, you should think of either a rectangular array or repeated addition.In some cases, it may help to think both ways.

We have used an “�” to denote multiplication. A dot “�” is also commonlyused. (Use of the dot is attributed to the German mathematician GottfriedWilhelm von Leibniz in 1698.) Parentheses are also used to denote multipli-cation. For example,

3 � 5 � 3 � 5 � �3� �5� � 3�5� � 15.

3 rows with 5 bills in each row;3 � 5 � 15

5 columns with 3 bills in each column;3 � 5 � 15

1

1 2 3 4 5

3

2

5

3

5

3

3 � 5.

We combine 3 setsof 5 dollar bills each.

The resulting set isa set of 15 dollar bills.

5

3 � 5 � 5 � 5 � 5 � 15

15

5

5

3 addends; each is 5

3 � 5

1.51.5 MULTIPLICATION AND AREAObjectivesMultiply whole numbers.

Estimate products byrounding.

Use multiplication in finding area.

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EXAMPLES Write a multiplication sentence that corresponds to each situation.

1. It is known that Americans drink 24 million gal of soft drinks per day( per day means each day). What quantity of soft drinks is consumedevery 5 days?

We draw a picture in which million gallons or we can simplyvisualize the situation. Repeated addition fits best in this case.

million gallons million gallons

2. One side of a building has 6 floors with 7 windows on each floor. How many windows are on that side of the building?

We have a rectangular array and can easily draw a sketch.

Do Exercises 1–3.

Repeated addition of 0 results in 0, so . When 1 is addedto itself a times, the result is a, so . We say that the number 1is the multiplicative identity.

EXAMPLE 3 Multiply: .

We have

Multiply the 7 ones by 2: .Multiply the 3 tens by 2: .

Add. 7 4

2 � 30 � 60 6 02 � 7 � 14 1 4

� 2 3 7

37 � 2

1 � a � a � 1 � a0 � a � a � 0 � 0

7 windows

6 floors

6 � 7 � 42

� 1205 � 24

24 milliongallons

24 milliongallons

24 milliongallons

24 milliongallons

24 milliongallons

� � � �

� 1

Write a multiplication sentence thatcorresponds to each situation.

1. Marv practices for the U.S. Openbowling tournament. He bowls 8 games each day for 7 days.How many games does he bowl altogether for practice?

2. A lab technician pours 75 mL (milliliters) of acid intoeach of 10 beakers. How muchacid is poured in all?

3. Checkerboard. A checkerboardconsists of 8 rows with 8 squaresin each row. How many squaresin all are there on acheckerboard?

Answers on page A-2

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We can simplify what we write by writing just one line for the products.

Multiply the ones by 2: 14 ones1 ten 4 ones. Write 4 in the ones column and 1 above the tens.

Multiply the 3 tens by 2 and add 1 ten:6 tens, 6 tens 1 ten 7 tens. Write 7 in the tenscolumn.

Try to write only this.

Do Exercises 4–7.

The fact that we can do this is based on a property called the distributive law.

THE DISTRIBUTIVE LAW


Applied to the example above, the distributive law gives us

.

EXAMPLE 4 Multiply:

Multiplying by 3

Multiplying by 40. (We write a 0 and then multiply 57 by 4.)

Adding to obtain the product


2 4 5 1

2 2 8 0 1 7 1

� 4 3 52

7

2 2 8 0

1 7 1

� 4 3 5

2

7

1 7 1

� 4 3 5

2

7

43 � 57.

2 � 37 � 2 � �30 � 7� � �2 � 30� � �2 � 7�

a � �b � c� � �a � b� � �a � c�.

7 4

� 2 31

7

7 4

� 2 ��2 � �3 tens� � 3

1

7

4

� 2 ��2 � �7 ones� � 3

1

7

Multiply.

4.

5.

6.

7.

Multiply.

8.

9.

Answers on page A-2

48 � 63

� 2 3 4 5

� 5 1 3 4 8

� 6 8 2 3

� 4 3 7

� 2 5 8

36


⎫⎪⎬⎪⎭

You may have learned that such a 0 does not have to be written. You may omit it if you wish. If you do omit it, remember, when multiplying by tens, to put the answer in the tens place.

2

2

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EXAMPLE 5 Multiply:

Multiplying 683 by 7



Adding


EXAMPLE 6 Multiply:

Note that

Multiplying by 6 onesMultiplying by 3 hundreds. (We write 00 and then multiply 274 by 3.)Adding


EXAMPLE 7 Multiply:

Note that

Multiplying by 6 tens. (We write 0 and then multiply 274 by 6.)

Multiplying by 3 hundreds. (We write 00 and then multiply 274 by 3.)

Adding


9 8 , 6 4 0

8 2 2 0 0 1 6 4 4 0

� 3 6 0 2 7 4

360 � 3 hundreds � 6 tens.

360 � 274.

8 3 , 8 4 4

8 2 2 0 0 1 6 4 4

� 3 0 6 2 7 4

306 � 3 hundreds � 6 ones.

306 � 274.

3 1 2 , 1 3 1

2 7 3 2 0 0 3 4 1 5 0

4 7 8 1

� 4 5 7 6

5

8

2

3

3 4 1 5 0 4 7 8 1

� 4 5 7 6

5

8

2

3

4 7 8 1

� 4 5 7 6

5

8

2

3

457 � 683. Multiply.

10. � 6 2

7 4 6

37


4 1

4 13 1

11. 245 � 837

Multiply.

12. � 3 0 6

4 7 2

13. 408 � 704

Multiply.

14. � 6 0 0 5

2 3 4 4

15. 16. � 7 4 0 0

2 3 4 4 � 8 3 0

4 7 2

17. 100 � 562

18. 1000 � 562

Answers on page A-2

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Check on your own that and that This illus-trates the commutative law of multiplication. It says that we can multiplytwo numbers in any order, and get the same answer.

EXAMPLE 8 Complete the following to illustrate the commutative law ofmultiplication:

.

We reverse the order of the multiplication:

.


To multiply three or more numbers, we group them so that we multiplytwo at a time. Consider and The parentheses tell what todo first:

We multiply 3 and 4, then 2.

We can also multiply 2 and 3, then 4:

Either way we get 24. It does not matter how we group the numbers. This il-lustrates that multiplication is associative:

EXAMPLE 9 Insert parentheses to illustrate the associative law ofmultiplication:

.

We regroup as follows:

.


THE COMMUTATIVE LAW OF MULTIPLICATION

For any numbers a and b,

.

THE ASSOCIATIVE LAWOF MULTIPLICATION


.

Caution!

Do not confuse the associative law with the distributive law. When youmultiply , do not use the 2 twice. When you multiply using the distributive law, use the 2 twice: .2 � 3 � 2 � 4

2 � �3 � 4�2 � �3 � 4�

a � �b � c� � �a � b� � c

a � b � b � a

6 � �2 � 5� � �6 � 2� � 5

6 � �2 � 5� � 6 � 2 � 5

a � �b � c� � �a � b� � c.

�2 � 3� � 4 � �6� � 4 � 24.

2 � �3 � 4� � 2 � �12� � 24.

�2 � 3� � 4.2 � �3 � 4�

6 � 9 � 9 � 6

�6 � 9 �

a � b � b � a,

37 � 17 � 629.17 � 37 � 629Complete the following to illustratethe commutative law ofmultiplication.

19.

20.

Insert parentheses to illustrate theassociative law of multiplication.

21.

22.

Multiply.

23.

24.

Answers on page A-2

5 � 1 � 3

5 � 2 � 4

�5 � 4� � 8 � 5 � 4 � 8

3 � �7 � 9� � 3 � 7 � 9

�2 � 6 �

�8 � 7 �

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25. Lawn Tractors. By rounding tothe nearest ten, estimate thecost to Leisure Lawn Care of 12 lawn tractors.

26. Estimate the product by firstrounding to the nearest ten andto the nearest hundred. Showyour work.

Answers on page A-2

� 2 4 5 8 3 7

39


Estimating Products by Rounding

EXAMPLE 10 Lawn Tractors. Leisure Lawn Care is buying new lawntractors that cost $1534 each. By rounding to the nearest ten, estimate thecost of purchasing 18 tractors.

Exact

2 7 , 6 1 2

� 1 8 1 5 3 4

Nearest ten

3 0 , 6 0 0

� 2 0 1 5 3 0

The lawn tractors will cost about $30,600.

Do Exercise 25.

EXAMPLE 11 Estimate the following product by first rounding to the near-est ten and to the nearest hundred:

Nearest ten

3 1 2 , 8 0 0 2 7 2 0 0 0

4 0 8 0 0

� 4 6 0 6 8 0

683 � 457.

Nearest hundred

3 5 0 , 0 0 0

� 5 0 0 7 0 0

Exact

3 1 2 , 1 3 1

2 7 3 2 0 0 3 4 1 5 0

4 7 8 1

� 4 5 7 6 8 3

Do Exercise 26.

Finding Area

The area of a rectangular region can be considered to be the number of squareunits needed to fill it. Here is a rectangle 4 cm (centimeters) long and 3 cmwide. It takes 12 sq cm (square centimeters) to fill it.

In this case, we have a rectangular array of 3 rows, each of which con-tains 4 squares. The number of square units is given by or 12. That is,

AREA

The area of a shape is a measure of its surface using square units.

A � l � w � 4 cm � 3 cm � 12 sq cm.4 � 3,

This is a squarecentimeter(a square unit).

1 cm

1 cm3 cm

4 cm

CALCULATORCORNER

Multiplying WholeNumbers To multiply wholenumbers on a calculator, weuse the and keys. Forexample, to find wepress .The calculator displays 611, so

Exercises: Use a calculatorto find each product.

1. 448

2. 21,970

3. 6380

4. 39,564

5.180,480

6.2,363,754 � 4 5 3

5 2 1 8

� 4 8 3 7 6 0

126�314�5 � 1276

845 � 26

56 � 8

13 � 47 � 611.

�74�31

13 � 47,��

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EXAMPLE 12 Professional Pool Table. The playing area of a standard pooltable has dimensions of 50 in. by 100 in. (There are rails 6 in. wide on the out-side that are not included in the playing area.) Find the playing area.

If we think of filling the rectangle with square inches, we have a rectan-gular array. The length and the width Thus the area A isgiven by the formula

Do Exercise 27.

A � l � w � 100 � 50 � 5000 sq in.

w � 50 in.l � 100 in.

50 in.

100 in.

27. Table Tennis. Find the area of a standard table tennis tablethat has dimensions of 9 ft by5 ft. 45 sq ft

Answer on page A-2

Study Tips LEARNING RESOURCES

� The Student’s Solutions Manual contains worked-out solutions to the odd-numbered exercises inthe exercise sets.

� An extensive set of videos supplements this text.These are available on CD-ROM by calling 1-800-282-0693.

� InterAct Math offers unlimited practice exercisesthat are correlated to the text. It is availableonline at www.interactmath.com.

� Tutorial software called MathXL Tutorials on CDis available with this text. If it is not available inthe campus learning center, you can order it bycalling 1-800-282-0693.

� The Addison-Wesley Math Tutor Center hasexperienced instructors available to help with the odd-numbered exercises. You can order thisservice by calling 1-800-824-7799.

� Extensive help is available online via MyMathLaband/or MathXL. Ask your instructor forinformation about these or visit MyMathLab.comand MathXL.com.

Are you aware of all the learning

resources that existfor this textbook?Many details are

given in the Preface.

40


To the instructor and thestudent: This section presenteda review of multiplication ofwhole numbers. Students whoare successful should go on toSection 1.6. Those who havetrouble should studydevelopmental unit M near theback of this text and thenrepeat Section 1.5.

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Write a multiplication sentence that corresponds to each situation.

41

Exercise Set 1.5

EXERCISE SET For Extra Help1.51.5 Student’sSolutionsManual


Math TutorCenter

InterActMath

MyMathLabMathXL

1. The Los Angeles Sunday Times crossword puzzle isarranged rectangularly with squares in 21 rows and 21 columns. How many squares does the puzzle have altogether?

2. Pixels. A digital phototgraph is stored using smallrectangular dots called pixels. How many pixels arethere in a digital photo that has 600 rows with 400 pixels in each row? 600 � 400 � 240,00021 � 21 � 441

3. A new soft drink beverage carton contains 8 cans, eachof which holds 12 oz. How many ounces are there in the carton?

4. There are 7 days in a week. How many days are there in18 weeks? 18 � 7 � 126

8 � 12 oz � 96 oz

5. Computer Printers. The HP Photosmart 8250 can print4800 1200 dots per square inch (dpi), optimized. Howmany dots can it print in one square inch?Source: www.hp.com

6. Computer Printers. The Epson Stylus Photo R1800 canprint 5760 1440 dots per square inch (dpi), optimized.How many dots can it print in one square inch?Source: www.epson.com

5760 � 1440 � 8,294,400

�

4800 � 1200 � 5,760,000

�

Multiply.

7.

870

8.

9600

9.

2,340,000

10.

56,000

� 7 0 8 0 0

� 1 0 0 0 2 3 4 0

� 9 6 1 0 0

� 1 0 8 7

11.

520

12.

348

13.

564

14.

684

� 9 7 6

� 6 9 4

� 4 8 7

� 8 6 5

15. 1527 16. 5642 17. 64,603 18. 31,4684�7867�7�9229�7 � 8063 � 509

19. 4770 20. 4680 21. 3995 22. 2958�34��87��47� �85�60�78�90�53�

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42


23.

46,080

24.

59,829

25.

14,652

26.

44,792

� 8 8 5 0 9

� 3 3 4 4 4

� 7 7 7 7 7

� 7 2 6 4 0

27.

207,672

28.

162,000

29.

