Whole Numbers and Introduction to Algebra - Fullerton …staff · Then we have to master the reasoning, or logic, ... shortcuts. In this book, watch ... 4 Chapter 1 Whole Numbers

Whole Numbers andIntroduction to Algebra1.1 UNDERSTANDING WHOLE NUMBERS 2

1.2 ADDING WHOLE NUMBER EXPRESSIONS 10

1.3 SUBTRACTING WHOLE NUMBER EXPRESSIONS 23

1.4 MULTIPLYING WHOLE NUMBER EXPRESSIONS 33

1.5 DIVIDING WHOLE NUMBER EXPRESSIONS 44

1.6 EXPONENTS AND THE ORDER OF OPERATIONS 55

HOW AM I DOING? SECTIONS 1.1–1.6 63

1.7 MORE ON ALGEBRAIC EXPRESSIONS 64

1.8 INTRODUCTION TO SOLVING LINEAR EQUATIONS 69

1.9 SOLVING APPLIED PROBLEMS USING SEVERAL OPERATIONS 82

CHAPTER 1 ORGANIZER 92

CHAPTER 1 REVIEW PROBLEMS 95

HOW AM I DOING? CHAPTER 1 TEST 101

Should I buy or lease a car?What are the benefits of

each? Which choice fits myneeds? Does leasing a car or

buying a car save me themost money? Turn to Putting

Your Skills to Work on page62 to find out.

1

CHAPTER

M01_TOBE7935_06_SE_C01.QXD 9/24/08 5:03 PM Page 1

2

Often we learn a new concept in stages. First comes learning the new terms andbasic assumptions. Then we have to master the reasoning, or logic, behind the newconcept. This often goes hand in hand with learning a method for using the idea.Finally, we can move quickly with a shortcut.

For example, in the study of stock investments, before tackling the question“What is my profit from this stock transaction?” you must learn the meaning of suchterms as stock, profit, loss, and commission. Next, you must understand how stockswork (reasoning/logic) so that you can learn the method for calculating your profit.After you master this concept, you can quickly answer many similar questions usingshortcuts.

In this book, watch your understanding of mathematics grow through thissame process. In the first chapter we review the whole numbers, emphasizingconcepts, not shortcuts. Do not skip this review even if you feel you have masteredthe material since understanding each stage of the concepts is crucial to learningalgebra. With a little patience in looking at the terms, reasoning, and step-by-stepmethods, you’ll find that your understanding of whole numbers has deepened,preparing you to learn algebra.

Understanding Place Values of Whole NumbersWe use a set of numbers called whole numbers to count a number of objects.

The whole numbers are as follows:

There is no largest whole number. The three dots indicate that the set of wholenumbers goes on forever. The numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called digits.The position or placement of the digit in a number tells the value of the digit. Forthis reason, our number system is called a place-value system. For example, look atthe following three numbers.

632 The “6” means 6 hundreds (600).61 The “6” means 6 tens (60).6 The “6” means 6 ones (6).

To illustrate the values of the digits in a number, we can use the followingplace-value chart. Consider the number 847,632, which is entered on the chart.

Á

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, Á

, ,

Hundre

d mill

ions

Ten m

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Mill

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Hundre

d thousa

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Ten th

ousands

Thousands

Hundre

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48 7 6 3 2

Place Value Chart

Periods Millions Thousands Ones

Student LearningObjectivesAfter studying this section, you willbe able to:

Understand place valuesof whole numbers.

Write whole numbers inexpanded notation.

Write word names for whole numbers.

Use inequality symbols with whole numbers.

Round whole numbers.

1.1 UNDERSTANDING WHOLE NUMBERS

The digit 8 is in the hundred thousands place.The digit 4 is in the ten thousands place.The digit 7 is in the thousands place.The digit 6 is in the hundreds place.The digit 3 is in the tens place.The digit 2 is in the ones place.

When we write very large numbers, we place a comma after every group ofthree digits, moving from right to left. These three-digit groups are called periods. Itis usually agreed that four-digit numbers do not have a comma, but numbers withfive or more digits do.


Section 1.1 Understanding Whole Numbers 3

NOTE TO STUDENT: Fully worked-outsolutions to all of the Practice Problemscan be found at the back of the textstarting at page SP-1

Writing Whole Numbers in Expanded NotationWe sometimes write numbers in expanded notation to emphasize place value. Thenumber 47,632 can be written in expanded notation as follows:

4 ten + 7 + 6 + 3 + 2thousands thousands hundreds tens ones

2+30+600+7000+40,000

Teaching Example 1 In the number63,752,481:

(a) In what place is the digit 6?

(b) In what place is the digit 2?

Ans: (a) ten millions (b) thousands

EXAMPLE 1 In the number 573,025:

(a) In what place is the digit 7? (b) In what place is the digit 0?

Solution

(a)

ten thousands

(b)

hundredsc

573, 0 25c

5 7 3,025

Practice Problem 1 In the number 3,502,781:

(a) In what place is the digit 5? hundred thousands

(b) In what place is the digit 0? ten thousands

EXAMPLE 2 Write 1,340,765 in expanded notation.

SolutionWe write 1 followed by a zero for each of the remaining digits.

We write 1 ,340,765 as We continue in this manner for each digit.

Since there is a zero in the thousands place, we do not write it as part of the sum.

Practice Problem 2 Write 2,507,235 in expanded notation.

2,000,000 + 500,000 + 7000 + 200 + 30 + 5

1 ,000,000 + 3 00,000 + 4 0,000 + 7 00 + 6 0 + 5 T

EXAMPLE 3 Jon withdraws $493 from his account. He requests the mini-mum number of bills in one-, ten-, and hundred-dollar bills. Describe the quantityof each denomination of bills the teller must give Jon.

Solution If we write $493 in expanded notation, we can easily describe thedenominations needed.

4 9 3400 + 90 + 3

hundred-dollarbills

ten-dollarbills

one-dollarbills

Practice Problem 3 Christina withdraws $582 from her account. Sherequests the minimum number of bills in one-, ten-, and hundred-dollar bills.Describe the quantity of each denomination of bills the teller must give Christina.

5 hundred-dollar bills, 8 ten-dollar bills, 2 one-dollar bills

Teaching Example 2 Write 509,637 inexpanded notation.

Ans: 500,000 + 9000 + 600 + 30 + 7

Teaching Example 3 Tom withdraws$642 from his account. He requests theminimum number of bills in one-, ten-, andhundred-dollar bills. Describe the quantityof each denomination of bills the tellermust give Tom.

Ans: 6 hundred-dollar bills, 4 ten-dollarbills, 2 one-dollar bills


Writing Word Names for Whole NumbersSixteen, twenty-one, and four hundred five are word names for the numbers 16, 21,and 405. We use a hyphen between words when we write a two-digit number greaterthan twenty. To write a word name, start from the left. Name the number in eachperiod, followed by the name of the period, and a comma. The last period name,“ones,” is not used.

4 Chapter 1 Whole Numbers and Introduction to Algebra

Understanding the ConceptThe Number ZeroNot all number systems have a zero. The Roman numeral system does not. Inour place-value system the zero is necessary so that we can write a number suchas 308. By putting a zero in the tens place, we indicate that there are zero tens.Without a zero symbol we would not be able to indicate this. For example, 38has a different value than 308. The number 38 means three tens and eight ones,while 308 means three hundreds and eight ones. In this case, we use zero as aplaceholder. It holds a position and shows that there is no other digit in that place.

Teaching Example 4 Write a word name foreach number.

(a) 73,024 (b) 2,462,108

Ans:(a) Seventy-three thousand, twenty-four

(b) Two million, four-hundred sixty-twothousand, one hundred eight

CAUTION: We should not use the word and in the word names for whole numbers.Although we may hear the phrase “three hundred and two” for the number 302, it isnot technically correct. As we will see later in the book, we use the word and for thedecimal point when using decimal notation.

Using Inequality Symbols with Whole NumbersIt is often helpful to draw pictures and graphs to help us visualize a mathematicalconcept. A number line is often used for whole numbers. The following number linehas a point matched with zero and with each whole number. Each number is equallyspaced, and the arrow at the right end indicates that the numbers go on forever.The numbers on the line increase from left to right.

“:”


1 2 3 4 7650

EXAMPLE 4 Write a word name for each number.

(a) 2135 (b) 300,460

Solution Look at the place-value chart on page 2 if you need help identifyingthe period.

(a) 2135 The number begins with 2 in the thousands place. The word name is

(b) 300,460 The number begins with 3 in the hundred thousands place.

Practice Problem 4 Write a word name for each number.

(a) 4006 four thousand, six (b) 1,220,032one million, two hundred twentythousand, thirty-two

We place a comma here to match the comma in the number.c

Three hundred thousand, four hundred sixty

We use a hyphen here.c

two thousand, one hundred thirty-five.

If one number lies to the right of a second number on the number line, itis greater than that number.

4 lies to the right of 2 on the number line because 4 is greater than 2.

1 2 3 4 650



A number is less than a given number if it lies to the left of that number on thenumber line.

3 lies to the left of 5 on the number line because 3 is less than 5.

The symbol means is greater than, and the symbol means is less than.Thus we can write

4 is greater than 2. 3 is less than 5.

The symbols and are called inequality symbols. The statements andare both correct. Note that the inequality symbol always points to the smaller

number.2 6 4

4 7 276

TT

4 7 2 3 6 5

67

1 2 3 4 650

14,792

13,000 14,000 15,000 16,000

EXAMPLE 5 Replace each question mark with the inequality symbol or 7 .6

(a) 1 ? 6 (b) 8 ? 7 (c) 4 ? 9 (d) 9 ? 4

Solution

(a)

1 is less than 6.

(b)

8 is greater than 7.

(c)

4 is less than 9.

(d)

9 is greater than 4.T

9 7 4T

4 6 9T

8 7 7T

1 6 6

Practice Problem 5 Replace each question mark with the inequality sym-bol or 7 .6

(a) 3 ? 2 (b) 6 ? 8 (c) 1 ? 7 (d) 7 ? 1 7667

Teaching Example 5 Replace each questionmark with the inequality symbol or .

(a) 5 ? 9 (b) 3 ? 0

(c) 0 ? 3 (d) 11 ? 6

Ans:(a) (b)

(c) (d) 11 7 60 6 3

3 7 05 6 9

76

Teaching Example 6 Rewrite usingnumbers and an inequality symbol.

(a) Four is greater than two.

(b) Six is less than ten.

Ans: (a) (b) 6 6 104 7 2

EXAMPLE 6 Rewrite using numbers and an inequality symbol.

(a) Five is less than eight. (b) Nine is greater than four.

Solution

(a) Five is less than eight. (b) Nine is greater than four.T T T9 7 4

T T T5 6 8

Remember, the inequality symbol always points to the smaller number.

Practice Problem 6 Rewrite using numbers and an inequality symbol.

(a) Seven is greater than two. (b) Three is less than four. 3 6 47 7 2

Rounding Whole NumbersWe often approximate the values of numbers when it is not necessary to know theexact values. These approximations are easier to use and remember. For example, ifour hotel bill was $82.00, we might say that we spent about $80. If a car cost $14,792,we would probably say that it cost approximately $15,000.

Why did we approximate the price of the car at $15,000 and not $14,000? Tounderstand why, let’s look at the number line.

The number 14,792 is closer to 15,000 than to 14,000, so we approximate the cost ofthe car at $15,000.



It would also be correct to approximate the cost at $14,800 or $14,790, sinceeach of these values is close to 14,792 on the number line. How do we know whichapproximation to use? We specify how accurate we would like our approximation tobe. Rounding is a process that approximates a number to a specific round-off place(ones, tens, hundreds, ). Thus the value obtained when rounding depends on howaccurate we would like our approximation to be. To illustrate, we round the price ofthe car discussed above to the thousands and to the hundreds place.

14,792 rounded to the nearest thousand is 15,000. The round-off place is thousands.

14,792 rounded to the nearest hundred is 14,800. The round-off place is hundreds.

We can use the following set of rules instead of a number line to round wholenumbers.

Á

PROCEDURE TO ROUND A WHOLE NUMBER1. Identify the round-off place digit.

2. If the digit to the right of the round-off place digit is:(a) Less than 5, do not change the round-off place digit.(b) 5 or more, increase the round-off place digit by 1.

3. Replace all digits to the right of the round-off place digit with zeros.

EXAMPLE 7 Round 57,441 to the nearest thousand.

Solution The round-off place digit is in the thousands place.

c_________5 ~7 , 4 4 1 1. Identify the round-off place digit 7.

2. The digit to the right is less than 5.

Do not change the round-off place digit.

3. Replace all digits to the right with zeros.

We have rounded 57,441 to the nearest thousand: 57,000. This means that 57,441 iscloser to 57,000 than to 58,000.

Practice Problem 7 Round 34,627 to the nearest hundred. 34,600

c_________

57,000()*

c________

EXAMPLE 8 Round 4,254,423 to the nearest hundred thousand.

Solution The round-off place digit is in the hundred thousands place.

c___________4, ~2 5 4, 4 2 3 1. Identify the round-off place digit 2.

2. The digit to the right is 5 or more.

Increase the round-off place digit by 1.

3. Replace all digits to the right with zeros.

We have rounded 4,254,423 to the nearest hundred thousand: 4,300,000.

Practice Problem 8 Round 1,335,627 to the nearest ten thousand. 1,340,000

c_________

4,300,000()*

c______

CAUTION: The round-off place digit either stays the same or increases by 1. Itnever decreases.


Teaching Example 7 Round 237,069 to thenearest ten thousand.

Ans: 240,000

Teaching Example 8 Round 14,075,829 tothe nearest hundred thousand.

Ans: 14,100,000


7

Verbal and Writing Skills1. Write the word name for

(a) 8002. Eight thousand two(b) 802. Eight hundred two(c) 82. Eighty-two(d) What is the place value of the digit 0 in the num-

ber eight hundred twenty? One

2. Write in words.

(a) Two is less than five.(b) Five is greater than two.(c) What can you say about parts (a) and (b)?

We can use the inequality symbols to show therelationship between 5 and 2 in two different ways.

5 7 22 6 5

1.1 EXERCISES

3. In the number 9865:

(a) In what place is the digit 8? Hundreds(b) In what place is the digit 5? Ones

4. In the number 23,981:

(a) In what place is the digit 2? Ten thousands(b) In what place is the digit 9? Hundreds


(a) In what place is the digit 4? Thousands(b) In what place is the digit 7? Hundred thousands


(a) In what place is the digit 9? Hundred thousands(b) In what place is the digit 1? Ten thousands

7. In the number 1,284,073:

(a) In what place is the digit 1? Millions(b) In what place is the digit 0? Hundreds

8. In the number 3,098,269:

(a) In what place is the digit 0? Hundred thousands(b) In what place is the digit 8? Thousands

Write in expanded notation.

9. 5876 10. 7632 11. 49214000 + 900 + 20 + 17000 + 600 + 30 + 25000 + 800 + 70 + 6

12. 3562 13. 867,301 14. 913,045900,000 + 10,000 + 3000 + 40 + 5800,000 + 60,000 + 7000 + 300 + 13000 + 500 + 60 + 2

15. Damian withdraws $562 from his account. He re-quests the minimum number of bills in one-, ten-, andhundred-dollar bills. Describe the quantity of eachdenomination of bills the teller must give Damian.5 hundred-dollar bills, 6 ten-dollar bills, and 2 one-dollar bills

16. Erin withdraws $274 from her account. She requests theminimum number of bills in one-, ten-, and hundred-dollar bills. Describe the quantity of each denomina-tion of bills the teller must give Erin.2 hundred-dollar bills, 7 ten-dollar bills, and 4 one-dollar bills

17. Describe the denominations of bills for $46:

(a) Using only ten- and one-dollar bills.4 ten-dollar bills and 6 one-dollar bills

(b) Using tens, fives, and only 1 one-dollar bill.4 ten-dollar bills, 1 five-dollar bill, and 1 one-dollarbill; Answers may vary.

18. Describe the denominations of bills for $96:

(a) Using only ten- and one-dollar bills.9 ten-dollar bills and 6 one-dollar bills

(b) Using tens, fives, and only 1 one-dollar bill.9 ten-dollar bills, 1 five-dollar bill, and 1 one-dollarbill; Answers may vary.

Write a word name for each number.

19. 6079Six thousand seventy-nine

20. 4032Four thousand thirty-two

21. 86,491Eighty-six thousand, four hundred ninety-one

22. 33,224Thirty-three thousand, two hundred twenty-four



PAY to theORDER of

MEMO

DOLLARS

DATE 20

Ellen Font22 Rose PlaceGarden Grove, CA 92641

2520

Mason BankCalifornia

Atlas InsuranceThree hundred seventy-nine and 00/100

379.00

23. Fill in the check with the amount $672. 24. Fill in the check with the amount $379.

PAY to theORDER of

MEMO

DOLLARS

DATE 20

James Hunt4 Platt St.Mapleville, RI 02839

2824


Hampton ApartmentsSix hundred seventy-two and 00/100

672.00

Type of Automobile 2008 MSRP

$30,620

2008 Ford Supercab XLT $27,595

$25,685

2008 Dodge Caravan SXT K $26,805

2008 Dodge Charger SXT G

2008 Ford Expedition XLT

Source: www.dodge.com; www.fordvehicles.com

Replace each question mark with the inequality symbol or 7 .6

25. 5 ? 7 26. 2 ? 1 27. 6 ? 8 28. 9 ? 6 7676

29. 13 ? 10 30. 10 ? 11 31. 9 ? 0 32. 0 ? 9 6767

33. 2131 ? 1909 34. 3011 ? 3210 35. 52,647 ? 616,000 36. 101,351 ? 101,251 7667

Rewrite using numbers and an inequality symbol.

Round to the nearest ten.

43. 45 50 44. 85 90 45. 661 660 46. 123 120

Round to the nearest hundred.

47. 63,854 63,900 48. 12,799 12,800 49. 823,042 823,000 50. 701,529 701,500

Round to the nearest thousand.

51. 38,431 38,000 52. 56,312 56,000 53. 143,526 144,000 54. 312,544 313,000

Round to the nearest hundred thousand.

55. 5,254,423 5,300,000 56. 1,395,999 1,400,000 57. 9,007,601 9,000,000 58. 3,116,201 3,100,000

37. Five is greater than two. 5 7 2 38. Seven is less than ten. 7 6 10

39. Two is less than five. 2 6 5 40. Six is greater than four. 6 7 4

Automobile Prices The table lists the 2008 sticker prices on some popular vehicles. Use this table to answer exercises 41and 42.

Replace the question mark with an inequality symbol to indicate the relationship between the prices of the vehicles.

41. Ford Expedition XLT ? Ford Supercab XLT 7 42. Dodge Caravan SXT K ? Dodge Charger SXT G 7

59. The Sun The diameter of the sun is approximately865,000 miles. Round this figure to the nearest tenthousand. 870,000 miles

60. Inches and Miles There are 3,484,800 inches in 55 miles.Round 3,484,800 to the nearest ten thousand.3,480,000 inches

One Step Further Round to the nearest hundred.

61. 16,962 17,000 62. 44,972 45,000



, ,

Hundre

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ions

Ten m

illio

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Milli

ons

Hundre

d thousa

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Ten th

ousands

Thousands

Hundre

ds

Tens

Ones

08 9 0 0 013 15 1 9 2,

Hundre

d billio

ns

Ten bill

ions

Billions

,

Hundre

d trill

ions

Ten tr

illio

ns

Trillio

ns

Very large numbers are used in some disciplines to measure quantities, such as distance in astronomy and the national debt inmacroeconomics. We can extend the place-value chart to include these large numbers.

63. Write 5,311,192,809,000 using the word name.Five trillion, three hundred eleven billion, one hundredninety-two million, eight hundred nine thousand

64. Round 5,311,192,809,000 to the nearest million.5,311,193,000,000

To Think About Sometimes to get an approximation we must round to the nearest unit, such as a foot, yard, hour, or minute.

65. Train Travel Time A train takes 3 hours and 50 min-utes to reach its destination. Approximately how manyhours does the trip take? 4 hours

66. Automobile Travel Time An automobile trip takes5 hours and 40 minutes. Approximately how manyhours does the drive take? 6 hours

67. Fence Measurements The Nguyens’ backyard has afence around it that measures 123 feet 5 inches. Ap-proximately how many feet of fencing do the Nguyenshave? 123 feet

68. Yardage Measurements Jessica has 15 yards 4 inchesof material. Approximately how many yards of mate-rial does Jessica have? 15 yards

1. Write 6402 in expanded notation. 2. Replace each question mark with the appropriatesymbol or .

(a) 0 ? 10

(b) 15 ? 10 7

6

766000 + 400 + 2

3. Round 154,627 to

(a) the nearest ten thousand 150,000

(b) the nearest hundred 154,600

4. Concept Check Explain how to round 8937 to thenearest hundred. Answers may vary

Quick Quiz 1.1

Classroom Quiz 1.1 You may use these problems to quiz your students’ mastery of Section 1.1.

1. Write 5301 in expanded notation. Ans: 2. Replace each question mark with the appropriate symbol or .

(a) 8 ? 0 Ans:(b) 2 ? 11 Ans: 6

7

76

5000 + 300 + 1

3. Round 3571 to

(a) the nearest thousand Ans: 4000(b) the nearest ten Ans: 3570

Finding a Study PartnerAttempt to make a friend in your class and become studypartners. You may find that you enjoy sitting together anddrawing support and encouragement from each other. Youmust not depend on a friend or fellow student to tutor you,do your work for you, or in any way be responsible for yourlearning. However, you will learn from each other as you seekto master the course. Studying with a friend and comparingnotes, methods, and solutions can be very helpful. And itmakes learning mathematics a lot more fun!

Exercises1. Exchange phone numbers with someone in class so you

can call each other whenever you are having difficultywith your studying.

2. Set up convenient times to study together on a regularbasis, to do homework, and to review for exams.


10

Using Symbols and Key Words for Expressing AdditionWhat is addition? We perform addition when we group items together. Consider thefollowing illustration involving the sale of bikes.

Bikes sold Saturday Bikes sold Sunday Total bikes sold

��

4 3 7 bikes��

is equal to

Teaching Example 1 Translate each Englishphrase using numbers and symbols.

(a) A number plus two

(b) Twelve more than a number

Ans: (a) (b) x + 12n + 2


1.2 ADDING WHOLE NUMBER EXPRESSIONS


Use symbols and key wordsfor expressing addition.

Use properties of additionto rewrite algebraic expressions.

Evaluate algebraic expressionsinvolving addition.

Add whole numbers whencarrying is needed.

Find the perimeters ofgeometric figures.

We see that the number 7 is the total of 4 and 3. That is, is an addi-tion fact. The numbers being added are called addends. The result is called the sum.

addend addend sum

In mathematics we use symbols such as in place of the words sum or plus. TheEnglish phrase “five plus two” written using symbols is Writing Englishphrases using math symbols is like translating between languages such as Spanishand French.

There are several English phrases that describe the operation of addition. Thefollowing table gives some of them and their translated equivalents written usingmathematical symbols.

“5 + 2.”“+”

4 + 3 = 7

4 + 3 = 7

English Phrase Translation into Symbols

Six more than nine

The sum of some number and seven

Four increased by two

Three added to a number

One plus a number 1 + x

n + 3

4 + 2

x + 7

9 + 6

When we do not know the value of a number, we use a letter, such as x, to representthat number. A letter that represents a number is called a variable. Notice that thevariables used in the table above are different. We can choose any letter as a vari-able. Thus we can represent “a number plus seven” by and so on. Combinations of variables and numbers such as and arecalled algebraic expressions or variable expressions.

a + 7x + 7x + 7, a + 7, n + 7, y + 7,

EXAMPLE 1 Translate each English phrase using numbers and symbols.

(a) The sum of six and eight (b) A number increased by four

Solution

6 x 4� �8

(a) The sum of six and eight (b) A number increased by four

Although we used the variable x to represent the unknown quantity in part (b),any letter could have been used.

Practice Problem 1 Translate each English phrase using numbers andsymbols.

(a) Five added to some number (b) Four more than five 5 + 4x + 5


Section 1.2 Adding Whole Number Expressions 11

Using Properties of Addition to Rewrite Algebraic ExpressionsMost of us memorized some basic addition facts. Yet if we study these sums, weobserve that there are only a few addition facts for each one-digit number that wemust memorize. For example, we can easily see that when 0 items are added to anynumber of items, we end up with the same number of items: and so on. This illustrates the identity property of zero: and 0 + a = a.a + 0 = a

5 + 0 = 5, 0 + 8 = 8,

EXAMPLE 2 Express 4 as the sum of two whole numbers. Write all possibil-ities. How many addition facts must we memorize? Why?

Solution Starting with we write all the sums equal to 4 and observe anypatterns.

4 + 0,

10

�

�

34

�

�

44

432

�

�

�

012

�

�

�

444

The numbers in this columnincrease by 1.

The last two rows of the patternare combinations of the samenumbers listed in the firstthree rows.

The numbers in this columndecrease by 1.

We need to learn only two addition facts for the number four: and The remaining facts are either a repeat of these or use the fact that when 0 is addedto any number, the sum is that number.

Practice Problem 2 Express 8 as the sum of two whole numbers. Write allpossibilities. How many addition facts must we memorize? Why?

We need to learn only four addition facts for the number 8: , and .The remaining facts are either a repeat of these or use the fact that when 0 is added to anynumber, the sum is that number.

In Example 2 we saw that the order in which we add numbers doesn’t affectthe sum. That is, and This is true for all numbers and leads usto a property called the commutative property of addition.

1 + 3 = 4.3 + 1 = 4

4 + 47 + 1, 6 + 2, 5 + 3

2 + 2.3 + 1

COMMUTATIVE PROPERTY OF ADDITION

Two numbers can be added in either order with the same result.

13 = 13

a + b = b + a 4 + 9 = 9 + 4

EXAMPLE 3 Use the commutative property of addition to rewrite each sum.

(a) (b) (c)

Solution

(a) (b) (c)

Notice that we applied the commutative property of addition to the expres-sions with variables n and x. That is because variables represent numbers, eventhough they are unknown numbers.

x + 3 = 3 + x7 + n = n + 78 + 2 = 2 + 8

x + 37 + n8 + 2

Teaching Example 2 Express 6 as the sumof two whole numbers. Write allpossibilities. How many addition factsmust we memorize? Why?

Ans: We need to learn three additionfacts for the number six: ,and . The remaining facts are eithera repeat of these, or use the identityproperty of zero.

3 + 35 + 1, 4 + 2

Teaching Tip Point out to studentsthat Example 2 shows how thinkingabout numbers and trying to under-stand mathematical concepts is ofteneasier than memorizing. Ask studentsto think about all those hours theyspent using flash cards to memorizeaddition facts.

Teaching Example 3 Use the commutativeproperty of addition to rewrite each sum.

(a) (b) (c)

Ans:(a) (b) (c) 15 + ax + 119 + 5

a + 1511 + x5 + 9



To simplify an expression like we find the sum of 8 and 1.

Simplifying is similar to rewriting the English phrase “8 plus 1 plus somenumber” as the simpler phrase “9 plus some number.” Since addition is commuta-tive, we can write this simplification as either or We choose to writethis sum as since it is standard to write the variable in the expression first.x + 9,

x + 9.9 + x

8 + 1 + x

8 + 1 + x = 9 + x or x + 9

8 + 1 + x,

Teaching Example 4 If , then

Ans: 532

n + 190 = ?190 + n = 532

Teaching Tip Students often confusethe rules for simplifying and (2x)(3x). Introducing the conceptthrough examples before a formalrule is less likely to confuse them. Itmakes sense to students that we can-not add an unknown quantity to anumber.

2 + 3x, 2x + 3x,

Practice Problem 3 Use the commutative property of addition to rewrite

each sum.

(a) (b) (c) 0 + 44 + 0w + 99 + w3 + xx + 3

EXAMPLE 4 If then

SolutionWhy? The commutative property states that the order in which we add numbers doesn’t affect the sum.

Practice Problem 4 If then

6075y + x = ?

x + y = 6075,

159 + 2566 = 2725

159 + 2566 = ?

2566 + 159 = 2725,

EXAMPLE 5 Simplify.

Solution To simplify, we find the sum of the known numbers.

We cannot add the variable n and the number 5 because n represents an un-known quantity; we have no way of knowing what quantity to add to the number 5.

Practice Problem 5 Simplify. 9 + x or x + 96 + 3 + x

= n + 5or

3 + 2 + n = 5 + n

3 + 2 + n

Addition of more than two numbers may be performed in more than onemanner. To add we can first add the 5 and 2, or we can add the 2 and 1first. We indicate which sum we add first by using parentheses. We perform the oper-ation inside the parentheses first.

In both cases the order of the numbers 5, 2, and 1 remains unchanged and the sumsare the same. This illustrates the associative property of addition.

5 + 2 + 1 = 5 + 12 + 12 = 5 + 3 = 8 5 + 2 + 1 = 15 + 22 + 1 = 7 + 1 = 8

5 + 2 + 1

ASSOCIATIVE PROPERTY OF ADDITION

When we add three or more numbers, the addition may be grouped in any way.