798,408

30.

224,900

� 6 5 0 3 4 6

� 9 3 6 8 5 3

� 3 7 5 4 3 2

� 4 0 8 5 0 9

31.

20,723,872

32.

28,319,616

33.

362,128

34.

38,858,112

� 6 0 6 4 6 4 0 8

� 1 0 4 3 4 8 2

� 3 1 7 2 8 9 2 8

� 3 2 2 4 6 4 2 8

35.

20,064,048

36.

15,818,370

37.

25,236,000

38.

35,978,400

� 7 8 9 0 4 5 6 0

� 4 5 0 0 5 6 0 8

� 2 3 3 0 6 7 8 9

� 4 0 0 8 5 0 0 6

39.

302,220

40.

106,145

41.

49,101,136

42.

22,421,949

� 3 4 4 9 6 5 0 1

� 6 2 2 4 7 8 8 9

� 2 9 9 3 5 5

� 3 4 5 8 7 6

Estimate the product by first rounding to the nearest ten. Show your work.

43. 44. 45. 46.

60 � 50 � 3000

� 5 4 6 3

30 � 30 � 900

� 2 9 3 4

50 � 80 � 4000

� 7 8 5 1

50 � 70 � 3500

� 6 7 4 5

Estimate the product by first rounding to the nearest hundred. Show your work.

47. 48. 49. 50.

800 � 400 � 320,000

� 4 3 4 7 8 9

400 � 200 � 80,000

� 1 9 9 4 3 2

400 � 300 � 120,000

� 2 9 9 3 5 5

900 � 300 � 270,000

� 3 4 5 8 7 6

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Exercise Set 1.5

51. Toyota Sienna. A pharmaceutical company buys aToyota Sienna for each of its 112 sales representatives.Each car costs $27,896 plus an additional $540 per carin destination charges.

a) Estimate the total cost of the purchase by roundingthe cost of each car, the destination charge, and thenumber of sales representatives to the nearesthundred. $2,840,000

b) Estimate the total cost of the purchase by roundingthe cost of each car to the nearest thousand and thedestination charge and the number of reps to thenearest hundred. $2,850,000

Source: Toyota

52. A travel club of 176 people decides to fly from LosAngeles to Tokyo. The cost of a round-trip ticket is $643.

a) Estimate the total cost of the trip by rounding thecost of the airfare and the number of travelers to thenearest ten. $115,200

b) Estimate the total cost of the trip by rounding thecost of the airfare and the number of travelers to thenearest hundred. $120,000

Find the area of each region.

53. 529,984 sq mi 54. 8385 sq yd129 yd

65 yd

728 mi

728 mi

55. 18 sq ft 56. 49 sq mi 57. 121 sq yd

11 yd

11 yd7 mi

7 mi

6 ft

3 ft

58. 144 sq cm 59.

144 sq mm

60. 4693 sq mi

247 mi

19 mi

48 mm3 mm

9 cm

16 cm

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61. Find the area of the region formed by the base lines on aMajor League Baseball diamond.

8100 sq ft

62. Find the area of a standard-sized hockey rink.

17,000 sq ft

200 ft

85 ft90 ft

90 ft

63. Describe a situation that corresponds to eachmultiplication:

64. Explain the multiplication illustrated in thediagram below.

0 5 10

5 5 5

3 � 5

15 20

DW4 � $150; $4 � 150.

DW

Add. [1.2b]

65.

12,685

66.

10,834

67.

427,477

68.

111,110

� 2 2 , 3 3 3 8 8 , 7 7 7

� 8 6 , 6 7 9 3 4 0 , 7 9 8

� 6 7 6

8 7 6 9 8 7 6

� 2 1 1 0 5 6 6 7 4 9 0 8

Subtract. [1.3d]

73. Round 6,375,602 to the nearest thousand. [1.4a]

6,376,000

74. Round 6,375,602 to the nearest ten. [1.4a]

6,375,600

SKILL MAINTENANCE

SYNTHESIS

69.1241 � 3 6 6 7

4 9 0 8 70.8889 � 9 8 7

9 8 7 6 71.254,119 � 8 6 , 6 7 9

3 4 0 , 7 9 8 72.66,444 � 2 2 , 3 3 3

8 8 , 7 7 7

77. An 18-story office building is box-shaped. Each floor measures 172 ft by 84 ft with a 20-ft by 35-ft rectangular area lost toan elevator, lobby, and stairwell. How much area is available as office space? 247,464 sq ft

75. What do the commutative laws of addition andmultiplication have in common?

76. Explain how the distributive law differs from theassociative law.DWDW

78. Computer Printers. Eva prints a 5 in. by 7 in. photo onher Epson R1800 printer (see Exercise 6). How many dotswill be printed when the printer produces the photo?

290,304,000 dots

79. Computer Printers. Adam’s HP 8250 is printing an 8 in. by 10 in. photo (see Exercise 5). How many dotswill be printed when the printer finishes the photo?

460,800,000 dots

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1.6 Division

Division

REPEATED SUBTRACTIONDivision of whole numbers applies to two kinds of situations. The first is re-peated subtraction. Suppose we have 20 notebooks in a pile, and we want tofind out how many sets of 5 there are. One way to do this is to repeatedly sub-tract sets of 5 as follows.

Since there are 4 sets of 5 notebooks each, we have

20 5 4.

Dividend Divisor Quotient

The division read “20 divided by 5,” corresponds to the figure above.We say that the dividend is 20, the divisor is 5, and the quotient is 4. We di-vide the dividend by the divisor to get the quotient.

We can also express the division as

EXAMPLE 1 Write a division sentence that corresponds to this situation.

A parent directs 3 children to share $24, with each child getting the sameamount. How much does each child get?

We think of an array with 3 rows. Each row will go to a child. How manydollars will be in each row?

24 3 � 8

3 rows with 8in each row

205

� 4 or 5�20 4

20 5 � 4

20 5,

�

20 notebooks

How many sets of 5notebooks each?

12

3

4

1.61.6 DIVISIONObjectivesWrite a division sentencethat corresponds to asituation.

Given a division sentence,write a related multiplicationsentence; and given amultiplication sentence,write two related divisionsentences.

Divide whole numbers.

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We can also think of division in terms of rectangular arrays. Consider againthe pile of 20 notebooks and division by 5. We can arrange the notebooks in arectangular array with 5 rows and ask, “How many are in each row?”

We can also consider a rectangular array with 5 notebooks in each col-umn and ask, “How many columns are there?” The answer is still 4.

In each case, we are asking, “What do we multiply 5 by in order to get 20?”

Missing factor Quotient

This leads us to the following definition of division.

DIVISION

The quotient where is that unique whole number c forwhich

EXAMPLE 2 Write a division sentence that corresponds to this situation.You need not carry out the division.

How many uniforms that cost $45 each can be purchased for $495?

We think of an array with 45 dollar bills in each row. The money in eachrow will buy one uniform. How many rows will there be?


495 45 �

45 in each row

How many rows?

…

…

…

…

a � b � c.b � 0,a b,

20 5 �� 205 �

4 notebooks in each row

5 rows5 notebooksin eachcolumn

20 � 5 � 4

1

2

3

4

5

4321

4 columns

Write a division sentence thatcorresponds to each situation. Youneed not carry out the division.

1. There are 112 students in acollege band, and they aremarching with 14 in each row.How many rows are there?

2. A college band is in arectangular array. There are 112 students in the band, andthey are marching in 8 rows.How many students are there in each row?

Answers on page A-2

46


⎫⎪⎪⎬⎪⎪⎭ ⎫⎪⎬⎪⎭

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Write a related multiplicationsentence.

3.

4.

Write two related division sentences.

5.

6.

Answers on page A-2

7 � 6 � 42

6 � 2 � 12

72 8 � 9

15 3 � 5

47

1.6 Division

Related Sentences

By looking at rectangular arrays, we can see how multiplication and divisionare related. The array of notebooks on the preceding page shows that

The array also shows the following:

The division is defined to be the number that when multiplied by 5gives 20. Thus, for every division sentence, there is a related multiplicationsentence.

Division sentence

Related multiplication sentence

EXAMPLE 3 Write a related multiplication sentence:

We have

Division sentence

Related multiplication sentence

The related multiplication sentence is


For every multiplication sentence, we can write related division sentences.

EXAMPLE 4 Write two related division sentences:

We have

This factor becomes a divisor.

7 � 56 8.

7 � 8 � 56

7 � 8 � 56.

12 � 2 � 6.

12 � 2 � 6.

12 6 � 2

12 6 � 2.

20 � 4 � 5

20 5 � 4

20 5

20 5 � 4 and 20 4 � 5.

5 � 4 � 20.

This factorbecomesa divisor.

8 � 56 7.

7 � 8 � 56

The related division sentences are and


8 � 56 7.7 � 56 8

To get the related multiplication sentence, we use

Dividend Quotient Divisor.��

By the commutative law of multiplication, there is alsoanother multiplication sentence: 12 � 6 � 2.

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Division of Whole Numbers

Before we consider division with remainders, let’s recall four basic facts aboutdivision.

DIVIDING BY 1

Any number divided by 1 is that same number:

DIVIDING A NUMBER BY ITSELF

Any nonzero number divided by itself is 1:

DIVIDENDS OF 0

Zero divided by any nonzero number is 0:

EXCLUDING DIVISION BY 0

Division by 0 is not defined. (We agree not to divide by 0.)

is not defined.

Why can’t we divide by 0? Suppose the number 4 could be divided by 0.Then if were the answer,

and since 0 times any number is 0, we would have

False!

Thus, would be some number such that So the onlypossible number that could be divided by 0 would be 0 itself.

But such a division would give us any number we wish, for we could say

because

because

because

All true!

We avoid the preceding difficulties by agreeing to exclude division by 0.

0 � 7 � 0.0 0 � 7

0 � 3 � 0;0 0 � 3

0 � 8 � 0;0 0 � 8

� 0 � 0.a �a 0

� 0 � 0.4 �

4 0 �

a0

0a

� 0, a � 0.

aa

� 1, a � 0.

a 1 �a1

� a.

⎫⎪⎬⎪⎭

Study Tips

EXERCISES

� Odd-numbered exercises.Usually an instructorassigns some odd-numbered exercises ashomework. When youcomplete these, you cancheck your answers at theback of the book. If youmiss any, check your workin the Student’s SolutionsManual or ask yourinstructor for guidance.

� Even-numbered exercises.Whether or not yourinstructor assigns theeven-numbered exercises,always do some on yourown. Remember, there areno answers given for tests,so you need to practicedoing exercises withoutanswers. Check youranswers later with a friendor your instructor.

48


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Divide by repeated subtraction.Then check.

7.

8.

9.

10.

Answers on page A-2

157 24

53 12

61 9

54 9

49

1.6 Division

Suppose we have 18 cans of soda and want to pack them in cartons of6 cans each. How many cartons will we fill? We can determine this by repeatedsubtraction. We keep track of the number of times we subtract. We stop whenthe number of objects remaining, the remainder, is smaller than the divisor.

EXAMPLE 5 Divide by repeated subtraction:

Subtracting 3 times

The remainder, 0, is smaller than the divisor, 6.

Thus,

Suppose we have 22 cans of soda and want to pack them in cartons of6 cans each. We end up with 3 cartons with 4 cans left over.

EXAMPLE 6 Divide by repeated subtraction:

Subtracting 3 times

Remainder 4

� 6 1 0 1 0

� 6 1 6 1 6

� 6 2 2

22 6.

1

2

3

4 left over

18 6 � 3.

0

� 6 6 6

� 6 1 2 1 2

� 6 1 8

18 6.

Check: 18 � 4 � 22.

3 � 6 � 18,

We can write this division as follows.

4

1 8 6�2 2

3

⎫⎪⎪⎪⎬⎪⎪⎪⎭

⎫⎪⎪⎪⎬⎪⎪⎪⎭

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Note that

Quotient � Divisor � Remainder � Dividend.

We write answers to a division sentence as follows.

Dividend Divisor Quotient Remainder


EXAMPLE 7 Divide and check:

The remainder is larger than the divisor.

The remainder is larger than the divisor.

The remainder is less than the divisor.

2

4 0 4 2

1 0 1 4

3 5 5 � 3 6 4 2

7 2 8

2

4 0 4 2

1 0 0 1 4 2

3 5 0 0

5 � 3 6 4 2

7 2 8

4 2

1 0 0

1 4 2

3 5 0 0

5 � 3 6 4 2

7 2

1 4 2

3 5 0 0

5 � 3 6 4 2

7

5 � 3 6 4 2

?

3642 5.

22 6 � 3 R 4

Divide and check.

11.

12.

13.

Answers on page A-2

5 � 5 0 7 5

6 � 8 8 5 5

4 � 2 3 9

50


Check:

The answer is 728 R 2.


3640 � 2 � 3642. 728 � 5 � 3640,

1. Find the number of thousands in the quotient.Consider 3 thousands 5 and think 3 5.Since 3 5 is not a whole number, move tohundreds.

2. Find the number of hundreds in the quotient.Consider 36 hundreds 5 and think 36 5.The estimate is about 7 hundreds. Multiply 700by 5 and subtract.

3. Find the number of tens in the quotient using142, the first remainder. Consider 14 tens 5and think 14 5. The estimate is about 2 tens.Multiply 20 by 5 and subtract. (If our estimatehad been 3 tens, we could not have subtracted150 from 142.)

4. Find the number of ones in the quotient using42, the second remainder. Consider 42 ones 5and think 42 5. The estimate is about 8 ones.Multiply 8 by 5 and subtract. The remainder, 2,is less than the divisor, 5, so we are finished.

You may have learnedto divide like this, notwriting the extra zeros.You may omit them ifdesired.

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We can summarize our division procedure as follows.

To divide whole numbers:

a) Estimate.b) Multiply.c) Subtract.

Sometimes rounding the divisor helps us find estimates.

EXAMPLE 8 Divide:

We mentally round 42 to 40.

Caution!

Be careful to keep the digits lined up correctly.

The answer is 212. Remember : If after estimating and multiplying you get anumber that is larger than the number from which it is being subtracted,lower your estimate.


0

8 4 8 4

4 2 0 5 0 4

8 4 0 0 4 2 � 8 9 0 4

2 1 2

8 4

4 2 0 5 0 4

8 4 0 0 4 2 � 8 9 0 4

2 1

5 0 4

8 4 0 0

4 2 � 8 9 0 4

2

8904 42.

Divide.

14.

15.

Divide.

16.

17.

Answers on page A-2

7 � 7 6 1 6

6 � 4 8 4 6

5 2 � 3 2 8 8

4 5 � 6 0 3 0

51

1.6 Division

Think: 89 hundreds 40. Estimate 2 hundreds. Multiply 200 � 42 and subtract.

Think: 50 tens 40. Estimate 1 ten. Multiply 10 � 42 and subtract.

Think: 84 ones 40. Estimate 2 ones. Multiply 2 � 42 and subtract.

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ZEROS IN QUOTIENTS

EXAMPLE 9 Divide:

Think: 4 tens 7. The tens digit of the quotient is 0.

The remainder, 6, is less than the divisor, 7.


Do Exercises 16 and 17 on the preceding page.

EXAMPLE 10 Divide:

We round 37 to 40.

The remainder, 9, is less than the divisor, 37.



9

1 4 8 0 1 4 8 9

7 4 0 0 3 7 � 8 8 8 9

2 4 0

9

1 4 8 0

1 4 8 9

7 4 0 0 3 7 � 8 8 8 9

2 4

1 4 8 9

7 4 0 0

3 7 � 8 8 8 9

2

8889 37.

6

3 5 4 1

6 3 0 0 7 � 6 3 4 1

9 0 5

4 1

6 3 0 0 7 � 6 3 4 1

9 0

4 1

6 3 0 0

7 � 6 3 4 1

9

6341 7.

Divide.

18.

19.

Answers on page A-2

5 6 � 4 4 , 8 4 7

2 7 � 9 7 2 4

52


Think: 63 hundreds 7. Estimate 9 hundreds. Multiply 900 � 7 and subtract.

Think: 41 ones 7. Estimate 5 ones. Multiply 5 � 7 and subtract.

Think: 37 � 40; 88 hundreds 40.Estimate 2 hundreds. Multiply 200 � 37 and subtract.

Think: 148 tens 40.Estimate 4 tens. Multiply 40 � 37 and subtract.

To the instructor and thestudent: This section presenteda review of division of wholenumbers. Students who aresuccessful should go on toSection 1.7. Those who havetrouble should studydevelopmental unit D near theback of this text and then repeatSection 1.6.

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1.6 Division

CALCULATOR CORNER

Dividing Whole Numbers: Finding Remainders To divide wholenumbers on a calculator, we use the and keys. For example, todivide 711 by 9, we press . The display reads

, so When we enter 453 15, the display reads . Note that the

result is not a whole number. This tells us that there is a remainder. Thenumber 30.2 is expressed in decimal notation. The symbol “.” is called adecimal point. (Decimal notation will be studied in Chapter 5.) The numberto the left of the decimal point, 30, is the quotient. We can use theremaining part of the result to find the remainder. To do this, first subtract30 from 30.2 to find the remaining part of the result. Then multiply thedifference by the divisor, 15. We get 3. This is the remainder. Thus,

The steps that we performed to find this result can besummarized as follows:

To follow these steps on a calculator, we press and write the number that appears to the left of the decimal

point. This is the quotient. Then we continue by pressing . The last number that appears is the remainder. In some

cases, it will be necessary to round the remainder to the nearest one.To check this result, we multiply the quotient by the divisor and then

add the remainder.

Exercises: Use a calculator to perform each division. Check the resultswith a calculator also.

1. 3 R 11

2. 28

3. 124 R 2

4. 131 R 18

5. 283 R 57

6. 843 R 1873 0 8 � 2 5 9 , 8 3 1

1 2 6 � 3 5 , 7 1 5

3817 29

6 � 7 4 6

1 9 � 5 3 2

92 27

450 � 3 � 453

30 � 15 � 450,

�51�

�03�

�51

354

0.2 � 15 � 3.

30.2 � 30 � .2,

453 15 � 30.2,

453 15 � 30 R 3.

30.2

711 9 � 79.79

�9117

�

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54


Write a division sentence that corresponds to each situation.



Math TutorCenter

InterActMath


1. Canyonlands. The trail boss for a trip intoCanyonlands National Park divides 760 lb (pounds) of equipment among 4 mules. How many pounds does each mule carry?

2. Surf Expo. In a swimwear showing at Surf Expo, a trade show for retailers of beach supplies, eachswimsuit test takes 8 min (minutes). If the show runs for 240 min, how many tests can be scheduled?

240 8 � 30760 4 � 190

3. A lab technician pours 455 mL of sulfuric acid into 5 beakers, putting the same amount in each. How much acid is in each beaker?

4. A computer screen is made up of a rectangular array ofpixels. There are 480,000 pixels in all, with 800 pixels ineach row. How many rows are there on the screen?

480,000 800 � 600455 5 � 91

Write a related multiplication sentence.

5.

, or

6.

, or

7.

, or

8.

, or32 � 32 � 132 � 1 � 32

32 1 � 32

22 � 1 � 2222 � 22 � 1

22 22 � 1

72 � 8 � 972 � 9 � 8

72 9 � 8

18 � 6 � 318 � 3 � 6

18 3 � 6

9.

, or

10.

, or

11.

, or

12.

, or28 � 1 � 2828 � 28 � 1

28 28 � 1

37 � 37 � 137 � 1 � 37

37 1 � 37

40 � 5 � 840 � 8 � 5

40 8 � 5

54 � 9 � 654 � 6 � 9

54 6 � 9

Write two related division sentences.

13.

;

14.

;

15.

;

16.

;12 � 48 44 � 48 12

4 � 12 � 48

1 � 37 3737 � 37 1

37 � 1 � 37

7 � 14 22 � 14 7

2 � 7 � 14

5 � 45 99 � 45 5

9 � 5 � 45

17. 18.

;

19.

;

20.

;43 � 43 11 � 43 43

1 � 43 � 43

6 � 66 1111 � 66 6

11 � 6 � 66

7 � 63 99 � 63 7

9 � 7 � 63

8 � 64 8

8 � 8 � 64

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Exercise Set 1.6

Divide, if possible. If not possible, write “not defined.”

21. 12 22. 6 23. 1 24. 13737

2323

54 972 6

25. 22 26. 56 27. Not defined 28. Not defined74 0160

561

22 1

Divide.

29. 55 R 2 30. 233 31. 108 32. 108 R 5869 8864 8699 3277 5

33. 307 34. 708 35. 753 R 3 36. 1012 R 29 � 9 1 1 06 � 4 5 2 13 � 2 1 2 44 � 1 2 2 8

37. 1703 38. 2009 39. 987 R 5 40. 517 R 38 � 4 1 3 99 � 8 8 8 83 � 6 0 2 75 � 8 5 1 5

41. 12,700 42. 1270 43. 127 44. 4264260 10127,000 1000127,000 100127,000 10

45.

52 R 52

46.

289 R 18

47.

29 R 5

48.

24 R 27

4 0 � 9 8 73 0 � 8 7 52 0 � 5 7 9 87 0 � 3 6 9 2

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56


49.

29

50.

55 R 2

51.

105 R 3

52.

107

7 � 7 4 98 � 8 4 31 0 2 � 5 6 1 21 1 1 � 3 2 1 9

53.

1609 R 2

54.

808 R 1

55.

1007 R 1

56.

1010 R 4

7 � 7 0 7 45 � 5 0 3 69 � 7 2 7 35 � 8 0 4 7

57.

23

58.

301 R 18

59.

107 R 1

60.

102 R 3

4 8 � 4 8 9 93425 327242 241058 46

61.

370

62.

210 R 3

63.

609 R 15

64.

803 R 21

3 6 � 2 8 , 9 2 92 8 � 1 7 , 0 6 73 6 � 7 5 6 32 4 � 8 8 8 0

65. 304 66. 984 67. 3508 R 2192 8 5 � 9 9 9 , 9 9 99 0 � 8 8 , 5 6 08 0 � 2 4 , 3 2 0

68. 2904 R 264 69. 8070 70. 70028 0 3 � 5 , 6 2 2 , 6 0 64 5 6 � 3 , 6 7 9 , 9 2 03 0 6 � 8 8 8 , 8 8 8

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Exercise Set 1.6

54 68 3672 122

84 33 2772 117

4 8 32 12

71. Is division associative? Why or why not? Give anexample.

72. Suppose a student asserts that “ becausenothing divided by nothing is nothing.” Devise anexplanation to persuade the student that the assertionis false.

0 0 � 0DWDW

SKILL MAINTENANCE

SYNTHESIS

73. The distance around an object is its .[1.2c]

74. A sentence like is called a(n) ; a sentence like is called a(n) .[1.4c]

75. For large numbers, are separated bycommas into groups of three, called .[1.1a]

76. Because changing the order of addends does not change thesum, addition is . [1.2b]

77. In the sentence the is 28.[1.6a]

78. In the sentence 10 and 1000 are calledand 10,000 is called the .

[1.5a]

79. The is the number from which anothernumber is being subtracted. [1.3a]

80. The sentence illustrates thelaw of multiplication. [1.5a]associative

3 � �6 � 2� � �3 � 6� � 2

minuend

productfactors10 � 1000 � 10,000,

dividend28 7 � 4,

commutative

periodsdigits

inequality31 � 33

equation10 � 3 � 7

perimeter associative

commutative

addends

factors

perimeter

dividend

quotient

product

minuend

subtrahend

digits

periods

equation

inequality

i VOCABULARY REINFORCEMENT

In each of Exercises 73–80, fill in the blank with the correct term from the given list. Some of the choices may notbe used and some may be used more than once.

83. Complete the following table. 84. Find a pair of factors whose product is 36 and:

a) whose sum is 13. 4, 9b) whose difference is 0. 6, 6c) whose sum is 20. 2, 18d) whose difference is 9. 3, 12

85. A group of 1231 college students is going to take busesfor a field trip. Each bus can hold only 42 students. Howmany buses are needed? 30 buses

86. Fill in the missing digits to make the equation true:

,386. 2; 5� 434,584,132 76

a b a � b a � b

81. Describe a situation that corresponds to thedivision . (See Examples 1 and 2.)

82. What is it about division that makes it moredifficult than addition, subtraction, or multiplication?DW

1180 295DW

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Solutions by Trial

Let’s find a number that we can put in the blank to make this sentence true:

.

We are asking “9 is 3 plus what number?” The answer is 6.


A sentence with is called an equation. A solution of an equation is anumber that makes the sentence true. Thus, 6 is a solution of

because is true.

However, 7 is not a solution of

because is false.


We can use a letter instead of a blank. For example,

We call n a variable because it can represent any number. If a replacement fora variable makes an equation true, it is a solution of the equation.

SOLUTIONS OF AN EQUATION

A solution is a replacement for the variable that makes the equationtrue. When we find all the solutions, we say that we have solved theequation.

EXAMPLE 1 Solve by trial.

We replace y with several numbers.

If we replace y with 13, we get a false equation:

If we replace y with 14, we get a false equation:

If we replace y with 15, we get a true equation:

No other replacement makes the equation true, so the solution is 15.

EXAMPLES Solve.

15 � 12 � 27.

14 � 12 � 27.

13 � 12 � 27.

y � 12 � 27

9 � 3 � n.

9 � 3 � 79 � 3 �

9 � 3 � 69 � 3 �

�

9 � 3 � 6

9 � 3 �

1.71.7 SOLVING EQUATIONS

Find a number that makes thesentence true.

1.

2.

3. Determine whether 7 is asolution of

4. Determine whether 4 is asolution of

Solve by trial.

5. 5

6.

7.

8.

Answers on page A-3

10 � t � 32

45 9 � y

x � 2 � 8

n � 3 � 8

� 5 � 9.

� 5 � 9.

� 2 � 7

8 � 1 �

ObjectivesSolve simple equations by trial.

Solve equations like

and98 � 2 � y.28 � x � 168,x � 28 � 54,

2.(7 plus what number is 22?) The solution is 15.

3.(63 is 3 times what number?)The solution is 21.

63 � 3 � x7 � n � 22

Do Exercises 5–8.

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Solving Equations

We now begin to develop more efficient ways to solve certain equations.When an equation has a variable alone on one side, it is easy to see the solu-tion or to compute it. When a calculation is on one side and the variable isalone on the other, we can find the solution by carrying out the calculation.

EXAMPLE 4 Solve:

To solve the equation, we carry out the calculation.

8 3 3 0

7 3 5 0 9 8 0

� 3 4 2 4 5

x � 245 � 34.

Solve.

9.

10.

11.

12.

Answers on page A-3

x � 6007 � 2346

4560 8 � t

x � 2347 � 6675

346 � 65 � y

x � 8330 x � 245 � 34

The solution is 8330.


Consider the equation

We can get x alone by writing a related subtraction sentence:

12 gets subtracted to find the related subtractionsentence.

Doing the subtraction

It is useful in our later study of algebra to think of this as “subtracting 12 onboth sides.” Thus

Subtracting 12 on both sides

Carrying out the subtraction

SOLVING

To solve subtract a on both sides.