14 = 14

13 + 1 = 4 + 10

1a + b2 + c = a + 1b + c2 14 + 92 + 1 = 4 + 19 + 12

Teaching Example 5 Simplify

Ans: 12 + x or x + 12

4 + 8 + x




Sometimes we must use both the associative and commutative properties ofaddition to rewrite a sum and simplify. In other words, we can change the order inwhich we add (commutative property) and regroup the addition (associative property)to simplify an expression.

EXAMPLE 6 Use the associative property of addition to rewrite the sumand then simplify.

SolutionThe associative property allows us to regroup.

Simplify:

Practice Problem 6 Use the associative property of addition to rewritethe sum and then simplify. w + 51w + 12 + 4

3 + 6 = 9. = x + 9

1x + 32 + 6 = x + 13 + 62

1x + 32 + 6

EXAMPLE 7 Use the associative and/or commutative property as necessaryto simplify the expression.

SolutionThe commutative property allows us to change the order of addition.

Regroup the sum using the associative property.

Simplify.

Write as

Practice Problem 7 Use the associative and/or commutative property asnecessary to simplify each expression.

(a) (b) x + 814 + x + 32 + 1x + 1012 + x2 + 8

n + 12.12 + n 5 + 1n + 72 = n + 12

= 12 + n

= 15 + 72 + n

5 + 1n + 72 = 5 + 17 + n2

5 + 1n + 72

Understanding the ConceptAddition Facts Made SimpleThere are many methods that can be used to add one-digit numbers. For exam-ple, if you can’t remember that but can remember that just add 1 to 14 to get 15.

7 + 7 = 14,7 + 8 = 15

7+8=7+(7+1)

=(7+7)+1

=14+1

=15

7+5=(2+5)+5

=2+(5+5)

=2+10

=12

Exercises

1. Use the fact that to add 2. Use the fact that to add 6 + 8 = 6 + 16 + 22 = 16 + 62 + 2 = 12 + 2 = 146 + 8.6 + 6 = 12

8 + 5 = 13 + 52 + 5 = 3 + 15 + 52 = 3 + 10 = 138 + 5.5 + 5 = 10

Another quick way to add is to use the sum since it is easy toremember. Let’s use this to add 7 + 5.

5 + 5 = 10,

Teaching Example 6 Use the associativeproperty of addition to rewrite the sumand then simplify.

Ans: a + 6

1a + 42 + 2

Teaching Example 7 Use the associativeand/or commutative property as necessaryto simplify the expression.

Ans: a + 15

4 + 1a + 8 + 32

Teaching Tip Many students need toreview addition and multiplicationfacts. They often resist since they feelthat they can use a calculator. This isa good time to let them know that inarithmetic and algebra, as well as inreal-life situations, we need to be ableto do calculations mentally withoutthe use of a calculator. Remind themof the following situations thatrequire quick mental math.

1. You often have just a few secondsto check the change you receive orto make change for others.

2. While shopping, you need to besure that you have enough moneyto pay for the cost of several items.

3. In arithmetic, you need to knowthe multiplication facts whenworking with fractions.

4. In algebra, there are often manysteps, thus using a calculator forevery step of the process is slowand cumbersome.

Emphasizing the importance ofknowing the addition facts makesstudents more responsive to shortcutsthat make the task easier. Afterdiscussing the Understanding theConcept, ask students to find othershortcuts.



Evaluating Algebraic Expressions Involving AdditionWe have already learned that when we do not know the value of a number, we des-ignate the number by a letter. We call this letter a variable. We use a variable to rep-resent an unknown number until such time as its value can be determined. Forexample, if 6 is added to a number but we do not know the number, we could write

where n is the unknown number.

If we were told that n has the value 9, we could replace n with 9 and then simplify.

Replace n with 9.Simplify by adding.

Thus has the value 15 when n is replaced by 9. This is called evaluating theexpression if n is equal to 9.n + 6

n + 6

15 9 + 6 n + 6

n + 6

EXAMPLE 8 Evaluate for the given values of x and y.

(a) x is equal to 6 and y is equal to 1(b) x is equal to 4 and y is equal to 2

Solution

x + y + 3

(a) Replace x with 6 and y with 1.

Simplify.

When x is equal to 6 and y is equalto 1, is equal to 10.

(b) Replace x with 4 and y with 2.

Simplify.

When x is equal to 4 and y is equalto 2, is equal to 9.x + y + 3

94 + 2 + 3

x + y + 3

x + y + 3

106 + 1 + 3

x + y + 3

Practice Problem 8 Evaluate for the given values of x and y.

(a) x is equal to 9 and y is equal to 3 18 (b) x is equal to 1 and y is equal to 7 14

x + y + 6

Adding Whole Numbers When Carrying Is NeededOf course, we are often required to add numbers that have more than a single digit.In such cases we must:

1. Arrange the numbers vertically, lining up the digits according to place value.2. Add first the digits in the ones column, then the digits in the tens column, then

those in the hundreds column, and so on, moving from right to left.

Sometimes the sum of a column is a multidigit number—that is, a numberlarger than 9. When this happens we evaluate the place values of the digits to findthe sum.

Teaching Example 8 Evaluate for the given values of m and n.

(a) m is equal to 3 and n is equal to 1

(b) m is equal to 5 and n is equal to 9

Ans: (a) 11 (b) 21

m + n + 7


To evaluate an algebraic expression, we replace the variables in the expressionwith their corresponding values and simplify.

An algebraic expression has different values depending on the values we useto replace the variable.



8 tens+1 ten+3 ones

68+ 25

6 tens 8 ones2 tens 5 ones8 tens 13 ones We cannot have two digits in the ones column,

so we must rename 13 as 1 ten and 3 ones.

9 tens+3 ones=93

68

68

1

+ 2593

1

+ 253

8 ones+5 ones=13 ones

Add 1 ten+6 tens+2 tens.

Soda

Coffee

21

577+ 841018

Iced tea357

We add 7+7+4=18. Since 18 equals 1 ten and 8 ones,we carry 1 ten placing a 1 at the top of the tens column.

We add 1+5+7+8=21. Since 21 tens equals 2hundreds and 1 ten, we carry 2 hundreds placing a 2 at thetop of the hundreds column.

We add 2+3+5=10. Since 10 hundreds equals 1thousand and zero hundreds, we write 0 in the hundredscolumn and 1 in the thousands column.

Type ofBeverage

Number ofResponses

Soda

Orange juice

Coffee

Iced tea

Milk

Other

577

475

84

357

286

91

EXAMPLE 9 Add.

Solution We arrange numbers vertically and add the digits in the ones columnfirst, then the digits in the tens column.

68 + 25

A shorter way to do this problem involves a process called “carrying.” Insteadof rewriting 13 ones as 1 ten and 3 ones we would carry the 1 ten to the tens columnby placing a 1 above the 6 and writing the 3 in the ones column of the sum.

Practice Problem 9 Add. 285247 + 38

Teaching Example 9 Add.

Ans: 324

249 + 75

Teaching Tip Tell students that if they need more practice adding whole numbers, they can find more exercises in Appendix D.

Teaching Example 10 Find the totalnumber of people whose responses weremilk, iced tea, or orange juice.

Ans: 1118

Often you must carry several times, by bringing the left digit into the nextcolumn to the left.

EXAMPLE 10 A market research company surveyed 1870 people to deter-mine the type of beverage they order most often at a restaurant. The results of thesurvey are shown in the table. Find the total number of people whose responseswere iced tea, soda, or coffee.

Solution We add whenever we must find the “total” amount.

A total of 1018 people responded ice tea, soda, or coffee.

357 + 577 + 84 = 1018



Finding the Perimeters of Geometric FiguresGeometry has a visual aspect that many students find helpful to their learning.Numbers and abstract quantities may be hard to visualize, but we can take pen inhand and draw a picture of a rectangle that represents a room with certain dimen-sions. We can easily visualize problems such as “what is the distance around the out-side edges of the room (perimeter)?” In this section we study rectangles, squares,triangles, and other complex shapes that are made up of these figures.

A rectangle is a four-sided figure like the ones shown here.

Practice Problem 10 Use the survey results from Example 10 to answer thefollowing: Find the total number of people whose responses were milk, orangejuice, or other. 852


44

11

11

Oppositesides areequal.

77

7

7

All sides of asquare areequal.

A rectangle has the following two properties:

1. Any two adjoining sides are perpendicular.2. Opposite sides are equal.

When we say that any two adjoining sides are perpendicular we mean that any twosides that join at a corner form an angle that measures 90 degrees (called a rightangle) and thus forms one of the following shapes.

When we say that opposite sides are equal we mean that the measure of a side isequal to the measure of the side across from it. When all sides of a rectangle are thesame length, we call the rectangle a square.

A triangle is a three-sided figure with three angles.

The distance around an object (such as a rectangle or triangle) is called theperimeter. To find the perimeter of an object, add the lengths of all its sides.

Teaching Tip The formulas forperimeter are not used here becausethe goal is not to ask students tomemorize formulas but to have themunderstand that perimeter is the dis-tance around an object. Students willbe introduced to perimeter formulasin a later section. If students areunfamiliar with the units of measurein either the U.S. or metric system,refer them to Appendix A.



Teaching Example 11 Find the perimeterof the rectangle.

Ans: 20 in.

3 in.

7 in.

215 ft

65 ft

150 ft

50 ft

65 ft

215 ft

65 ft

150 ftWe cross off 65 ftsince inside lengthsare not includedin the perimeter.

50 ft

? ft? ft 65 ft

215 ft

65 ft

150 ft

This side is 65 ftbecause the shaded

figure is a square.

50 ft

115 ft

65 ft

This side equals 50 � 65or 115 ft because oppositesides of a rectangle havethe same length.

65 ft

EXAMPLE 11 Find the perimeter of the triangle. (The abbreviation “ft”means feet).

5 ft 5 ft

7 ft

Solution We add the lengths of the sides to find the perimeter.

The perimeter is 17 ft.

Practice Problem 11 Find the perimeter of the square. 60 ft

5 ft + 5 ft + 7 ft = 17 ft5 ft 5 ft

7 ft

15 ft

If you are unfamilar with the value, meaning, and abbreviations for the metricand U.S. units of measure, refer to Appendix A, which contains a brief summary ofthis information.

EXAMPLE 12 Find the perimeter of the shape consisting of a rectangle and asquare.

Solution We want to find the distance around the figure. We look only at theoutside edges since dashed lines indicate inside lengths.

Now we must find the lengths of the unlabeled sides. The shaded figure is asquare since the length and width have the same measure. Thus each side of theshaded figure has a measure of 65 ft.

Teaching Example 12 Find the perimeter of the shape consisting of a rectangle anda square.

Ans: 62 m

Teaching Tip After completingExample 12, explain to studentsthat we often do not have all theinformation we need to solveproblems in real-life situations.We must use reasoning and logicskills to find this information, justas we did in Example 12.

8 m

5 m13 m

5 m




Teaching Tip After discussing theDeveloping Your Study Skills, explainto students that we sharpen our brainwhen we learn math because wedevelop the ability to reason andthink logically. Tell students that welearn logic and basic number facts indifferent parts of the brain. Thus, welearn algebra and geometry in differ-ent parts of the brain—algebra in theleft part and geometry in the rightpart. Since some of us learn mainlyon the left side of the brain and oth-ers on the right, we usually have apreference to either geometry oralgebra. Then explain that this classincludes both topics to enhance bothsides of the brain. Exercising bothsides of the brain sharpens us nowand may even keep our brains func-tional in our old age.

Next, we add the length of the six sides to find the perimeter.

The perimeter is 660 ft.

Practice Problem 12 Find the perimeter of the shape consisting of a rectan-gle and a square. 450 ft

150 ft + 115 ft + 215 ft + 65 ft + 65 ft + 50 ft = 660 ft

155 ft

30 ft40 ft

30 ft

125 ft

Understanding the ConceptUsing Inductive Reasoning to Reach a ConclusionWhen we reach a conclusion based on specific observations, we are usinginductive reasoning. Much of our early learning is based on simple cases of induc-tive reasoning. If a child touches a hot stove or other appliance several times andeach time he gets burned, he is likely to conclude, “If I touch something that ishot, I will get burned.” This is inductive reasoning. The child has thought aboutseveral actions and their outcomes and has made a conclusion or generalization.

The following is an illustration of how we use inductive reasoning in mathematics.

Find the next number in the sequence 10, 13, 16, 19, 22, 25, 28, . . .

We observe a pattern that each number is 3 more than the preceding number:, and so on. Therefore, if we add 3 to 28, we conclude

that the next number in the sequence is 31.

Exercise

1. For each of the following find the next number by identifying the pattern.(a) 8, 14, 20, 26, 32, 38, . . . 44

(b) 17, 28, 39, 50, 61, . . . 72

For more practice, complete exercises 89–94 on page 22.

10 + 3 = 13; 13 + 3 = 16

Preparing to Learn AlgebraMany people have learned arithmetic by memorizing facts andproperties without understanding why the facts are true orwhat the properties mean. Learning strictly by memorizationcan cause problems. For example:

• Many of the shortcuts in arithmetic do not work inalgebra.

• Memorizing does not help one develop reasoning andlogic skills, which are essential to understanding algebraconcepts.

• Memorization can eventually cause memory overload.Trying to remember a collection of unrelated facts cancause you to become anxious and discouraged.

In this book you will see familiar arithmetic topics. Do not skipthem, even if you feel that you have mastered them. Theexplanations will probably be different from those you havealready seen because they emphasize the underlying concepts.If you don’t understand a concept the first time, be patient andkeep trying. Sometimes, by working through the material youwill see why it works. Read all the Understanding the Conceptboxes in the book since they will help you learn mathematics.

Exercise1. Write in words why the commutative property of addition

reduces the amount of memorization necessary to learnaddition facts.Since the order in which we add numbers does not affect theoutcome, we need only learn one addition fact for every two sums.


19

Verbal and Writing Skills Write in words.

1.Ten plus a number. Answers may vary.10 + x 2.

Some number plus four. Answers may vary.n + 4

1.2 EXERCISES

3. Write in your own words the steps you must performto find the answer to the following problem. Evaluate

if x is equal to 9.Replace x with 9, and then add 9 and 6.x + 6

4. Explain why the following statement is true. Ifthen

The commutative property allows us to change the order ofaddition without changing the value of the sum.

z + y + x = 105.x + y + z = 105,

State what property is represented in each mathematical statement.

5.Associative property of addition12 + 32 + 4 = 2 + 13 + 42 6.

Commutative property of addition4 + 1x + 32 = 4 + 13 + x2

Translate using numbers and symbols.

7. A number plus twom + 2

8. Two added to a numberm + 2

9. The sum of five and yor y + 55 + y

10. The sum of eight and xor x + 88 + x

11. Some number added to twelve12 + m or m + 12

12. Twelve more than a numbery + 12

13. A number increased by sevenm + 7

14. A number plus foury + 4

Use the commutative property of addition to rewrite each sum.

15. 16. 17. 18.x + 55 + x

x + 33 + x

6 + yy + 6

a + 55 + a

19. If then 3758

216 + 3542 = ?3542 + 216 = 3758, 20. If then 8947

156 + 8791 = ?8791 + 156 = 8947,

21. If then 12

n + 5 = ?5 + n = 12, 22. If then 31

x + 8 = ?8 + x = 31,

Simplify.

23. 24. 25.n + 129 + 3 + n

a + 8a + 5 + 3

x + 6x + 4 + 2

26. 27. 28.x + 3x + 3 + 0

x + 2x + 0 + 2

y + 87 + 1 + y

Use the associative property of addition to rewrite each sum, then simplify.

29. 30. 31.12 + n9 + 13 + n2

x + 61x + 42 + 2

x + 31x + 22 + 1

32. 33. 34.a + 101a + 42 + 6

n + 111n + 32 + 8

7 + x2 + 15 + x2

Use the associative and/or commutative property as necessary to simplify each expression.

35. 36. 37.n + 712 + n2 + 5

y + 51y + 12 + 4

x + 151x + 42 + 11

38. 39. 40.a + 85 + 13 + a2

x + 98 + 11 + x2

x + 914 + x2 + 5



41. 42. 43.a + 1313 + a + 22 + 8

n + 64 + 1n + 22

n + 913 + n2 + 6

x represents the number of productivityunits earned.

y represents the number of years of employment.

44. 45. 46.n + 1512 + n + 82 + 5

x + 1615 + x + 72 + 4

x + 1416 + x + 42 + 4

Yearly Bonus Pay For exercises 55 and 56, use the table and the formula to calculate the yearlybonus for MJ Industry employees.

Rb

Bonus = x + y + 250

Bonus � x � y � 250

Add. For more practice, refer to Appendix D.

57.

38

58.

83

59.

279

60.

388

331+ 57

236+ 43

71+ 12

15+ 23

61.

70

62.

54

63.

344

64.

635

3087

245+ 75

1058

133+ 98

33116

+ 4

321120

+ 7

65.729

66.766

67.884

68.760562 + 65 + 133281 + 64 + 539531 + 217 + 18236 + 467 + 26

47. Evaluate for the given values of y.

(a) y is equal to 3 10(b) y is equal to 8 15

y + 7 48. Evaluate for the given values of n.

(a) n is equal to 4 12(b) n is equal to 7 15

n + 8

49. Evaluate if x is 6 and y is 13.19

x + y 50. Evaluate if a is 6 and b is 9.15

a + b

51. Evaluate if a is 9, b is 15, and c is 12.36

a + b + c 52. Evaluate if x is 11, y is 18, and z is 15.44

x + y + z

53. Evaluate if n is 26 and m is 44.83

n + m + 13 54. Evaluate if x is 32 and y is 44.97

x + y + 21

55. Calculate the yearly bonus for

(a) Mary McCab. $442(b) Leo J. Cornell. $435

56. Calculate the yearly bonus for

(a) Julio Sanchez. $415(b) Jamal March. $393

EmployeeName

ProductivityUnits Earned

Mary McCab

Julio Sanchez 150

180

EmployeeNumber

00316

00315

Years ofEmployment

12

15

Leo J. Cornell

Jamal March 125

17500318

00317

10

18

69.84427287 + 732 + 423 70.

40033366 + 152 + 485

71.929,217922,876 + 54 + 1287 + 5000 72.

842,659836,147 + 99 + 2413 + 4000

73.391,8503107 + 9063 + 54 + 379,626 74.

998,0642902 + 9050 + 12 + 986,100



Applications Exercises 75–78 Answer each question.

75. Checking Account Angelica’s check register indi-cates the deposits and debits (checks written or ATMwithdrawals) for a 1-month period.

76. Checking Account The bookkeeper for the SpauldingAppliance Company examined the following recordfrom the company account for the month of March.

Date Deposits Debits

12/3/09 $159

12/9/09 $63

12/13/09 $241

12/15/09 $121

12/22/09 $44

Date Deposits Debits

3/6/08 $3477

3/9/08 $120

3/13/08 $3500

3/15/08 $4614

3/22/08 $1388

(a) What is the total of the deposits made to Angeli-ca’s checking account? $400

(b) What is the total of the debits made to Angelica’schecking account? $228

(a) What is the total of the deposits in this timeperiod? $8091

(b) What is the total of the debits in this time period?$5008

77. Apartment Expenses The rent on an apartment was$875 per month. To move in, Charles and Vincentwere required to pay the first and last month’s rent, asecurity deposit of $500, a connection fee with theutility company of $24, and a cable T.V. installationfee of $35. How much money did they need to moveinto the apartment? $2309

78. Car Expenses Shawnee found that for a 6-monthperiod, in addition to gasoline, she had the followingcar expenses: insurance, $562; repair to brakes, $276;and new tires, $142. If gasoline for her car cost $495for this time period, what was the total amount shespent on her car? $1475

Find the perimeter of each rectangle.

79. 36 in. 80. 16 in.

3 ft 8 ft

6 in.

4 in. 4 in.3 ft

8 ft

8 ft

13 in.

5 in.

7 in.

1 in.

81. 12 ft 82. 32 ft

Find the perimeter of each square.

Find the perimeter of each triangle.

83. 14 in. 84. 19 ft

Find the perimeters of the shapes made of rectangles and squares.

85. 58 ft 86. 112 ft

12 ft

6 ft

11 ft

4 ft

7 ft

7 ft

24 ft

17 ft25 ft

8 ft



To Think About For each of the following, find the next number in the sequence by identifying the pattern.

87. 1120 in. 88. 730 in.

310 in.

120 in.

190 in.

100 in.

150 in.

205 in.

55 in.150 in.

20 in.

140 in.

89. 1, 3, 5, 7, 9, 11, 13, . . .15

90. 2, 4, 6, 8, 10, 12, . . .14

91. 0, 5, 10, 15, 20, 25, . . .30

92. 24, 31, 38, 45, 52, 59, 66, . . .73

93. 7, 16, 25, 34, 43, . . .52

94. 12, 25, 38, 51, 64, . . .77

1. Use the associative and/or commutative property asnecessary to simplify each expression.

(a)

(b)

2. Evaluate if m is 25 and n is 8. 46m + n + 13

x + 10 or 10 + x2 + 11 + x + 72a + 1314 + a2 + 9

3. Find the perimeter of the shape consisting of a rectan-gle and a square. 490 in.

4. Concept Check

(a) When we carry, what is the value of the 1 that isplaced above the 9?

(b) When we carry, what is the value of the 1 that isplaced above the 3? Answers may vary

31915

+ 28423

Quick Quiz 1.2

175 in.

40 in.

135 in.

30 in.

40 in.


1. Use the associative and/or commutative property asnecessary to simplify each expression.

(a) Ans:

(b) Ans:

2. Evaluate if a is 13 and b is 29. Ans: 54

3. Find the perimeter of the shape made of rectangles andsquares. Ans: 312 ft.

a + b + 12

n + 12 or 12 + n1 + 15 + n + 62x + 1115 + x2 + 6

129 ft

14 ft115 ft

13 ft

14 ft


23

Understanding Subtraction of Whole NumbersWhat is subtraction? We do subtraction when we take objects away from a group.When we subtract we find “how many are left.” The symbol used to indicate sub-traction is called a minus sign We illustrate below.

� one left

There are three parts to a subtraction problem: minuend, subtrahend, anddifference.

Subtraction is defined in terms of addition. Thus each subtraction sentence has arelated addition sentence. For example, to find the value of we think ofthe number that when added to 2 gives 3.

Subtraction sentence

Related addition sentence

Therefore we can use related addition sentences to help with subtraction. To sub-tract we can think 12 = ? + 8.12 - 8 = ? ,

� �3 = 1 + 2

�

3 - 2 = 1

3 - 2

T T T

minuend subtrahend difference

3 - 2 = 1

�

3 � 2 � 1

T

1=

T

2T

-

T

3

Three take away two

“- .”

Observe the pattern in the following subtraction problems: Each time you subtract the next larger whole number, the

result decreases by 1. We can use this subtraction pattern to subtract mentally.6 - 2 = 4; 6 - 3 = 3.

6 - 0 = 6; 6 - 1 = 5;

Increase numbersin this column by 1.

Decrease numbersin this column by 1.

1.3 SUBTRACTING WHOLE NUMBER EXPRESSIONS


Understand subtraction ofwhole numbers.

Use symbols and key wordsfor expressing subtraction.

Evaluate algebraicexpressions involvingsubtraction.

Subtract whole numbers with two or more digits.

Solve applied problemsinvolving subtraction of whole numbers.

EXAMPLE 1 Subtract.

(a) (b) (c) (d)

Solution(a) (b) (c) (d)

Practice Problem 1 Subtract.

(a) 3 (b) 3 (c) 18 (d) 018 - 1818 - 06 - 35 - 2

15 - 15 = 015 - 0 = 157 - 2 = 59 - 5 = 4

15 - 1515 - 07 - 29 - 5

EXAMPLE 2 Use this fact to find

Solution Since we know we can use subtraction patterns tofind

Practice Problem 2 Use this fact to find 546600 - 54.600 - 50 = 550.

800 - 53 = 747

800 - 52 = 748

800 - 51 = 749

800 - 50 = 750

800 - 53.800 - 50 = 750,

800 - 53.800 - 50 = 750.

Teaching Example 1 Subtract.

(a) (b)

(c) (d)

Ans: (a) 5 (b) 0 (c) 3 (d) 4

4 - 09 - 6

11 - 118 - 3


Teaching Example 2 Usethis fact to find

Ans: 298

300 - 52.350 - 50 = 300.



Using Symbols and Key Words for Expressing SubtractionThere are several English phrases to describe the operation of subtraction. The fol-lowing table presents some English phrases and their translated equivalents writtenusing mathematical symbols.

Teaching Tip Students often confusestatements that represent the symbols

and Emphasize the differencebetween the two. “Is less than”includes the word is and representsthe inequality symbol “Less than”does not include the word is and rep-resents the subtraction symbol Then have studentstranslate the following:

1. Five is less than x.

2. Five less than x.

3. Sharon’s share of the winningsis less than Justin’s share.

4. $15 less than the original price.

Ans:

1. 2.

3. 4. P - 15S 6 J

x - 55 6 x

- .

6 .

- .6


The difference of three and x

Eight minus a number

Two subtracted from seven

A number decreased by four

Five less than nine 9 - 5

n - 4

7 - 2

8 - n

3 - x

CAUTION: Math symbols are not always written in the same order as the words inthe English phrase. Notice that when we translate the phrases “less than” or “sub-tracted from,” the math symbols are not written in the same order as they are readin the statement.

EXAMPLE 3 Translate using numbers and symbols.

(a) The difference between five and x (b) Four less than seven

Solution

5 � x

(a) The difference between five and x

7 � 4

(b) Four less than seven

Practice Problem 3 Translate using numbers and symbols.

(a) The difference of nine and n (b) x minus three(c) x subtracted from eight 8 - x

x - 39 - n

CAUTION: Note that the symbol means “is less than” while the symbol means“less than.” Therefore in Example 3b we use the minus symbol , not the inequalitysymbol .

When we use the phrase “less than” and “subtracted from,” the order in which wewrite the numbers in the subtraction is reversed. It is important to write these num-bers in the correct order because, in general, subtraction is not commutative.In other words, is not the same as To show this, let’s see whathappens when we change the order of the numbers in subtraction.

You have $30 in your checking account and write a check for $20;your balance will be $10.You have $20 in your checking account and write a check for $30;you will be overdrawn!

Obviously, the results are not the same. We summarize as follows.

$20 - $30

$30 - $20

20 - 30.30 - 20

6

-

-6

SUBTRACTION IS NOT COMMUTATIVEIf a and b are not the same number, then

a - b does not equal b - a. 30 - 20 does not equal 20 - 30.

Teaching Example 3 Translate usingnumbers and symbols.

(a) A number minus one

(b) Six decreased by a number

(c) Eight subtracted from a number

Ans: (a) (b) (c) x - 86 - nx - 1

Teaching Tip Although studentsmay understand that they may overlook the fact thatsubtraction is not commutative whenworking with variable expressions.Remind students not to write as y - x.

x - y

3 - 2 Z 2 - 3,


Section 1.3 Subtracting Whole Number Expressions 25

Evaluating Algebraic Expressions Involving SubtractionRecall from Section 1.2 that to evaluate an expression we replace the variables inthe expression with the given values and simplify.

EXAMPLE 4 Evaluate for the given values of x.

(a) x is equal to 2 (b) x is equal to 4

Solution

(a)Replace x with 2.Simplify.

When x is equal to 2, is equal to 5.

7 - x

5 7 - 2 7 - x

7 - x

Practice Problem 4 Evaluate for the given values of n.

(a) n is equal to 3 5 (b) n is equal to 6 2

8 - n

(b)Replace x with 4.Simplify.

When x is equal to 4, is equal to 3.

7 - x

3 7 - 4 7 - x

Subtracting Whole Numbers with Two or More DigitsOften, we cannot subtract mentally, especially if the numbers being subtractedinvolve more than one digit. In this case we follow the same procedure as we didin addition, except we subtract digits instead of adding them. Therefore, we must:

1. Arrange the numbers vertically.2. Subtract the digits in the ones column first, then the digits in the tens column,

then those in the hundreds column, and so on, moving from right to left.

Many times, however, a digit in the lower number (subtrahend) is greater thanthe digit in the upper number (minuend) for a particular place value, as illustratedbelow.

7 2-3 8

8 7 2

When this happens we must rename 72 using place values so we can subtract.

2 ones

2 ones7 tens

6 tens 10 ones

EXAMPLE 5 Subtract.

SolutionWe cannot subtract 8 ones from 2 ones, so we rewrite 7 tens as “tens and ones.”

Thus A shorter way to do this is called borrowing. Instead of rewriting

as , we would borrow 1 ten from the 7 tens by6 tens + 12 ones7 tens + 2 ones

72 - 38 = 34.

12 ones - 8 ones = 4 ones; 6 tens - 3 tens = 3 tens

10 ones + 2 ones = 12 ones 6 tens + 12 ones-3 tens + 8 ones

3 tens + 4 ones

72- 38

34

7 tens6 tens

+ 2 ones10 ones

-3 tens + 8 ones72

- 38

72 - 38

Teaching Example 4 Evaluate forthe given values of .

(a) is equal to 4

(b) is equal to 15

Ans: (a) 21 (b) 10

x

x

x25 - x


Ans: 14

43 - 29



Sometimes we cannot borrow from the digit directly to the left because thisdigit is 0. In this case we borrow from the next nonzero digit to the left of the 0, asillustrated in the next example.