If we can get an equation in a form with the variable alone on one side,we can “see” the solution.

EXAMPLE 5 Solve:

We have


t � 26.

t � 0 � 26

t � 28 � 28 � 54 � 28

t � 28 � 54

t � 28 � 54.

x � a � b,

x � a � b

x � 15.

x � 0 � 15

x � 12 � 12 � 27 � 12

x � 15.

x � 27 � 12

x � 12 � 27.

Study Tips

STAYING AHEAD

Try to keep one section aheadof your syllabus. If you studyahead of your lectures, youcan concentrate on what isbeing explained in them,rather than trying to writeeverything down. You can thenwrite notes on only specialpoints or questions related towhat is happening in class.

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To check the answer, we substitute 26 for t in the original equation.

Check:

TRUE

The solution is 26.


EXAMPLE 6 Solve:

We have


65 plus n minus 65 is

Check:

TRUE

The solution is 117.

Do Exercise 15.

EXAMPLE 7 Solve:

We have


The check is left to the student. The solution is 686.


We now learn to solve equations like Look at

We can get n alone by writing a related division sentence:

96 is divided by 8.

Doing the division

It is useful in our later study of algebra to think of the preceding as “dividingby 8 on both sides.” Thus,

Dividing both sides by 8

8 times n divided by 8 is n. n � 12.

8 � n

8�

968

n � 12.

n � 96 8 �968

8 � n � 96.

8 � n � 96.

x � 686.

7381 � x � 7381 � 8067 � 7381

7381 � x � 8067

7381 � x � 8067.

182 182 ? 65 � 117

182 � 65 � n

117 � n.

0 � n. 117 � 0 � n

182 � 65 � 65 � n � 65

182 � 65 � n

182 � 65 � n.

54 26 � 28 ? 54

t � 28 � 54

Solve. Be sure to check.

13.

14.

15. Solve: Be sure to check.


16.

17.

Answers on page A-3

8172 � h � 2058

4566 � x � 7877

155 � t � 78.

77 � m � 32

x � 9 � 17

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SOLVING

To solve divide both sides by a.

EXAMPLE 8 Solve:

We have


Check:

TRUE

The solution is 24.

EXAMPLE 9 Solve:

We have




EXAMPLE 10 Solve:

We have



EXAMPLE 11 Solve:

We have




n � 83.

n � 56

56�

464856

n � 56 � 4648

n � 56 � 4648.

y � 78.

14 � y

14�

109214

14 � y � 1092

14 � y � 1092.

578 � t.

5202

9�

9 � t9

5202 � 9 � t

5202 � 9 � t.

240 10 � 24 ? 240

10 � x � 240

x � 24.

10 � x

10�

24010

10 � x � 240

10 � x � 240.

a � x � b,

a � x � b Solve. Be sure to check.

18.

19.

20.


21.

22.

Answers on page A-3

n � 48 � 4512

18 � y � 1728

5152 � 8 � t

144 � 9 � n

8 � x � 64

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62


Solve by trial.



Math TutorCenter

InterActMath



5. 29 6. 7 7. 0 8. 016 � t � 1612 � 12 � m15 � t � 2213 � x � 42

9. 8 10. 7 11. 14 12. 18162 � 9 � m112 � n � 86 � x � 423 � x � 24

13.

1035

14.

1794

15.

25

16.

16

w � 256 16t � 125 523 � 78 � y45 � 23 � x

17.

450

18.

3340

19.

90,900

20.

14,712

5678 � 9034 � tx � 12,345 � 78,5559007 � 5667 � mp � 908 � 458

21.

32

22.

24

23.

143

24.

247

741 � 3 � t715 � 5 � z4 � y � 963 � m � 96

25.

79

26.

37

27.

45

28.

36

53 � 17 � w61 � 16 � y20 � x � 5710 � x � 89

29.

324

30.

851

31.

743

32.

141

9 � x � 12695 � x � 37154 � w � 34046 � p � 1944

33.

37

34.

36

35.

66

36.

66

z � 67 � 133x � 78 � 14456 � p � 9247 � n � 84

1. 14 2. 25 3. 0 4. 856 m � 7y � 17 � 0x � 7 � 18x � 0 � 14

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63

Exercise Set 1.7

37. 15 38. 55 39. 48 40. 49784 � y � 16624 � t � 13660 � 12 � n165 � 11 � n

41. 175 42. 112 43. 335 44. 369438 � x � 807567 � x � 902x � 221 � 333x � 214 � 389

45. 104 46. 320 47. 45 48. 7520 � x � 150040 � x � 180019 � x � 608018 � x � 1872

49.

4056

50.

959

51.

17,603

52.

959

9281 � 8322 � y8322 � 9281 � x9281 � 8322 � t2344 � y � 6400

53.

18,252

54.

23

55.

205

56.

95

233 � x � 22,13558 � m � 11,89010,534 458 � q234 � 78 � y

57. Describe a procedure that can be used to convertany equation of the form to a related divisionequation.

58. Describe a procedure that can be used to convertany equation of the form to a relatedsubtraction equation.

a � b � cDW

a � b � cDW

59. Write two related subtraction sentences:[1.3b]

60. Write two related division sentences: [1.6b]

8 � 48 66 � 48 8;

6 � 8 � 48.8 � 15 � 77 � 15 � 8;

7 � 8 � 15.

Use � or � for to write a true sentence. [1.4c]

61. 123 789 � 62. 342 339 � 63. 688 0 � 64. 0 11 �

Divide. [1.6c]

65.

142 R 5

66.

142

67.

334

68.

334 R 11

1 7 � 5 6 8 91 7 � 5 6 7 81278 91283 9

Solve.

71. 347 72. 29348,916 � x � 14,332,38823,465 � x � 8,142,355

SKILL MAINTENANCE

SYNTHESIS

69. Give an example of an equation in which a variableappears but for which there is no solution. Then explainwhy no solution exists.

70. Is it possible for an equation to have manysolutions? If not, explain why, and if so, explain how towrite such equations.

DWDW

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64


A Problem-Solving Strategy

Applications and problem solving are the most important uses of mathe-matics. To solve a problem, we first look at the situation and try to translatethe problem to an equation. Then we solve the equation. We check to see if thesolution of the equation is a solution of the original problem. We are using thefollowing five-step strategy.

FIVE STEPS FOR PROBLEM SOLVING

1. Familiarize yourself with the situation.

a) Carefully read and reread until you understand what you arebeing asked to find.

b) Draw a diagram or see if there is a formula that applies to thesituation.

c) Assign a letter, or variable, to the unknown.

2. Translate the problem to an equation using the letter or variable.3. Solve the equation.4. Check the answer in the original wording of the problem.5. State the answer to the problem clearly with appropriate units.

EXAMPLE 1 International Adoptions. The top ten countries of origin forUnited States international adoptions in 2005 and 2004 are listed in the tablebelow. Find the total number of adoptions from China, South Korea, Ethiopia,and India in 2005.

Source: U.S. Department of State

RANK

2005 2004

INTERNATIONAL ADOPTIONS

NUMBERCOUNTRYOF ORIGIN

123456789

10

China (mainland)RussiaGuatemalaSouth KoreaUkraineKazakhstanEthiopiaIndiaColombiaPhilippines

790646393783821755441323291271231

NUMBERCOUNTRYOF ORIGIN

China (mainland)RussiaGuatemalaSouth KoreaKazakhstanUkraineIndiaHaitiEthiopiaColombia

7044586532641716826723406356289287

1.81.8 APPLICATIONS AND PROBLEM SOLVING

ObjectiveSolve applied problemsinvolving addition,subtraction, multiplication,or division of whole numbers.

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1. Familiarize. We can make a drawing or at least visualize the situation.

7906 � 821 � 323 � 291

Since we are combining numbers of adoptions, addition can be used.First, we define the unknown. We let the total number of adoptionsfrom China, South Korea, Ethiopia, and India.

2. Translate. We translate to an equation:

3. Solve. We solve the equation by carrying out the addition.

9 3 4 1

� 2 9 1 3 2 3 8 2 1

72

91

01

6

7906 � 821 � 323 � 291 � n.

n �

fromIndia

fromEthiopia

from South Korea

fromChina

Refer to the table on the precedingpage to answer Margin Exercises 1–3.

1. Find the total number ofadoptions from China, Russia,Kazakhstan, and India in 2004.

2. Find the total number ofadoptions from Russia,Guatemala, South Korea,Ukraine, and Colombia in 2005.

3. Find the total number ofadoptions from the top tencountries in 2005.

Answers on page A-3

9341 � n

7906 � 821 � 323 � 291 � n

4. Check. We check 9341 in the original problem. There are many ways inwhich this can be done. For example, we can repeat the calculation. (Weleave this to the student.) Another way is to check whether the answer isreasonable. In this case, we would expect the total to be greater than thenumber of adoptions from any of the individual countries, which it is. Wecan also estimate by rounding. Here we round to the nearest hundred.

Since we have a partial check. If we had an estimate like4800 or 7500, we might be suspicious that our calculated answer is incor-rect. Since our estimated answer is close to our calculation, we are furtherconvinced that our answer checks.

5. State. The total number of adoptions from China, South Korea, Ethiopia,and India in 2005 is 9341.

Do Exercises 1–3.

9300 � 9341,

� 9300

7906 � 821 � 323 � 291 � 7900 � 800 � 300 � 300

Study Tips TIME MANAGEMENT

Time is the most critical factor

in your success in learning

mathematics. Havereasonable expecta-

tions about the time you need to

study math.

� Juggling time. Working 40 hours per week and taking 12 credit hours is equivalent toworking two full-time jobs. Can you handle such a load? Your ratio of number of workhours to number of credit hours should be about 40�3, 30�6, 20�9, 10�12, or 5�14.

� A rule of thumb on study time. Budget about 2–3 hours for homework and study forevery hour you spend in class each week.

� Scheduling your time. Use the Student Organizer card to make an hour-by-hourschedule of your typical week. Include work, school, home, sleep, study, and leisuretimes. Try to schedule time for study when you are most alert. Choose a setting thatwill enable you to focus and concentrate. Plan for success and it will happen!

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EXAMPLE 2 Checking Account Balance. The balance in Tyler’s checkingaccount is $528. He uses his debit card to buy the Roto Zip Spiral Saw Comboshown in this ad. Find the new balance in his checking account.

1. Familiarize. We first make a drawing or at least visualize the situation.We let the new balance in his account. This gives us the following:

2. Translate. We can think of this as a “take-away” situation. We translateto an equation.

528 � 129 � M

3. Solve. This sentence tells us what to do. We subtract.

11

3 9 9

� 1 2 9 54

21

818

Newbalanceis

Moneyspentminus

Moneyin the

account

New balance

Take away

$129

$528

M �

$12900NOW

Source: Reproduced with permission from the copyright owner (©2004 by Robert Bosch Tool Corporation). Further reproductions strictly prohibited.

4. Checking Account Balance.The balance in Heidi’s checkingaccount is $2003. She uses herdebit card to buy the same RotoZip Spiral Saw Combo, featuredin Example 2, that Tyler did.Find the new balance in herchecking account.

Answer on page A-3

66


⎫⎪⎬⎪⎭ ⎫⎪⎬⎪⎭ ⎫⎪⎬⎪⎭

399 � M

528 � 129 � M

4. Check. To check our answer of $399, we can repeat the calculation. We note that the answer should be less than the original amount, $528,which it is. We can add the difference, 399, to the subtrahend, 129:

We can also estimate:

5. State. Tyler has a new balance of $399 in his checking account.

Do Exercise 4.

528 � 129 � 530 � 130 � 400 � 399.

129 � 399 � 528.

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In the real world, problems may not be stated in written words. You muststill become familiar with the situation before you can solve the problem.

EXAMPLE 3 Travel Distance. Abigail is driving from Indianapolis to SaltLake City to interview for a news anchor position. The distance from Indi-anapolis to Salt Lake City is 1634 mi. She travels 1154 mi to Denver. How muchfarther must she travel?

1. Familiarize. We first make a drawing or at least visualize the situation.We let the remaining distance to Salt Lake City.

2. Translate. We see that this is a “how much more” situation. We translateto an equation.

1154 � x � 1634

3. Solve. To solve the equation, we subtract 1154 on both sides:

x � 480.

1154 � x � 1154 � 1634 � 1154

1154 � x � 1634

Totaldistance

of tripisDistance

to goplus

Distancealreadytraveled

IOWANEBRASKA

KANSAS

COLORADO

UTAH

WYOMING

ILLINOIS

INDIANA

MISSOURIKENTUCKY

IndianapolisDenver

Salt Lake City

Distance to gox Distance already traveled

1154 mi

Total distance = 1634 mi

x �

5. Digital Camera. William has$228. He wants to purchase thedigital camera shown in the adbelow. How much more does he need?

Answer on page A-3

67


⎫⎪⎬⎪⎭ ⎫⎪⎬⎪⎭ ⎫⎪⎬⎪⎭

4 8 0

� 1 1 5 4 1 6

5

313

4

4. Check. We check our answer of 480 mi in the original problem. Thisnumber should be less than the total distance, 1634 mi, which it is. We canadd the difference, 480, to the subtrahend, 1154: Wecan also estimate:

The answer, 480 mi, checks.

5. State. Abigail must travel 480 mi farther to Salt Lake City.

Do Exercise 5.

� 400 � 480.

1634 � 1154 � 1600 � 1200

1154 � 480 � 1634.

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EXAMPLE 4 Total Cost of Chairs. What is the total cost of 6 Logan sidechairs from Restoration Hardware if each one costs $210?

1. Familiarize. We first make a drawing or at least visualize the situation.We let the cost of 6 chairs. Repeated addition works well in this case.

2. Translate. We translate to an equation.

6 � $210 � T

3. Solve. This sentence tells us what to do. We multiply.

1 2 6 0

� 6 2 1 0

Totalcostis

Cost ofeach chairtimes

Numberof chairs

$ 210 $ 210 $ 210 $ 210 $ 210 $ 210

T �

6. Total Cost of Gas Grills. Whatis the total cost of 14 gas grills,each with 520 sq in. (squareinches) of total cooking surfaceand porcelain cast-iron cookinggrates, if each one costs $398?

Answer on page A-3

68


⎫⎪⎬⎪⎭ ⎫⎪⎬⎪⎭ ⎫⎬⎭

1260 � T

6 � 210 � T

4. Check. We have an answer, 1260, that is much greater than the cost ofany individual chair, which is reasonable. We can repeat our calculation.We can also check by estimating:

The answer checks.

5. State. The total cost of 6 chairs is $1260.

Do Exercise 6.

EXAMPLE 5 Truck Bed Cover. The dimensions of a fiberglass truck bedcover for a pickup truck are 79 in. by 68 in. What is the area of the cover?

79 in.

68 in.

6 � 210 � 6 � 200 � 1200 � 1260.

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1. Familiarize. The truck bed cover is a rectangle that measures 79 in. by68 in. We let the area of the cover, in sq in. (square inches), and usethe area formula

2. Translate. Using this formula, we have

3. Solve. We carry out the multiplication.

5 3 7 2

4 7 4 0 6 3 2

� 6 8 7 9

A � length � width � l � w � 79 � 68.

A � l � w.A �

7. Bed Sheets. The dimensions ofa flat sheet for a queen-size bedare 90 in. by 102 in. What is thearea of the sheet?

Answer on page A-3

69


A � 5372

A � 79 � 68

4. Check. We repeat our calculation. We also note that the answer is greaterthan either the length or the width, which it should be. (This might not bethe case if we were using fractions or decimals.) The answer checks.

5. State. The area of the truck bed cover is 5372 sq in.

Do Exercise 7.

EXAMPLE 6 Cartons of Soda. A bottling company produces 3304 cans of soda. How many 12-can cartons can be filled? How many cans will be left over?

1. Familiarize. We first make a drawing. We let the number of 12-cancartons that can be filled. The problem can be considered as repeatedsubtraction, taking successive sets of 12 cans and putting them into n cartons.

2. Translate. We translate to an equation.

3304 12 � n

Numberof

cartonsis

Numberin eachcarton

dividedby

Numberof

cans

1

2

n

12-can cartons

•••

How many cartons?

How many cans are left over?

n �

⎫⎪⎬⎪⎭ ⎫⎪⎬⎪⎭ ⎫⎪⎬⎪⎭

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3. Solve. We solve the equation by carrying out the division.

4

6 0 6 4

8 4 0 9 0 4

2 4 0 0 1 2 � 3 3 0 4

2 7 5

8. Cartons of Soda. The bottlingcompany in Example 6 also uses6-can cartons. How many 6-can cartons can be filled with2269 cans of cola? How manywill be left over?

Answer on page A-3

70


275 R 4 � n

3304 12 � n

4. Check. We can check by multiplying the number of cartons by 12 andadding the remainder, 4:

5. State. Thus, 275 twelve-can cartons can be filled. There will be 4 cansleft over.

Do Exercise 8.

EXAMPLE 7 Automobile Mileage. The Pontiac G6 GT gets 21 miles to the gallon (mpg) in city driving. How many gallons will it use in 3843 mi of city driving?Source: General Motors

1. Familiarize. We first make a drawing. It is often helpful to be descriptiveabout how we define a variable. In this case, we let the number of gallons (“g” comes from “gallons”).

2. Translate. Repeated addition applies here. Thus the following multi-plication applies to the situation.

21 � g � 3843

3. Solve. To solve the equation, we divide by 21 on both sides.

g � 183

21 � g

21�

384321

21 � g � 3843

Number ofmiles to

driveisNumber of

gallons neededtimes

Number ofmiles per

gallon

3843 mi to drive

21 mi 21 mi 21 mi 21 mi

g �

3300 � 4 � 3304.

12 � 275 � 3300,

⎫⎪⎬⎪⎭ ⎫⎪⎪⎬⎪⎪⎭ ⎫⎪⎬⎪⎭

0

6 3 6 3

1 6 8 0 1 7 4 3

2 1 0 0 2 1 � 3 8 4 3

1 8 3

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9. Automobile Mileage. ThePontiac G6 GT gets 29 miles tothe gallon (mpg) in highwaydriving. How many gallons willit take to drive 2291 mi ofhighway driving?Source: General Motors

Answer on page A-3

71


⎫⎪⎪⎬⎪⎪⎭⎫⎪⎪⎬⎪⎪⎭ ⎫⎪⎪⎬⎪⎪⎭

⎫⎪⎪⎬⎪⎪⎭⎫⎪⎪⎬⎪⎪⎭ ⎫⎪⎪⎬⎪⎪⎭

⎫⎪⎪⎬⎪⎪⎭⎫⎪⎪⎬⎪⎪⎭ ⎫⎪⎪⎬⎪⎪⎭

4. Check. To check, we multiply 183 by 21:

5. State. The Pontiac G6 GT will use 183 gal.

Do Exercise 9.

Multistep Problems

Sometimes we must use more than one operation to solve a problem, as in thefollowing example.

EXAMPLE 8 Aircraft Seating. Boeing Corporation builds commercialaircraft. A Boeing 767 has a seating configuration with 4 rows of 6 seats acrossin first class and 35 rows of 7 seats across in economy class. Find the total seat-ing capacity of the plane.Sources: The Boeing Company; Delta Airlines

1. Familiarize. We first make a drawing.

2. Translate. There are three parts to the problem. We first find the num-ber of seats in each class. Then we add.

First class: Repeated addition applies here. Thus the following multi-plication corresponds to the situation. We let the number of seats infirst class.

4 � 6 � F

Economy class: Repeated addition applies here. Thus the following multiplication corresponds to the situation. We let the number ofseats in economy class.

35 � 7 � E

We let the total number of seats in both classes.

F � E � T

Total number of seats on planeis

Number of seatsin economy classplus

Number of seatsin first class

T �

Total numberisSeats in each rowtimesNumber of rows

E �

Total numberisSeats in each rowtimesNumber of rows

F �

Economy class:35 rows of 7 seats

First class:4 rows of

6 seats

21 � 183 � 3843.

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3. Solve. We solve each equation and add the solutions.

24 � F

4 � 6 � F

10. Aircraft Seating. A Boeing 767used for foreign travel has threeclasses of seats. First class has 3 rows of 5 seats across;business class has 6 rows with 6 seats across and 1 row with 2 seats on each of the outsideaisles. Economy class has 18 rows with 7 seats across. Find the total seating capacity of the plane.Sources: The Boeing Company; Delta Airlines

Answer on page A-3

72


245 � E

35 � 7 � E

4. Check. To check, we repeat our calculations. (We leave this to the student.) We could also check by rounding, multiplying, and adding.

5. State. There are 269 seats in a Boeing 767.

Do Exercise 10.

As you consider the following exercises, here are somewords and phrases that may be helpful to look for when youare translating problems to equations.

269 � T

24 � 245 � T

F � E � T

KEY WORDS, PHRASES, AND CONCEPTS

ADDITION (�)

add

added to

sum

total

plus

more than

increased by

SUBTRACTION (�)

subtract

subtracted from

difference

minus

less than

decreased by

take away

how much more

missing addend

MULTIPLICATION (�)

multiply

multiplied by

product

times

of

repeated addition

rectangular arrays

DIVISION (�)

divide

divided by

quotient

repeated subtraction

missing factor

split into equalquantities

Economy class:18 rows of 7 seats

First class:3 rows of 5 seats

Business class:6 rows of 6 seats…

…with 2 seats on each outside aisle

Sentences containing the subtraction phrases “less than” and “subtractedfrom” can be difficult to translate correctly. Since subtraction is not commu-tative (for example, is not the same as ), the order in which thesubtraction is written is important. Listed below are some sample problemsand correct translations.

2 � 55 � 2

Bryan weighs 98 lb. This is 7 lb less than Cory’s weight. How much does Cory weigh?

Let Cory’s weight.

After payroll deductions of $89 were subtracted from her paycheck, Gina brought home $255. How much was her paycheck before deductions?

Let Gina’s paycheck. p � 89 � 255

p �

98 � c � 7c �

PROBLEM TRANSLATION

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The goal of these matchingquestions is to practice step (2),Translate, of the five-step problem-solving process. Translate eachword problem to an equation andselect a correct translation fromequations A–O.

A.

B.

C.

D.

E.

F.

G.

H.

I.

J.

K.

L.

M.

N.

O.

Answers on page A-3

52 � n � 60

15 195 � n

69 � n � 76

n � 52 � 8

15 � 195 � n

30 � n � 10,860

n � 423 69

n � 10,860 � 300

69 � n � 423

15 � n � 195

30 � 10,860 � n

423 � 73 � 76 � 69 � n

73 � 76 � 69 � n

69 � n � 76

8 � 52 � n

Translatingfor Success

6. Hourly Rate. Miller AutoRepair charges $52 an hour forlabor. Jackson Auto Care charges$60 per hour. How much moredoes Jackson charge than Miller?

7. College Band. A college bandwith 195 members marches in a15-row formation in the home-coming halftime performance.How many members are in each row?

8. Cleats Purchase. A professionalfootball team purchases 15 pairsof cleats at $195 a pair. What isthe total cost of this purchase?

9. Loan Payment. Kendraborrows $10,860 for a new boat.The loan is to be paid off in 30 payments. How much is eachpayment (excluding interest)?

10. College Enrollment. At thebeginning of the fall term, thetotal enrollment in LakeviewCommunity College was 10,860.By the end of the first twoweeks, 300 students withdrew.How many students were thenenrolled?

1. Brick-mason Expense. Acommercial contractor isbuilding 30 two-unit condo-miniums in a retirementcommunity. The brick-masonexpense for each building is$10,860. What is the total cost ofbricking the buildings?

2. Heights. Dean’s sons are on thehigh school basketball team.Their heights are 73 in., 69 in.,and 76 in. How much taller isthe tallest son than the shortestson?

3. Account Balance. You have$423 in your checking account.Then you deposit $73 and useyour debit card for purchases of$76 and $69. How much is left inyour account?

4. Purchasing Camcorder. Acamcorder is on sale for $423.Jenny has only $69. How muchmore does she need to buy thecamcorder?

5. Purchasing Coffee Makers. Sarapurchases 8 coffee makers forthe newly remodeled bed-and-breakfast inn that she manages.If she pays $52 for each coffeemaker, what is the total cost ofher purchase?

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Solve.



Math TutorCenter

InterActMath


Longest Broadway Run. In January 2006, The Phantom of the Opera became the longest-running broadway show. The bargraph below lists the five broadway shows with the greatest number of performances as of January 9, 2006. Use the graph forExercises 1–4.

Longest Broadway Runs

Nu

mb

er o

f per

form

ance

s

Cat

s

Th

e P

han

tom

of th

e O

per

a

Les

Mis

erab

les

A C

hor

us

Lin

e

Oh

! Cal

cutt

a!4000

30002000

1000

50006000

7000

8000 7486 74856680

6137 5959

Source: League of American Theatres and Producers, Inc.

1. What was the total number of performances of all fiveshows? 33,747 performances

2. What was the total number of performances of the topthree shows? 21,651 performances

3. How many more performances of The Phantom of theOpera were there than performances of A Chorus Line?

1349 performances

4. How many more performances of Les Misérables werethere than performances of Oh! Calcutta!?

721 performances

5. Boundaries Between Countries. The boundary betweenmainland United States and Canada including the GreatLakes is 3987 miles long. The length of the boundarybetween the United States and Mexico is 1933 miles.How much longer is the Canadian border? 2054 milesSource: U.S. Geological Survey

6. Caffeine. Hershey’s 6-oz milk chocolate almond barcontains 25 milligrams of caffeine. A 20-oz bottle ofCoca-Cola has 32 more milligrams of caffeine than theHershey bar. How many milligrams of caffeine does the20-oz bottle of Coca-Cola have? 57 milligramsSource: National Geographic, “Caffeine,” by T. R. Reid, January 2005

7. Carpentry. A carpenter drills 216 holes in a rectangulararray in a pegboard. There are 12 holes in each row.How many rows are there? 18 rows

8. Spreadsheets. Lou works as a CPA. He arranges 504 entries on a spreadsheet in a rectangular array thathas 36 rows. How many entries are in each row?

14 entries

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Bachelor’s Degree. The line graph below illustrates data about bachelor’s degrees awarded to men and women from 1970 to2003. Use this graph when answering Exercises 9–12.