Ans: 178

504 - 326

Teaching Tip Encourage students toread this section carefully even ifborrowing is a familiar topic. Thegoal in this section is to understandthe concept of borrowing. Referstudents who need more practicewith subtraction to Appendix D.


crossing out the 7 and placing 6 above the 7. Then we would cross out the 2 andplace 12 above the 2.

76

212

- 3 83 4

6 tens � 12 ones

7 tens � 2 ones

We subtract: 12 ones � 8 ones � 4 ones.

We subtract: 6 tens � 3 tens � 3 tens.

Practice Problem 5 Subtract. 4793 - 46

EXAMPLE 6 Subtract.

SolutionWe must borrow since we cannot subtract 6 ones from 4 ones.

We cannot borrow a ten since there are 0 tens, so we must borrow from 3 hundreds.

304- 146

304 - 146

29

3041410

-146158

We subtract: 14-6=8; 9-4=5; 2-1=1.

3 hundreds

2 hundreds

2 hundreds

2 hundreds

0 tens

10 tens

10 ones

14 ones

4 ones

4 ones

4 ones9 tens

9 tens

Practice Problem 6 Subtract. 325603 - 278

Understanding the ConceptMoney and BorrowingConverting money (changing $100 bills to $10 and $1 bills) illustrates the processof borrowing. To see this, let’s look at the following:

A cashier in a gift shop must give a customer $11 change for a purchase. Sincethe cashier is out of small bills and has only 3 hundred-dollar bills in the register,she must ask another cashier to convert a hundred-dollar bill to tens and ones.

3 hundreds

2 hundreds 9 tens

10 ones

1 hundred-dollar billis converted to 9 tensand 10 ones

The cashier now has 2 hundreds, 9 tens, and 10 ones and can give the customer$11 change.

$300

2300

109

- 11289

- 11

twohundreds

ninetens

tenones



We can check our subtraction problems using the related addition problems.For example, to check that we verify that 7 = 5 + 2.7 - 2 = 5,

Teaching Example 7 Subtract and check your answer.

Ans: 1964

6002 - 4038

Exercises

1. What happens when we must borrow from 0? That is, when subtractingwhy must we change the middle 0 to 9, then borrow 1 from the first

nonzero whole number to the left of it? Explain. We can borrow only from aplace value that has a nonzero whole number. For example, in $400 there are only hundred-dollar bills to break down (borrow from).

2. Explain why changing 1 ten-dollar bill to 10 one-dollar bills is similar to bor-rowing in subtraction.

When we change the ten-dollar bill to 10 one-dollar bills, we have 0 ten-dollar bills and 10 one-dollar bills. This is similar to borrowing: .2 10

0 10

400 - 68,

76

09

09

414

- 3 6 7 53 3 2 9

We cannot subtract 5 from 4, so we must change700 to 699 to borrow 10 ones.

Then we add: 10 ones+4 ones=14 ones.

We subtract: 14-5=9; 9-7=2; 9-6=3; 6-3=3.

EXAMPLE 7 Subtract and check your answer.

Solution

7004 - 3675

Check your answer.

Subtraction Check by Addition

It checks.3 6 7 5

+ 3 3 2 97 0 0 4

7 0 0 4- 3 6 7 5

3 3 2 9

Practice Problem 7 Subtract and check your answer. 37698006 - 4237

Solving Applied Problems Involving Subtraction of Whole Numbers

Key words and phrases found in applied problems often help determine whichoperations should be used for computations. Subtraction is often used in real-lifeproblems when we are comparing more than one amount. Often we want to knowhow much more or how much less one amount is than another. Subtraction is alsonecessary when we want to know how much is left or when the problem uses the keywords or phrases for subtraction, such as difference, minus, subtracted from, de-creased by, or less than. When we solve applied problems it is a good idea to use thefollowing three steps in the problem-solving process.

Step 1. Understand the problem. Draw pictures. Look for key words and phrases tohelp you determine what operations should be used.

Step 2. Calculate and state the answer. Perform all calculations and answer thequestion asked in the problem.

Step 3. Check your answer. You may use a different method to find the answer, oryou may estimate to see if your answer is reasonable.



How many more calico bass were caught off the wharf on May 30 than on May 31?

Solution Understand the problem. The key phrase “how many more” indicatesthat we subtract.

Calculate and state the answer. We subtract: the number of calico bass caught onMay 30 minus the number of calico bass caught on May 31.

311 - 133

EXAMPLE 8 Fish counts of calico bass caught out of a local sportfishingwharf on the last three days of May are given in the table.

3 10

111

- 1 3 38

We borrow 1 ten.

We cannot subtract 3 tens from 0 tens so we borrow again.

3 10

111

102

- 1 3 387

We borrow: write 3 hundreds as 2 hundreds and 10 tens.

Thus, 178 more fish were caught on May 30 than on May 31.1

Number of CalicoBass Caught

232311133

May 29May 30May 31

We leave the check to the student.

Practice Problem 8 Use the information in Example 8 to answer the fol-lowing question. How many fewer fish were caught off the wharf on May 31 thanon May 29? 99


Teaching Example 8 How many more fishwere caught off the wharf on May 30 thanon May 29?

Ans: 79

EXAMPLE 9 Find the perimeter of the shape consisting of rectangles.

70 in.

20 in.

60 in.

100 in.

Teaching Example 9 Find the perimeter ofthe shape consisting of rectangles.

Ans: 90 ft

25 ft

7 ft

14 ft

20 ft



To find thisside subtract70 in. � 20 in.� 50 in.

To find thisside subtract100 in. � 60 in.� 40 in.

70 in.

50 in.

20 in.

60 in.

40 in.

100 in.

70 in.

?

20 in.

60 in.

?

100 in.

40 in.

18 in.

19 in.8 in.

Solution To find the perimeter we must find the distance around the figure.Therefore, we must find the measures of the unlabeled sides.

Next we add the lengths of the six sides.

The perimeter is 340 in.

Practice Problem 9 Find the perimeter of the shape consisting ofrectangles. 118 in.

50 in. + 100 in. + 70 in. + 40 in. + 20 in. + 60 in. = 340 in.

Class Attendance and the Learning CycleDid you know that an important part of the learning processhappens in the classroom? People learn by reading, writing,listening, verbalizing, and seeing. These activities are all partof the learning cycle and always occur in class.

• Listening and seeing: hearing and watching theinstructor’s lecture

• Reading: reading the information on the board and inhandouts

• Verbalizing: asking questions and participating in classdiscussions

• Writing: taking notes and working problems assigned inclass

The Learning Cycle

Attendance in class completes the entire learning cycle once.Completing assignments activates the entire learning cycleone more time:

• Reading class notes and the text• Writing your homework• Listening to other students and talking with them about

your strategies

Keep in mind that you must pay attention and participate tolearn. Just being there is not enough.

Reading

" Writing

c T

Seeing ; Verbalizing ; Listening


30

Verbal and Writing Skills

Write using words.

1. Six minus x. Answers may vary.6 - x 2. Two subtracted from ten. Answers may vary.10 - 2

Fill in the blank.

3. The key phrase “how many more” indicates the operation .

Answer true or false.

4. The English phrase “five less than x” written using symbols is False

Subtract.

5 - x.

subtraction

1.3 EXERCISES

5. 3 6. 2 7. 4 8. 48 - 46 - 25 - 37 - 4

9. 6 10. 3 11. 1 12. 14 - 38 - 79 - 69 - 3

13. 15 14. 29 15. 0 16. 015 - 1520 - 2029 - 015 - 0

Translate using symbols.

Evaluate for the given values of n.9 - n

31. If n is equal to 4 5 32. If n is equal to 6 3 33. If n is equal to 9 0 34. If n is equal to 1 8

Evaluate for the given values of x.x - 2

35. If x is equal to 9 7 36. If x is equal to 5 3 37. If x is equal to 3 1 38. If x is equal to 9 7

Subtract and check. For more practice, refer to Appendix D.

39. 62 40. 73 41. 33 42. 3576 - 4156 - 2399 - 2697 - 35

43. 16 44. 19 45. 54 46. 3873 - 3572 - 1856 - 3783 - 67

17. If find using subtractionpatterns. 97

700 - 603700 - 600 = 100, 18. If find using subtractionpatterns. 94

900 - 806900 - 800 = 100,

19. If find using subtractionpatterns. 95

300 - 205300 - 200 = 100, 20. If find using subtractionpatterns. 95

800 - 705800 - 700 = 100,

21. Nine minus two 9 - 2 22. Three decreased by a number 3 - a

23. The difference of eight and y 8 - y 24. The difference of three and a number 3 - n

25. Ten subtracted from seventeen 17 - 10 26. Seven subtracted from a number x - 7

27. A number decreased by one n - 1 28. Eight minus two 8 - 2

29. Two less than some number a - 2 30. Nine less than twelve 12 - 9

47. 678 48. 219 49. 457 50. 671700 - 29500 - 43761 - 542873 - 195

51. 5065 52. 8067 53. 5116 54. 42188801 - 45835301 - 1858721 - 6548912 - 3847

55.

8,679

56.

25,335

57.

105,377

58.

727,589

796,020- 68,431

164,300- 58,923

29,002- 3,667

15,107- 6,428



Find the perimeter of each shape consisting of rectangles.

59. 36 ft 60. 76 in. 61. 84 ft 62. 116 in.

11 ft

3 ft 2 ft

7 ft

20 in.

8 in.

18 in.

7 in.

25 ft

9 ft

6 ft

17 ft

38 in.

17 in.

20 in.9 in.

� � � �

Applications63. Checking Account Fill in the balances in Pedro’s

check register.64. Whale Population Decline Although the Interna-

tional Whaling Commision has banned commercialwhaling since 1987, several countries still hunt whales.As a result, the number of whales continues to de-cline, as shown in the following chart.

(a) Which species has had the largest decline inpopulation? the Blue Whale

(b) What is the decline in the total whale population?451,500

Sun

Sun

MoonMoon

Earth

Check Number Amount Balance $1364

# 123 $238 $1126

# 124 $137 $ 989

# 125 $ 69 $ 920

# 126 $ 98 $ 822

# 127 $369 $ 453 Approximate Population Years EarlierSpecies

Population in the World’s Oceans 2000

Blue 275,000 5,000

Bowhead 60,000 8,500

Humpback 150,000 20,000

Source: Orange County Register

65. Sun vs. Moon Diameter The moon is about 400 timessmaller than the sun.

66. Earth vs. Moon Diameter If the moon were next toEarth it would be like a tennis ball next to a basketball.

The diameter of the sun is approximately 865,000 miles,and the diameter of the moon is approximately2160 miles. How many more miles is the diameter ofthe sun than that of the moon? 862,840 miles

The approximate polar diameter of Earth (distancethrough Earth from North Pole to South Pole) is7900 miles. The diameter of the moon is approximately2160 miles. Find the difference in the diameters ofEarth and the moon. 5740 miles

Roller Coasters When a roller coaster descends at high speeds, the force exerted ona rider’s body by the roller coaster becomes less than that of gravity, producing asensation of weightlessness. Then when the roller coaster hits the bottom and eithershoots up or turns sharply, a g-force is exerted on the rider’s body for a fraction of a second. This g-force can be stronger than the one felt by astronauts during a space-shuttle launch.



For exercises 67–70, refer to the bar graphs, which display the top speeds and maximum drops for some of the most popularroller coaster rides.

Cumulative Review Replace each question mark with an inequality symbol.

U.S. Roller Coasters

0

50

100

150

200

250

350

400

300

450

Max

imum

dro

p (i

n fe

et)

Colossus

115

Ghostrider

108

Goliath

255

MagnumXL-200

195

MillenniumForce

300 300

Supermanthe Escape

0

25

50

75

100

125

Top

spee

d (i

n m

ph)

62 56

8572

92100

Colossus Ghostrider Goliath MagnumXL-200

MillenniumForce

Supermanthe Escape

Source: www.ultimaterollercoaster.com

67. How much faster is the top speed of Supermanthe Escape than that of Goliath?15 mph faster

68. How much slower is the top speed of Ghostriderthan that of Superman the Escape?44 mph slower

69. How much less is the maximum drop of MagnumXL-200 than that of Millennium Force?105 ft less

70. How much greater is the maximum drop ofSuperman the Escape than that of Colossus?185 ft more

To Think About71. For what value(s) of x and y will

When the values of x and y are equal

Translate using symbols, then evaluate.

72. Eight minus y, if y is equal to 38 - y, 5

x - y = y - x?

73. [1.1.4] 5,117,206 ? 13,842 7 74. [1.1.4] 2,386,702 ? 117,401 7

Add.

75. [1.2.4] Hours Worked Edward worked in the super-market 120 hours in May, 135 in June, and 105 in July.How many hours did he work in the three-monthperiod? 360 hours

76. [1.2.4] Pet Supply Purchases Drew bought a dog for$430. He returned to the store the next day to pur-chase the following items for the dog: bed, $32; leash,$12; dog food, $28; and dog treats, $6. How much didDrew pay for the dog and all the supplies? $508

1. Translate the following using numbers and symbols.

(a) Five subtracted from a number

(b) A number decreased by 7

(c) Eight less than a number

2. Subtract and check.

(a) 6779

(b) 408,795601,307 - 192,512

14,062 - 7283

n - 8

n - 7

n - 5

3. Jose’s salary was $2860 per month at his former job. Hisnew job pays a salary of $3270 per month. How muchmore per month will Jose earn at his new job? $410

4. Concept Check Explain why when we subtractwe change 8 to 7 in the borrowing process.

Answers may vary800 - 35,

Quick Quiz 1.3


1. Translate the following using numbers and symbols.

(a) A number subtracted from 8 Ans:

(b) Two less than a number Ans:

(c) Four decreased by a number Ans:


(a) Ans: 4761

(b) Ans: 210,769502,401 - 291,632

11,055 - 6294

4 - n

n - 2

8 - n

3. Leroy received 2 bids to put new plumbing in his home. The first bid was for $4822 and the second bid was for $3788. How muchmore is the first bid than the second? Ans: $1034


33

Understanding Multiplication of Whole NumbersMultiplication of whole numbers can be thought of as repeated addition. For exam-ple, suppose that a small parking lot has 4 rows of parking spaces with 8 spaces ineach row. How many parking spaces are in the lot?

To get the total we add 8 four times, or we can use ashortcut: 4 rows of 8 is the same as 4 times 8, which equals 32. This is multiplication,a shortcut for repeated addition. When numbers are large, multiplication is easierthan addition, but for smaller numbers, you can—if you are stuck—do a multiplica-tion problem by working the equivalent addition problem.

The illustration of the parking lot is an example of an array, a rectangular figurethat consists of rows and columns. Since the parking lot has 4 rows and 8 columns, itis a 4 by 8 array (always write the rows first). We can use dots, squares, or any figureto represent the elements of an array.

8 + 8 + 8 + 8 = 32,

1.4 MULTIPLYING WHOLE NUMBER EXPRESSIONS


Understand multiplicationof whole numbers.

Use symbols and key words forexpressing multiplication.

Use multiplication propertiesto simplify numerical andalgebraic expressions.

Multiply two several-digitnumbers.

Solve applied problemsinvolving multiplicationof whole numbers.

8 spaces

4 rows

4 times 8 � 32

8

8

8

8

32

�

�

�

3

12 items

412 items

3

4

EXAMPLE 1 Draw two arrays that represent the multiplication 3 times 4.

Solution There are two arrays consisting of twelve items that represent themultiplication 3 times 4. One array has 4 rows and 3 columns, and the other onehas 3 rows and 4 columns.

Practice Problem 1 Draw two arrays that represent the multiplication 5times 3.

It is often helpful to use arrays for real-life multiplication problems.


Teaching Example 1 Draw two arrays thatrepresent the multiplication 4 times 5.

Ans:

Teaching Example 2 A neighbor’s child setup a lemonade stand with 3 different drinksto sell: lemonade, water, and tea. Each drinkis available in 2 sizes: small and medium.

(a) Set up an array that describes allpossible drinks the child can sell.

(b) Determine how many types of drinksthe child can sell.

Ans:(a) (b) 6

3

15 items

515 items

3

5

EXAMPLE 2 L&M’s Print Shop makes business cards in 3 colors: white,beige, and light blue. The shop has 4 types of print to choose from: boldface, italic,fine line, and Roman.

(a) Set up an array that describes all possible business cards that can be made.(b) Determine how many different types of cards can be made.

SL SW ST

ML MW MT



Teaching Example 3 Identify the productand the factors in each equation.

(a) (b)

Ans: (a) 2 and n are the factors and 30 isthe product.

(b) a and b are the factors and 12 isthe product.

ab = 122n = 30

Solution(a) We set up a 4 by 3 array where each row corresponds to a type of print and each

column corresponds to a color. Each item in the array represents one possiblebusiness card.

Italic

Boldface

Fine line

Roman

White Beige Light blue

Jesse WillettesSales Manager(312) 123-5462












(b) We have a 4 by 3 array that corresponds to the multiplication 4 times 3, or 12 business cards.

Practice Problem 2 A manufacturer makes 3 different types of bikes: dirt,racer, and road. Each type comes in 5 different colors: red, blue, green, pink, andblack.

(a) Set up an array that describes all possible bikes that can be made. (see margin)

(b) Determine how many different bikes can be made. 15

(a) Dirt Racer Road

Red Dirt; Red Racer; Red Road; Red

Blue Dirt; Blue Racer; Blue Road; Blue

Green Dirt; Green Racer; Green Road; Green

Pink Dirt; Pink Racer; Pink Road; Pink

Black Dirt; Black Racer; Black Road; Black

factor factor product

6 # 8 = 48T T T

ab means a # b

6a means 6 # a

4152 1425 4 # 54 * 5 142152 4 * 5

Using Symbols and Key Words for Expressing MultiplicationIn mathematics there are several ways of indicating multiplication. We write themultiplication problem 4 times 5 as illustrated in the margin.

If two variables a and b are multiplied, we indicate this by writing ab, with nosymbol between the a and b. If a number is multiplied by a variable, we write thenumber first with no symbol between the number and the variable. Thus 6a indi-cates “six times a number.”

The numbers or variables we multiply are called factors. The result of the mul-tiplication is called the product.

EXAMPLE 3 Identify the product and the factors.

(a) (b)

Solution

(a) 5 and 4 are the factors and 20 is the product.(b) 3 and x are factors and 12 is the product.

3x = 125142 = 20

Practice Problem 3 Identify the product and the factors in each equation.

(a) (b) xy = z9 # 7 = 63

(a) 9 and 7 are the factors and 63 isthe product.

(b) x and y are the factors and z is the product.

The word product is also used to indicate the operation of multiplication.There are several other English phrases used to describe multiplication. The follow-ing table gives some English phrases and their translated equivalents written usingmathematical symbols.


Section 1.4 Multiplying Whole Number Expressions 35


(a) The product of x and y

(b) Twice a number

Ans: (a) xy (b) 2n

Using Multiplication Properties to Simplify Numerical and Algebraic Expressions

Like addition, multiplication is commutative. By this we mean that the order inwhich we multiply factors does not change the product. We use an array to illustratethis fact.

3 by 4 array 4 by 3 array


The product of two and three 2(3) or

The product of x and y xy

Six times a number 6x

Double a number 2x

Twice a number 2x

Triple a number 3x

2 # 3


(a) The product of four and a number (b) Triple a number

Solution


(a) Double a number 2n (b) Two times a number 2n

3 # n4 # n =4n =3n

(a) The product of four and a number (b) Triple a number

3 rows

4 columns

4 rows

3 columns

objects objects

Both arrays represent multiplication of 3 and 4; and illustrat-ing that multiplication is commutative.

Multiplication is also associative, meaning that we can regroup the factorswhen multiplying and the product does not change.

We state these properties as follows.

4132 = 12,3142 = 12

4132 = 123142 = 12

COMMUTATIVE PROPERTY OF MULTIPLICATIONChanging the order of factors does notchange the product. 30 = 30

5162 = 6152 ab = ba

Changing the grouping of factorsdoes not change the product.

42 = 42

21122 = 7162 17 # 32 # 2 = 7 # 13 # 22

1ab2c = a1bc2ASSOCIATIVE PROPERTY OF MULTIPLICATION

In addition to these properties there are two other properties of multiplica-tion. The identity property of 1 states that when any number is multiplied by 1, the



Teaching Example 5 Multiply.

Ans: 180

6 # 5 # 2 # 3

Teaching Tip Students initially have ahard time with the fact that whenmultiplying variable expressions thecoefficient is not attached to thevariable. If students initially write outthe multiplications using the familiar

symbol, they easily see that thecommutative property allows them to rearrange terms and multiplycoefficients. Later, when they learnhow to combine like terms, revisit thisexample to emphasize why it is okayto multiply coefficients of unliketerms but not okay to add them.

#

product is that number: The multiplication property of 0 statesthat when any number is multiplied by 0, the product is

We list a few other facts that can help us with multiplication.

1. Multiplying by 2 is the same as doubling a number.2. Multiplying by 5 is the same as repeatedly adding 5, which is easy since all the

numbers end with 0 or 5: 5, 10, 15, 20, 25, 3. Multiplying any number by 10 can be done simply by attaching a 0 to the end

of that number.

Á .

0: a # 0 = 0; 2 # 0 = 0.a # 1 = a; 2 # 1 = 2.

Teaching Example 6 Simplify.

Ans: 105x

5(3)(x # 7)

3(10)=30 4(10)=40 5(10)=50

160

�

�

16 10

� 4 4 2 5

� 4 2 4 5

To multiply 16(10), write 16 and attach a zero at the end.

4 4 � 16; 2 5 � 10

Use the commutative property to change the order of factors sothat one factor is 10.

# # #

# # #

#

# #

We can use these properties and facts to make multiplication of several num-bers easier.

EXAMPLE 5 Multiply.

Solution

4 # 2 # 4 # 5

Practice Problem 5 Multiply.

(a) 0 (b) 302 # 3 # 1 # 52 # 6 # 0 # 3

We follow the same process with algebraic expressions.

EXAMPLE 6 Simplify.

Solution It may help to rewrite expressions using familiar notation: the multi-plication symbol

21321n # 72 = 42n

= 16 # 72 # n

= 6 # 17 # n2 = 6 # 1n # 72

21321n # 72 = 2 # 3 # 1n # 72# .

2132 1n # 72

Rewrite using familiar notation.

Multiply

Change the order of factors.

Regroup.

Multiply and write in standard notation: 42 # n = 42n.

2 # 3 = 6.

Practice Problem 6 Simplify.

(a) 12x (b) 40n21421n # 5241x # 32

Understanding the ConceptMemorizing of Multiplication FactsIf we think of multiplication as repeated addition, very little memorization isneeded to learn the multiplication facts. Once we know the 2, 5, and 10 timestables, which are fairly easy to learn, we can get the rest using the methods thatfollow.

Teaching Tip Students may focus onthe properties more if they see howthey can use them to simplify operations of whole numbers andreduce memorization, rather than justto write 2(3) as 3(2). Have studentstry the following

1. 2.

Ans:

1.

2. = 9 # 10 = 90

3 # 5 # 3 # 2 = 13 # 32 # 12 # 52 = 10 # 13 = 130

2 # 13 # 5 = 12 # 52 # 13

3 # 5 # 3 # 22 # 13 # 5



Multiplying Two Several-Digit NumbersThe numbers 10, 100, 200, and 2000 have trailing zeros (zeros at the end). We canmultiply these numbers fairly easily. For example, to find 3 times 300 we use repeatedaddition: We see that to find 3(300) we need only multiplythe nonzero digits (numbers that are not equal to zero) and attach the number oftrailing zeros to the right side of the product.

300 + 300 + 300 = 900.

Teaching Example 7 Multiply.

Ans: 413,000

(826)(500)

For example, from the 5 times table we can get the 4 and 6 times tables asfollows. To find 4(7) we think

Similarly, from the 10 times table we can get the 9 times table, and from the2 times table we can get the 3 times table.

Exercise

1. Use the techniques discussed to find each product.(a) 3(7) (b) 4(8)(c) 6(8) (d) 9(8) 10182 - 8 = 725182 + 8 = 48

5182 - 8 = 322172 + 7 = 21

35 + 7 = 42

T

T

17 + 7 + 7 + 7 + 72$'''%'''&

+ 7 7 + 7 + 7 + 7 + 7 + 7$''''%''''&

= 42

5172 + 7 is the same as 6172 35 - 7 = 28

b

T

$''''%'''&

$'''%'''&17 + 7 + 7 + 7 + 7 2 7 + 7 + 7 + 7 = 28

5172 - 7 is the same as 4172

EXAMPLE 7 Multiply. (547)(600)

Solution Since the number 600 has trailing zeros, we use the method statedabove. We multiply the nonzero digits and attach the trailing zeros to the right sideof the product.

Bring down the trailing zeros.

place the 2 here and carry the 4.

Then add the carried digit: Place the8 here and carry the 2.

Then add the carried digit: 30 + 2 = 32.6152 = 30.

24 + 4 = 28.6142 = 24.

6172 = 42;

;

(547)(600) = 328,200

! ! !

52447

* 600328200

Practice Problem 7 Multiply. 436(700) 305,200

How can we multiply numbers with several digits when there are no trailingzeros? Consider the multiplication Recall that in expanded notation

or Thus We can use the expanded notationto see how to multiply large numbers using a condensed form.

2 # 23 = 213 + 202.3 + 20.23 = 20 + 32 # 23.



NOTE TO STUDENT: Fully worked-outsolutions to all of the Practice Problems can be found at the back of the text starting at page SP-1

Teaching Example 8 Multiply. 631(28)

Ans: 17,668

We see that we can multiply simply by calculating and usingthe condensed form.

2 # 202 # 32 # 23

Teaching Example 9 Multiply. 1354(206)

Ans: 278,924

Solving Applied Problems Involving Multiplicationof Whole Numbers

One of the most important steps in solving a word problem is determining what op-eration(s) we must perform to find the answer. Applied problems that require themultiplication operation often state key words such as times and product, deal with

Expanded Notation Process Condensed Form

= 6 + 40 = 46

= 2 # 3 + 2 # 20

= 13 + 32 + 120 + 202 = 13 + 202 + 13 + 202

2 # 23 = 2 # 13 + 202 To multiply we

can add twice.

We regroup.

3 + 3 = 2 # 3; 20 + 20 = 2 # 20

13 + 2022 # 13 + 202,

2 # 3 = 62 # 20 = 40

23* 2

46

EXAMPLE 8 Multiply. 857(43)

Solution To multiply 857(43), we multiply or using the condensed form.

857(3) + 857(40)857(3 + 40)

857� 432571

3428036,851

Multiply: 3(857)=2571.

To find the product 40(857)=34,280,we multiply 4(857) and add one trailing zero.

Add.

The products and are called partial products.


34, 2802571

EXAMPLE 9 Multiply. 3679(102)

Solution3679

* 1027358

00000 367900375,258

Multiply: 2(3679).

Multiply: 0(3679), and attach 1 trailing zero.

Multiply: 100(3679), or 1(3679) and attach 2 trailing zeros.

Add.

We can eliminate the trailing zeros in the partial products if we line up the par-tial products correctly.

3 6 7 9* 1 0 27 3 5 8

00 0 0 367 9 375,2 5 8

(3679)(102) = 375,258


Place the 8 under the 2.





Teaching Tip Remind students that acalculator may be able to performcalculations, but we must determinewhich operation to do.

Teaching Example 10 Meri works 3 hoursa day for 7 days straight on a project forschool. How many hours did Meri spendon the project?

Ans: 21 hours

arrays (rows and columns), or represent situations involving repeated addition.When reading a word problem, look for this information so that you can easilydetermine that you must perform the multiplication operation to solve the problem.

Remember to use the following three steps in the problem-solving process.

Step 1. Understand the problem.Step 2. Calculate and state the answer.Step 3. Check your answer.

EXAMPLE 10 Jessica drove an average speed of 60 miles per hour for 7 hours(per hour means each hour). How far did she drive?

Solution Understand the problem. We draw a diagram and see that this is asituation that involves repeated addition, which indicates that we multiply.

1 hour 1 hour 1 hour p

60 miles60 miles60 miles and so on p

Calculate and state the answer.

Miles driven each hourNumber of hours drivenTotal miles driven

Check. From the diagram we can see that in 3 hours Jessica drove 180 milesThus in 6 hours she drove 360 miles

Now, since she drove 60 miles the seventh hour we add

Practice Problem 10 Drew earns $9 per hour as a retail clerk. How muchwill he earn if he works 30 hours? $270

360 + 60 = 420 miles.1180 miles + 180 miles2.160 + 60 + 602.

420* 7 60

EXAMPLE 11 An apartment building is 4 stories high with 6 apartments oneach floor. How many apartments are in the apartment building?

Solution Understand the problem. We draw a picture and see that this situationdeals with an array and thus requires that we multiply.

Number ofstories � 4

Number ofapartments � 6

Calculate and state the answer. We have a 4 by 6 array. To find the total number ofitems in the array, we multiply There are 24 apartments in the building.

Check. We can use repeated addition and add 6 four times: We get the same result.

Practice Problem 11 Allen is building a brick wall. The wall will be 12bricks high and 30 bricks long. How many bricks will Allen need to build thewall? 360 bricks

6 + 6 + 6 + 6 = 24.