75

Exercise Set 1.8

9. Find the total number of bachelor’s degrees awarded in1970 and the total number awarded in 2003.

792,316 degrees; 1,348,503 degrees

10. Determine how many more bachelor’s degrees wereawarded in 2003 than in 1970. 556,187 degrees

11. How many more bachelor’s degrees were awarded towomen than to men in 2003? 202,345 degrees

12. How many more bachelor’s degrees were awarded tomen than to women in 1970? 109,878 degrees

100,000

200,000

300,000

400,000

500,000

600,000

700,000

800,000

'03'95 '00'90'85'80'75'70Year

Nu

mb

er o

f deg

rees

Bachelor's Degrees

Source : U.S. Department of Education

451,097

Women Men

341,219

775,424

573,079

13. Median Mortgage Debt. The median mortgage debt in2004 was $48,388 more than the median mortgage debtin 1989. The debt in 1989 was $39,802. What was themedian mortgage debt in 2004? $88,190Source: Federal Reserve Board Survey of Consumer Finances

14. Olympics in Athens. In the first modern Olympics inAthens, Greece, in 1896, there were 43 events. In the2004 Summer Olympics, also in Athens, there were 258 more events than in 1896. How many events werethere in 2004?Source: USA Today research; The Olympic Games:Athens 1896–Athens 2004

301 events

15. Longest Rivers. The longest river in the world is theNile in Egypt at 4180 mi. The longest river in the UnitedStates is the Missouri–Mississippi–Red Rock at 3710 mi.How much longer is the Nile? 470 miSource: Time Almanac 2006

16. Speeds on Interstates. Recently, speed limits oninterstate highways in many Western states were raisedfrom 65 mph to 75 mph. By how many miles per hourwere they raised? 10 mph

17. There are 24 hr (hours) in a day and 7 days in a week.How many hours are there in a week? 168 hr

18. There are 60 min in an hour and 24 hr in a day. Howmany minutes are there in a day? 1440 min

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19. Crossword. The USA Today crossword puzzle is arectangle containing 15 rows with 15 squares in eachrow. How many squares does the puzzle havealtogether? 225 squares

20. Pixels. A computer screen consists of smallrectangular dots called pixels. How many pixels arethere on a screen that has 600 rows with 800 pixels ineach row? 480,000 pixels

Pixel1 2 3 4 5 6 7 8 9 10 11 12 13

14 15 16

17 18 19

20 21 22

23 24

25 26 27 28 29 30 31 32 33

34 35 36

37 38 39

40 41

42 43 4544

46 47

48 49 50 51 52 53 54 55

56 57 58

59 60 61

62 63 64

21. Refrigerator Purchase. Gourmet Deli has a chain of 24 restaurants. It buys a commercial refrigerator foreach store at a cost of $1019 each. Find the total cost ofthe purchase. $24,456

22. Microwave Purchase. Bridgeway College isconstructing new dorms, in which each room has asmall kitchen. It buys 96 microwave ovens at $88 each.Find the total cost of the purchase. $8448

23. “Seinfeld” Episodes. “Seinfeld” was a long-runningtelevision comedy with 177 episodes. A local stationpicks up the syndicated reruns. If the station runs 5 episodes per week, how many full weeks will passbefore it must start over with past episodes? How manyepisodes will be left for the last week?

35 weeks; 2 episodes left over

24. Laboratory Test Tubes. A lab technician separates a vialcontaining 70 cc (cubic centimeters) of blood into testtubes, each of which contains 3 cc of blood. How manytest tubes can be filled? How much blood is left over?

23 test tubes; 1 cc left over

25. Automobile Mileage. The 2005 Hyundai Tucson GLSgets 26 miles to the gallon (mpg) in highway driving.How many gallons will it use in 6136 mi of highwaydriving? 236 galSource: Hyundai

26. Automobile Mileage. The 2005 Volkswagen Jetta (5 cylinder) gets 24 miles to the gallon (mpg) in citydriving. How many gallons will it use in 3960 mi of citydriving? 165 galSource: Volkswagen of America, Inc.

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Boeing Jets. Use the chart below in Exercises 27–32 to compare the Boeing 747 jet with its main competitor, the Boeing 777 jet.

232 ft 209 ft

Source: Eclat Consulting; Boeing

BOEING 747–400

Passenger capacityNonstop flight distanceCruising speedGallons of fuel used per hour

Costs (to fly one hour):CrewFuel

4168826 miles

567 mph3201

$1948$2867

BOEING 777–200

Passenger capacityNonstop flight distanceCruising speedGallons of fuel used per hour

Costs (to fly one hour):CrewFuel

3685210 miles

615 mph2021

$1131$1816

77

Exercise Set 1.8

27. The nonstop flight distance of the Boeing 747 jet is howmuch greater than the nonstop flight distance of theBoeing 777 jet? 3616 mi

28. How much larger is the passenger capacity of theBoeing 747 jet than the passenger capacity of theBoeing 777 jet? 48 passengers

29. How many gallons of fuel are needed for a 4-hour flightof the Boeing 747? 12,804 gal

30. How much longer is the Boeing 747 than the Boeing 777? 23 ft

31. What is the total cost for the crew and fuel for a 3-hourflight of the Boeing 747 jet? $14,445

32. What is the total cost for the crew and fuel for a 2-hourflight of the Boeing 777 jet? $5894

33. Car Payments. Dana borrows $5928 for a used car. Theloan is to be paid off in 24 equal monthly payments.How much is each payment (excluding interest)?

$247

34. Loan Payments. A family borrows $7824 to build asunroom on the back of their home. The loan is to bepaid off in equal monthly payments of $163 (excludinginterest). How many months will it take to pay off the loan? 48 mo

35. High School Court. The standard basketball court usedby high school players has dimensions of 50 ft by 84 ft.

a) What is its area? 4200 sq ftb) What is its perimeter? 268 ft

36. NBA Court. The standard basketball court used bycollege and NBA players has dimensions of 50 ft by 94 ft.

a) What is its area? 4700 sq ftb) What is its perimeter? 288 ftc) How much greater is the area of an NBA court than a

high school court? (See Exercise 35.) 500 sq ft

84 ft

50 ft

94 ft 50 ft

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37. Clothing Imports and Exports. In the United States, theexports of clothing in 2003 totaled $2,596,000,000 whilethe imports totaled $31,701,000,000. How much morewere the imports than the exports?Source: U.S. Bureau of the Census, Foreign Trade Division

$29,105,000,000

38. Corn Imports and Exports. In the United States, theexports of corn in 2003 totaled $2,264,000,000 while theimports totaled $130,000,000. How much more were theexports than the imports? $2,134,000,000Source: U.S. Bureau of the Census, Foreign Trade Division

39. Colonial Population. Before the establishment of theU.S. Census in 1790, it was estimated that the Colonialpopulation in 1780 was 2,780,400. This was an increaseof 2,628,900 from the population in 1680. What was theColonial population in 1680? 151,500Source: Time Almanac, 2005

40. Deaths by Firearms. Deaths by firearms totaled 30,242in 2002. This was a decrease of 9353 from the deaths byfirearms in 1993. How many deaths by firearms werethere in 1993? 39,595 deathsSource: Centers for Disease Control and Prevention, National Center forHealth Statistics Mortality Report online, 2005

41. Hershey Bars. Hershey Chocolate USA makes small,fun-size chocolate bars. How many 20-bar packages can be filled with 11,267 bars? How many bars will beleft over? 563 packages; 7 bars left over

42. Reese’s Peanut Butter Cups. H. B. Reese Candy Co.makes small, fun-size peanut butter cups. The companymanufactures 23,579 cups and fills 1025 packages. Howmany cups are in a package? How many cups will be left over? 23 cups per package; 4 cups left over

43. Map Drawing. A map has a scale of 64 mi to the inch.How far apart in reality are two cities that are 6 in. aparton the map? How far apart on the map are two citiesthat, in reality, are 1728 mi apart? 384 mi; 27 in.

44. Map Drawing. A map has a scale of 150 mi to the inch.How far apart on the map are two cities that, in reality,are 2400 mi apart? How far apart in reality are two citiesthat are 13 in. apart on the map? 16 in.; 1950 mi

ILLINOIS

OHIOINDIANA

WV

VA

NCTENNESSEE

MISSOURI

ARKANSAS

LexingtonLouisville

64

64

55

75

65

40

0 150

Miles per inch

SOUTH DAKOTA

MINNESOTAMONTANA

CANADA

Bismarck Fargo

83

5

83 281

212

94

94

29

0 64

Miles per inch

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79

Exercise Set 1.8

45. Crossword. The Los Angeles Times crossword puzzle isa rectangle containing 441 squares arranged in 21 rows. How many columns does the puzzle have?

21 columns

46. Sheet of Stamps. A sheet of 100 stamps typically has 10 rows of stamps. How many stamps are in each row?

10 stamps

47. Copies of this book are generally shipped from thewarehouse in cartons containing 24 books each. Howmany cartons are needed to ship 1355 books?

56 full cartons; 11 books left over. If 1355 books areshipped, it will take 57 cartons.

48. According to the H. J. Heinz Company, 16-oz bottles ofcatsup are generally shipped in cartons containing 12 bottles each. How many cartons are needed to ship528 bottles of catsup?

44 cartons

49. Elena buys 5 video games at $64 each and pays for themwith $10 bills. How many $10 bills does it take?

32 $10 bills

50. Pedro buys 5 video games at $64 each and pays forthem with $20 bills. How many $20 bills does it take?

16 $20 bills

51. You have $568 in your bank account. You use your debitcard for $46, $87, and $129. Then you deposit $94 backin the account after the return of some books. Howmuch is left in your account? $400

52. The balance in your bank account is $749. You use yourdebit card for $34 and $65. Then you make a deposit of$123 from your paycheck. What is your new balance?

$773

Weight Loss. Many Americans exercise for weight control.It is known that one must burn off about 3500 calories inorder to lose one pound. The chart shown here details howmuch of certain types of exercise is required to burn 100calories. Use this chart for Exercises 53–56.

To burn off 100 calories,you must:

• Run for 8 minutes at a brisk pace, or• Swim for 2 minutes at a brisk pace, or• Bicycle for 15 minutes at 9 mph, or• Do aerobic exercises for 15 minutes.

53. How long must you run at a brisk pace in order to loseone pound? 280 min, or 4 hr 40 min

54. How long must you swim in order to lose one pound?

70 min, or 1 hr 10 min

55. How long must you do aerobic exercises in order to loseone pound? 525 min, or 8 hr 45 min

56. How long must you bicycle at 9 mph in order to loseone pound? 525 min, or 8 hr 45 min

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57. Bones in the Hands and Feet. There are 27 bones ineach human hand and 26 bones in each human foot.How many bones are there in all in the hands and feet?

106 bones

58. Subway Travel. The distance to Mars is about303,000,000 miles. The number of miles that people inthe United States traveled on subways in 2003approximately equaled 22 round trips to Mars. Whatwas the total distance traveled on subways?Source: American Public Transportation Association, NASA

13,332,000,000 mi59. Index Cards. Index cards of dimension 3 in. by 5 in. are

normally shipped in packages containing 100 cardseach. How much writing area is available if one uses thefront and back sides of a package of these cards?

3000 sq in.

60. An office for adjunct instructors at a community collegehas 6 bookshelves, each of which is 3 ft wide. The officeis moved to a new location that has dimensions of 16 ftby 21 ft. Is it possible for the bookshelves to be put sideby side on the 16-ft wall?

No, because 18 ft of wall is needed61. In the newspaper article “When Girls Play, Knees

Fail,” the author discusses the fact that female athleteshave six times the number of knee injuries that maleathletes have. What information would be needed if youwere to write a math problem based on the article?What might the problem be?Source: The Arizona Republic, 2/9/00, p. C1

62. Write a problem for a classmate to solve. Designthe problem so that the solution is “The driver still has329 mi to travel.”

DWDW

Round 234,562 to the nearest: [1.4a]

63. Hundred. 234,600 64. Ten. 234,560 65. Thousand. 235,000

Estimate the computation by rounding to the nearest thousand. [1.4b]

66. 22,000 67. 16,00028,430 � 11,9772783 � 4602 � 5797 � 8111

68. 8000 69. 40005800 � 21002100 � 5800

75. Speed of Light. Light travels about 186,000 (miles per second) in a vacuum as in outer space. In iceit travels about 142,000 , and in glass it travelsabout 109,000 . In 18 sec, how many more mileswill light travel in a vacuum than in ice? than in glass?

792,000 mi; 1,386,000 mi

76. Carney Community College has 1200 students. Eachprofessor teaches 4 classes and each student takes 5 classes. There are 30 students and 1 teacher in eachclassroom. How many professors are there at CarneyCommunity College? 50 professors

mi�secmi�sec

mi�sec

Estimate the product by rounding to the nearest hundred. [1.5b]

70. 320,000 71. 720,000 72. 46,800,00010,362 � 4531887 � 799787 � 363

SKILL MAINTENANCE

SYNTHESIS

73. Karen translates a problem into a multiplicationequation, whereas Don translates the same probleminto a division equation. Can they both be correct?Explain.

74. Of the five problem-solving steps listed at thebeginning of this section, which is the most difficult for you? Why?

DWDW

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Write exponential notation.

1.

2.

3.

4.

Evaluate.

5. 6.

7. 8.

Answers on page A-3

2583

102104

10 � 10 � 10 � 10

10 � 10

5 � 5 � 5 � 5 � 5

5 � 5 � 5 � 5

81


Writing Exponential Notation

Consider the product Such products occur often enough thatmathematicians have found it convenient to create a shorter notation, calledexponential notation, for them. For example,

is shortened to exponent

base4 factors

We read exponential notation as follows.

The wording “seven squared” for comes from the fact that a square withside s has area A given by

An expression like is read “three times five squared,” or “three timesthe square of five.”

EXAMPLE 1 Write exponential notation for

Exponential notation is 5 is the exponent.10 is the base.

EXAMPLE 2 Write exponential notation for

Exponential notation is

Do Exercises 1–4.

23.

2 � 2 � 2.

105.

10 � 10 � 10 � 10 � 10.

3 � 52

s

s

A � s 2

A � s2.72

343 � 3 � 3 � 3

3 � 3 � 3 � 3.

1.91.9 EXPONENTIAL NOTATION AND ORDER OF OPERATIONS

ObjectivesWrite exponential notationfor products such as

Evaluate exponentialnotation.

Simplify expressions usingthe rules for order ofoperations.

Remove parentheses withinparentheses.

4 � 4 � 4.

NOTATION WORD DESCRIPTION

“three to the fourth power,” or “the fourth power of three”

“five cubed,” or “the cube of five,” or “five to the third power,” or “the third power of five”

“seven squared,” or “the square of seven,” or “seven to the second power,” or “the second power of seven”

72

53

34

⎧ ⎪ ⎨ ⎪ ⎩

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Exponents are often used for units of area. For example, 10 square feetcan be written 10 Lengths are measured in units like ft, cm, and in. andareas are measured in units like and We can think of multiplyingunits like we multiply numbers. For example, the area of a square with sidesof length 5 cm can be thought of as

Knowing that we must multiply two units of length to get a unit of area canserve as a check when calculating the area of a figure.

Evaluating Exponential Notation

We evaluate exponential notation by rewriting it as a product and computingthe product.

EXAMPLE 3 Evaluate:

EXAMPLE 4 Evaluate:


Simplifying Expressions

Suppose we have a calculation like the following:

How do we find the answer? Do we add 3 to 4 and then multiply by 8, or do wemultiply 4 by 8 and then add 3? In the first case, the answer is 56. In the sec-ond, the answer is 35. We agree to compute as in the second case.

Consider the calculation

What do the parentheses mean? To deal with these questions, we must makesome agreement regarding the order in which we perform operations. Therules are as follows.

RULES FOR ORDER OF OPERATIONS

1. Do all calculations within parentheses ( ), brackets [ ], or braces { }before operations outside.

2. Evaluate all exponential expressions.3. Do all multiplications and divisions in order from left to right.4. Do all additions and subtractions in order from left to right.

It is worth noting that these are the rules that computers and most scien-tific calculators use to do computations.

7 � 14 � �12 � 18�.

3 � 4 � 8.

54 � 5 � 5 � 5 � 5 � 625

54.

103 � 10 � 10 � 10 � 1000

103.

� 25 cm2.

� �5 � 5� � �cm � cm� � �5 cm� � �5 cm�

Area � s2 � �5 cm�2

in2.cm2,ft2,ft2.

Simplify.

9.

10.

11.

12.

Answers on page A-3

75 5 � �83 � 14�

25 � 26 � �56 � 10�

104 4 � 4

93 � 14 � 3

82


Caution!

does not mean 10 � 3.103

CALCULATORCORNER

Exponential NotationMany calculators have a or key for raising a base toa power. To find , forexample, we press

or .The result is 4096.

Exercises: Use a calculatorto find each of the following.

1. 243

2. 15,625

3. 20,736

4. 2048211

124

56

35

�3xy61�3

yx61

163xy

yx

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EXAMPLE 5 Simplify:

There are no parentheses or exponents, so we start with the third step.

Doing all multiplications and divisions in order from left to right

EXAMPLE 6 Simplify:

Carrying out operations inside parentheses

Doing all multiplications and divisions

Doing all additions and subtractions


EXAMPLE 7 Simplify and compare: and

We have

We can see that and represent different num-bers. Thus subtraction is not associative.


EXAMPLE 8 Simplify:

Carrying outoperations insideparentheses

Doing all multiplicationsand divisions in order fromleft to right

Doing all additions andsubtractions in order fromleft to right

Do Exercise 15.

EXAMPLE 9 Simplify:

Carrying out operationsinside parentheses



� 5

� 10 2

� 5 � 2 2

15 3 � 2 �10 � 8� � 15 3 � 2 2

15 3 � 2 �10 � 8�.

� 7

� 14 � 4 � 3

7 � 2 � �12 � 0� 3 � �5 � 2� � 7 � 2 � 12 3 � 3

7 � 2 � �12 � 0� 3 � �5 � 2�.

�23 � 10� � 923 � �10 � 9�

�23 � 10� � 9 � 13 � 9 � 4.

23 � �10 � 9� � 23 � 1 � 22;

�23 � 10� � 9.23 � �10 � 9�

� 68

� 98 � 30

7 � 14 � �12 � 18� � 7 � 14 � 30

7 � 14 � �12 � 18�.

� 4

16 8 � 2 � 2 � 2

16 8 � 2. Simplify and compare.

13. and

14. and

15. Simplify:

Simplify.

16.

17.

18.

Answers on page A-3

95 � 2 � 2 � 2 � 5 �24 � 4�

� 2330 5 � 2 � 10 � 20 � 8 � 8

� �16 � 25 � 3�5 � 5 � 5 � 26 � 71

9 � 4 � �20 � 4� 8 � �6 � 2�.

28 � �13 � 11��28 � 13� � 11

�64 32� 264 �32 2�

83


⎫⎪⎬⎪⎭

⎫⎬⎭

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EXAMPLE 10 Simplify:

Subtracting inside parentheses

Adding inside parentheses

Evaluating exponential expressions

Doing all multiplications and divisionsin order from left to right

Subtracting



There are no parentheses. Evaluatingexponential expressions

Doing all multiplications and divisions inorder from left to right

Do Exercise 22.

� 32

� 4 � 8

26 24 � 23 � 64 16 � 8

26 24 � 23.

� 1

� 6 � 5

� 2 � 3 � 5

� 16 8 � 3 � 5

� 42 23 � 3 � 5

� 42 �1 � 1�3 � 3 � 5

42 �10 � 9 � 1�3 � 3 � 5

42 �10 � 9 � 1�3 � 3 � 5.Simplify.

19.

20.

21.

22. Simplify:

Answers on page A-3

23 � 25 26.

81 � 32 � 2 �12 � 9�

�1 � 3�3 � 10 � 20 � 82 � 23

53 � 26 � 71 � �16 � 25 � 3�

84


⎫⎬⎭

⎫⎬⎭

CALCULATOR CORNER

Order of Operations To determine whether a calculator isprogrammed to follow the rules for order of operations, we can enter asimple calculation that requires using those rules. For example, we enter

. If the result is 11, we know that the rules fororder of operations have been followed. That is, the multiplication

was performed first and then 3 was added to produce a resultof 11. If the result is 14, we know that the calculator performs operationsas they are entered rather than following the rules for order of operations.That means, in this case, that 3 and 4 were added first to get 7 and thenthat sum was multiplied by 2 to produce the result of 14. For suchcalculators, we would have to enter the operations in the order in whichwe want them performed. In this case, we would press

.Many calculators have parenthesis keys that can be used to enter an

expression containing parentheses. To enter for example, wepress . The result is 35.

Exercises: Simplify.

1. 49 2. 85

3. 36 4. 0

5. 73 6. 49�4 � 3�215 � 7 � �23 � 9�44 64 � 432 � 92 3

80 � 50 1084 � 5 � 7

�)3�4(5

5�4 � 3�,

�3

�2�4

4 � 2 � 8

�2�4�3

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AVERAGESIn order to find the average of a set of numbers, we use addition and thendivision. For example, the average of 2, 3, 6, and 9 is found as follows.

The number of addends is 4.

Divide by 4.

The fraction bar acts as a pair of grouping symbols so

is equivalent to

Thus we are using order of operations when we compute an average.

AVERAGE

The average of a set of numbers is the sum of the numbers divided bythe number of addends.

EXAMPLE 12 Average Number of Career Hits. The number of career hitsof five Hall of Fame baseball players are given in the bar graph below. Find theaverage number of career hits of all five.

The average is given by

Thus the average number of career hits of these five Hall of Fame players is 3118.

Do Exercise 23.

Removing Parentheses within Parentheses

When parentheses occur within parentheses, we can make them differentshapes, such as [ ] (also called “brackets”) and { } (also called “braces”). All ofthese have the same meaning. When parentheses occur within parentheses,computations in the innermost ones are to be done first.

3419 � 3255 � 3110 � 2930 � 28765

�15,590

5� 3118.

Career Hits

3419

Mel Ott

3255

Jake Beckley

3110Dave Winfield

2930

Eddie Murray

2876

Carl Yastrzemski

Sources: Associated Press; Major League Baseball

�2 � 3 � 6 � 9� 4.2 � 3 � 6 � 9

4

Average �2 � 3 � 6 � 9

4�

204

� 5

23. World’s Tallest Buildings. Theheights, in feet, of the fourtallest buildings in the world aregiven in the bar graph below.Find the average height of thesebuildings.

Answer on page A-3

Taip

ei 1

01, T

aipe

i,

Taiw

an

Petr

onas

Tow

ers 1

& 2

,

Kual

a Lu

mpu

r, M

alay

sia

Sear

s Tow

er, C

hica

goJin

Mao

Bui

ldin

g,

Shan

ghai

World’s Tallest Buildings

Hei

ght (

in fe

et)

1300

12001100

1000

14001500

1600

17001800

1670

1483 14501381

Source: Council on Tall Buildings and Urban Habitat,Lehigh University, 2004

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Doing the calculations in theinnermost parentheses first

Doing the multiplication in thebrackets

Subtracting

Dividing


Doing the calculations in theinnermost parentheses first

Again, doing the calculations inthe innermost parentheses

Subtracting inside the braces


Adding


� 41

� 8 � 33

� 16 2 � 33

� 16 2 � �40 � 7�

� 16 2 � �40 � �13 � 6�16 2 � �40 � �13 � �4 � 2��

16 2 � �40 � �13 � �4 � 2��.

� 1

� 4 4

� �25 � 21 �11 � 7�

� �25 � 7 � 3 �11 � 7��25 � �4 � 3� � 3 �11 � 7�

�25 � �4 � 3� � 3 �11 � 7�.Simplify.

24.

25.

Answers on page A-3

� �31 � 10 � 2��18 � �2 � 7� 3

9 � 5 � �6 �14 � �5 � 3��

Study Tips TEST PREPARATION

� Make up your own test questions as you study. After you have done your homeworkover a particular objective, write one or two questions on your own that you thinkmight be on a test. You will be amazed at the insight this will provide.

� Do an overall review of the chapter, focusing on the objectives and the examples. Thisshould be accompanied by a study of any class notes you may have taken.

� Do the review exercises at the end of the chapter. Check your answers at the back ofthe book. If you have trouble with an exercise, use the objective symbol as a guide togo back and do further study of that objective.

� Call the AW Math Tutor Center at 1-888-777-0463 if you need extra help.

� Take the chapter test at the end of the chapter. Check the answers and use theobjective symbols at the back of the book as a reference for where to review.

� Ask former students for old exams. Working such exams can be very helpful andallows you to see what various professors think is important.

� When taking a test, read each question carefully and try to do all the questions thefirst time through, but pace yourself. Answer all the questions, and mark those torecheck if you have time at the end. Very often, your first hunch will be correct.

� Try to write your test in a neat and orderly manner. Very often, your instructor tries togive you partial credit when grading an exam. If your test paper is sloppy anddisorderly, it is difficult to verify the partial credit. Doing your work neatly can easesuch a task for the instructor.

You are probably readyto begin preparing

for your first test. Hereare some test-taking

study tips.

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Write exponential notation.

87

Exercise Set 1.9

EXERCISE SET For Extra Help1.91.9 Student’sSolutionsManual


Math TutorCenter

InterActMath

MyMathLabMathXL

1. 2. 3. 4. 13313 � 13 � 13525 � 5252 � 2 � 2 � 2 � 2343 � 3 � 3 � 3

5. 6. 7. 8. 141 � 1 � 1 � 110310 � 10 � 1010210 � 10757 � 7 � 7 � 7 � 7

Evaluate.

9. 49 10. 125 11. 729 12. 100102935372

13. 20,736 14. 100,000 15. 121 16. 21663112105124

Simplify.

17. 22 18. 36 19. 2052 � �40 � 8��12 � 6� � 1812 � �6 � 4�

20. 4 21. 100 22. 1�1000 100� 101000 �100 10��52 � 40� � 8

23. 1 24. 16 25. 49�2 � 5�2256 �64 4��256 64� 4

26. 29 27. 5 28. 8125�32 � 27�3 � �19 � 1�3�11 � 8�2 � �18 � 16�222 � 52

29. 434 30. 383 31. 4183 � 7 � 623 � 18 � 2016 � 24 � 50

32. 66 33. 88 34. 4090 � 5 � 5 � 210 � 10 � 3 � 410 � 7 � 4

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35. 4 36. 48 37. 30317 � 20 � �17 � 20�82 � 8 � 243 8 � 4

38. 20 39. 20 40. 643 � 8 � 5 � 86 � 10 � 4 � 101000 25 � �15 � 5�

41. 70 42. 34 43. 2953 � �2 � 8�2 � 5 � �4 � 3�2144 4 � 2300 5 � 10

44. 311 45. 32 46. 3362 � 34 3342 � 82 227 � �10 � 3�2 � 2 � �3 � 1�2

47. 906 48. 83 49. 626 � 11 � �7 � 3� 5 � �6 � 4�72 � 20 � 4 � �28 � 9 � 2�103 � 10 � 6 � �4 � 5 � 6�

53. 32 54. 1627 25 � 24 2223 � 28 26

55. Find the average of $64, $97, and $121.

$94

56. Find the average of four test grades of 86, 92, 80, and 78.

84

Simplify.

59. 110 60. 1072 6 � �2 � �9 � �4 � 2��8 � 13 � �42 �18 � �6 � 5��

61. 7 62. 58�92 � �6 � 4� 8 � �7 � �8 � 3��14 � �3 � 5� 2 � �18 �8 � 2�

57. Find the average of 320, 128, 276, and 880.

401

58. Find the average of $1025, $775, $2062, $942, and $3721.

$1705

50. 688 � 9 � �12 � 8� 4 � �10 � 7� 51. 102120 � 33 � 4 �5 � 6 � 6 � 4� 52. 6880 � 24 � 15 �7 � 5 � 45 3�

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89

Exercise Set 1.9

63. 544 64. 9�18 2� � ��9 � 9 � 1� 2 � �5 � 20 � �7 � 9 � 2��82 � 14� � ��10 � 45 5� � �6 � 6 � 5 � 5�

69. Consider the problem in Example 8 of Section 1.8.How can you translate the problem to a single equationinvolving what you have learned about order ofoperations? How does the single equation relate to howwe solved the problem?