4 # 6 = 24.

Teaching Example 11 An office has 4 filingcabinets. Each filing cabinet has 6 drawers.How many drawers are there for files?

Ans: 24 drawers



Translate the symbols into words.

1. (a) 4x Four times a number(b) ab The product of a and b

40

2. (a) 7y Seven times a number(b) xy The product of x and y

Shapes may vary.

Ice Cream Toppings The Ice Cream Palace has 8 flavorsof ice cream: vanilla, French vanilla, chocolate, strawberry,coffee, pecan, chocolate chip, and mint chip. There are 5toppings for the ice cream: fudge, cherry, candy sprinkle,caramel, and nut.

13. How many different one-topping single scoop ice creamdishes can you order? 40

14. If the Ice Cream Palace increases the number of fla-vors to 10, how many different one-topping ice creamdishes can you order? 50

1.4 EXERCISES

Draw two arrays that represent each product.

3. 2 times 3

State what property is represented in each mathematical statement.

4. 4 times 2

Shapes may vary.

5.Associative property of multiplication316 # 52 = 13 # 625 6.

Commutative property of multiplication316 # 52 = 16 # 52 # 3

Fill in each box to complete each problem.

7. 8.

= � 60 �x = 4 # 5 # � 3 � # x

4 # 513x2

= � 24 �y = 3 # 4 # � 2 � # y

3 # 412y2 9. 10.

= � 24 �y = 4 # 3 # 2 # � y � = 4 # � y � # 3 # 2

14y2 # 3 # 2

= � 24 �a = 3 # 4 # 2 # � a � = 3 # � a � # 4 # 2

13a2 # 4 # 2

11. Shirt and Tie Anthony has 4 ties: brown, black, gray,and dark blue, and 3 shirts: white, pink, and blue.

(a) Set up an array that shows all the possible outfitsthat Anthony can make.

12. Carpet and Window Blinds Gerry has a choice of 4carpet colors: beige, gray, blue, and light brown; and3 colors of blinds: white, pale blue, and rose.

(a) Set up an array that shows all the possible colorcombinations of carpet and blinds that Gerry canchoose from.

(b) How many different outfits are possible?4 times 3, or 12 different outfits

(b) How many different combinations are possible?4 times 3, or 12 combinations

White Pink Blue

Brown Brown, White Brown, Pink Brown, Blue

Black Black, White Black, Pink Black, Blue

Gray Gray, White Gray, Pink Gray, Blue

Dark blue Dark blue, White Dark blue, Pink Dark blue, Blue

White Pale blue Rose

Beige Beige, White Beige, Pale blue Beige, Rose

Gray Gray, White Gray, Pale blue Gray, Rose

Blue Blue, White Blue, Pale blue Blue, Rose

Lightbrown

Light brown,White

Light brown,Rose

Light brown,Pale blue



Identify the factors and the product in each equation.

15.6, 3: factors; 18: product

16.4, 7: factors; 28: product

17.22, x: factors; 88: product

18.7, a: factors; 49: product7a = 4922x = 884172 = 286132 = 18


19. Seven times a number 7x 20. A number times five 5x

21. Triple a number 3x 22. Double a number 2x

23. The product of six and a number 6x 24. The product of a and b ab

Use what you have learned about the properties of multiplication to answer each question.

25. If and then 0y = ?x = 6,x # y = 0 26. If and then 0b = ?a = 2,a # b = 0

27. If then 401x # y2z = ?x1y # z2 = 40, 28. If then 301a # b2 # c = ?b1a # c2 = 30,

Multiply. See Example 5.

29.180

30.80

31.240

32.120152142132122122132182152142152122122132162122152

33.0

34.0

35.160

36.1203 # 2 # 4 # 54 # 2 # 4 # 59 # 0 # 3 # 72 # 4 # 6 # 0

Simplify.

37. 48b 38. 35b 39. 40z 40. 24x41x # 6251z # 82715b2816b241. 56a 42. 18a 43. 14c 44. 40x518 # x2217 # c231a # 6281a # 7245. 90x 46. 30z 47. 0 48. 001521z # 92912210 # y2513212 # z29(2)(x # 5)

49. 18b 50. 28x 51. 30y 52. 72y6 # 314y22 # 315y271421x # 12613211 # b253. 126x 54. 40a 55. 90y 56. 60a413a2 # 5315y2 # 612a25 # 416x23 # 7

Multiply. For more practice, refer to Appendix D.

57. 5733 58. 7408 59. 4214 60. 243061405271602281926291637261. 119,400 62. 289,000 63. 475,800 64. 261,3008711300279316002578150023981300265.

5168

66.

2754

67.

1888

68.

2992

44* 68

32* 59

81* 34

76* 68

69.

47,432

70.

63,460

71.

39,130

72.

23,828

322* 74

455* 86

668* 95

847* 56

73.248,508

74.127,032

75.176,688

76.126,83112012631409143226321201235417022

77.7,674,728

78.2,575,568

79.2,516,022

80.5,068,126900215632300618372445615782832419222

81.9,782,456

82.13,980,945

83.61,711,000

84.172,492,00086,24612000261,71111000223,1091605212,10718082


Applications85. Total Weekly Pay A restaurant cook earns $8 per

hour and works 40 hours per week. Calculate thecook’s total pay for the week. $320


86. Airplane Travel Distance An airplane travels for 6hours at an average speed of 450 miles per hour. Howfar does it travel? 2700 miles

87. Orange Trees in a Grove An orange grove has 15rows of trees with 25 trees in each row. How many or-ange trees are in the grove? 375 trees

88. Flowers in a Garden John plants 6 rows of plants inhis garden. Each row contains 12 small plants. Howmany plants does he have? 72 plants

89. Spelling Books Purchased East Gate Academy pur-chased 327 spelling workbooks at $12 per book. Whatwas the total cost of the workbooks?327 * 12 = $3924

90. Yards Rushed A football player averages 116 yardsper game rushing. At this average, how many rushingyards will be gained in a 9-game season?116 * 9 = 1044 yd

91. Hotel Curtain Purchase A five-story hotel has 40rooms on each floor. The owners are purchasing 25boxes of curtains at a discount. If there are 10 sets ofcurtains in each box, can the owners replace one set ofcurtains in every room of the hotel? Why or why not?Yes, because there are 200 rooms and 250 sets of curtains

92. Tiles Needed Robert will be laying tile on sections ofboth floors of a two-story store. He has determinedthat each floor will require 50 rows of tile with 35 tilesin each row. Robert ordered 46 boxes of tiles at a dis-count. If there are 75 tiles in each box, will Robert haveenough tiles to complete the job? Why or why not?No, because he needs 3500 tiles but only ordered 3450 tiles

Temperatures in Various Cities Use the bar graph to answer exercises 93 and 94.

93. (a) What was the high temperature in Portland onJanuary 15, 2007? 30°F

(b) If the high temperature reading in Honolulu,Hawaii, was two times the high temperature inAlbany, New York, what was the high in Honoluluthat day? 74°F

94. (a) What was the high temperature in Boston onJanuary 15, 2007? 41°F

(b) On January 15, 2007, the high temperature read-ing in Buffalo, New York, was two times the hightemperature in Burlington, Vermont. What wasthe high in Buffalo that day? 60°F Burlington,

VermontPortland,

MaineBoston,Mass.

Albany,New York

Anchorage,Alaska

20�

24�

28�

32�

36�

40�

44�

Hig

h te

mpe

ratu

re r

eadi

ng, 1

-15-

2007

(deg

rees

Fah

renh

eit)

303030

41

37

25

Source: www.almanac.com

One Step Further Simplify.

95. 40abc 96. 48xyz 97. 240abc413a212b2110c218x212y213z22a14b215c298. 90xyz 99. 28xyz 100. 80abc8a(5b)2cx(4y)7z312x213y215z2



To Think About Multiplication facts can be listed in a table such as shown here. For example, the product of 9 and 3 isplaced where row 9 and column 3 meet.

101. Fill in the multiplication table using the followingstep-by-step directions.

(a) Use the multiplication property of zero to fill inthe second row. Now, use the commutative prop-erty to fill in the second column.

(b) Use the identity property of 1 to fill in the thirdrow. Now, use the commutative property to fill inthe third column.

(c) Complete the 2 times table: andso on. Place the products in the fourth row. Now,use the commutative property to place the prod-ucts in the fourth column.

(d) Complete the 5 times table: andso on. Place the products in the seventh row.Now, use the commutative property to place theproducts in the seventh column.

(e) How many multiplication facts are blank in thetable? 36

5 # 1, 5 # 2, 5 # 3,

2 # 1, 2 # 2, 2 # 3,

(f) Since the 0, 1, 2, and 5 times tables are fairly sim-ple to learn, what does this process tell you aboutthe amount of memorization necessary to learnall the multiplication facts? There are only a fewblank spaces left on the table, so there are not manymultiplication facts to learn.

0 1 2 3 4 5 6 7 8 9

0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9

2 0 2 4 6 8 10 12 14 16 18

3 0 3 6 15

4 0 4 8 20

5 0 5 10 15 20 25 30 35 40 45

6 0 6 12 30

7 0 7 14 35

8 0 8 16 40

9 0 9 18 45

Cumulative Review102. [1.2.4] Add. 428,990426,862 + 2128 103. [1.3.4] Subtract. 68587000 - 142

104. [1.1.5] Round to the nearest thousand. 826,540827,000

105. [1.1.5] Round to the nearest ten thousand. 168,406,000168,410,000

106. [1.3.5] Electric Bill Julio’s electric bill for April was$97. If he planned a budget that included electricityexpenses of $120 a month, how much less was the billthan the budget allotment? $23 less

107. [1.3.5] Distance Traveled Mary Ann is planning todrive 920 miles to her sister’s house over a two-dayperiod. If she stays at a hotel 455 miles from herhouse the first night, how far must she drive the sec-ond day? 465 miles

1. Translate using numbers and symbols. The product ofsix and a number 6n

2. Multiply. (1610)(105) 169,050

3. Simplify by multiplying. 30ab 4. Concept Check Explain what to do with the zeros whenyou multiply Answers may vary546 * 2000

3a12b2152

Quick Quiz 1.4

1. Translate using numbers and symbols. The product of fourand a number Ans: 4n

2. Multiply. Ans: 219,45712051211072Classroom Quiz 1.4 You may use these problems to quiz your students’ mastery of Section 1.4.

3. Simplify by multiplying. Ans: 40xy2x(5y)(4)


44

1 bouquet

1 bouquet

1 bouquet

1 bouquet

4 bouquets can be made.

3 - 3 = 06 - 3 = 3

9 - 3 = 612 - 3 = 9

When is division necessary to solve real-life problems? How do I divide whole num-bers? Both these questions are answered in this section. It is just as important toknow when a situation requires division as it is to know how to divide. Even if weuse a calculator, we must know when the situation requires us to divide.

Understanding Division of Whole NumbersSuppose we wanted to display 12 roses in bouquets of 3. To determine the numberof bouquets we can make, we count out 12 roses and repeatedly take out sets of 3.


Understand division of wholenumbers.

Use symbols and key wordsfor expressing division.

Master basic division facts.

Perform long division withwhole numbers.

Solve applied problemsinvolving division of wholenumbers.

1.5 DIVIDING WHOLE NUMBER EXPRESSIONS

By repeatedly subtracting 3, we found how many groups of 3 are in 12. In mathe-matics we express this as division:

The symbols used for division are We can write a division prob-lem in any of the following ways:

c a c Q

3 divided into 12 equals 4 12 divided by 3 equals 4

4 3�12

12 , 3 = 4

12>3 = 4

12= 4

3

� , , , >, .

12 divided by 3 equals 4 .

EXAMPLE 1 Write the division statement that corresponds to the followingsituation. You need not carry out the division.

180 chairs in an auditorium are arranged so that there are 12 chairs in eachrow. How many rows of chairs are there?

Solution We draw an array with 12 columns.

12 chairs for 2nd row

12 chairs for 3rd row

12 chairs for 1st row

How many rows for 180 chairs? …

12

?

We want to know how many groups of 12 are in 180. The division statement thatcorresponds to this situation is 180 , 12.

Teaching Example 1 Write the divisionstatement that corresponds to thefollowing situation. You do not need tocarry out the division.

The concert hall has 144 chairsarranged in rows of 9 chairs. How manyrows of chairs are there?

Ans: 144 , 9


Section 1.5 Dividing Whole Number Expressions 45

Practice Problem 1 Write the division statement that corresponds to thefollowing situation. You need not carry out the division.

John has $150 to spend on paint that costs $15 per gallon. How many gallons ofpaint can John purchase? 150 , 15

We also divide when we want to split an amount equally into a certain numberof parts. For example, if we split the 12 roses into 3 equal groups, how many roseswould be in each group? There would be 4 roses in each group.

The division statement that represents this situation is

12 divided by 3 equals 4 or 12 , 3 = 4.

12roses

…

…

…


EXAMPLE 2 Write the division statement that corresponds to the followingsituation. You need not carry out the division.

120 students in a band are marching in 5 rows. How many students are in eachrow?

Solution We draw a picture. We want to split 120 into 5 equal groups.

120students

…5 rows

Practice Problem 2 Write the division statement that corresponds to thefollowing situation. You need not carry out the division.

Rita would like to donate $170 to 5 charities, giving each charity an equalamount of money. How much money will each charity receive? 170 , 5

Using Symbols and Key Words for Expressing DivisionWhen referring to division we sometimes use the words quotient, divisor, anddividend to identify the three parts in a division problem.

There are also several phrases to describe division. The following table givessome English phrases and their mathematical equivalents.

quotientdivisor�dividend

Teaching Example 2 Write the divisionstatement that corresponds to thefollowing situation. You do not need tocarry out the division.

An 18-hole golf course is 5220 yards inlength. Although the difficulty level variesfrom hole to hole, the distance betweeneach hole is the same. What is the distancebetween holes?

Ans: 5220 , 18

Teaching Tip Throughout all mathematics courses from BasicMathematics to Calculus, the wordsdivisor, dividend, and quotient areused extensively. Stress to studentsthat in any given division problemthey need to be able to recognizewhich number is the divisor, whichis the dividend, and which is thequotient.


n divided by six

The quotient of seven and thirty-five

The quotient of thirty-five and seven

Fifteen items divided equally among five groups

Fifteen items shared equally among five groups 15 , 5

15 , 5

35 , 7

7 , 35

n , 6

The division statement that corresponds to this situation is 120 , 5.




(a) The quotient of x and three

(b) A number n divided by five

Ans: (a) (b) n , 5x , 3

Mastering Basic Division FactsBy looking at rectangular arrays we can see how multiplication and division arerelated. Earlier we saw that the number of items in an array is equal to the numberof We can use this fact to find how many groups of2 are in 6. That is, 6 , 2 = ?

rows * the number of columns.



(a) The quotient of forty-six and two (b) The quotient of two and forty-six

Solution

2 , 4646 , 2

(a) The quotient of forty-six and two (b) The quotient of two and forty-six

Practice Problem 3 Translate using symbols.

(a) The quotient of twenty-six and three(b) The quotient of three and twenty-six 3 , 26

26 , 3

Understanding the ConceptThe Commutative Property and DivisionExample 3 illustrates that the order in which we write the numbers in the divisionis different when we use the phrases

“the quotient of 46 and 2” and “the quotient of 2 and 46.”

It is important to write these numbers in the correct order, as illustrated below.

The division statement:

The situation: $2 divided equally among 46 people

The division statement:

The situation: $46 divided equally between 2 people

We can see that these are not the same situations; thus in general, division is notcommutative:

Exercise

1. Can you think of one case where

When then a , b = b , aa = b,

a , b = b , a?

2 , 46 Z 46 , 2.

46 , 2

2 , 46

2 , 4646 , 2

Teaching Tip Because it is so obvious,students often don’t pay attentionwhen we explain that Stress that it may not be so obviouswhen dealing with algebraicexpressions. For example, whensimplifying we must becareful not to use the commutativeproperty and write Remind students that order matterswhen we divide and when wesubtract.

y , x , 2z.

x , y , 2z

46 , 2 Z 2 , 46.

2

3 6

From the array we see that there are3 rows, thus there are 3 groups of 2 in 6.

6 , 2 = 3

6 ? 2

The number ofitems in array

The numberof rows

The numberof columns

=

=

*

*

These are calledrelated sentences.

We see that the answer to the division is that number which when mul-tiplied by 2 yields 6. We can use this fact when we divide.

To find 6 , 2 = ? , think 6 = ? * 2.

6 , 2



EXAMPLE 4 Divide.

SolutionThink,

Practice Problem 4 Divide. 321 , 7

18 = 6 # 3 18 , 3 = 618 = ? # 3. 18 , 3 = ?

18 , 3 Teaching Example 4 Divide.

Ans: 4

32 , 8

What about division by 0? Zero can be divided by any nonzero number, butdivision by zero is not possible. To see why, suppose that we could divide by zero.Then Let us represent “some number” by .

Ifthen The related multiplication sentence.

Which would mean Since any number times 0 equals 0.

That is, we would obtain which we know is not true. Therefore, our assump-tion that is wrong. Thus we conclude that we cannot divideby zero. We say division by 0 is undefined. It is helpful to remember the followingbasic concepts.

7 , 0 = some number7 = 0,

7 = 0 7 = 0 * ?

7 , 0 = ?

?7 , 0 = some number.

DIVISION PROBLEMS INVOLVING THE NUMBER 1 AND THE NUMBER 0

1. Any nonzero number divided by itself is 1

2. Any number divided by 1 remains unchanged

3. Zero may be divided by any nonzero number; the result is always zero

4. Zero can never be the divisor in a division problem are undefined.

are impossible to determine.b0 , 0,

00

, and 0�0a3 , 0, 30

, and 0�3

a0 , 4 = 0, 04

= 0, and 04�0b .

a29 , 1 = 29, 291

= 29, and 29

1�29b .

a7 , 7 = 1, 77

= 1, and 17�7b .

EXAMPLE 5 Divide.

(a) (b) (c)

Solution(a) 0 divided by any nonzero number is equal to 0.

(b) Zero can never be the divisor in a division problem. is undefined.

(c) Any number divided by itself is 1.

Practice Problem 5 Divide.

(a) 1 (b) Undefined (c) 003

3 , 03 , 3

16 , 16 = 19 , 09 , 0

0 , 9 = 0

1616

9 , 00 , 9

Teaching Example 5 Divide.

(a) (b) (c)

Ans:(a) 0 (b) Undefined (c) 1

12 , 1250

05

Performing Long Division with Whole NumbersSuppose that we want to split 17 items equally between 2 people.

8 items 8 items 1 item

M01_TOBE7935_06_SE_C01.QXD 9/23/08 11:44 AM Page 47

EXAMPLE 6 Divide and check your answer.

Solution We guess that is close to 38.

66�38-36

6 * 6

38 , 6


Each person would get 8 items with 1 left over. We call this 1 the remainder (R) and write

We use related multiplication sentences and the division symbol whendivision involves large numbers, or remainders. For example,

8 R12�17 -16 1

8 R2�17

?2�17

?2�17

� 17 , 2 = 8 R1.

Thus to divide, we guess the quotient and check by multiplying the quotient bythe divisor. If the guess is too large or too small, we adjust it and continue theprocess until we get a remainder that is less than the divisor.

Our guess, 6, is placed here.

Check: 36 must be less than 38.6 * 6 = 36;

Since we do not need to adjust our guess to a smaller number.36 6 38,

6 R26�38-36

2We subtract: 38-36=2.

Check: 2 must be less than 6. We write R2 in the quotient.

Since we do not need to adjust our guess to a larger number.To verify that this is correct, we multiply the divisor by the quotient, then add

the remainder:

2 6 6,

6R2

Multiply

38 = 38

+ 238�38

6 * 6 = 366 Then add the remainder.

Practice Problem 6 Divide and check your answer. 7 R143 , 6

38 , 6 = 6 R2

Teaching Example 6 Divide and check youranswer.

Ans: 7 R6

62 , 8


Let’s see what we do if our guess is either too large or too small.


Solution First guess (too large):

293 , 41

841�293

-328

too large

Check: 41(8)=328; Our guess is too largeso we must adjust.

Guess: 41 times what number is close to 293? 8We write 8 in the quotient.


Ans: 8 R5

149 , 18

17 , 2 = ?

Think: two times what number is close to or equal to 17?2 # ? = 17;

which is close to 17, so we have a remainder.2 # 8 = 16,

We subtract and get a remainder 1.



Guess: We try 6.

but 47 is not less than 41.Our guess is too small so we must adjust.

too small

641�293

-24647

Check: 41(6)=246; 246 is less than 293,

Second guess (too small):

Third guess:

Guess: We try 7.7 R641�293

-2876

Check: 41(7)=287; 287 is less than 293, and 6 is lessthan 41. We do not need to adjust our guess, and 6 isthe remainder. We write R6 in the quotient.

We verify that the answer is correct:

Practice Problem 7 Divide and check your answer. 9 R30354 , 36

293 , 41 = 7 R6

1divisor # quotient2 + remainder = dividend141 # 72 + 6 = 293


Solution Accurate guesses can shorten the division process. If we consideronly the first digit of the divisor and the first two digits of the dividend, it is easier toget accurate guesses.

First set of steps:

70�3672

5�

- 35017

Guess: We look at 7 and 36 to make our guess.

7 times what number is close to 36? 5

Check: 5(70)=350.

350 is less than 367, and

17 is less than 70. We do not adjust our guess.

70 3672

Second set of steps: We bring down the next number in the dividend: 2. Thenwe continue the guess, check, and adjust process until there are no more numbersin the dividend to bring down.


Ans: 24 R14

1478 , 61

32 is the remainder because there are no more numbers to bring down.

Check:

Practice Problem 8 Divide and check your answer. 32 R5180�2611

1divisor # quotient2 + remainder = dividend170 # 522 + 32 = 3672.

3672 , 70 = 52 R32


52 R32�3672

-350

-14032

Guess: We look at 7 and 17 to make our guess. We try 2.

Check: 2(70)=140; 140 is less than 172.

Check: 32 is less than 70.

70

172



Teaching Tip Emphasize the impor-tance of the zero in the quotient ofExample 9. Explain that “56 goes into29 zero times.” Therefore we mustput a zero in the quotient.


Ans: 804 R17

25,745 , 32


Check: 6(56)=336; 336 is less than 338.

6056�33897

-33629 Check: 2 is less than 56.

We bring down the 9. Since 56 cannot be divided into 29, wewrite 0 in the quotient.


Solution First set of steps:

33,897 , 56


Check: 5(56)=280; 280 is less than 297, and

17 is less than 56.

605 R1756�33897

-336297

-28017

Second set of steps: We bring down the 7.

17 is the remainder because there are no more numbers to bring down.

Practice Problem 9 Divide and check your answer. 40314,911 , 37

CAUTION: In Example 9 we placed a zero in the quotient because 56 did not di-vide into 29. You must remember to place a zero in the quotient when this happens,otherwise you will get the wrong answer. There is a big difference between 65 and605, so be careful.

Solving Applied Problems Involving Divisionof Whole Numbers

As we have seen, there are various key words, phrases, and situations that indicatewhen we must perform the division operation. Knowing these can help us solve real-life applications.

EXAMPLE 10 Twenty-six students in Ellis High School entered their class project in a contest sponsored by the Falls City Baseball Association. The classwon first place and received 250 tickets to the baseball play-offs. The teacher gaveeach student in the class an equal number of tickets, then donated the extra ticketsto a local boys and girls club. How many tickets were donated to the boys and girlsclub?

Solution Understand the problem. Since we must split 250 equally among26 students, we divide.


Since there are 16 tickets left over, 16 tickets are donated to the boys and girls club.

Check. 126 # 92 + 16 = 250.

9 R1626�250 234 16

Teaching Example 10 Jason works on adairy farm and one of his jobs is to packeggs in cartons that hold twelve eggs eachto prepare the eggs for shipment. AfterJason packaged all the eggs one day, theowner of the dairy farm instructed him todonate any leftover eggs to the localshelter. If Jason had 2659 eggs to place incartons, how many eggs did he have leftover to donate to the shelter?

Ans: 7 eggs



Teaching Tip After completingSection 1.5, students are prepared tocomplete the Putting Your Skills toWork on page 62 without muchassistance. Divide the class intogroups to start this activity as ahomework assignment, allowing someclass time to complete it. It should benoted that the goal of Putting YourSkills to Work is to teach studentshow to

1. Develop plans.

2. Make decisions.

3. Develop and stay within a budget.

4. Work as a team (identify anddelegate tasks).

5. Write and present proposalsor estimates.

The Putting Your Skills to Worksections have been designed todevelop these skills gradually.

Understanding the ConceptConclusions and Inductive ReasoningIn Section 1.2 we saw how to use inductive reasoning to find the next number in asequence. How accurate is inductive reasoning? Do we always come to the rightconclusion? Conclusions arrived at by inductive reasoning are always tentative.They may require further investigation to avoid reaching the wrong conclusion.For example, inductive reasoning can result in more than one probable next num-ber in a list as illustrated below.

Identify 2 different patterns and find the next number forthe following sequence: 1, 2, 4, . . .

Notice that and . Using a pattern of multiplying the precedingnumber by 2, the next number is . For the second pattern we see that

and . Using a pattern of adding consecutive counting num-bers, the next number is

To know for sure which answer is correct, we would need more informationsuch as more numbers in the sequence to verify the pattern. You should alwaystreat inductive reasoning conclusions as tentative, requiring further verification.

Exercise

1. Identify 2 different patterns and find the next number for the followingsequence: 1, 1, 2, . . .Adding consecutive whole numbers 0, 1, 2, . . . , we obtain 4 for the next number. Multiplying by consecutive counting numbers 1, 2, 3, . . . , we obtain 6 for the next number.

For more practice, complete exercises 55–62 on pages 53–54.

4 + 3 = 7.2 + 2 = 41 + 1 = 2

4 # 2 = 82 # 2 = 41 # 2 = 2

Why Is Homework Necessary?You learn mathematics by practicing, not by watching. Yourinstructor may make solving a mathematics problem lookeasy, but to learn the necessary skills you must practice themover and over again, just as your instructor once had to do.There is no other way. Learning mathematics is like learninghow to play a musical instrument or to play a sport. You mustpractice, not just observe, to do well. Homework provides thispractice. The amount of practice varies for each person. Themore problems you do, the better you get.

Many students underestimate the amount of time eachweek that is required to learn math. In general, two to three

hours per week per unit is a good rule of thumb. This meansthat for a three-unit class you should spend six to nine hoursa week studying math. Spread this time throughout the week,not just in a few sittings. Your brain gets overworked just asyour muscles do!

Exercise1. Start keeping a log of the time that you spend studying

math. If your performance is not up to your expectations,increase your study time.

Practice Problem 10 Twenty-two players on a recreational basketball teamwon second place in a tournament sponsored by Meris and Mann 3DMax MovieTheater. The team won 100 movie passes and divided these passes equally amongplayers on the team. The extra passes were donated to a local children’s home.How many passes were donated to the children’s home? 12 movie passes


52

Verbal and Writing Skills Write the division statement that corresponds to each situation. You need not carry out thedivision.

1.5 EXERCISES

1. 220 paintings are arranged in rows so that 4 paintingsare in each row. How many rows of paintings arethere? 220 , 4

2. In the school gym, 320 chairs must be arranged inrows with 16 chairs in each row. How many rows ofchairs are there? 320 , 16

Howmanyrows?

…

16 chairs

3. 225 tickets to the Dodgers’ first game of the year willbe distributed equally among n people. 225 , n

4. A dinner bill totaling $n was split among 5 people.$n , 5

5. For the division problem which wording iscorrect? There may be more than one right answer.

(a) 3 divided by 15 (b) 15 divided by 3(c) 3 divided into 15 (d) 15 divided into 3(b) and (c)

15 , 3, 6. For the division problem , which wording iscorrect? There may be more than one right answer.

(a) 6 divided by 18 (b) 18 divided by 6(c) 6 divided into 18 (d) 18 divided into 6(b) and (c)

18 , 6


7. Twenty-seven divided by a number 27 , x 8. Eight divided by a number 8 , a

9. Forty-two dollars divided equally among six people42 , 6

10. Sixty-three jelly beans divided equally among threechildren 63 , 3

11. The quotient of thirty-six and six 36 , 6 12. The quotient of forty-four and eleven 44 , 11

15. 1 16. 1 17. 005

15 , 1542 , 42

13. The quotient of three and thirty-six 3 , 36 14. The quotient of eleven and forty-four 11 , 44

Divide.

Divide and check your answer. Refer to Appendix D for more practice.

18. 0 19. Undefined 20. Undefined29 , 017 , 00

77

21. 6 R4 22. 6 R6 23. 371 24. 6215�31057�259760 , 958 , 9

29. 21 R2 30. 39 R3 31. 306 R3 32. 201 R432�643619�581720�78330�632

25. 448 R2 26. 674 R2 27. 42 R8 28. 93 R3186320

126830

6�40463�1346



Applications45. Computer Conference Tickets The 14 members of the

Carver High School Chess Club team won first placein a tournament sponsored by the Carver ConventionCenter. The chess team won 60 tickets to the World-wide Computer Conference. The team decided to di-vide the tickets equally among all 14 team membersand to donate the extra tickets to the PTA. How manytickets were donated to the PTA? 4 tickets

46. Entertainment Event Tickets The 21 members of theLaurel High School track team won first place in atournament sponsored by the Laurel Recreation Cen-ter. The team won 75 tickets to the county fair. Theteam decided to divide the tickets equally among all21 team members and to donate the extra tickets tothe homeless shelter. How many tickets were donatedto the shelter? 12 tickets

14 squares

1 inch

37. 703 R4 38. 306 R2 39. 340 R11 40. 370 R2743,317 , 11711,571 , 3412,854 , 4218,985 , 27

41. 508 R33 42. 906 R48 43. 515 R101 44. 756 R2121,945 , 2970,141 , 136123,264 , 136113,317 , 223

47. Restaurant Bill The bill for dinner, including tip, atLido’s Restaurant was $85. If 5 people split the billevenly, how much did each person have to pay? $17

48. Banquet Ticket Price The members of the Elks Clubare planning a banquet. The cost of the entire ban-quet will be $1071. If 63 members plan to attend, howmuch should the ticket price be to cover the cost ofthe banquet? $17

49. Travel Allowance JoAnn received a travel allowanceof $1050 from her employer for food and lodging. Ifher business trip takes 6 days, how much moneyshould she budget each day so that she will not goover her total travel allowance? $175 per day

50. Cow Pasture Capacity A rancher plans to have 250square feet of pasture for each cow on his field. If thearea of the field is 156,250 square feet, how manycows should the rancher allow on the field? 625 cows

51. Photographing Deer A photographer sets a tele-photo lens so that she can be twice as far away fromher subject as she would with a regular lens. She is tak-ing pictures of deer that are 124 feet from her camera.How far from the deer would the photographer haveto be to get the same shot with a regular lens? 62 ft

52. Cross-Stitch Pattern Janice is making a cross-stitchpattern on 14-count material. This means that thereare 14 squares to the inch. If Janice’s pattern is98 squares across, how many inches wide will it be?7 in.

53. How many 41-cent stamps can be bought with1300 cents? 31 stamps

54. A young toy-car collector has 218 miniature cars. Hebought carrying cases to store these cars. Each carry-ing case holds 15 cars. He plans to give his youngerbrother any cars that won’t fill up a case.

(a) How many cases can he fill completely? 14 cases

(b) How many cars will he give to his brother? 8 cars

33. 48 R11 34. 54 R5 35. 72 R1 36. 84 R61350 , 161369 , 191301

24140329



Identify two patterns and find the next number for each of the following.

62. 1, 4, 8, . . .Multiplying by the pattern 4, 2, 4, . . . , we get 32.Adding successive counting numbers starting with 3, we get 13.

61. 0, 1, 4, . . .Adding the pattern 1, 3, 1, 3 . . . , we get 5. Multiplying thepattern we get3 * 3 = 9.

0 * 0 = 0, 1 * 1 = 1, 2 * 2 = 4,

Complete each of the following.

63. (a) 4(b) 16(c) What can you say about division and the associative

property? The property does not apply to division.

32 , 14 , 22132 , 42 , 2 64. (a) 4(b) 16(c) What can you say about division and the associative

property? The property does not apply to division.

48 , 16 , 22148 , 62 , 2

Cumulative Review65. [1.2.1] Translate into symbols. Seven plus x equals

eleven. 7 + x = 1166. [1.3.4] Subtract. 9461060 - 114

67. [1.4.4] Multiply. 4031 (202) 814,262 68. [1.1.5] Round 556,432 to the nearest thousand.556,000

69. [1.3.5] Distance Traveled Leo wanted to make a1389-mile trip in 3 days to visit his aunt. He drove 430miles the first day and 495 miles the second day. Howfar does Leo have to drive the third day to reach hisdestination? 464 miles

70. [1.3.5] Truck Purchase Price The total cost of thetruck Ranak purchased, including tax and license, is$29,599. If the dealer gave Ranak $6200 for his caras a trade-in and Ranak put $5500 down, what is thebalance owed on the truck? $17,899

1. Translate using numbers and symbols.

(a) the quotient of fourteen and seven

(b) the quotient of seven and fourteen

2. Divide. 408 R415,916 , 39

7 , 14

14 , 7

3. A school district receives a grant for $5,484,000 to bedistributed equally among its three junior colleges.How much does each college receive? $1,828,000

4. Concept Check

Explain the next 2 steps for this division problem.Answers may vary

213�2645 26 04

Quick Quiz 1.5


(a) the quotient of twenty and two Ans:

(b) the quotient of two and twenty Ans:

2. Divide. Ans: 302 R710,577 , 35

2 , 20

20 , 2


3. A group of five investors are planning to purchase a smallbusiness for $6,225,000. If the purchase price is split equallyamong the five investors, how much will each investor payfor his or her share of the purchase? Ans: $1,245,000

To Think About For each of the following, find the next number in the sequence by identifying the pattern.

55. 5, 15, 45, 135, . . . 405 56. 4, 16, 64, 256, . . . 1024 57. 3, 4, 7, 12, 19, 28, 39, . . . 52

58. 0, 2, 6, 12, 20, . . . 30 59. 7, 9, 10, 12, 13, 15, 16, . . . 18 60. 1, 6, 8, 13, 15, 20, . . . 22


55

We do not multiply the base 3 by the exponent 5. The 5 just tells us how many3’s are in the repeated multiplication.

If a whole number or variable does not have an exponent visible, the exponentis understood to be 1.

9 = 91 and x = x1

Writing Whole Numbers and Variables in Exponent FormRecall that in the multiplication problem the number 3 is calleda factor. We can write the repeated multiplication using a shorter nota-tion, because there are five factors of 3 in the repeated multiplication. We saythat is written in exponent form. is read “three to the fifth power.”3535

35,3 # 3 # 3 # 3 # 3

3 # 3 # 3 # 3 # 3 = 243


Write whole numbers andvariables in exponent form.

Evaluate numerical andalgebraic expressions inexponent form.

Use symbols and key wordsfor expressing exponents.

Follow the order of operations.

1.6 EXPONENTS AND THE ORDER OF OPERATIONS

EXPONENT FORMThe small number 5 is called an exponent. Whole number exponents, exceptzero, tell us how many factors are in the repeated multiplication. The number3 is called the base. The base is the number that is multiplied.

EXAMPLE 1 Write in exponent form.

(a) (b) (c) 7 (d)

Solution(a) (b)

(c) (d) or

Note, it is standard to write the number before the variable in a term. Thus is written

Practice Problem 1 Write in exponent form.

(a) n (b) or

(c) (d) or 83x2x2 # 83x # x # 8 # 8 # 8585 # 5 # 5 # 5 # 5 # 5 # 5 # 562y462 # y46 # 6 # y # y # y # yn1

34y3.y334

34y3y # y # y # 3 # 3 # 3 # 3 = y3 # 34,7 = 71

4 # 4 # 4 # x # x = 43 # x2 or 43 x22 # 2 # 2 # 2 # 2 # 2 = 26

y # y # y # 3 # 3 # 3 # 34 # 4 # 4 # x # x2 # 2 # 2 # 2 # 2 # 2

Teaching Example 1 Write in exponentform.

(a) (b) 9

(c)

(d)

Ans:(a) (b)

(c) (d) 96b562x3

91124

b # b # b # b # b # 9 # 9 # 9 # 9 # 9 # 9

6 # 6 # x # x # x

12 # 12 # 12 # 12

EXAMPLE 2 Write as a repeated multiplication.

(a) (b)

Solution(a) (b)

Practice Problem 2 Write as a repeated multiplication.

(a) (b) 1 # 1 # 1 # 1 # 1 # 1 # 117x # x # x # x # x # xx6

65= 6 # 6 # 6 # 6 # 6n3

= n # n # n

65n3

Teaching Example 2 Write as a repeatedmultiplication.

(a)

(b)

(c)

Ans:(a)

(b)

(c) 5 # a # a

x # x # x # x

11 # 11 # 11 # 11 # 11

5a2

x4

115

3 # 3 # 3 # 3 # 3 = 35

3 appears as a factor 5 times. The base is 3.

The exponent is 5.



Evaluating Numerical and Algebraic Expressionsin Exponent Form

To evaluate, or find the value of, an expression in exponent form, we first write theexpression as repeated multiplication, then multiply the factors.

EXAMPLE 3 Evaluate each expression.

(a) (b) (c)

Solution(a)(b)

We do not need to write out this multiplication because repeated multiplica-tion of 1 will always equal 1.

(c)

Practice Problem 3 Evaluate each expression.

(a) 64 (b) 8 (c) 1001028143

24= 2 # 2 # 2 # 2 = 16

19= 1

33= 3 # 3 # 3 = 27

241933


Teaching Example 3 Evaluate eachexpression.

(a) (b) (c)

Ans:(a) 81 (b) 625 (c) 128

275492

Sometimes we are asked to express an answer in exponent form and othertimes to find the value of (evaluate) an expression. Therefore, it is important thatyou read the question carefully and express the answer in the correct form.

Large numbers are often expressed using a number in exponent form that hasa base of 10: and so on. Let’s look for a pattern to find an easy wayto evaluate an expression when the base is 10.

Notice that when the exponent is 1 there is 1 trailing zero; when the exponent is 2there are 2 trailing zeros; when it is 3 there are 3 trailing zeros; and so on. Thus tocalculate a power of 10, we write 1 and attach the number of trailing zeros named bythe exponent.

10 2 = (10)(10) = 1 00 10 4 = (10)(10)(10)(10) = 1 0,000

10 1 = 1 0 10 3 = (10)(10)(10) = 1 000

101, 102, 103, 104

Evaluate 53: 53= 5 # 5 # 5 = 125.

Write 5 # 5 # 5 in exponent form: 5 # 5 # 5 = 53.

Teaching Example 4 Evaluate .

Ans: 1000

103

Write 1.

10,000,000

The exponent is 7; attach 7 trailing zeros.

10‡=10,000,000

EXAMPLE 4 Evaluate

Solution

107.

Practice Problem 4 Find the value of 100,000105.

To evaluate the expression when x is equal to 4, we replace the variable xwith the number 4 and find the value of We can write the state-ment “x is equal to 4” using math symbols “x = 4.”

42: 42= 4 # 4 = 16.

x2


Section 1.6 Exponents and the Order of Operations 57

EXAMPLE 5 Evaluate for

Solution

x = 3.x3

Replace x with 3.

Write as repeated multiplication, then multiply. 3 # 3 # 3 = 27

x3 (3)3

When is equal to 27.

Practice Problem 5 Evaluate for 64y = 8.y2

x = 3, x3

Using Symbols and Key Words for Expressing ExponentsHow do you say or We can say “10 raised to the power 2,” or “5 raised to thepower 3,” but the following phrases are more commonly used.

53 ?102

If the value of the exponent is 2, we say the base is squared.

is read “six squared.”

If the value of the exponent is 3, we say the base is cubed.

is read “six cubed.”

If the value of the exponent is greater than 3, we say that the base is raised tothe (exponent)th power.

is read “six to the fifth power.”65

63

62

EXAMPLE 6 Translate using symbols.

(a) Five cubed (b) Seven squared (c) y to the eighth power

Solution(a) (b) (c) y to the

Practice Problem 6 Translate using symbols.

(a) Four to the sixth power (b) x cubed (c) Ten squared 10 2x346

eighth power = y8Seven squared = 72Five cubed = 53

Following the Order of OperationsIt is often necessary to perform more than one operation to solve a problem. Forexample, if you bought one pair of socks for $3 and 4 undershirts for $5 each, youwould multiply first and then add to find the total cost. In other words, the order inwhich we performed the operations (order of operations) was multiply first, thenadd. However, the order of operations may not be as clear when dealing with a mathstatement. When we see the problem written as understanding what to docan be tricky. Do we add, then multiply, or do we multiply before adding? Let’swork this calculation both ways.

Add First Multiply First

Wrong! Correct3 + 4152 = 3 + 20 = 233 + 4152 = 7152 = 35

3 + 4152

Teaching Example 5 Evaluate for .

Ans: 16

a = 2a4

Teaching Example 6 Translate usingsymbols.

(a) four to the fifth power

(b) three cubed

(c) x squared

Ans: (a) (b) (c) x23345

Since can be written 23 is correct. Thuswe see that the order of operations makes a difference. The following rule tellswhich operations to do first: the correct order of operations. We call this a list ofpriorities.

3 + 15 + 5 + 5 + 52 = 3 + 20,3 + 4152


2 # 32= 18

= 2 # 9

# 2 # 32= 2 # 9 Identify: The highest priority is exponents. Calculate: 3 3=9.

Replace: 3¤ with 9.

Identify: Multiplication is last. Calculate: 2 9=18.Replace: 2 9 with 18.

#

#


ORDER OF OPERATIONSFollow this order of operations.

Do first

Do last

1. Perform operations inside parentheses.

2. Simplify any expression with exponents.

3. Multiply or divide from left to right.

4. Add or subtract from left to right.

parentheses : exponents : multiply or divide : add or subtract

‹

Teaching Tip Many students use thefollowing statement to help themmemorize the order of operations:Please Excuse My Dear Aunt Sally.Emphasize that they must alsoremember to complete operations asthey read, when choosingbetween multiplication/division oraddition/subtraction.

AS L : R

MD L : R

E

P

L : R,

Now, following the order of operations, we can clearly see that to findwe multiply and then add. You will find it easier to follow the order of

operations if you keep your work neat and organized, perform one operation at atime, and follow the sequence identify, calculate, replace.

1. Identify the operation that has the highest priority.2. Calculate this operation.3. Replace the operation with your result.

3 + 4152,

EXAMPLE 7 Evaluate.

Solution

23- 6 + 4

23- 6 + 4 = 6

= 2 + 4

= 8 - 6 + 4

# #

23- 6 + 4 = 8 - 6 + 4 Identify: The highest priority is exponents.

Calculate: 2 2 2=8. Replace: 2‹ with 8.

Identify: Subtraction has the highest priority.Calculate: 8-6=2. Replace: 8-6 with 2.

Identify: Addition is last. Calculate: 2+4=6.

Replace: 2+4 with 6.

Note that addition and subtraction have equal priority. We do the operationsas they appear, reading from left to right. In Example 7 the subtraction appearsfirst, so we subtract before we add.

Practice Problem 7 Evaluate. 632+ 2 - 5


Teaching Example 7 Evaluate.

Ans: 11

4 + 32- 2

EXAMPLE 8 Evaluate.

Solution

2 # 32

CAUTION: does not equal We must follow the rules for the order ofoperations and simplify the exponent before we multiply; otherwise, we will getthe wrong answer.

Practice Problem 8 Evaluate. 324 # 23

3262!2 # 32


Ans: 200

2 # 102



EXAMPLE 9 Evaluate.

Solution We always perform the calculations inside the parentheses first.Once inside the parentheses, we proceed using the order of operations.

4 + 316 - 222 - 7 = 3 = 3 = 10 - 7 = 4 + 6 - 7 = 4 + 31 2 2 - 7 = 4 + 316 - 4 2 - 7

4 + 316 - 22 2 - 7

4 + 316 - 222 - 7 Teaching Example 9 Evaluate.

Ans: 58

3 + 2(52+ 3) - 1

Teaching Tip It is important thatstudents have a positive experiencewith all new topics they learn. Forthis reason, combined with the factthat students find the order ofoperations difficult, problemsinvolving brackets and nestedparentheses will be introduced laterin the book. In addition, the order ofoperations will be revisited often inthe text.


Ans: 3

(12 - 6 , 2)

(8 - 5)

Within the parentheses, exponents have the highestpriority: We must finish all operations inside theparentheses, so we subtract:

The highest priority is multiplication:

Add first:

Subtract last: 10 - 7 = 3.

4 + 6 = 10.

3 # 2 = 6.

6 - 4 = 2.

22= 4.

Practice Problem 9 Evaluate. 262 + 7110 - 3 # 22 - 4

As we stated earlier, it is easier to follow the order of operations if we keepour work neat and organized, perform one operation at a time, and follow thesequence: identify, calculate, replace.

EXAMPLE 10 Evaluate.

Solution We rewrite the problem as division and then follow the order ofoperations.

We perform operations inside parentheses first.

Divide.

Practice Problem 10 Evaluate.

214 + 8 , 2217 - 32

8 , 4 = 26 , 3 = 2; 5 - 1 = 4. 16 + 2 2 , 4

16 + 6 , 32 , 15 - 12

16 + 6 , 3215 - 12

Reviewing for an ExamReviewing for an exam enables you to connect concepts youlearned over several classes. Your review activities shouldcover all the components of the learning cycle.

The Learning Cycle

1. Reread your textbook. Make 3-by-5 study cards as follows.• Write the name of the new term or rule on the front of

the card. Then write the definition of the term or therule on the back.

• Write sample examples on the front of the card and thesolutions on the back.

• Periodically use these cards as flash cards and quizyourself, or study with a classmate.

2. Reread your notes. Study returned homework and quizzesand redo problems you got wrong.

Reading

" Writing

c T

Seeing ; Verbalizing ; Listening

3. Read the Chapter Organizer and solve some of the reviewproblems at the end of the chapter. Check your answersand redo problems you got wrong.

4. After you finish the exercises in Section 1.6, complete theHow Am I Doing? Sections 1.1–1.6. Complete this as if itwere the real exam. Do not refer to notes or to the textwhile completing the exercises. Then check your answers.The problems you missed are the type of problems thatyou should get help with and review before the exam.

5. Start reviewing several days before the exam so that youhave time to get help if you need it.

It is not a good idea to complete all six steps at one time.For best results, complete each step at a separate sitting andstart the process early so that you are done at least three daysbefore the exam.

Exercise1. Can you think of other ways of preparing for an exam that

include activities in the learning cycle? Answers may vary:form study groups, work on computer tutorials, . . . .


60

1. Write in words the question being asked by the equa-tion What number squared is equal to 16?n2

= 16.2. Write in words the question being asked by the equa-

tion What number cubed is equal to 27?x3= 27.


1.6 EXERCISES

Write each product in exponent form.

3. 4. 5. 6. z2z # za5a # a # a # a # a424 # 4232 # 2 # 2

7. 4 8. y 9. 10. 939 # 9 # 9343 # 3 # 3 # 3y141

Write each product in exponent form.

11. 12. 13. 22z52 # 2 # z # z # z # z # z32x33 # 3 # x # x # x52a35 # 5 # a # a # a

14. 15. 16. 72x1y27 # 7 # x # y # y53y2x25 # 5 # 5 # y # y # x # x32y43 # 3 # y # y # y # y

17. 18. x5 # 72 or 72x5x # x # x # x # x # 7 # 7n5 # 92 or 92n5n # n # n # n # n # 9 # 9

Write as a repeated multiplication.

19. (a) (b) 20. (a) (b) x # xx27 # 7 # 7 # 7 # 7 # 776y # y # y # y # yy57 # 7 # 773

Evaluate.

21. 8 22. 27 23. 25 24. 3662523323

25. 1 26. 1 27. 49 28. 9327211116

29. 256 30. 729 31. 10 32. 5511019344

33. 125 34. 16 35. 1,000,000 36. 10,0001041062453

37. for 25 38. for 27 39. for 1 40. for 1b = 1b14a = 1a4y = 3y3x = 5x2

Translate using numbers and exponents.

41. Seven to the third power 42. Three cubed 3373

43. Nine squared 44. Four to the seventh power 4792

Evaluate.

45. 5 46. 13 47. 5172+ 5 - 33 # 5 - 23 # 4 - 7

48. 212 49. 45 50. 164 # 225 # 3263+ 4 - 8

51. 8 52. 64 53. 2152- 7 + 34 # 422 # 22

54. 63 55. 13 56. 325 + 3 # 99 + 2 # 243- 8 + 7

57. 19 58. 79 59. 2540 , 5 * 2 + 328 + 17 + 4329 + 16 + 22260. 13 61. 16 62. 113 * 12 , 4 + 22 * 15 , 5 + 1062

, 6 * 2 + 1

63. 6 64. 29 65. 518 + 4 , 2215 - 3233

+ 6 , 322+ 8 , 4



To Think About79. Fred wanted to evaluate He multiplied

3 times 6 to get 18. What is wrong with his reasoning?What is the correct answer? He should have multiplied3 times 2 first and then added 4 to get 10.

3 # 2 + 4. 80. Sara wanted to evaluate She squared 8 to get 64.What is wrong with her reasoning? What is the cor-rect answer? She should have squared 4 first and thenmultiplied by 2 to get 32.

2 # 42.

81. Multiply: Doyou see a pattern that might suggest a quick way tomultiply a number by a power of 10? Explain. Theexponent on the 10 determines the number of trailing zeroswe attach to the number.

21 # 101; 21 # 102; 21 # 103; 21 # 104. 82. Multiply: Do you see apattern that might suggest a quick way to multiply 10by a power of 10? Explain. We add exponents to deter-mine the exponent of the product.

101 # 102; 101 # 103; 101 # 104.

Cumulative Review83. [1.2.4] Add. 68414079 + 2762 84. [1.3.4] Subtract. 84238900 - 477

85. [1.4.4] Multiply. (387)(196) 75,852 86. [1.4.2] Translate using symbols. The product of twoand some number. 2x

1. Write the product in exponent form.

(a) (b)

2. Evaluate.

(a) 16 (b) 11524555 # 5 # 5 # 5 # 593x29 # 9 # 9 # x # x

3. Evaluate. 3 4. Concept Check Explain in what order you would dothe steps to evaluate Answers may vary

50 + 3 * 52, 25.

22+ 2(10 , 2) - 11

Quick Quiz 1.6

1. Write the product in exponent form.

(a) Ans: (b) Ans:

2. Evaluate.

(a) Ans: 27 (b) Ans: 111133747 # 7 # 7 # 762a36 # 6 # a # a # a


3. Evaluate. Ans: 2433- 2(12 , 4 + 2) + 7

One Step Further75. 1932 # 6 - 4143

- 5 # 222 + 3 76. 5263 # 4 - 5132+ 4 # 232 + 5

66. 2 67. 68. 1116 - 42136 , 6 * 22

44

= 113 + 12

112 , 6 * 2215 + 15 , 5219 - 52

69. 100 70. 72 71. 1959 - 411 + 5 # 22 + 43 + 415 # 2 + 82 - 37 + 513 # 4 + 72 - 2

72. 16 73. 53 74. 762 + 1213 # 2 + 12 - 106 + 214 # 5 + 92 - 1188 - 312 + 6 # 42 + 6

77. 3412 # 5 - 3133- 2 # 322 + 1 78. 4242 # 5 - 3152

+ 2 # 422 + 3



AUTOMOBILE LEASING VS. PURCHASELouvy has his eye on a brand new car. He thinks heshould lease the car because his best friend Tranh hasa car lease and says he can get the same deal forLouvy. On the other hand, Louvy’s girlfriend Alliesays it is always better to buy the car and finance it bytaking out a loan.

Louvy does some research and finds that it is not atall simple. While a lease offers lower monthly pay-ments, at the end of the lease period you are left withnothing, except expenses.

Look at the following comparison for lease vs. buyfor the car Louvy is considering.

• You don’t want major repairs risk• You want a lower monthly payment

IF you lease a car:

• You pay only that portion of the vehicle you use• There may be mileage restrictions, hidden fees, or

security deposits• You can sometimes find a lease with no down

payment• You pay sales tax only on monthly fees• You usually must keep the car for the entire lease

period, or pay a heavy penalty

You may wish to buy a car if:

• You intend to keep it a long time• You want to be debt-free after a time• You qualify for a very low interest rate• The long term cost is more important than the

lower monthly payment

IF you buy a car:

• You pay for the entire vehicle• You pay the sales tax on the entire price of the car• There are no hidden mileage costs, except in wear

and tear on the vehicle• You can sell or trade the car during the period of

the loan

Use Math to Save Money

Lease Purchase

Automobile price $23,000.00 $23,000.00

Interest rate 6% 6%

Length of loan 36 months 36 months

Down payment $1000.00 $1000.00

Residual (value of car youare turning in, amount you pay if you wish to purchase it) $11,000.00 Not applicable

Monthly payment $388.06 $669.28

1. How much would Louvy pay over the entire lengthof the loan?

2. How much would Louvy pay over the entirelength of the lease?

3. If Louvy decided to buy the car at the end of thelease, he would have to pay $11,000.00 in additionto his lease cost. What would that bring the totalcost of that car up to?

Making It Personal for You

4. How much can you afford per month for pay-ments for a car? How much would you pay ininsurance, taxes, and gas? Answers may vary

5. Would you prefer to lease or buy? Why?Answers may vary

Facts You Should Know

You may wish to lease a car if:

• You want a new vehicle every 2–3 years• You don’t drive an excessive number of miles each

year

$14,970.16 + $11,000.00 = $25,970.16

$1000 + 36 * $388.06 = $14,970.16

$1000 + 36 * $669.28 = $25,094.08


Sections 1.1–1.6

1.

2.

3. 17,200,000

4. (a) (b)

5. 20

6. 10,105

7. 50 in.

8.

9. 33,222

10. 2x

11. 40y

12. 298,746

13. 72 rooms

14.

15. 503 R1

16.

17. 64

18. 18

19. 5

33n4

144 , x

11 - x

x + 12a + 9

6

9000 + 60 + 2

63

How are you doing with your homework assignments in Sections 1.1 to 1.6? Do youfeel you have mastered the material so far? Do you understand the concepts you havecovered? Before you go further in the textbook, take some time to do each of the fol-lowing problems.

1.11. Write in expanded notation. 9062

2. Replace the question mark with the appropriate inequality symbol or 16 ? 22

3. Round 17,248,954 to the nearest hundred thousand.

1.24. Use the associative and/or commutative properties of addition, then simplify.

(a) (b)

5. Evaluate if x is equal to 9 and y is equal to 11.

6. Add.

7. Find the perimeter of the following shape madeup of two rectangles.

1.38. Translate using numbers and symbols. Eleven

decreased by a number


1.410. Translate using numbers and symbols. Double a number

11. Simplify. 12. Multiply.

13. A small hotel is 6 stories high with 12 rooms on each floor. How many roomsare in the hotel?

1.514. Translate using numbers and symbols. The quotient of 144 and x

15. Divide.

1.616. Write the product in exponent form.

17. Evaluate.

Evaluate.

18.

19. 12 + 102 + 12 , 6 - 32

2 # 32

43

n # n # n # n # 3 # 3 # 3

362,664721

12371212621421y # 52

39,204 - 5982

9532 + 251 + 322

x + y

16 + x + 42 + 216 + a2 + 3

7 .6

Now turn to page SA-2 for the answers to each of these problems. Each answer alsoincludes a reference to the objective in which the problem is first taught. If you missedany of these problems, you should stop and review the Examples and Practice Prob-lems in the referenced objective. A little review now will help you master the materialin the upcoming sections of the text.

9 in.11 in.

8 in.

6 in.�


64

In this section we will see how to translate some new types of phrases into symbols.We will also use the skills we learned in previous sections to simplify and evaluatealgebraic expressions.

Using Symbols and Key Words for Expressing Algebraic Expressions

When we translate phrases into numbers and symbols we must take care to preservethe order of operations indicated by the phrase. When a phrase contains key wordsfor more than one operation, the phrases sum of or difference of indicate that theseoperations must be placed within parentheses so that they are completed first. Weillustrate below.

Three times the difference of five and two Three times five minus two

The phrase difference of indicates thatwe must place within parentheses.

We must include parentheses when we see the phrases sum of or difference of orwe will get the wrong answer: but 3 # 5 - 2 = 15 - 2 = 13.3 # 15 - 22 = 3 # 3 = 9

5 - 2

T T T T T

3 # 5 - 2T T T T

3 # 15 - 22

1.7 MORE ON ALGEBRAIC EXPRESSIONS


Use symbols and key wordsfor expressing algebraicexpressions.

Evaluate algebraic expressionsinvolving multiplication anddivision.

Use the distributive propertyto simplify numerical andalgebraic expressions.


(a) Two times x plus seven (b) Two times the sum of x and seven

Solution(a) Two times x plus seven

2x + 7

T T T T T

2 # x + 7

(b) Two times the sum of and T T T T

2 # 1 x + 7 2sevenx

The key phrasesum of indicatesthat isplaced withinparentheses.

x + 721x + 72


(a) Five times y plus three (b) Three times the sum of m and two

(a) (b) 3(m + 2)5y + 3

Evaluating Algebraic Expressions Involving Multiplication and Division

We evaluate variable expressions involving multiplication and division just as wedid expressions involving addition and subtraction. For example, to evaluate 2n if nis equal to 5, we replace the variable in the expression with 5 and then simplify:2n : 2152 = 10.

EXAMPLE 2 Evaluate for

Solution

= 3

=

217

=

118 + 327

12 a + 32

7=

12 # 9 + 327

a = 9.12a + 32

7

We replace with .

We multiply first.

Next, we complete operations within the parentheses.

We divide.

9a


(a) Two times x minus seven

(b) Two times the difference of x and 7

Ans: (a) (b) 2(x - 7)2x - 7

Teaching Example 2 Evaluate for.

Ans: 5

x = 3

(6x + 7)

5


Section 1.7 More on Algebraic Expressions 65

Practice Problem 2

Evaluate for 2y = 2.15y - 42

3

(b)

= 7

=

142

=

142- 222

1x2- 22y

EXAMPLE 3 Evaluate.

(a) for and (b) for and

Solution We replace each variable with the indicated value and then followthe order of operations to simplify.

(a)

= 30 = 9 + 15 + 6 = 3 # 3 + 3 # 5 + 63x + 3y + 6

y = 2x = 41x2

- 22y

y = 5x = 33x + 3y + 6

We replace x with 3 and y with 5.We multiply first.We add last.

We replace x with 4 and y with 2.

We square 4 first then subtract:

We divide last: 14 , 2 = 7.

16 - 2 = 14.42= 16,

Practice Problem 3 Evaluate.

(a) for and 30 (b) for and 2b = 3a = 2(a3

- 2)b

n = 3m = 75m - 2n + 1

Using the Distributive Property to Simplify Numerical and Algebraic Expressions

A property that is often used to simplify and multiply is the distributive property.This property states that we can distribute multiplication over addition or subtrac-tion. The following example will help you understand what we mean by this.

“4 times ” is written We can find this product using repeatedaddition.

4(n + 7) = 4n + 284 # 7+4n

= 1n + n + n + n2 + 17 + 7 + 7 + 72∂ ∂

41n + 72 = 1n + 72 + 1n + 72 + 1n + 72 + 1n + 7241n + 72.1n + 72

We write asrepeated addition.

We change the order ofaddition and group then’s and 7’s together.We have 4 n’s plus 4 7’s.

41n + 72

A shorter way to do this is to distribute the 4 by multiplying each number orvariable inside the parentheses by 4.

4 # 7

4 # n

# #4(n+7)=4(n+7)=4 n+4 7=4n+28

We can state the distributive property as follows.

DISTRIBUTIVE PROPERTYIf a, b, and c are numbers or variables, then

We distribute a over addition and subtraction by multiplying every number orvariable inside the parentheses by a. Then we simplify the result.

a1b + c2 = ab + ac and a1b - c2 = ab - ac


(a) for and

(b) for and

Ans: (a) 16 (b) 4

y = 3x = 2(x3

+ 4)

y

b = 2a = 94 + 2a - 3b




EXAMPLE 4 Use the distributive property to simplify.

Solution

31x - 22

3 # x

3 # 2

Simplify.

# #

Multiply 3 times x.

Multiply 3 times 2.

3(x-2)=3x-6

3(x-2)=3(x-2)=3 x-3 2

Practice Problem 4 Use the distributive property to simplify.

(a) (b)

CAUTION: We must only use the distributive property if the numbers or variablesinside the parentheses are separated by a or sign. We do not use the distribu-tive property when the numbers or variables inside the parentheses are separated bymultiplication or division symbols. Thus, we can use the distributive property inExample 4 because in the expression the x and the 2 are separated by a

sign. We cannot use the distributive property for the expression 3(2x) because the2 and the x are not separated by a or sign. Thus while3(2x) = 3 # 2 # x = 6x.

31x - 22 = 3x - 6-+

-

31x - 22

-+

4y + 1241y + 322x - 1021x - 52

EXAMPLE 5 Simplify.

Solution First we use the distributive property and then we simplify.

= 2y + 6 = 2y + 2 + 4

2 1y + 12 + 4 = 2 # y + 2 # 1 + 4

21y + 12 + 4

We use the distributive propertyto multiply

We simplify: 2 + 4 = 6.

21y + 12.

Practice Problem 5 Simplify. 7y + 2371y + 32 + 2

Getting the Most from Your Study TimeDid you know that there are many things you can do to increaseyour learning when you study? If you use the followingstrategies, you can improve the way you study and learn morewhile studying less.

1. Read the material and review your class notes on thesame day as your class meets.

2. Do homework in more than one sitting so that you arefresh for the later problems, which are usually thehardest.

3. Check your answer only after you complete a problem. Putan * beside any problem that you get wrong or don’t knowhow to start.

4. Follow up wrong answers. Check your work for errors or lookin the book for a similar problem. Compare your solutionwith the book’s and, if necessary, rework the problem usingthe book’s solution as a guide. Use this process to solve the

problems you didn’t know how to start. If you still can’tsolve a problem, reread the section or ask for help.

5. Revisit * problems. After finishing the assignment, workanother problem that is like each * problem. In this text,an even-numbered problem is similar to the precedingodd-numbered problem.

6. Review or rewrite your notes at the end of each week. Worka few problems in the sections covered since the last test.Review past tests periodically, especially if you are havingdifficulty or having trouble remembering earlier material.

It is important that you realize that completing yourhomework assignment and studying your homework areseparate activities. Activity 4 describes the process ofcompleting your homework, whereas activity 5 describes theprocess of studying your homework. For best results, thesetwo activities should be done at different times.

Teaching Example 4 Use the distributiveproperty to simplify.

Ans: 3a - 3

3(a - 1)

Teaching Example 5 Simplify.

Ans: 4m + 18

10 + 4(m + 2)


67

Verbal and Writing SkillsState what property is represented in each mathematical statement.

1.7 EXERCISES

1.Distributive property of multiplication over addition513 + 42 = 5 # 3 + 5 # 4 2.

Distributive property of multiplication over subtraction516 - 42 = 5 # 6 - 5 # 4

3. (a) False. We only use the distribu-tive property when the terms inside the parentheses arebeing added or subtracted.

(b) True. The terms 3 and yare separated by a sign, so we can use the distribu-tive property.

+

813 + y2 = 8 # 3 + 8 # y

813y2 = 8 # 3 # 8 # y 4. (a) False. We only use the distribu-tive property when the terms inside the parentheses arebeing added or subtracted.

(b) True. The terms 2 and xare separated by a sign, so we can use the distribu-tive property.

+

412 + x2 = 4 # 2 + 4 # x

412x2 = 4 # 2 # 4 # x

5. 21x + 12 = 2 # � x � + 2 # � 1 � 6. 31y + 22 = 3 # � y � + 3 # � 2 �

7. 61y - 32 = 6 # � y � - 6 # � 3 � 8. 81x - 12 = 8 # � x � - 8 # � 1 �

9. Six times y plus two 6y + 2 10. Four times x plus three 4x + 3

11. Seven times four minus one 7 # 4 - 1 12. Eleven times five minus two 11 # 5 - 2

13. Four times the sum of three and nine 413 + 92 14. Nine times the sum of four and six 914 + 6215. Triple the sum of y and six 31y + 62 16. Double the sum of x and one 21x + 1217. Eight times the difference of four and y 814 - y2 18. Five times the difference of six and x 516 - x2

19. (a) Four times two plus seven(b) Four times the sum of two and seven

412 + 72 = 36

4 # 2 + 7 = 15 20. (a) Eight times six plus one(b) Eight times the sum of six and one

816 + 12 = 56

8 # 6 + 1 = 49

21. (a) Four times three minus one(b) Four times the difference of three and one

413 - 12 = 8

4 # 3 - 1 = 11 22. (a) Two times seven minus one(b) Two times the difference of seven and one

217 - 12 = 12

2 # 7 - 1 = 13

23. (a) Twelve times one plus three(b) Twelve times the sum of one and three

1211 + 32 = 48

12 # 1 + 3 = 15 24. (a) Nine times four plus one(b) Nine times the sum of four and one

914 + 12 = 45

9 # 4 + 1 = 37

25. for and 38

b = 6a = 24a + 5b 26. for and 22

n = 5m = 43m + 2n 27. for and 60

y = 2x = 98x - 6y

Are the following true or false? Explain your answers.

Fill in each box with the correct number or variable.


Mixed Practice Exercises 19–24Translate using numbers and symbols, then simplify.

Evaluate for the given values.

28. for and 62y = 5x = 89x - 2y 29. for 5x = 111x + 42

330. for 4y = 131y + 72

5

31. for and

7

b = 3a = 51a2

- 42b

32. for and

10

n = 3m = 61m2

- 62n

33. for and

6

y = 2x = 21x3

+ 42y

34. for and

6

y = 6x = 31x3

+ 92y

35. for and

2

b = 5a = 21a2

+ 62b

36. for and

2

m = 7n = 31n2

+ 52m



61. for and 21x = 2y = 6yx2- 3 62. for and 49b = 3a = 5ab2

+ 4

65. (a) Add four times.

(b) Multiply using the distributive property.

(c) What do you notice about the answers in (a) and(b)? The answers are the same.

41x + 22 = 4x + 841x + 221x + 22 + 1x + 22 = 4x + 8

1x + 22 + 1x + 22 +1x + 22 66. (a) Add three times.

(b) Multiply using the distributive property.

(c) What do you notice about the answers in (a) and(b)? The answers are the same.

31x + 42 = 3x + 1231x + 421x + 42 = 3x + 12

1x + 42 + 1x + 42 +1x + 42

67. [1.4.3] Simplify. 64x81221x # 42 68. [1.2.3] Evaluate if x is 2. 64 + x

69. [1.2.3] Evaluate if x is 1 and y is 3. 8x + y + 4 70. [1.3.4] Subtract. 15382001 - 463

63. for and 15b = 2a = 51a2

- 32 + 23

b64. for and 2b = 7a = 31a3

- 42 - 32

b

37. for 7y = 161y - 22

238. for 5y = 181y - 32

339. for and

29n = 7m = 24m + 3n

40. for and 44

y = 6x = 45x + 4y 41. for and

5

y = 4x = 51x2

- 52y

42. for and

3

y = 11x = 61x2

- 32y

43. 4x + 441x + 12 44. 2x + 221x + 12 45. 3n - 1531n - 5246. 6n - 2461n - 42 47. 3x - 1831x - 62 48. 4x - 1241x - 3249. 4x + 1641x + 42 50. 5x + 4551x + 92 51. 2x + 1721x + 62 + 5

52. 4x + 1441x + 22 + 6 53. 2y + 721y + 12 + 5 54. 7y + 1071y + 12 + 3

55. 4x + 1841x + 32 + 6 56. 3x + 1131x + 22 + 5 57. 9y + 691y + 12 - 3

58. 5y + 351y + 12 - 2 59. 3x + 231x + 12 - 1 60. 6x + 361x + 12 - 3

Use the distributive property to simplify.

One Step FurtherEvaluate for the given values.

To Think About

Cumulative Review


Double the sum of n and five

2. Use the distributive property to simplify.

6y + 96(y + 1) + 32(n + 5)

3. Evaluate.

(a) if and 17

(b) if and 2

4. Concept Check Simplify , then evaluatefor . Compare results and state the dif-

ference in the process to simplify and to evaluate.Answers may vary

x = 25(x + 1)5(x + 1)

y = 7x = 4(x2

- 2)y

y = 4x = 33x + 2y

Quick Quiz 1.7

1. Translate using numbers and symbols.Two times the sum of x and three.Ans:

2. Use the distributive property to simplify.

Ans: 2a + 152(a + 6) + 3

2(x + 3)


3. Evaluate.

(a) if and Ans: 34

(b) if and Ans: 4n = 8m = 6(m2

- 4)

n

b = 5a = 14a + 6b


69

1.8 INTRODUCTION TO SOLVING LINEAR EQUATIONS

Combining Like TermsIn algebra we often deal with terms such as 4y or 7x. What do we mean by terms?

A term is a number, a variable, or a product of a number and one or more vari-ables. Terms are separated from other terms in an expression by a sign or a sign.Often a term has a number factor and a variable factor. The number factor is calledthe coefficient.

-+


Combine like terms.

Translate English statementsinto equations.

Solve equations using basicarithmetic facts.

Translate and solve equations.

numerical partof term

variable partof term

7x

A term that has no variable is called a constant term, and a term that has a variableis called a variable term.

What do the expressions 3n and 4x mean? 3n is the term that represents the sumand 4x is the term that represents the sum As we saw

earlier, 3n and 4x also indicate multiplication: 3 times n and 4 times x.

We count three n’s added. We count four x’s added.x + x + x + x

T

4x

n + n + n

T

3n

x + x + x + x.n + n + n,

constant term variable terms

9+3n+4x

Practice Problem 1 Write a term that represents each of the following.

(a) Four n’s 4n (b) 3y

(c) Eight 8 (d) One y y

y + y + y

EXAMPLE 1 Write a term that represents each of the following.

(a) Two y’s (b)(c) Seven (d) One x

Solution(a) Two (b)(c) (d) One or xx = 1xSeven = 7

a + a + a + a = 4ay’s = 2y

a + a + a + a

We see many examples of adding and subtracting quantities that are like quan-tities, as shown in the following example.

However, we cannot combine things that are not the same:

Similarly, in algebra we cannot combine terms that are not like terms. Liketerms are terms that have identical variable parts. For example, in the expression

the terms 8x and 2x are called like terms since they have the samevariable parts. They are both counting x’s.8x + 6b + 2x,

7 trucks - 4 feet 1cannot be done!2

3 feet + 7 feet = 10 feet 7 trucks - 2 trucks = 5 trucks

Teaching Example 1 Write a term thatrepresents each of the following.

(a) Six (b) Three x’s

(c) Two y’s (d)

Ans: (a) 6 (b) (c) (d) 5z2y3x

z + z + z + z + z




7ab and 2ab are like terms; the variable parts, ab, are the same.

7ab+4a+2ab+3y

There are no terms like 4a;none have the exact variable part, a.

There are no terms like 3y;none have the exact variable part, y.

Expression

The variable parts are the same.

Like Terms

8x+6b+2x 8x 2x

There are no like terms for 6b, since none of the terms have exactly the same vari-able part as 6b.

The numerical part of a term is called the coefficient of a term. The coefficienttells you how many you have of whatever variable follows. To combine like terms,we either add or subtract the coefficients of like terms.

COMBINING LIKE TERMSTo combine like terms, add or subtract the numerical coefficients of like terms.The variable parts stay the same.

6x + 4x = 10x 8y - 2y = 6y

EXAMPLE 2 Identify the like terms.

Solution

7ab + 4a + 2ab + 3y

Practice Problem 2 Identify the like terms.

2mn and 4mn

2mn + 5y + 4mn + 6n

EXAMPLE 3 Identify like terms, then combine like terms.

(a) (b) (c)

Solution

4xy + 8y + 2xy9m - m - 83x + 7y + 2x

There are no terms like 7y;none have the exact variable part.

(a) We identify and group like terms.

“Three x’s plus two x’s” can be restated as13 + 22x’s; “three plus two x’s.”

+ + = +

= 13 + 22x +

+ + = 1 + 2 +

13 + 22x = 5x3x

3x 3x

7y 7y

7y7y 7y

5x2x

2x2x

(b) = 9 m - 1 m - 8

9m - m - 8 = 9m - 1m - 8 Write the numerical coefficient 1.

Think: “nine m’s minus one m equals 8 m’s.”

Note that the term m does not have a visible numerical coefficient. We canwrite “1” as the numerical coefficient since 1m = 1 # m = m.

9m - m - 8 = 8 m - 8

Teaching Example 2 Identify the like terms.

Ans: and x are like terms; and are like terms.

6xy5xy3x

5xy + 3x + 4y + x + 6xy

Teaching Example 3 Identify like terms,then combine like terms.

(a) (b)

(c)

Ans: (a) (b)

(c) 7m + 3mn

3x - 25a + 8b

3m + 4m + 3mn

4x - x - 22b + 5a + 6b

Teaching Tip Once students learn howto combine like terms, they have ahard time with the fact that whenmultiplying variable expressions theycan multiply coefficients of unliketerms. As a result, they consider theexpression (2x)(4y) to be simplified.To help students see the differencebetween multiplying expressions andcombining like terms, revisit themethod used to multiply expressionsusing the familiar symbol This willhelp students remember that thecommutative property allows themto rearrange terms and multiplycoefficients of unlike terms.

# .


Section 1.8 Introduction to Solving Linear Equations 71

2x � y

x � 4y

3x � 2y

Teaching Example 4 Write the perimeter ofthe rectangular figure as an algebraicexpression and simplify.

Ans: 10x + 6y

3x � 2y

2x � y

(c) We identify and group like terms.

Add the numerical coefficients of liketerms.

We write 8y as a separate term. We cannot combine it with 6xy since the vari-able parts are not the same.

4xy + 8y + 2xy = 6xy + 8y

= 14 + 22xy + 8y

4xy + 8y + 2xy = 14xy + 2xy2 + 8y

Practice Problem 3 Identify like terms, then combine like terms.

(a) (b)(c) No like terms; 7x + 3y + 3z7x + 3y + 3z

5y + 6x4y + 5x + y + x5ab + 4a2ab + 4a + 3ab

EXAMPLE 4 Write the perimeter of the rectangular figure as an algebraicexpression and simplify.

Practice Problem 4 Write the perimeter of the triangular figure in themargin as an algebraic expression and simplify. 6x + 7y

2a � 3b

4a � 7b

Solution Since the figure is a rectangle, opposites sides are equal.

We add all sides to find the perimeter:

= 12a + 20b

= 12a + 2a + 4a + 4a2 + 13b + 3b + 7b + 7b212a + 3b2 + 14a + 7b2 + 12a + 3b2 + 14a + 7b2

Translating English Statements into EquationsTwo expressions separated by an equals sign is called an equation. When we use anequals sign we are indicating that two expressions are equal in value.1=2,

2a � 3b

4a � 7b

2a � 3b

4a � 7b

We must combine like terms.

We use the associative andcommutative properties tochange the order of additionand regroup.We combine like terms.

The algebraic expression for the perimeter is 12a + 20b.

The value of thisexpression is 8.

The value of thisexpression is 8.

2+6=8

Some English phrases for the symbol are

is is the same as equals is equal to the result is

=

�

�



Solving Equations Using Basic Arithmetic FactsSuppose we ask the question “Three plus what number is equal to nine?” The an-swer to this question is 6, since three plus six is equal to nine. The number 6 is calledthe solution to the equation and is written “the value of x is 6.” Inother words, an equation is like a question, and the solution is the answer to thisquestion.

x = 6:3 + x = 9

EXAMPLE 5 Translate each English sentence into an equation.

(a) Three subtracted from what number is equal to ten?(b) Five times what number is the same as thirty-five?(c) Kari’s savings decreased by $100 equals $500.

Solution

(a) Three subtracted from what number is equal to ten?

(b) Five times what number is the same as thirty-five?

(c) We let x represent Kari’s savings, the unknown value.Kari’s savings decreased by $100 equals $500.

T T T T T

x - $100 = $500x - $100 = $500

T T T T T5 # n = 35

5n = 35

n - 3 = 10

Practice Problem 5 Translate each English sentence into an equation.

(a) Four times what number is the same as seven?(b) Three subtracted from what number is equal to nine?(c) The number of baseball cards in a collection plus 20 new cards equals 75 cards.

n + 20 = 75

x - 3 = 9

4x = 7

c

c

c

English Phrase Math Symbols

Question EquationThree plus what number is equal to nine?

Answer to the Question SolutionThree plus six is equal to nine. x = 6

3 + x = 9

The solution to an equation must make the equation a true statement. For example,if 6 is a solution to we must get a true statement when we evaluate theequation for

We replace the variable with 6, then simplify.

We get a true statement. 9 = 9

3 + 6 = 9

3 + x = 9

x = 6.3 + x = 9,

To solve an equation we must find a value for the variable in the equation thatmakes the equation a true statement.


Teaching Tip Many students think thatto learn mathematics they must mem-orize a set of rules, and become over-whelmed and frustrated when theycan’t remember which rule to use. Asa result, students don’t think aboutwhat they are doing. When solvingequations, students become overlyconcerned with “What rule should Iuse?” By realizing that an equationis a question that we must find theanswer to, rather than just math witha set of rules, students understandand think about what they are doing.Explain that as equations becomemore difficult, we must use rules tohelp us solve them.

Teaching Example 5 Translate each Englishsentence into an equation.

(a) What number squared is equal totwenty-five?

(b) The sum of what number and ten isequal to fifty?

(c) The quotient of eighteen and whatnumber is equal to six?

Ans: (a) (b)

(c)18x

= 6

x + 10 = 50x2= 25



EXAMPLE 6 Is 2 a solution to

Solution If 2 is a solution to when we replace x with the valuewe will get a true statement.

“Six minus what number equals nine?”

Replace the variable with 2 and simplify.

This is a false statement.

Since is not a true statement, 2 is not a solution to .6 - x = 94 = 9

4 � 9

6 - 2 � 9

6 - x = 9

x = 26 - x = 9,

6 - x = 9?

Practice Problem 6 Is 5 a solution to Nox + 8 = 11?

EXAMPLE 7 Solve the equation and check your answer.

Solution To solve the equation we answer this question:

“Three plus what number is equal to ten?”

Using addition facts we see that the answer, or solution, is 7.To check the solution, we replace n with the value and verify that we get

a true statement.

Check: Write the equation.


✓ Verify that we get a true statement.

Since we get a true statement, the solution to is 7 and is written .n = 73 + n = 10

10 = 10

3 + 7 � 10

3 + n = 10

n = 7

3 + n = 10,

3 + n = 10

Practice Problem 7 Solve the equation and check your answer.

n = 5

4 + n = 9


Solution To solve the equation we answer this question:

“Nine times what number equals forty-five?”

The answer or solution is 5 and is written To check the answer, we replace n with the value and verify that we get

a true statement.

Check:


✓ Verify that this is a true statement.

Thus the solution to is 5 and is written n = 5.9n = 45

45 = 45

91 5 2 � 45

9 n = 45

n = 5n = 5.

9n = 45,

9n = 45


x = 8

6x = 48


Teaching Example 6 Is 7 a solution to?

Ans: Yes

15 - x = 8

Teaching Example 7 Solve the equation.

Ans: “Twelve subtracted from whatnumber equals zero?” a = 12

a - 12 = 0

Teaching Example 8 Solve the equation.

Ans: “Four times what number equalstwenty-eight?” n = 7

4n = 28



We may need to combine like terms before we solve an equation.


Solution “Six divided by what number equals 3?”

Using division facts, we see that the answer or solution is 2 and is written

Check:


✓ Verify that this is a true statement.

Thus is the solution.x = 2

3 = 3

6

x = 3 : 6

2

� 3

x = 2.

6x

= 3

6x

= 3


Sometimes, we must first use the associative and commutative properties tosimplify an equation and then find the solution.

x = 8

x4 = 2

EXAMPLE 10 Simplify using the associative and commutative properties andthen find the solution to the equation

Solution First we simplify.

Commutative property

Associative property

Simplify.

Next we solve

“What number plus 6 is equal to 9?”

We leave the check to the student.The solution to the equation is 3 and is written n = 3.

n = 3

n + 6 = 9

n + 6 = 9.

n + 6 = 9

n + 15 + 12 = 9

1n + 52 + 1 = 9

15 + n2 + 1 = 9

15 + n2 + 1 = 9.

Practice Problem 10 Simplify using the associative and commutative prop-erties and then find the solution to the equation x = 313 + x2 + 1 = 7.

EXAMPLE 11 Simplify by combining like terms and then find the solution tothe equation

Solution

Write the equation.

Write n as 1n.

Add numerical coefficients of like terms.

Think: “Six times what number equals eighteen?” 3.

n = 3

6n = 18

11 + 52n = 18

1 n + 5n = 18

n + 5n = 18

n + 5n = 18.

Teaching Example 9 Solve the equationand check your answer.

Ans: “What number divided by two equalsfour?” x = 8

x2 = 4

Teaching Example 10 Simplify by using theassociative and commutative propertiesand then find the solution to the equation

.

Ans: x = 5

7 + (x + 2) = 14

Teaching Tip Emphasize the properway to state the solution; not3. Compare this to answering a ques-tion using the statement: “The solu-tion is 3,” instead of just saying “3.”

n = 3,

Teaching Example 11 Simplify bycombining like terms and then find thesolution to the equation .

Ans: x = 9

2x + 4x = 54




Teaching Tip Students are often con-fused about the difference betweenevaluating an expression and solvingan equation. The Understanding theConcept compares the two. Discussthe difference periodically in the early chapters.

Teaching Example 12 Translate, then solve.The product of five and some numberequals fifteen.

Ans: n = 3

Translating and Solving EquationsIn many real-life applications we must translate an English statement into an equa-tion and then solve the equation.

Practice Problem 11 Simplify by combining like terms and then find thesolution to the equation n = 5n + 3n = 20.

EXAMPLE 12 Translate, then solve. Double what number is equal toeighteen?

Solution Double what number is equal to eighteen?

Translate.Use multiplication facts to find n.

n = 9

T T T T2 # n = 18

Practice Problem12 Translate, then solve. What number times five isequal to twenty? n = 4

Understanding the ConceptEvaluate or Solve?Do you know the difference between evaluating the expression 8x when x is 3and solving the equation

• Evaluate an expression. We replace the variable in the expression with thegiven number and then perform the calculation(s).Evaluate 8x when x is 3.

• Solve an equation. We find the value of the variable that makes the equation atrue statement—that is, the solution to the equation.Solve: .

We can illustrate this idea with the following situations.

1. Evaluating(a) Fact. You are given directions to the Lido Movie Theater.(b) Evaluate. You follow these directions to the movie theater.

2. Solving(a) Fact. You know the address of the theater.(b) Solve. You must find the directions yourself.

In summary, an equation has an equals sign, and an expression does not. Wefind the solutions to equations, and we evaluate expressions as directed.

Exercise

1. Can you think of other real-life situations that illustrate the difference betweenevaluating and solving? Answers may vary

x = 28x = 16

8 # 3 = 24

8x = 16?



Improving Your Test-Taking Skills

Step 1 Write key facts on your test. As soon as you get yourtest, find a blank area to write down any importantstrategies, formulas, or key facts.

An ideal place to write these facts is on the blankback page of the test. If there are no blank pages orareas on the test, be sure to ask the instructor if youcan have a blank piece of paper for this purpose.Having this information easily accessible shouldlessen your anxiety and help you focus on the type ofproblem-solving techniques you need.

Step 2 Scan the test and work problems that are easy first.Quickly glance at each question on the test, placingan * beside the ones you feel confident you cancomplete. Then complete these problems first. Thiswill help build your confidence.

Step 3 Keep track of the time as you complete the rest of the test.Determine how much time is left so you can plan thestrategy for the rest of the test. This plan should includedetermining how many minutes you should spend oneach of the remaining unanswered questions so thatyou can finish the test in the time that is left. Forexample, if there are 10 questions remaining on the testand 30 minutes left, you should try to spend no morethan 3 minutes on each of the remaining questions.

Step 4 Complete the rest of the problems on the test. Completethe remaining problems on the test, starting with the

ones you feel most confident about. If you get stuckon a problem, stop working on it and move on toanother one. Do not spend too much time trying tocomplete one problem; leave it and move on!

Step 5 Relax periodically. If you start to feel anxious at anytime during the test, take a few moments to relax.Close your eyes, place yourself in a comfortableposition in your chair, breathe deeply, and take amoment to think about something pleasant. Next,think positive thoughts such as “I will answer thequestion to the best of my ability and will not worryabout what I have forgotten or do not understand.Instead I will show that I can master what I dounderstand.”

You may think that taking a few minutes awayfrom the test to relax is wasting time. This is not true.You will perform better if you are relaxed.

Step 6 Revisit the problems you are not sure of or did notcomplete. Try to rework the problems that youstruggled with earlier. You may recall how to completethese problems once you have completed the majorityof the test.

Step 7 Review the entire test to check for careless errors. Takewhatever time is left to review all your work. Check forcareless errors and be sure that you have followed alldirections properly.


77

Verbal and Writing SkillsTranslate the mathematical symbols using words.

1.8 EXERCISES

1. 7x Seven times x, or the product of seven and x 2. 5x Five times x, or the product of 5 and x

3. Eight times what number equals 40?8x = 40 4. Five times what number equals thirty?5x = 30

5. Can we add Why or why not? No, becausethe variable parts, x and y, are not the same.

2x + 3y? 6. Can we add Why or why not? No, becausethe variable parts x and xy are not the same.

6x + 3xy?

7. In the expression 6x is called a termand 5 is called a term. constant

variable 6x + 5, 8. When two expressions are separated by an equals sign,we call it an . equation

9. The numerical part of 8x is and is called theof the term. coefficient

8 10. The numerical part of x is and is called theof the term. coefficient

1

11. Rewrite y with a coefficient: . 1y 12. In the expression 12x and 9x are calledterms. like

12x + 9x,

17. 3x + 5xy + � 4x � = 7x + 5xy 18. 9a + 2ab + � 2ab � = 9a + 4ab

23. 5x and 2x; 3y and 7y5x + 3y + 2x + 8m + 7y 24. 6m and 7m; 4b and 4b6m + 4b + 7m + 3x + 4b

25. 2 mn and 4mn2mn + 3y + 4mn + 6 26. 3xy and 2xy6x + 3xy + 9 + 2xy

Fill in the blanks.

Fill in each box to complete each problem.

13. 7x + 3� x � = 10x 14. 10x - 2� x � = 8x 15. 3xy + � 4xy � = 7xy 16. � 2ab � + 4ab = 6ab

19. Three x’s 3x 20. Six y’s 6y 21. 4aa + a + a + a 22. 5xx + x + x + x + x

27. 9x7x + 2x 28. 16x12x + 4x 29. 8y9y - y 30. 6m7m - m

31.11x3x + 2x + 6x 32.

15a5a + 3a + 7a 33.

11x + 6a8x + 4a + 3x + 2a 34.

11y + 6b9y + 2b + 2y + 4b

35.9xy + 4b6xy + 4b + 3xy 36.

12ab + 5x3ab + 5x + 9ab 37.

15xy + 3x + 96xy + 3x + 9 + 9xy 38.

7mn + 6m + 15mn + 6m + 1 + 2mn

39.7ab + 912ab - 5ab + 9 40.

9xy + 312xy - 3xy + 3 41.

17xy + 1014xy + 4 + 3xy + 6 42.

17ab + 813ab + 6 + 4ab + 2

Write a term that represents each expression.

Identify like terms.

Combine like terms.



43.

18x + 14y

44.

12a + 14b

45.

28a + 14b

46.

24x + 22y

Write the perimeter of each triangle as an algebraic expression, then simplify.

47.

6x + 8y

48.

4a + 8b

Translate to an equation. Do not solve the equation.

49. Five plus what number equals sixteen? 5 + x = 16 50. When twenty-four is added to a number, the result isfifty. 24 + x = 50

51. What number times three equals thirty-six? 3x = 36 52. What number times two is equal to forty? 2x = 40

53. If a number is subtracted from forty-five, the result issix. 45 - x = 6

54. If a number is subtracted from twelve, the result istwo. 12 - n = 2

55. Twenty-five divided by what number is equal to five?25n

= 5 or 25 , n = 5

56. Twenty-two divided by what number is equal toeleven? 22

n= 11 or 22 , n = 11

57. Let J represent James’ age. James’ age plus 12 yearsequals 25. J + 12 = 25

58. Let S represent Sherie’s checking account balance.Sherie’s checking account balance plus $14 equals$56. S + 14 = 56

59. Let C represent Chuong’s monthly salary. Chuong’smonthly salary decreased by $50 equals $1480.C - 50 = 1480

60. Let P represent the price of the ticket. The price ofthe ticket decreased by $5 equals $16.P - 5 = 16

61. Is 4 a solution to the equation No9 - x = 3? 62. Is 3 a solution to the equation No5 - x = 3?

63. Is 15 a solution to the equation Yesx + 4 = 19? 64. Is 20 a solution to the equation Yesx + 6 = 26?

Write the perimeter of each rectangular figure as an algebraic expression, then simplify.

4x � 7y

5x

6a � 5b

2b

8a � 2b

6a � 5b

x

5x � 2y6y

9x � 7y

3x � 4y

a

6b3a � 2b

Answer yes or no.

65. x = 4x + 5 = 9 66. x = 6x + 4 = 10 67. n = 811 - n = 3 68. n = 313 - n = 10

69. x = 6x - 6 = 0 70. x = 2x - 2 = 0 71. x = 112 + x = 13 72. x = 421 + x = 25

73. x = 525 - x = 20 74. n = 244 - n = 42 75. x = 28x = 16 76. y = 27y = 14

Solve and check your answer.

�

�

�

�

�

�



107. Find the missing side of the following triangle if theperimeter is 170 feet. x = 70 ft

108. Find the missing side of the following triangle if theperimeter is 110 yards. x = 50 yd

77. y = 34y = 12 78. x = 79x = 63 79. x = 78x = 56 80. y = 310y = 30

81. y = 1515y

= 1 82. x = 1212x

= 1 83. x = 714x

= 2 84. x = 1020x

= 2

Simplify using the associative and/or commutative property, then find the solution. Check your answer.

Simplify by combining like terms, then find the solution. Check your answer.

91.n = 33n + n = 12 92.

n = 36n + n = 21 93.

x = 17x - 2x - x = 4 94.

y = 23y + y + 2y = 12

85. x = 31x + 12 + 3 = 7 86. x = 21x + 62 + 5 = 13 87. y = 12 + 17 + y2 = 10

88. x = 213 + x2 + 2 = 7 89. n = 23 + 1n + 52 = 10 90. x = 22 + 18 + x2 = 12

95.x = 45x = 20 96.

x = 2

30x

= 15 97.x = 1516 - x = 1 98.

n = 438 - n = 34

99.a = 101 + 14 + a2 = 15 100.

x = 316 + x2 + 1 = 10 101.

x = 58x - 5x - x = 10 102.

y = 24y + y + 2y = 14

Mixed Practice Exercises 95–102Solve.

For the following English sentences,

(a) Translate into an equation. (b) Solve the equation.

103. Four plus what number equals eight?(a) (b) x = 44 + x = 8

104. Three added to what number equals nine?(a) (b) x = 63 + x = 9

105. Three times what number is equal to nine?(a) (b) x = 33x = 9

106. Four times what number is equal to twelve?(a) (b) n = 34n = 12

One Step Further

50 ft 50 ft

x

30 yd 30 yd

x

Simplify by combining like terms.

109. 6x2+ 186 + 12x2

+ 52 + 17 + 3x22 + x2 110. 13x2+ 1712 + 8x22 + 9 + 14x2

+ 62 + x2

111. (a)(b) (2x)(5y) 10xy

5x + 5y2x + 3x + 5y 112. (a)(b) (5x)(6y) 30xy

9x + 6y5x + 4x + 6y

113. (a)(b) (5a)(6y) 30ay

7a + 6y5a + 6y + 2a 114. (a)(b) (6a)(7y) 42ay

9a + 7y6a + 7y + 3a

Simplify.

� �



To Think AboutRunning Speeds Compared Use the bar graph to answerexercises 115 and 116.

115. (a) How fast can a domestic cat run if it is as fast as agrizzly bear? 30 miles per hour

(b) The speed of a lion is two times the speed of anelephant. How fast is the elephant? 25 mph

116. (a) Which animal is faster, a zebra or a Cape huntingdog? Cape hunting dog

(b) The speed of a cheetah is two times the speed ofa rabbit. How fast is the rabbit? 35 mph

(a) Addition

(b) Subtraction

(c) Multiplication

(d) Division

117. [1.5.2] Split equally between

118. [1.4.2] Find the number of items in an array

119. [1.2.1] Find the total

120. [1.3.2] How much less b

a

c

d

Cumulative ReviewMatch each operation described by the phrase in the right column with the appropriate operation listed in the left column.Place the correct letter in the blank space.

0

2530

40455055

657075

35

60

App

roxi

mat

e m

axim

um s

peed

of a

nim

als

(in

mph

)

Zebra Lion Cheetah Capehuntingdog

Grizzlybear

Source: The World Almanac for Kids

1. Combine like terms. 2. Solve each equation and check your answer.

(a)

(b)

(c) y = 88y + y = 72

x = 14 + (x + 7) = 12

a = 510a

= 23ab + 4a + 12ab + 4a + 1 + ab

3. Translate into an equation and solve.

(a) The product of three and what number is equal toeighteen?

(b) Let D represent Dave’s age. Dave’s age increasedby seven is equal to twenty-one.

4. Concept Check Explain the difference in the processyou must use to complete (a) and (b).

(a) Combine like terms.

(b) Solve. Answers may vary

3x + x + 2x = 12

3x + x + 2x

D + 7 = 21; D = 14

3x = 18; x = 6

1. Combine like terms.

Ans:

2. Solve each equation and check your answer.

(a) Ans:

(b) Ans:

(c) Ans: x = 33 + (x + 5) = 11

a = 44a - 2a = 8

x = 216x

= 811m + 13n2m + 6n + 9m + 7n

Quick Quiz 1.8

Classroom Quiz 1.8. You may use these problems to quiz your students’ mastery of Section 1.8.

3. Translate into an equation and solve.

(a) What number divided by two equals eight? Ans:

(b) Let R represent Randy’s savings account balance. Randy’s savings account balance increased by $20 equals $70Ans: R + 20 = 70; R = 50

x

2= 8; x = 16



Positive ThinkingSome people convince themselves that they cannot learnmathematics. If you are concerned that you may havedifficulty in this course, it is time to reprogram your thinking.Replace those negative thoughts with positive ones such as

“I can learn mathematics if I work at it. I will give this math class my best shot.”

You will be pleasantly surprised at the difference thispositive attitude makes!

Now, some might wonder how a change in attitude canmake a difference. Well, the approach and features in thisbook can be the keys to your success. Here are a few featuresthat you should pay special attention to:

• Developing Your Study Skills boxes offer tips on how tostudy. If you know how to study, learning will be far easier.

• Understanding the Concept boxes help you understandwhat you are doing and why you are doing it. These

explanations and other descriptions relate the concepts toother topics such as algebra and applications.

• Exercises develop the fundamentals on which theconcepts are built. Many students do not learn becausethey miss certain building blocks. For example, how cansomeone learn long division if he or she doesn’t knowmultiplication facts?

If you take this advice to heart, you’ll be off to a goodstart. Keep up the good work.

Exercises1. Name a few things that you plan to do in this class to help

you be successful. Answers may vary2. Write in words two positive thoughts about mathematics

and/or your ability to complete this course.Answers may vary


82

Solving Applied Problems Involving EstimationOften, it is not necessary to know the exact sum or difference; in this case we canestimate. Estimating is also helpful when it is necessary to do mental calculations.There are many ways to estimate, but in this book we use the following rule to makeestimations.


Solve applied problemsinvolving estimation.

Solve applied problemsinvolving charts and diagrams.

Use the Mathematics Blueprintfor Problem Solving.

1.9 SOLVING APPLIED PROBLEMS USING SEVERAL OPERATIONS

To estimate a sum or difference, round each number to the same round-offplace and then find the sum or difference.

EXAMPLE 1 Some sample sale prices for 2008 Ford motor vehicles arelisted below.

(a) Estimate the difference in the price between a Mustang V6 Premium Coupeand a Ford Focus S Coupe by rounding each price to the nearest thousand.

(b) Calculate the exact difference in cost. Is your estimate reasonable?

Manufacturers Suggested Retail Priceson 2008 Ford Vehicles

$28,345

Taurus SEL $23,995

$14,695

Mustang V6 Premium Coupe $21,225

Focus S Coupe

F-150 XLT Supercab Flairside

Source: www.fordvehicles.com

Solution Understand the problem. The information we need to solve the prob-lem is listed in the table.

(a) Calculate and state the answer. To estimate, we round each number to thethousands place.

Exact Value Rounded ValueThe price of the Mustang V6 Premium Coupe: 21,225 21,000The price of the Focus S Coupe: 14,695 15,000:

:

We subtract the rounded figures to estimate the difference in the cost of thetwo vehicles.

The estimated difference in price is $6000.(b) We subtract the original figures to find the exact difference in the cost of the

two vehicles.

The exact difference in price is $6530.The estimated difference in price, $6000, is close to the exact difference, $6530,so our estimate is reasonable. Note that if you round each number to the near-est hundred instead of to the nearest thousand your estimate will not bewrong, just a little closer to the exact amount. When we estimate we want tomake calculations with numbers that are easy to work with. In this case, it is

$21,225 - $14,695 = $6530

21,000 - 15,000 = 6000

Teaching Tip You may want to tellstudents that there are other ways toestimate. For example, one methodrounds each number so that there isone nonzero digit. Then calculationsare performed. Explain that althoughthis method can result in estimationsthat are not as close to the actualanswer as the one stated in this book,many times such a ballpark figure isfine.

Teaching Example 1 Use the sale priceslisted in Example 1 to answer thefollowing.

(a) Estimate the difference in pricebetween a Taurus SEL and a Focus SCoupe. Round to the nearestthousand.

(b) Calculate the exact difference in cost.Is your estimate reasonable?

Ans: (a) $9000 (b) $9300. Yes, theestimate is reasonable.


Section 1.9 Solving Applied Problems Using Several Operations 83

NOTE TO STUDENT: Fully worked-outsolutions to all of the Practice Problems can be found at the back of the text starting at page SP-1

AAARestaurantSupply

INVOICE NO.

SOLD TO

QUANTITY PRICE PERITEM

TOTAL COSTPER ITEM

Total Cost for all Items

TYPE OF PURCHASE

CASH C.O.D. CREDIT REPRESENTATIVE COMMENTS DATE SHIPPED DATE OF INVOICE

Thank You

easier to subtract numbers rounded to the thousands place than to the hun-dreds place; that is, subtracting is easier than subtracting21,200 - 14,700.

21,000 - 15,000

Practice Problem 1 Use the sale prices listed in Example 1 to answer thefollowing.

(a) Estimate the difference in price between an F-150 XLT and a Taurus SEL.(b) Calculate the exact difference in cost. Is your estimate reasonable?

(a) $4000 (b) $4350; the estimate is reasonable.

Solving Applied Problems Involving Charts and DiagramsHow much material do I need to fence my yard? How much gasoline will I need formy trip? How much profit did my business make? One important use of mathemat-ics is to answer these types of questions. In this section we combine problem-solvingskills with the mathematical operations of addition, subtraction, multiplication, anddivision to solve everyday problems. We follow the three-step problem-solvingprocess discussed earlier in the chapter when solving applied problems. We restatethe three steps for your review.

Step 1. Understand the problem. Read the problem and organize the information.Use pictures and charts to help you see facts more clearly.

Step 2. Calculate and state the answer. Use arithmetic and algebra to find theanswer.

Step 3. Check the answer. Use estimation and other techniques to test your answer.

EXAMPLE 2 The three owners of the Pizza Palace redecorated their busi-ness. The items purchased are listed on the invoice in the margin. If the cost of thesepurchases excluding tax was divided equally among the owners, how much did eachowner pay?

Solution Understand the problem. Read the problem carefully and study theinvoice. Then fill in the invoice.

Calculate and state the answer. Multiply to get the total cost per item. Then placethese amounts on the invoice.

Now, divide the total cost by 3 to find the amount that each owner paid:Each owner paid $2153.6459 , 3 = 2153.

AAARestaurantSupply

INVOICE NO.

SOLD TO


TOTAL COSTPER ITEM


TYPE OF PURCHASE


Thank You

9 tables � 9(335) � $301554 chairs � 54(26) � $14042 ovens � 2(1020) � $2040

$6459

Place these amountson the table.

Add to find thetotal cost.

Teaching Example 2 Four partners in acatering business purchased additionalsupplies so that they could expand theirbusiness. The items purchased are listed onthe following invoice. If the cost of thesepurchases excluding tax was dividedequally among the partners, how muchdid each partner pay?

Ans: $3105

AAARestaurantSupply

INVOICE NO.

SOLD TO


TOTAL COSTPER ITEM


TYPE OF PURCHASE


Thank You



Using the Mathematics Blueprint for Problem SolvingTo solve a word problem that contains many facts and requires several steps, it ishelpful to organize the information and then plan the process that you will use. Thisis very similar to what we do when we use a daily planner or date book to organizeand plan our days. A Mathematics Blueprint for Problem Solving will be used toorganize the information in a word problem and plan the method to solve it. Fromthe blueprint we will be able to see clearly the three steps for solving problems inreal-life situations: understand the problem; calculate and state the answer; andthen check the answer.

Check your answer. We estimate and compare the estimate with our calculatedanswer.

Practice Problem 2 The two owners of a Chinese restaurant redecoratedtheir place of business. The items purchased are listed on the invoice in the margin.If the cost of these purchases was divided equally between the owners, excludingtax, how much did each owner pay? $2910

Round: Number of Items Price per Item Find Total Cost

$65002 # 1000 = 2000$1020 : $10002 : 2

50 # 30 = 1500$26 : $ 3054 : 5010 # 300 = $3000$335 : $ 3009 : 10

Divide the estimated total by 3:

Our estimate of $2167 per owner is close to our exact calculation of $2153. Thusour exact answer is reasonable.

2166 R2 L $21673�6500

EXAMPLE 3 A frequent-flyer program offered by many major airlines tofirst-class passengers awards 3 frequent-flyer mileage points for every 2 miles flown.When customers accumulate a certain number of frequent-flyer points, they can cashthem in for free air travel, ticket upgrades, or other awards. How many frequent-flyer points would a customer accumulate after flying 3500 miles in first class?

Solution Understand the problem. Sometimes, drawing charts or pictures can helpus understand the problem as well as plan our approach to solving the problem.

How many groups of 2’s are in 3500?

We organize the information and make our plan in the Mathematics Blueprint.

3 points + 3 points Á= ? points

2 miles + 2 miles Á= 3500 miles

A customer isawarded 3frequent-flyerpoints for every2 miles flown.

Determine howmany frequent-flyerpoints a customerearns after flying3500 miles.

1. Divide 3500 by 2.

2. Multiply 3 timesthe numberobtained in step 1.

Frequent-flyer points aredetermined by thenumber of milesflown.

SOLD TO


TOTAL COSTPER ITEM


TYPE OF PURCHASE


INVOICE NO.

Thank You

Teaching Tip Explain to students thatapplied problems sometimes haveinformation organized in charts,diagrams, or graphs. Therefore, itmay not be necessary to use theblueprint. Emphasize that when thisis not the case, organizing the infor-mation and planning the process inthe Mathematics Blueprint will makethe problem-solving process mucheasier.

Teaching Example 3 Use the informationin Example 3 to determine how manyfrequent-flyer points a customer wouldaccumulate after flying 8300 miles.

Ans: 12,450 points

Key Points to Remember

How Do IProceed?

What Am IAsked to Do?

Gather the Facts



NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1


Step 1. We divide to find how many groups of 2 are in 3500.

Step 2. We multiply 1750 times 3 to find the total points earned.

The customer would earn 5250 points.

Check. If the customer earned 4 points (instead of 3) for every 2 miles traveled, wecould just double the mileage to find the points earned.

2 miles earns 4 points (double 2).3500 miles earns 7000 points (double 3500).

Since the customer earned a little less than 4 points, the total should be less than7000. It is: The customer also earned more points than miles trav-eled (3 points for every 2 miles), so the total points should be more than the totalmiles traveled. It is: Our answer is reasonable.5250 7 3500.

5250 6 7000.

1750 # 3 = 5250 points

3500 , 2 = 1750

Practice Problem 3 Use the information in Example 3 to determine howmany frequent-flyer points a customer would accumulate if she flew 4500 miles.

6750 points

EXAMPLE 4 Koursh was offered two different jobs: a 40-hour-a-week storemanagement position that pays $12 per hour and an executive secretary positionpaying a monthly salary of $2600. Which job pays more per year?

Solution Understand the problem. We organize the information in the Mathe-matics Blueprint.

Gather the Facts

What Am IAsked to Do? How Do I Proceed?


The storemanagementposition pays $12 perhour for 40 hours.

The secretary’sposition pays $2600per month.

Determine whichjob pays a highersalary per year.

1. Calculate the manager’s weekly pay.

2. Multiply the result of step 1 by 52 weeks to find yearly pay.

3. Multiply the secretary’s pay by 12 months to find yearlypay.

4. Compare both salaries.

I must find yearlypay:

52 weeks = 1 year12 months = 1 year

Calculate and state the answer. From the information organized in the blueprint,we can write out a process to find the answer.

Pay for 1 week(management)Pay for 1 year(management)Pay for 1 year(secretary)

The yearly pay is $24,960 for the management position and $31,200 for the secre-tary’s position. The secretary’s position pays more per year.

$2600 * 12 = $31,200

$480 * 52 = $24,960

$12 * 40 = $480

Teaching Example 4 Trang works 40 hoursa week and earns $15 an hour as a payrollclerk. She is considering accepting a joboffer to work as an assistant officemanager earning a salary of $2400 permonth. Which job pays more per year?

Ans: The payroll-clerk job pays more peryear.



Check the answer. We estimate the manager’s pay per year by rounding $12 perhour to $10 and 52 weeks to 50 weeks.

We estimate the secretary’s pay per year by rounding 12 months per year to 10.

Since the secretary position pays more. ✓$26,000 7 $20,000,

$10 * $2600 = $26,000 per year

$10 * 40 hr = $400 per week; $400 * 50 weeks = $20,000 per year

Practice Problem 4 Emily is a salesperson for A&E Appliance. For the lasttwo years she has averaged about 7 sales per week, and she is paid solely oncommission—$55 per sale. The store manager has decided to offer all salespersonsthe options of accepting a salary of $1770 per month or remaining on commission.If Emily continues to maintain her past sales record, which option will earn hermore money per year? $1770 per month

The Day Before the ExamIf you have been following the advice in the other DevelopingYour Study Skills boxes, you should be almost ready for theexam. If you have not read these boxes, reading them nowwill provide valuable advice.

The day before the exam is not a good time to startreviewing. Use this day to skim your chapter review, homework,quizzes, and other review material. On this day you can fine-tune what you already know and review what you are unsure of.Starting your review early reduces anxiety so that you can think

clearly during the test. Often, low test scores are related tohigh anxiety. Plan ahead so that you can relax on the day of theexam.

A few days before the exam, complete the How Am IDoing? test at the end of the chapter. Take this test as if itwere the real exam. Do not refer to notes or to the text whilecompleting the test. Grade the test. The problems youmissed on this test are the type of problems that you shouldget help with and review the day before the exam.


87

7/17/2009Sale Transaction

Satin enamelpaint, 5 gal.

TotalTaxTotal Sale

$81

362214

Paint brushesDrop cloths, 2Paint roller and tray

THANK YOU

ApplicationsEstimate each of the following.

1.9 EXERCISES

1. Supplies Purchased Emma purchased supplies topaint her kitchen, family room, and dining room. Thefollowing store receipt indicates the supplies shepurchased.

2. Supplies Purchased Julio Arias bought schoolbooksand school supplies. The following store receipt indi-cates his purchases.

(a) Excluding tax, estimate how much money Emmaspent by rounding each amount to the nearestten. $150

(b) Find the total money spent. $153(c) Is your estimate reasonable? Yes

(a) Excluding tax, estimate how much money Juliospent by rounding each amount to the nearestten. $180

(b) Find the total money spent. $180(c) Is your estimate reasonable? Yes

3. Mileage onVehicle The Arismendi family took a scenicdrive across the country. From Phoenix, Arizona, theydrove east 597 miles the first day, 512 miles the sec-ond day, 389 miles the third day, and 310 miles the fourthday. Round each amount to the nearest hundred andthen estimate how many more miles the Arismendifamily drove the first two days than the last two days.400 miles

4. Mileage on Vehicle Jay drove his Toyota Tundra truck14,200 miles the first year he owned it, 15,980 the sec-ond year, 8,100 the third year, and 14,950 the fourthyear. Round each amount to the nearest thousand andthen estimate how many more miles Jay drove histruck the first two years than the second two years.7000 miles

5. Restaurant Remodeling The 5 owners of Mei’sRestaurant remodeled their business. They bought7 tables, 20 chairs, and 2 crystal light fixtures. Thecost of these purchases was divided equally amongthe owners. Excluding tax, how much did each ownerpay? $1185

6. Appliances Purchased Last weekend, May’s Appli-ance Store sold 10 washing machines, 5 dryers, and20 dishwashers to a private college. The college willdivide the expense for upgrades equally among the200 students in the college apartments by chargingeach student a one-time assessment fee. How muchwill the assessment fee be for each student? $71

INVOICE NO.


INVOICE INVOICE NO.

SOLD TO


TOTAL COSTPER ITEM


TYPE OF PURCHASE



11. (a) The Torches tagged 6 of the Shooters’ players,and succeeded in pulling and hanging the Shoot-ers’ flag. The Torches had 5 players remaining atthe end of the match. How many points did theTorches have at the end of the match?

(b) In another match, the Shooters tagged all of theTorches’ players. They succeeded in pulling theTorches’ flag but were not able to hang it beforetime ran out. The Shooters had 1 player left atthe end of the match. How many points did theShooters have at the end of the match?3172 + 22 + 1112 = 44 points

6132 + 22 + 50 + 5112 = 95 points

12. (a) In the second half of the tournament the Alphateam was able to tag 5 players on the Greyhounds’team and succeeded in pulling and hanging theGreyhounds’ flag. The Alphas had 3 players re-maining at the end of the match. How many pointsdid the Alpha team have at the end of the match?

(b) In the final match of the tournament, the Grey-hounds were able to tag all of the players on theAlpha team. The Greyhounds had 5 players re-maining at the end of the match, but were unableto succeed in pulling and hanging the flag of theother team before time ran out. How many pointsdid the Greyhounds receive in this match?7132 + 5112 = 26

5132 + 22 + 50 + 3112 = 90 points

7. Entertainment Event Tickets Dave and his familywent to the Middletown Amusement Park. They pur-chased 2 adult tickets, 4 child tickets, and 1 senior cit-izen ticket. How much did they spend on the tickets?$53

8. Entertainment Event Tickets Janice and her friendswent to an outdoor jazz concert. They purchased4 adult, 6 student, and 2 child tickets. How muchmoney did they spend on concert tickets? $138

9. Fencing Needed John Tulson wants to put a fencearound the back and sides of his property (see thediagram). How many feet of fence must he purchase?400 ft

10. Ceiling Molding Needed Rosa would like to putmolding along the edge of the ceiling in her kitchen(see the diagram). How many feet of molding will sheneed? 26 ft

Paintball In many paintball tournaments there are 100 points possible in each match of a round of play.There are 7 playerson each team and when a player is tagged with a paintball they are out of the match. The team with the higher point total atthe end of each round of play moves on to the next round. Use the chart below to answer exercises 11 and 12.

Middletown Ticket Prices

AdultChild

Senior citizen

–––

$$$

1357

(under 12 years)(over 55 years)

150 ft

20 ft

20 ft 25 ft

25 ft80 ft

Outdoor Jazz Concert Ticket Prices

AdultChild

Student discount

–––

$$$

1789

(under 12 years)(college ID required)

12 ft2 ft6 ft

Point Chart

Tagging the opposing team players with paintballs 3 points/player

Pulling the flag of the opposing team 22 points

Hanging the flag of the opposing team 50 points

Players left in the game at the end of play 1 point/player

Solve each problem involving charts and diagrams.

� �



13. Hourly/Overtime Wages A restaurant cook earns $8 per hour for the first 40 hours worked and $12 per hour forovertime (hours worked in addition to the 40 hours a week). Last week the cook worked 52 hours. Calculate thecook’s total pay for that week. $464

(a) Fill in the Mathematics Blueprint for Problem Solving.(b) Calculate and state the answer.

14. Apartment Expenses Four roommates share equally the following expenses for their apartment: $920 for rent, $96for utilities, and $56 for the telephone. How much is each roommate’s monthly share? $268

(a) Fill in the Mathematics Blueprint for Problem Solving.(b) Calculate and state the answer.

Gather the Facts What Am I Asked to Do? How Do I Proceed? Key Points to Remember

Cook earns: $8 per hour for 40 hrs; $12 per hour forovertime; Hours worked 52=

Calculate the cook’s totalpay.

1. Multiply to findbase pay.

2. Multiply to findovertime pay.

3. Add results from steps 1and 2.

$12 * 12

$8 * 40 Overtime hours are hoursworked in addition to40 hours a week.

The expenses must beshared equally by 4roommates.

1. Add to find the sum of allexpenses.

2. Divide the result in step 1by 4.

Calculate each roommate’sshare of monthly expenses.

Apartment expenses: rent—$920 utilities—$96 telephone—$56

Gather the Facts Key Points to RememberHow Do I Proceed?What Am I Asked to Do?

15. Business Profit T. B. Etron’s Company made $68,542last year. The expenses for that year were $14,372.

(a) How much profit did the company make? $54,170(b) If the two owners divided the profits equally, how

much money did each owner receive? $27,085

16. PTA Raffle The R. L. Saunders High School PTAsold $2568 in raffle tickets. The expenses for theprizes were $1062.

(a) How much profit did the PTA make? $1506(b) If the profits were divided equally among three

clubs, how much money did each club receive?$502

17. Fishing Trip Expenses Carlos and three of his friendswill equally share the expenses for a 2-day fishing tripthey plan to take this summer. The boat rental will be$450 per day, and they estimate that the gasoline willcost $50 per day. If the total cost of food and bait forthe entire 2-day trip is $200, what will be each person’sshare of the total expenses? $300

18. Comparing Earning Sara’s current job as a computertechnician at ComTec pays a salary of $2200 permonth. BLM Accountants offered her a program-mer’s position that pays $14 per hour for a 40-hourweek. Which job pays more per year?BLM Accountants

19. Bus Pass vs. Daily Rate Round-trip bus fare is $2.Justin rides the bus 5 days a week to work and 2 nightsa week to school. He can buy a pass at school that al-lows 6 months of unlimited bus rides for $400. IfJustin only rides the bus round-trip to work andschool, is it cheaper for Justin to buy the pass or topay each time he rides the bus?Pay each time he rides

20. Salary vs. Commission Myra sells new membershipsfor a Total Flex Fitness Center chain. She is paid onlyon commission—$35 for each new membership. Forthe last three years she has signed up an average of 11new members per week. She has been offered an al-ternative pay option—a salary of $1800 per month.Which pay option pays more per year?Salary option



Promotional Points The L&M Clothing chain offered customers the following incentive to use their L&M charge card.

21. Inheritance Proceeds Glenda has inherited $6000,which will be distributed in two equal payments. Shewill receive the first half of the inheritance now andthe remainder of the inheritance in 6 months. Glendaplans to invest the first payment in a certificate of de-posit (CD) at her bank. She will distribute the secondpayment equally among her three children.

(a) How much will Glenda invest in a CD? $3000(b) How much will each child receive? $1000

22. Stamp Collection Lester donated one-half of the2500 stamps in his collection to the local senior citizengroup. He distributed the remaining stamps in his col-lection equally among his five grandchildren.

(a) How many stamps did Lester donate to the seniorcitizen group? 1250

(b) How many stamps from Lester’s collection dideach grandchild receive? 250

Grocery Store Purchases Al’s Grocery Store gave customers the following incentive to shop at the store.

23. Jesse made three purchases at Al’s Grocery Store during the month of June: $30,$240, and $170.

(a) How many points did Jesse earn during the month of June?(b) How many discount dollars did Jesse earn?(a)

(b) $6 in discounts

24. Marsha made three purchases at Al’s Grocery Store during the month of June:$230, $140, and $180.

(a) How many points did Marsha earn during the month of June?(b) How many discount dollars did Marsha earn?(a)

(b) $8 in discounts80 , 10 = 8;55 pts. + 25 pts. 1$230 purchase2 = 80 pts.230 + 140 + 180 = 550, 550 , 50 = 11 : 11 # 5 pts. = 55 pts.

65 , 10 = 6 R5;40 pts. + 25 pts. 1$240 purchase2 = 65 pts.30 + 240 + 170 = 440, 440 , 50 = 8 R40 : 8 # 5 pts. = 40 pts.

25. Alyssa made two purchases at L&M Clothing during the month of January: $170and $260.

(a) How many points did Alyssa earn during the month of January?(b) How many discount dollars did Alyssa earn?(a)

(b) in discounts

26. Ian made three purchases at L&M Clothing during the month of January: $80,$160, and $220.

(a) How many points did Ian earn during the month of January?(b) How many discount dollars did Ian earn?(a)

(b) in discounts140 , 25 = 5 R15. 5 # $5 = $2590 pts. + 50 pts. 1$220 purchase2 = 140 pts.80 + 160 + 220 = 460, 460 , 50 = 9 R10 : 9 # 10 pts. = 90 pts.

130 , 25 = 5 R5. 5 # $5 = $2580 pts. + 50 pts. 1$260 purchase2 = 130 pts.170 + 260 = 430, 430 , 50 = 8 R30 : 8 # 10 pts. = 80 pts.

Earn points every time you

use your L & M charge card.

Earn 10 points for every $50charged during the month

of January plus anadditional 50 points for anysingle purchase over $200.

Cash in your points!

25 points earn you a$5 discount in February.

To Think About27. Credit Card Debt Repayment A $5000 debt on a credit card will take 32 years to repay if only the minimum monthly

payment is made. This debt will cost the borrower about $7800 in interest. The borrower could be out of debt in 3 yearsby paying $175 per month. Find the amount of interest paid at the end of 3 years. $1300

Writing numbers in exponent form can often help us identify a pattern in a sequence of numbers. For example, the sequence1, 8, 27, 64, 125, . . . can be written as and we see that the next number would be or 216.

Identify a pattern and then find the next number in the following sequences.

6313, 23, 33, 43, 53, Á

28. 4, 16, 36, 64, 100, . . . 22, 42, 62, Á , 122= 144 29. 9, 25, 49, 81, 121, . . . 32, 52, 72, Á , 132

= 169



30. 31.

Find the next two figures that would appear in the sequence.

, , , ,

,

, , ,

,

Cumulative Review32. [1.4.3] Multiply. 1204 # 3 # 2 # 5 33. [1.8.3] Solve. x = 56x = 30 34. [1.8.3] Solve. x = 3x + 9 = 12

1. The Belmont High Service Club organized a charityjazz festival. Tickets to the festival were $20 foradults, $15 for students, and $7 for children under12 years old. The service club sold 350 adult tickets,200 student tickets, and 47 children tickets.

(a) Find the total income from the sale of tickets.$10,329

(b) If the expenses for the festival were $3400, howmuch profit did the Club make for the charityevent? $6929

2. Dairy Cows A dairy cow produces an average of7 gallons of milk a day. If a farmer has a herd of35 cows, how much milk will they produce in 1 day? In1 week? 245 gal in 1 day; 1715 gal in 1 week

3. Luis is opening his new accounting office and mustorder furniture for the front office. He found a sale ata local store that is offering free delivery and no salestax. Luis must order 2 filing cabinets, 1 desk, 1 officechair, 2 bookcases, 6 guest chairs, 2 end tables, and1 coffee table. The prices are listed in the table below.Round each price to the nearest hundred and thenestimate the total price of the furniture. $4900

4. Concept Check At the end of January, Sahara had$200 left in her vacation savings account and $1000 inher household savings account. Each month for thenext six months, Sahara plans to put $100 in her vaca-tion account and $200 in her household account. Inaddition, she plans to split her $900 tax return equallybetween both accounts. Explain how to determineif Sahara will have enough money in her vacationaccount at the end of six months to take a $1500vacation. Answers may vary

1. Melissa can earn 5 points for every hundred dollars shecharges on her MasterCard. Melissa made the followingcharges in the month of June: $542, $47, $149, and $286.How many points did Melissa earn in the month of June?Ans:

2. Jesse’s savings account had a balance of $3050. Over the last3 months he made deposits of $93, $133, and $220. Last weekhe made two withdrawals of $50 and $76. If Jesse puts one-half of his savings into a certificate of deposit, how much willhe have left in his savings account? Ans: $16851024 , 100 = 10 R24, 10 # 5 = 50 points

Quick Quiz 1.9


Filing Cabinet $467 Guest Chair $197

Desk $765 End Table $317

Office Chair $255 Coffee Table $421

Bookcase $299

3. Window Blind Prices Refer to the chart of custom window blind prices. A window blind size of means the blind is 30 incheswide and 36 inches high. We state the width first.

(a) Determine the total cost to purchase the following window blinds:two of size one of size and two of size Ans: $1933

(b) Estimate the cost of the window blinds in (a) by rounding to thenearest ten. Ans: $1940

(c) Find the difference between your estimate and the exact amounts.Ans: $7

48 * 42.36 * 48,30 * 36,

30 * 36

Width to:

Height to:

$282 $316 $353 $387 $423

297 335 373 413 452

313 354 397 438 48148–

42–

36–

48fl42fl36fl30fl24fl


92

Chapter 1 Organizer

Writing word namesfor whole numbers,p. 4.

Name the number in each period followed by thename of the period and a comma. The last periodname, “ones,” is not used.

1millions2 1thousands2 1ones2xxx xxx xxx

Write in words. 34,218,316

Thirty-four million, two hundred eighteenthousand, three hundred sixteen

Replace ? with the inequality symbol or

9 7 3 15 6 20

9 ? 3 15 ? 20

7 .6Inequality symbols,p. 4.

Rounding wholenumbers, pp. 5, 6.

1. Identify the round-off place.2. If the digit to the right of the round-off place is

(a) Less than 5, do not change the round-off placedigit.

(b) 5 or more, increase the round-off place digitby 1.

3. In either case, replace all digits to the right of theround-off place digit with zeros.

Round 27,468 to the nearest hundred.

2 7, 4 6 8

2 7, 5 0 0

The round-off place digit is 4.

The digit to the right is 5 or more.

Increase the round-off place digit.

Replace digits to the right with zero.

Using properties ofaddition to simplify,pp. 11, 12.

The commutative property states that we can changethe order of numbers when adding.

The associative property states that we can regroupnumbers when adding.

We use both of these properties to simplifyexpressions.

Use the commutative and/or associativeproperty as necessary to simplify.

Change the order

of addition.

Regroup.

Add. = 5 + n or n + 5

= 13 + 22 + n

= 3 + 12 + n2 3 + 1n + 22

3 + 1n + 22

Adding whole numbers,p. 14.

Starting with the far right column, add each columnseparately. If a two-digit sum occurs, carry the first digitover to the next column to the left.

Add.

32812

15673

+ 5616

382 + 156 + 73 + 5

Finding the perimeter,p. 16.

The perimeter is the distance around an object. We addthe lengths of all sides to find the perimeter.

Find the perimeter of the shape consisting ofrectangles.

1. We find the length of the unlabeled sides.unlabeled vertical sideunlabeled horizontal side

2. We add the lengths of all sides.

6 + 7 + 3 + 8 + 9 + 15 = 48 m

7 m + 8 m = 15 m;3 m + 6 m = 9 m;

7 m

6 m

8 m

3 m

Subtracting wholenumbers, p. 25.

Starting with the right column, subtract each columnseparately. If necessary, borrow a unit from the columnto the left and bring it to the right as a “10.”

Subtract.

2 6,5 5

15 8

7 111

- 4, 8 3 22 1, 7 4 9

26,581 - 4832

and are called inequality symbols. The symbolmeans is greater than, and means is less than.

The inequality symbol always points to the smallernumber.

“6”“7”76

Topic Procedure Examples


Multiplying wholenumbers, p. 37.

Multiply the top factor by the ones digit, then by thetens digit, then by the hundreds digit of the lowerfactor. Add the partial products.

Multiply.

567* 2384536

17011134134,946

567 # 238

Order of operations,p. 58.

1. Perform the operations inside parentheses.2. Then simplify any expressions with exponents.3. Then do multiplication and division in order from

left to right.4. Then do addition and subtraction in order from left

to right.

Evaluate.

Raise to a power first.

Then multiply and divide from left to right.

Then add and subtract from left to right.

13 - 3 = 10

8 + 5 - 3

8 + 1 # 5 - 3

8 + 16 , 16 # 5 - 3

23+ 16 , 42 # 5 - 3

Distributive property,p. 65.

To multiply and we distribute the aby multiplying every number or variable inside theparentheses by a and then simplifying.

a1b - c2a1b + c2 Simplify.

4 # x + 4 # 2 = 4x + 8

41x + 22

Long division, p. 47. We guess the quotient and check by multiplying thequotient by the divisor. We adjust our guess if it is toolarge or too small and continue the process until we geta remainder less than the divisor.

Divide.

51 R325�1278

-12528

-253

1278 , 25

Using symbols and keywords for expressingalgebraic expressions,p. 64.

When a phrase contains key words for more than oneoperation, the phrases sum of or difference of indicatethat these operations must be placed withinparentheses.

Translate using numbers and symbols.Six times x minus two.

Six times the difference of x and two.

61x - 22

6x - 2

Exponents, p. 55. is written in exponent form. The exponent is 3, andthe base is 2. Thus is read “two to the third power”and means that there are 3 factors of 2.

2323 Write in exponent form.

Find the value of and

33= 3 # 3 # 3 = 27 72

= 7 # 7 = 49

72.33

44 x2

4 # 4 # 4 # 4 # x # x

Key words for division,p. 45.

The key words or phrases that represent division aredivided by, shared equally, divided equally, andquotient.


Nine divided by

The quotient of fifteen and five:

The quotient of five and fifteen: 5 , 15

15 , 5

n: 9 , n

Properties ofmultiplication, p. 35.

We can regroup and change the order of numberswhen we multiply since multiplication is associativeand commutative.

Simplify.Change the order of multiplication and regroup.

Multiply. 12 # 32 # x = 6x12 # 32 # x

21x # 32

Chapter 1 Organizer 93

Key words formultiplication, p. 34.

The key words or phrases that represent multiplicationare times, product of, double, and triple.


Double a number: Triple a number:

2x 3x

The product of Six times x:

two and three: 6x2 # 3


(Continued on next page)



Solving equations,p. 72.

We solve simple equations using basic arithmetic facts.To check, we replace the variable with the number tosee if it makes the equation a true statement.

Solve.

(a)

(b)

x = 3

27x

= 9 : 273

= 9

n = 7 6n = 42 : 6 # 7 = 42

Translating and solvingequations, p. 75.

We translate the English sentence into math symbolsand then solve for the variable.

What number subtracted from twelve is equal tofour?

12 - 8 = 4 : x = 8

12 - x = 4

Simplifying andsolving equations,p. 74.

We simplify using the commutative and associativeproperties before we solve equations.

Solve.

n = 2

9 + n = 11

12 + 72 + n = 11

2 + 17 + n2 = 11

2 + 1n + 72 = 11

Estimating, p. 82. We round each number to the same round-off placebefore using the number in calculations.

A Ford dealership reduced the price of a FordTaurus from $23,995 to $22,950 for the LaborDay weekend sale. Estimate the savings on theTaurus. We round $23,995 to $24,000 and thesale price of $22,950 to $23,000. To estimate thesavings, we subtract.

$24,000 - $23,000 = $1000 savings

Evaluating expressions,p. 75.Addition, p. 14.Subtraction, p. 25.Multiplication, p. 64.Division, p. 64.

To evaluate an expression, we replace the variable inthe expression with the given value and then simplify.

Evaluate.

(a) for and

(b) for

15 - 122

=

42

= 2

x = 51x - 12

2

= 6 + 8 = 143 # 2 + 2 # 4

b = 4a = 23a + 2b

Combining like terms,p. 69.

We either add or subtract numerical parts of like terms.The variable part stays the same.

Combine like terms.

7xy + 3x

2xy + 3x + 5xy

Using the Mathematics Blueprint for Problem Solving, p. 84When solving problems, students often find it helpful to complete the following steps. You will not use all the steps all the time.Choose the steps that best fit the condition of the problem.

Understand the problem.

(a) Read the problem carefully and then draw a picture or chart.(b) Think about the facts you are given and what you are asked for.(c) Use the Mathematics Blueprint for Problem Solving to organize your work.

Calculate and state the answer. Perform the necessary calculations and state the answer, including the units of measure.

Check.

(a) Estimate your answer and check it with the value calculated to see if your answer is reasonable, or(b) Repeat the calculations working the problem a different way.


Procedure for Solving Applied Problems


EXAMPLE

The Austin Department of Housing purchased several computer stations for a total cost of $7850 and bought software to update theirsystem for $1055. The department had $9450 in their supply budget for the quarter. After this purchase, how much money was left forsupplies?

Understand the problem.

Chapter 1 Review Problems 95

Calculate and state the answer. Calculate the cost of the computers and the software.

Check. We estimate the amount of money left in the budget.

The estimated balance of $500 is close to the answer, $545. We determine that the answer is reasonable.

$7850 rounds to:$1055 rounds to:

Estimated cost of supplies:

$7900+ 1100

$9000 $9450 rounds to:

Estimate of money left:

$9500- 9000

$ 500

$7850+ 1055

Cost of supplies: $8905 $9450

- 8905Money left: $ 545

Cost of purchases:Computers: $7850Software: $1055

Money available:$9450

Find the amount ofmoney left after thepurchases.

1. Find the total cost of thecomputers and software.

2. Subtract this amountfrom the money availablein the supply budget.

Be sure to copy all thefacts correctly.

Gather the Facts

What Am I Asked to Do?

How Do IProceed?


Chapter 1 Review Problems

VocabularyWrite the definition for each word.

1. (1.2) Rectangle: A four-sided figure with adjoining sides that are perpendicular and opposite sides that are equal

2. (1.2) Square: A rectangle with all sides equal

3. (1.2) Right angle: An angle that measures 90 degrees

4. (1.2) Triangle: A three-sided figure with three angles

5. (1.2) Perimeter: The distance around an object

6. (1.4) Factors: The numbers or variables we multiply

7. (1.8) Term: A number, a variable, or a product of a number and one or more variables

8. (1.8) Constant term: A term that has no variable

9. (1.8) Coefficient: The number factor in a term

10. (1.8) Like terms: Terms that have identical variable parts

11. (1.8) Equation: Two expressions separated by an equals sign



Replace each question mark with the inequality symbol or 7 .6

Section 1.112. In the number 175,493:

(a) In what place is the digit 7? Ten thousands

(b) In what place is the digit 5? Thousands

13. In the number 458,013:(a) In what place is the digit 8? Thousands

(b) In what place is the digit 0? Hundreds

Write each number in expanded notation.

14. 7694 7000 + 600 + 90 + 4 15. 5831 5000 + 800 + 30 + 1

Fill in each check for the amount indicated.

16. $341 17. $187

PAY to theORDER of

MEMO

DOLLARS

DATE 20

Laura Mason7 Hampton AveHuntington Beach, CA 92628

2823


Three hundred forty-one and 00/100

341.00 PAY to theORDER of

MEMO

DOLLARS

DATE 20

Al and Ellen Ward2323 Flor Pl.Delton, Michigan 49015

2824

Farmers BankBattle Creek, Michigan

One hundred eighty-seven and 00/100

187.00

18. 2 ? 8 19. 12 ? 0 20. 11 ? 12 676

Rewrite using numbers and an inequality symbol.

21. Six is greater than one. 6 7 1 22. Three is less than five. 3 6 5

Round to the nearest hundred.

23. 61,269 61,300 24. 382,240 382,200

Round to the nearest hundred thousand.

25. 6,365,534 6,400,000 26. 8,118,701 8,100,000

Section 1.2Translate using symbols.

27. Seven more than a number 28. The sum of some number and five 29. A number increased by fourx + 4n + 5x + 7

Use whichever properties (associative and/or commutative) are necessary to simplify each expression.

30. 31. 32. n + 1112 + n2 + 9n + 51n + 42 + 116 + x7 + 19 + x2

Evaluate.

36. if y is equal to 9 11 37. if x is equal to 9 14 38. if x is equal to 11 and y isequal to 15 26x + yx + 52 + y

33. 34. 35. x + 813 + x + 42 + 1x + 1015 + x + 32 + 27 + n5 + 1n + 22

Add.

39. 90258398 + 372 + 255 40. 18,65117,456 + 213 + 982



Answer the following.

41. College Enrollment A private college has 1434 fresh-men, 1596 sophomores, 1423 juniors, and 1565 seniors.How many students are attending the college?6018 students

42. Find the perimeter of the shape made up of two rec-tangles (measured in meters).66 m

Section 1.3Translate using numbers and symbols.

5 m

8 m

7 m

13 m

43. Eight decreased by a number 44. The difference of a number and six 45. Ten subtracted from a numberx - 10n - 68 - n

Evaluate.

46. if x is equal to 3 58 - x 47. if y is equal to 15 6y - 9

48. if x is equal to 2 57 - x 49. if y is equal to 10 212 - y

Subtract and check your answer.

50. 5545 51. 3159 52. 24,116 53.130,528137,405 - 687729,104 - 49889021 - 58628502 - 2957

Golf Championships A professional golf player’s earnings from winning six golf championships are listed on the bargraph. Use the graph to answer exercises 54 and 55.

0

200,000

2004 2005 2006Year

2007 2008 2009

400,000

600,000

800,000

1,200,000

1,000,000

Pri

ze (

in d

olla

rs)

200,

000

450,

000

522,

000

100,

000 72

0,00

0

900,

000

54. How much more did the player earn in the 2009championship than in the 2006 championship?$378,000 more

55. How much less did the player earn in the 2005 cham-pionship than in the 2008 championship?$270,000 less

Section 1.4Identify the factors in each equation.

56. 4, x4x = 32 57. x, yxy = z

Translate the phrase using numbers and symbols.

58. Triple a number 3x

Translate the mathematical symbols to words.

59. Seven times what number equals 63?7y = 63

�



Multiply.

60. 07 # 2 # 3 # 0 61. 605 # 3 # 2 # 2

Simplify.

62. 42y 63. 30x 64. 70y 65. 24x31221x # 4221521y # 7231521x # 2261y # 72

Multiply.

66. 631(52) 32,812 67. (416)(2000) 832,000 68. (4251)352 1,496,352 69.

5,800,872

6424* 903

Solve each applied problem.

70. Miles Per Gallon Lisa has a truck that averages17 miles per gallon on the highway. Approximatelyhow far can she travel if she has 18 gallons of gas in hertank? 306 mi

71. Doors Replaced J&R Doors & Windows is replacingall the interior doors in a 6-unit apartment complexthat has 21 apartments in each unit. If each apartmenthas 4 interior doors, how many doors will need to bereplaced? 504 doors

Section 1.5Write the division that corresponds to each situation. You need not carry out the division.

72. 300 desks are arranged so that 20 desks are in eachrow. How many rows are there? 300 , 20

73. A $500 prize is divided equally between n people. Howmuch will each person receive? $500 , n

Translate each phrase using numbers and symbols.

74. Five divided by a number 5 , y 75. The quotient of a number and thirteen n , 13

76. Divide. Undefined10 , 0 77. Divide. 133 , 33

Divide.

78. 451 79. 567 80. 80 R52485 , 313�17014�1804

81. 50 R6 82. 401 R35 83. 603 R6510,144

846369,757 , 9221456 , 29

Solve each applied problem.

84. Fundraiser Proceeds The Dalton City Music Clubfund-raising committee raised $447. The club dividedthe funds equally between four youth groups and de-posited the rest of the funds in their club account.How much money did the club deposit in their clubaccount? $3

85. Loan Payments Lisa wishes to pay off a loan of $3528in 24 months. How large will her monthly payments be?$147

Section 1.6Write each product in exponent form.

86. 23 n22 # 2 # 2 # n # n 87. or 53z4z4

53z # z # z # z # 5 # 5 # 5



Write as a repeated multiplication.

88. x # x # xx3 89. 6 # 6 # 6 # 6 # 665

Evaluate.

90. 1000 91. 16 92. 819224103

Translate each phrase using numbers, symbols, and exponents.

93. Six cubed 63 94. x to the fifth power x5

Evaluate.

95. 5 96. 5 97. 205 # 22115 + 25 , 52 , 18 - 426 + 24 , 8 - 22

Section 1.7Translate using numbers and symbols.

98. (a) Three times x plus two(b) Three times the sum of x and two 31x + 22

3x + 2 99. (a) Four times x minus five(b) Four times the difference of x and 5. 41x - 52

4x - 5

Translate using numbers and symbols, then simplify.

100. (a) Three times seven plus one(b) Three times the sum of seven and one

101. Evaluate if x is equal to 3. 26

102. Evaluate if m is equal to 8 and n is equal to 2. 222m + 3n

x3- 1

317 + 12 = 24

3 # 7 + 1 = 22


103. 104. 105. 3x + 831x + 12 + 54x - 441x - 125x + 551x + 12

Section 1.8Combine like terms.

106. 9x2x + x + 6x 107. 11x + 6y5x + 6y + 6x

108. 5xy + 13y3xy + 5y + 2xy + 8y 109. Find an expression that represents the perimeter.10x + 10y 2x � 4y

3x � y

�

Solve and check your answer.

110. x = 7x + 2 = 9 111. n = 410 - n = 6

Simplify using the associative and/or commutative property and then find the solution.

112. x = 413 + x2 + 1 = 8 113. n = 12 + 1n + 72 = 10

Solve each equation and check your answer.

114. x = 39x = 27 115. x = 315x

= 5



Simplify by combining like terms and then find the solution.

116. n = 212n - n = 22 117. y = 2y + 3y + 2y = 12

For the following questions:

(a) Translate the English statement into an equation. (b) Solve.

118. What number subtracted from eighteen equals three?(a) (b) x = 1518 - x = 3

119. What number increased by five equals eleven?(a) (b) x = 6x + 5 = 11

120. Triple what number is equal to twelve?(a) (b) x = 43 # x = 12

Section 1.9121. Office Supply Purchases Joseph must purchase sup-

plies for his home office. The catalog from the supplywarehouse lists the following prices for the items heplans to purchase. Round each amount to the nearestten and then estimate the amount of money he willpay for these supplies. $310

Price List

Statistical calculator

Color inkjetprint cartridge

Four-drawermetal file cabinet

Computer chair

$25

26

87

156

You may want to use the Mathematics Blueprint for Problem Solving for exercises 122–124.

Gather the Facts

What Am I Asked to Do?

How Do IProceed?


122. Payroll Deductions A teacher’s assistant receives atotal salary per month of $3560. Deducted from hispaycheck are taxes of $499, social security of $218,and retirement of $97. What is the total of his checkafter the deductions? $2746

123. Savings Account Balance Jean’s savings accounthad a balance of $5021. Over the course of a year, shemade deposits of $759, $2534, and $532. She alsomade withdrawals of $799, $533, and $88.(a) What was her ending balance? $7426

(b) If Jean divides her savings equally into twomoney market accounts, how much money willthere be in each account? $3713

124. Ceiling Molding Purchase Ruth Ann must buy crown molding for the ceilings in her living and dining rooms. Theliving room measures 20 feet by 25 feet, and the dining room measures 15 feet by 18 feet. If crown molding costs $3a linear foot, how much will it cost Ruth Ann to purchase the molding for both rooms? $468


1.

2. (a)

(b)

3. (a) 3000

(b) 2900

4. (a)

(b)

(c)

5. 14,695

6. 244,888,278

7. (a) 538

(b) 12,279

8. 30 ft

9. 16y

10. (a) 134,784

(b) 261,999

11. (a) 41

(b) 120 R3

12. (a)

(b) 10n

(c)

(d)

(e)

13. (a)

(b)

14.

15.

16. (a) 20

(b) 77

17. 65

8x + 10

3y + 12

3m + 5 + 6mn

7xy + 2y - 2

61x + 9273

y4

n - 7

7 + n or n + 7

7 + y or y + 7

11 + x or x + 11

5 7 0

7 7 2

1000 + 500 + 20 + 5

Note to Instructor: The Chapter 1 Testfile in the TestGen program provides al-gorithms specifically matched to theseproblems so you can easily replicate thistest for additional practice or assessmentpurposes.

101

How Am I Doing? Chapter 1 Test

Remember to use your Chapter Test Prep Video CD to see the worked-out solutions tothe test problems you want to review.

1. Write 1525 in expanded notation.

2. Replace each question mark with the appropriate symbol or (a) 7 ? 2 (b) 5 ? 0

3. The total population of a small town is 2925. Round this population figure to(a) The nearest thousand (b) The nearest hundred

4. Use the commutative and/or associative property of addition and then simplify.(a) (b) (c)

5. Add. 6. Add.

7. Subtract and check your answer.(a) (b)

8. Find the perimeter of the figure made of rectangles.

9. Multiply.

10. Multiply.(a) (432)(312) (b)

11. Divide and check your answer.(a) (b)

12. Translate using numbers and symbols.(a) Seven subtracted from a number (b) The product of ten and a number(c) y to the fourth power (d) 7 cubed(e) Six times the sum of x and nine

13. Combine like terms.(a) (b)


14. 15.

16. Evaluate.(a) if x is equal to 16 and y is equal to 4(b) if

17. Write in exponent form. 6 # 6 # 6 # 6 # 6

a = 9a2- 4

2x - 3y

81x + 12 + 231y + 42

2m + 5 + m + 6mn3xy + 2y + 4xy - 2

5523 , 46492 , 12

2031* 129

21421y # 22

20,105- 7 826

613 - 75

244,869,201 + 19,07712,389 + 4 + 2302

1 + 1n + 22 + 45 + y + 23 + 18 + x2

7 .6

9 ft

6 ft

1 ft

7 ft

�


18. (a) 125

(b) 100,000

19. 0

20. 41

21. 30

22. (a)

(b)

(c)

(d)

(e)

23.

24. (a)

(b)

25. (a)

(b)

26. (a) $15,340

(b) $7990

27. 12

28. $1140

29. $1290

30. (a) $1600

(b) $300

31. 7500 points

x = 4

x - 3 = 1

x = 12

x , 6 = 2

B - 155 = 275

n = 4

b = 11

x = 9

x = 8

x = 6

18. Evaluate.(a) (b)

Evaluate.

19. 20. 21.

22. Solve each equation and check your answer.

(a) (b) (c)

(d) (e)

Translate into an equation.

23. Let B represent Fred’s checking account balance. Fred’s checking accountbalance decreased by $155 equals $275.

Translate and solve for problems 24 and 25:(a) Translate into an equation. (b) Solve.

24. What number divided by six equals two?

25. Three subtracted from what number equals one?

26. Tickets to a play were $25 for adults and $18 for children. 412 adult tickets weresold, and 280 children’s tickets were sold.(a) Find the total income from the sale of tickets.(b) If the expenses for the play were $7350, how much profit was made?

27. A restaurant sells 4 kinds of sandwiches: turkey, roast beef, veggie, and ham.Customers have a choice of 3 types of bread: wheat, white, or rye. How manydifferent sandwiches are possible?

28. A store clerk receives a total salary per month of $1540. Deducted from herpaycheck are taxes of $265, social security of $78, and retirement of $57. Whatis the total of her check after the deductions?

29. The rent on an apartment was $525. To move in, Fred was required to pay thefirst and last month’s rent, a security deposit of $200, and a telephone installa-tion fee of $40. How much money did he need to move into the apartment?

30. Sylvia kept the following record of her living expenses for the month of Febru-ary 2005:

(a) Round each amount to the nearest hundred and then estimate Sylvia’smonthly expenses for February.

(b) If her take-home (net) income for February was $1921, estimate howmuch money was left after all the expenses were paid.

31. A frequent-flyer program offered by many major airlines to first-class passengersawards 3 frequent-flyer mileage points for every 2 miles flown. When customersaccumulate a certain number of frequent-flyer points, they can cash in thesepoints for free air travel, a ticket upgrade, or other awards. How many frequent-flyer points would Elizabeth accumulate if she flew 5000 miles in first-class?

9n - n = 325 + 1b + 22 = 18

x + 3x = 36x

4= 27 + x = 13

3 # 2 + 417 - 1262- 7 + 3 # 424 , 4 - 2 # 3

10553

Rent $790 Car payment 210

Phone and utilities 114 Gas and insurance 187

Food 318



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