70. Consider the expressions and .Are the parentheses necessary in each case? Explain.

�3 � 4�29 � �4 � 2�DWDW

67. 27��18 � 2 � 6 � �40 �17 � 9�� 48 � 13 � 3 � ��50 � 7 � 5� � 2�

68. 234�19 � 24�5 � �141 47�2

71. 452 72. 835 73. 13 74. 371554 � 42 � y7 � x � 914197 � x � 5032x � 341 � 793

75. 2342 76. 77. 25 78. 10010,000 � 100 � t25 � t � 62548986000 � 1102 � t3240 � y � 898

Solve. [1.8a]

Solve. [1.7b]

79. Colorado. The state of Colorado is roughly the shapeof a rectangle that is 273 mi by 382 mi. What is its area?

104,286

80. On a four-day trip, a family bought the followingamounts of gasoline for their motor home:

23 gallons, 24 gallons,26 gallons, 25 gallons.

How much gasoline did they buy in all? 98 gal

mi2

Each of the answers in Exercises 83–85 is incorrect. First find the correct answer. Then place as many parentheses as needed inthe expression in order to make the incorrect answer correct.

83. 84. 7; 12 �4 � 2� �3 � 2� � 212 4 � 2 � 3 � 2 � 224; 1 � 5 � �4 � 3� � 361 � 5 � 4 � 3 � 36

85. 7; 12 �4 � 2� � 3 � 2 � 412 4 � 2 � 3 � 2 � 4

86. Use one occurrence each of 1, 2, 3, 4, 5, 6, 7, 8, and 9 and any of the symbols �, �, �, , and ( ) to represent 100.

�1 � 2 � 3� � �6 � 7� � �4 � 5� � �8 � 9� � 100

SKILL MAINTENANCE

SYNTHESIS

65. 7084 � ��200 � 50 5� � ��35 7� � �35 7� � 4 � 3� 66. 67515�23 � 4 � 2�3 �3 � 25�

81. Is it possible to compute an average of severalnumbers on a calculator without using parentheses orthe key?

82. Is the average of two sets of numbers the same asthe average of the two averages? Why or why not?DW

�

DW

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Review Exercises

1. What does the digit 8 mean in 4,678,952? [1.1a]

8 thousands

Write standard notation. [1.1c]

9. Four hundred seventy-six thousand, five hundredeighty-eight 476,588

10. e-books. The publishing industry predicted that salesof digital books would reach two billion, four hundredthousand by 2005. 2,000,400,000Source: Andersen Consulting

The review that follows is meant to prepare you for a chapter exam. It consists of two parts. The first part, ConceptReinforcement, is designed to increase understanding of the concepts through true/false exercises. The second part isthe Review Exercises. These provide practice exercises for the exam, together with references to section objectives soyou can go back and review. Before beginning, stop and look back over the skills you have obtained. What skills inmathematics do you have now that you did not have before studying this chapter?

90


Summary and Review11

i CONCEPT REINFORCEMENT

Determine whether the statement is true or false. Answers are given at the back of the book.

1. The product of two natural numbers is always greater than either of the factors. False

2. Zero divided by any nonzero number is 0. True

3. Each member of the set of natural numbers is a member of the set of whole numbers. True

4. The sum of two natural numbers is always greater than either of the addends. True

5. Any number divided by 1 is the number 1. False

6. The number 0 is the smallest natural number. False

The review exercises that follow are for practice. Answers are given at the back of the book. If you miss an exercise, restudy theobjective indicated in red next to the exercise or direction line that precedes it.

2. In 13,768,940, what digit tells the number of millions?[1.1a] 3

Write expanded notation. [1.1b]

3. 2793

2 thousands 7 hundreds9 tens 3 ones�

��

4. 56,078

5. 4,007,101 4 millions 0 hundred thousands0 ten thousands 7 thousands 1 hundred 0 tens1 one, or 4 millions 7 thousands 1 hundred 1 one��

��

Write a word name. [1.1c]

6. 67,819 Sixty-seventhousand, eight hundrednineteen

7. 2,781,427 Two million,seven hundred eighty-one thousand,four hundred twenty-seven

8. 1,065,070,607, the population of India in 2004.Source: U.S. Bureau of the Census, International Database

1 billion, sixty-five million, seventy thousand, six hundred seven

4. 5 ten thousands 6 thousands 0 hundreds 7 tens8 ones, or 5 ten thousands 6 thousands 7 tens 8 ones��

��

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91

Summary and Review: Chapter 1

Add. [1.2b]

11. 14,2727304 � 6968

35. Write a related multiplication sentence: [1.6b]

or 56 � 7 � 856 � 8 � 7,

56 8 � 7.

36. Write two related division sentences: [1.6b]

13 � 52 44 � 52 13;

13 � 4 � 52.

Divide. [1.6c]

37. 12 R 3 38. 580 1663 5

39. 913 R 3 40. 384 R 13073 87 � 6 3 9 4

41. 4 R 46 42. 544266 796 0 � 2 8 6

43. 452 44. 50081 4 � 7 0 , 1 1 23 8 � 1 7 , 1 7 6

45. 438952,668 12

Solve. [1.7b]

46. 8 47. 4547 � x � 9246 � n � 368

48. 58 49. 024 � x � 241 � y � 58

50. Write exponential notation: [1.9a] 434 � 4 � 4.

Evaluate. [1.9b]

51. 10,000 52. 3662104

Simplify. [1.9c, d]

53. 658 � 6 � 17

54. 23310 � 24 � �18 � 2� 4 � �9 � 7�

55. 567 � �4 � 3�2

56. 327 � 42 � 32

57. 260�80 16� � ��20 � 56 8� � �8 � 8 � 5 � 5�

58. Find the average of 157, 170, and 168. 165

12. 66,02427,609 � 38,415

13. 21,7882703 � 4125 � 6004 � 8956

14.

98,921

� 9 7 , 4 9 59 1 , 4 2 6

15. Write a related addition sentence: [1.3b]

or 10 � 4 � 610 � 6 � 4,

10 � 6 � 4.

16. Write two related subtraction sentences: [1.3b]

3 � 11 � 88 � 11 � 3;

8 � 3 � 11.

Subtract. [1.3d]

17. 51488045 � 2897

18. 16899001 � 7312

19. 22746003 � 3729

20.

17,757

� 1 9 , 6 4 8� 3 7 , 4 0 5

Round 345,759 to the nearest: [1.4a]

21. Hundred. 345,800 22. Ten. 345,760

23. Thousand. 346,000 24. Hundred thousand.

300,000

Estimate the sum, difference, or product by first rounding tothe nearest hundred. Show your work. [1.4b], [1.5b]

25. 26.

38,700 � 24,500 � 14,200

38,652 � 24,549

41,300 � 19,700 � 61,000

41,348 � 19,749

27. 400 � 700 � 280,000396 � 748

Use � or � for to write a true sentence. [1.4c]

28. 67 56 � 29. 1 23 �

Multiply. [1.5a]

30. 5,100,000 31. 6,276,8007846 � 80017,000 � 300

32. 506,748 33. 27,589587 � 47726 � 698

34.

5,331,810

� 8 6 4 28 3 0 5

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Solve. [1.8a]

59. Workstation. Natasha has $196 and wants to buy acomputer workstation for $698. How much more doesshe need? $502

60. Taneesha has $406 in her checking account. She is paid$78 for a part-time job and deposits that in herchecking account. How much is then in her account?

$484

61. Lincoln-Head Pennies. In 1909, the first Lincoln-headpennies were minted. Seventy-three years later, thesepennies began to be minted with a decreased coppercontent. In what year was the copper content reduced?

1982

62. A beverage company packed 228 cans of soda into 12-cancartons. How many cartons did they fill? 19 cartons

63. An apple farmer keeps bees in her orchard to helppollinate the apple blossoms so more apples will beproduced. The bees from an average beehive canpollinate 30 surrounding trees during one growingseason. A farmer has 420 trees. How many beehives does she need to pollinate them all? 14 beehivesSource: Jordan Orchards, Westminster, PA

64. An apartment builder bought 13 gas stoves at $425 each and 13 refrigerators at $620 each. What was thetotal cost? $13,585

65. A family budgets $7825 for food and clothing and $2860for entertainment. The yearly income of the family was$38,283. How much of this income remained after thesetwo allotments? $27,598

66. A chemist has 2753 mL of alcohol. How many 20-mLbeakers can be filled? How much will be left over?

137 beakers filled; 13 mL left over

68. Write a problem for a classmate to solve. Designthe problem so that the solution is “Each of the144 bottles will contain 8 oz of hot sauce.” [1.8a]

A vat contains 1152 oz of hot sauce. If 144 bottles are to befilled equally, how much will each bottle contain? Answersmay vary.

DW

69. Is subtraction associative? Why or why not? [1.3d]

Answer to Exercise 69 can be found on page IA-1.

DW

67. Olympic Trampoline. Shown below is an Olympictrampoline. Find the area and the perimeter of thetrampoline. [1.2c], [1.5c] 98 42 ftSource: International Trampoline Industry Association, Inc.

14 ft

7 ft

ft2;

SYNTHESIS

70. Determine the missing digit d. [1.5a]

d � 8 8 0 3 6

� d 2 9 d

71. Determine the missing digits a and b. [1.6c]

b � 4a � 8, 2 b 1 � 2 3 6 , 4 2 1

9 a 1

72. A mining company estimates that a crew must tunnel2000 ft into a mountain to reach a deposit of copper ore.Each day the crew tunnels about 500 ft. Each nightabout 200 ft of loose rocks roll back into the tunnel.How many days will it take the mining company toreach the copper deposit? [1.8a] 6 days

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1. In the number 546,789, which digit tells the number of hundred thousands?

93

Test: Chapter 1

Chapter Test11

2. Write expanded notation: 8843. 3. Write a word name: 38,403,277.

Solve.

20. Hostess Ding Dongs®. Hostess packages its DingDong® snack products in 12-packs. It manufactures22,231 cakes. How many 12-packs can it fill? How manywill be left over?

21. Largest States. The following table lists the five largeststates in terms of their land area. Find the total landarea of these states.

22. Pool Tables. The Hartford™ pool table made byBrunswick Billiards comes in three sizes of playing area,50 in. by 100 in., 44 in. by 88 in., and 38 in. by 76 in.

a) Find the perimeter and the area of the playing area ofeach table.

b) By how much playing area does the large tableexceed the small table?

Source: Brunswick Billiards

23. Voting Early. In the 2004 Presidential Election, 345,689Nevada voters voted early. This was 139,359 more thanin the 2000 election. How many voted early in Nevada in 2000?Source: National Association of Secretaries of State

Add.

4. 5. 6. 7.� 4 3 1 2

6 2 0 3

� 4738

1 2� 1 7 , 9 0 2

4 5 , 8 8 9� 3 1 7 8

6 8 1 1

Subtract.

8. 9. 10. 11.� 1 7 , 8 9 2

2 3 , 0 6 7� 2 0 5 9

8 9 0 7� 1 9 3 5

2 9 7 4� 4 3 5 3

7 9 8 3

Multiply.

12. 13. 14. 15.� 7 8 8

6 7 8� 3 7

6 5� 6 0 0

8 8 7 6� 9

4 5 6 8

Divide.

16. 17. 18. 19. 4 4 � 3 5 , 4 2 88 9 � 8 6 3 3420 615 4

Alaska 571,951

Texas 261,797

California 155,959

Montana 145,552

New Mexico 121,356

STATEAREA

(In Square Miles)

Source: U.S. Department of Commerce, U.S. Bureau of the Census

Work It Out!Chapter Test Video

on CD

For Extra Help

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24. A sack of oranges weighs 27 lb. A sack of apples weighs32 lb. Find the total weight of 16 bags of oranges and43 bags of apples.

25. A box contains 5000 staples. How many staplers can befilled from the box if each stapler holds 250 staples?

Solve.

26. 27. 28. 29. 381 � 0 � a38 � y � 532169 13 � n28 � x � 74

Round 34,578 to the nearest:

30. Thousand. 31. Ten. 32. Hundred.

Estimate the sum, difference, or product by first rounding to the nearest hundred. Show your work.

33. 34. 35.� 4 8 9

8 2 4� 2 3 , 6 4 9

5 4 , 7 5 1� 5 4 , 7 4 6

2 3 , 6 4 9

Use � or � for to write a true sentence.

36. 34 17 37. 117 157

38. Write exponential notation: 12 � 12 � 12 � 12.

Evaluate.

39. 40. 41. 25210573

Simplify.

42. 43. 44. �25 � 15� 5102 � 22 235 � 1 � 28 4 � 3

45. 46. 24 � 24 128 � ��20 � 11� � ��12 � 48� 6 � �9 � 2��

47. Find the average of 97, 98, 87, and 86.

48. An open cardboard container is 8 in. wide, 12 in. long,and 6 in. high. How many square inches of cardboardare used?

49. Use trials to find the single-digit number a for which

359 � 46 � a 3 � 25 � 72 � 339.

50. Cara spends $229 a month to repay her student loan. Ifshe has already paid $9160 on the 10-yr loan, how manypayments remain?

51. Jennie scores three 90’s, four 80’s, and a 74 on her eightquizzes. Find her average.

SYNTHESIS

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Whole Numbers - Columbia Southern University · CHAPTER 1: Whole Numbers ... Copyright © 2008 by Pearson Education, Inc. EXAMPLE 7 American Red Cross. In 2003, ... and Canada Source: