1.6 Division - McGraw Hill Higher Education trombones 4 19 trombones $55 11 $5 When one denominate number is divided by another, the result will get the units of the dividend over

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Division1.6

1.6 OBJECTIVES

1. Use the language of division2. Write a division problem as repeated subtraction3. Divide whole numbers

71

We will now examine a fourth arithmetic operation, division. Just as multiplication was re-peated addition, division is repeated subtraction. Division asks how many times one num-ber is contained in another.

48 40 32 24 16 8�8 �8 �8 �8 �8 �8

40 32 24 16 8 0

Example 1

Dividing by Using Subtraction

Joel needs to set up 48 chairs in the student union for a concert. If there is room for 8 chairsper row, how many rows will it take to set up all 48 chairs?

This problem can be solved by subtraction. Each row subtracts another 8 chairs.

Because 8 can be subtracted from 48 six times, there will be 6 rows.

This can also be seen as a division problem

648 � 8 � 6 or or � 6

No matter which method we use, we call the 48 the dividend, the 8 the divisor, and the 6the quotient.

48

88B48

C H E C K Y O U R S E L F 1

Carlotta is creating a garden path made of bricks. She has 72 bricks. Each row will

have 6 bricks in it. How many rows can she make?

72 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS

To solve a problem that requires division, you must first set up the problem as a divisionstatement. The next example will illustrate this.

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Units AnalysisWhen dividing a denominate number by an abstract number, the result will getthe units of the denominate number. Here are a couple of examples

76 trombones � 4 � 19 trombones

$55 � 11 � $5

When one denominate number is divided by another, the result will get theunits of the dividend over the units of the divisor.

144 miles � 6 gallons � 24 miles/gallon (which we read as “miles per gallon”)

$120 � 8 hours � 15 dollars/hour (“dollars per hour”)

Writing a Division Statement

Write a division statement that corresponds to the following situation. You need not do thedivision.

The staff at the Wok Inn Restaurant splits all tips at the end of each shift. Yesterday’sevening shift collected a total of $224. How much should each of the seven employees getin tips?

$224 � 7 employees (note that the units for the answer will be “dollars per employee”)

Example 2

In the previous section, we used a rectangular array of stars to represent multiplication.These same arrays can represent division. Just as 3 � 4 � 12 and 4 � 3 � 12, so is it truethat 12 � 3 � 4 and 12 � 4 � 3.


Write a division statement that corresponds to the following situation. You neednot do the division.

All nine sections of basic math skills at SCC (Sum Community College) are full.There are a total of 315 students in the classes. How many students are in each class?What are the units for the answer?

DIVISION SECTION 1.6 73©

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1 M

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NOTE For a division problemto check, the product of thedivisor and the quotient mustequal the dividend.

NOTE Because 36 � 9 � 4, wesay that 36 is exactly divisible by 9.

NOTE Notice that theremainder must be smaller thanthe divisor or we could subtractagain.

� 4 � 3 � 12

or

12 � 3 � 4

or

12 � 4 � 3

� � 3 � 4 � 12�

This relationship allows us to check our division results by doing multiplication.

Example 3

Checking Division by Using Multiplication

3(a) Check: 7 � 3 � 21

(b) 48 � 6 � 8 Check: 6 � 8 � 48

7B21

In our examples so far, the product of the divisor and the quotient has been equal to thedividend. This means that the dividend is exactly divisible by the divisor. That is not alwaysthe case. Let’s look at another example using repeated subtraction.

Dividing by Using Subtraction, Leaving a Remainder

How many times is 5 contained in 23?

23 18 13 8� 5 � 5 � 5 � 5

18 13 8

23 is not exactly divisible by 5. The “left over” 3 is called the remainder in the division.To check the division operation when a remainder is involved, we have the followingrule:

3

Example 4

We see that 5 is contained 4times in 23, but 3 is “left over.”


Complete the division statements, and check your results.

(a) (b) 28 � 7 �9B45

Dividend � divisor � quotient � remainder

Definitions: Remainder


How many times is 7 contained in 38?


Checking Division by a Single-Digit Number

Using the work of the previous example, we can write

4with remainder 3

To apply our previous rule, we have

Divisor Quotient

Dividend 23 � 5 � 4 � 3 Remainder

23 � 20 � 3

23 � 23 The division checks.

5B23

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Example 5

NOTE Another way to writethe result is

4 r3 The “r” stands forremainder.5B23

NOTE Notice that themultiplication is done beforethe 3 is added.

We must be careful when 0 is involved in a division problem. There are two specialcases.

Example 6

The first case involving zero occurs when we are dividing into zero.

Dividing into Zero

0 � 5 � 0 because 0 � 5 � 0.

Our second case illustrates what happens when 0 is the divisor. Here we have a specialproblem.

Example 7

Dividing by Zero

8 � 0 � ? This means that 8 � 0 � ?

Can 0 times some number ever be 8? From our multiplication facts, the answer is no! Thereis no answer to this problem, so we say that 8 � 0 is undefined.

1. 0 divided by any whole number (except 0) is 0.

2. Division by 0 is undefined.

Rules and Properties: Division and Zero


Evaluate . Check your answer.7B38


(a) 0 � 7 � (b) 0 � 12 �

Subtracting 160 is just a shortcut forsubtracting eight 20 times.

Adding 20 and 2 gives us the quotient, 22.�

Dividing by a Single-Digit Number

Divide 176 by 8.

Because 20 eights are 160, we know that there are at least 20 eights in 176.

Step 1 Write

20

20 eights 16016

After subtracting the 20 eights, or 160, we are left with 16. There are 2 eights in 16, and sowe continue.

Step 2 22220

16016

2 eights 160

Subtracting the 2 eights, we have a 0 remainder. So 176 � 8 � 22

8B176

8B176

DIVISION SECTION 1.6 75

It is easy to divide when small whole numbers are involved, because much of the work canbe done mentally. In working with larger numbers, we turn to a process called long divi-sion. This is a shorthand method for performing the steps of repeated subtraction.

To start, let’s look at an example in which we subtract multiples of the divisor.

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NOTE With larger numbers,repeated subtraction is just tootime-consuming to be practical.

Example 8

When we place 5 as the tens digit,we really mean 5 tens, or 50.

Example 9

Dividing by a Single-Digit Number

Divide 358 by 6.

The dividend is 358. We look at the first digit, 3. We cannot divide 6 into 3, and so we lookat the first two digits, 35. There are 5 sixes in 35, and so we write 5 above the tens digit ofthe dividend.

56B358

The next step is to simplify this repeated-subtraction process one step further. The resultwill be the long-division method.


Decide whether each problem results in 0 or is undefined.

(a) 9 � 0 (b) 0 � 9 (c) 0 � 15 (d) 15 � 0


Verify the results of Example 8, using multiplication.


Now multiply 5 � 6, place the product below 35, and subtract.

5

3005

Because the remainder, 5, is smaller than the divisor, 6, we bring down 8, the ones digit ofthe dividend.

5

30058

Now divide 6 into 58. There are 9 sixes in 58, and so 9 is the ones digit of the quotient. Mul-tiply 9 � 6 and subtract to complete the process.

59

3058544

To check: 358 � 6 � 59 � 4

6B358

6B358

6B358

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We have actually subtracted 50 sixes(300) from 358.

We now have:358 � 6 � 59 r4

NOTE Because the 4 is smallerthan the divisor, we have aremainder of 4.

NOTE Verify that this is trueand that the division checks.

7NOTE Think: 4B29

Long division becomes a bit more complicated when we have a two-digit divisor. It is nowa matter of trial and error. We round the divisor and dividend to form a trial divisor and atrial dividend. We then estimate the proper quotient and must determine whether our esti-mate was correct.

Dividing by a Two-Digit Number

Divide

Round the divisor and dividend to the nearest ten. So 38 is rounded to 40, and 293 isrounded to 290. The trial divisor is then 40, and the trial dividend is 290.

Now look at the nonzero digits in the trial divisor and dividend. They are 4 and 29. Weknow that there are 7 fours in 29, and so 7 is our first estimate of the quotient. Now let’s seeif 7 works.

7

26627

The remainder, 27, is less than the divisor, 38, and so the process is complete.

293 � 38 � 7 r27

Check: 293 � 38 � 7 � 27 You should verify that this statement is true.

38B293

38B293

Example 10

Your estimate

Multiply 7 � 38. The product, 266, isless than 293, and so we can subtract.


Divide 7B453.


Because this process is based on estimation, we can’t expect our first guess to always beright.

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Dividing by a Two-Digit Number

Divide

Rounding to the nearest ten, we have a trial divisor of 50 and a trial dividend of 430.Looking at the nonzero digits, how many fives are in 43? There are 8. This is our first es-

timate.

8

432

We adjust the quotient downward to 7. We can now complete the division.

7

37850

We have

428 � 54 � 7 r50

Check: 428 � 54 � 7 � 50

54B428

54B428

54B4288

NOTE Think: 5B43

Example 11

Too large

We multiply 8 � 54. Do you see what’s wrong? Theproduct, 432, is too large. We can’t subtract. Ourestimate of the quotient must be adjusted downward.

We have to be careful when a 0 appears as a digit in the quotient. Let’s look at an exam-ple in which this happens with a two-digit divisor.

Dividing with Large Dividends

Divide

32B9871

Example 12

NOTE Our divisor, 32, willdivide into 98, the first twodigits of the dividend.

C H E C K Y O U R S E L F 1 0

Divide.

57B482


Divide.

63B557


Rounding to the nearest ten, we have a trial divisor of 30 and a trial dividend of 100. Think,“How many threes are in 10?” There are 3, and this is our first estimate of the quotient.

3

962

Bring down 7, the next digit of the dividend.

30

9627

We continue by multiplying by 0. After subtraction, we bring down 1, the last digit of thedividend.

30

962700271

Another problem develops here. We round 32 to 30 for our trial divisor, and we round 271to 270, which is the trial dividend at this point. Our estimate of the last digit of the quotientmust be 9.

309

962700271288 Too large

We can’t subtract. The trial quotient must be adjusted downward to 8. We can now completethe division.

308

96270027125615

9871 � 32 � 308 r15

Check: 9871 � 32 � 308 � 15

32B9871

32B9871

32B9871

32B9871

32B9871

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Everything seems fine so far!

Now do you see the difficulty? We cannot divide32 into 27, and so we place 0 in the tens placeof the quotient to indicate this fact.


Divide.

43B8857


Because of the availability of the handheld calculator, it is rarely necessary that people findthe exact answer when performing long division. On the other hand, it is frequently impor-tant that one be able to either estimate the result of long division, or confirm that a givenanswer (particularly from a calculator) is reasonable. As a result, the emphasis in this sec-tion will be to improve your estimation skills in division.

Let’s divide a four-digit number by a two-digit number. Generally, we will round thedivisor to the nearest ten and the dividend to the nearest hundred.

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Example 13

Example 14

Estimating the Result of a Division Application

The Ramirez family took a trip of 2394 miles (mi) in their new car, using 77 gallons (gal)of gas. Estimate their gas mileage (mi/gal).

Our estimate will be based on dividing 2400 by 80.

30

They got approximately 30 mi/gal.

80B2400

Estimating the Result of a Division Application

Charles purchases a new car for $8574. Interest charges will be $978. He agrees to makepayments for 4 years. Approximately what should his payments be?

First, we find the amount that Charles owes:

$8574 � $978 � $9552

Now, to find the monthly payment, we divide that amount by 48 (months). To estimate thepayment, we’ll divide $9600 by 50 months.

192

The payments will be approximately $192 per month.

50B9600

As before, we may have to combine operations to solve an application of the mathemat-ics you have learned.


Troy flew a light plane on a trip of 2844 mi that took 21 hours (h). What was his

approximate speed in miles per hour (mi/h)?


One $10 bag of fertilizer will cover 310 square feet. Approximately what would it

cost to cover 2200 square feet?


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C H E C K Y O U R S E L F A N S W E R S

1. 12 2. 315 students � 9 classes; students per class 3. (a) 5; 9 � 5 � 45;

(b) 4; 7 � 4 � 28 4. 5 5. 5 with remainder 3 6. (a) 0; (b) 0

7. (a) undefined; (b) 0; (c) 0; (d) undefined 8. 8 � 22 � 176

9. 64 with remainder 5 10. 8 with remainder 26 11. 8 with remainder 53

12. 205 with remainder 42 13. 140 mi/h 14. $70

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Exercises

1. If 48 � 8 � 6, 8 is the _______, 48 is the _______, and 6 is the _______.

2. In the statement , 9 is the _______, 5 is the _______, and 45 is the _______.

3. Find 36 � 9 by repeated subtraction.

4. Find 40 � 8 by repeated subtraction.

5. Stefanie is planting rows of tomato plants. She wants to plant 63 plants with 9 plantsper row. How many rows will she have?

6. Nick is designing a parking lot for a small office building. He must make room for42 cars with 7 cars per row. How many rows should he plan for?

Divide the following. Identify the correct units for the quotient.

7. 36 pages � 4 8. $96 � 8

9. 4900 km � 7 10. 360 gal � 18

11. 160 miles � 4 hours 12. 264 ft � 3 sec

13. 3720 hours � 5 months 14. 560 calories � 7 grams

Divide using long division, and check your work.

15. 54 � 9 16. 21 � 3 17.

18. 19. 20. 56 � 8

21. 22. 40 � 9 23.

24. 25. 57 � 8 26. 74 � 8

27. 0 � 5 28. 5 � 0 29. 4 � 0

30. 0 � 12 31. 0 � 6 32. 18 � 0

6B51

9B655B43

4B327B63

6B42

5B459

1.6

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

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10.

11.

12.

13.

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15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

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30. 31.

32.

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Divide.

33. 34. 35.

36. 37. 38.

39. 40. 41.

42. 43. 44.

45. 46. 47.

48. 49. 50.

51. 52. 53.

54. 55. 56.

57. 58. 59.

60. 61. 62.

63. 64.

Solve the following applications.

65. Counting. Ramon bought 56 bags of candy. There were 8 bags in each box. Howmany boxes were there?

66. Capacity. There are 32 students who are taking a field trip. If each car can hold4 students, how many cars will be needed for the field trip?

67. Packaging. There are 63 candy bars in 7 boxes. How many candy bars are in eachbox?

871B4321763B3071

38B789228B854753B8729

42B790227B933534B8748

53B348045B236767B939

23B53454B37248B892

4B34,0933B13,4217B8923

4B73218B32514B4351

5B43,2878B22,1539B3527

8B54385B49388B3136

7B3468B2934B232

3B1629B785B83

ANSWERS

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

63. 64.

65.

66.

67.

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75. Telephone calls. The records of an office show that 1702 calls were made in 1 day. Ifthere are 37 phones in the office, how many calls were placed per phone?

76. Television costs. A television dealer purchased 23 sets, each the same model, for$5267. What was the cost of each set?

77. Computers. A computer printer can print 340 lines per minute (min). How long willit take to complete a report of 10,880 lines?

78. Distance. A train traveled 1364 mi in 22 h. What was the speed of the train? Hint: Speed is the distance traveled divided by the time.

79. Bonuses. A company distributes $16,488 in year-end bonuses. If each of the 36employees receives the same amount, what bonus will each receive?

80. Complete the following number cross.

Across1. 48 � 43. 1296 � 86. 2025 � 58. 4 � 59. 11 � 11

12. 15 � 3 � 11114. 144 � (2 � 6)16. 1404 � 618. 2500 � 519. 3 � 5

Down1. (12 + 16) � 22. 67 � 34. 744 � 125. 2600 � 137. 6300 � 12

10. 304 � 211. 5 � (161 � 7)13. 9027 � 1715. 400 � 2017. 9 � 5

Estimate the result in the following division problems. (Remember to round divisors tothe nearest ten and dividends to the nearest hundred.)

81. 810 divided by 38 82. 458 divided by 18


ANSWERS

75.

76.

77.

78.

79.

80.

81.

82.

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84.

84

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6 7

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Solve the following applications.

91. Gas mileage. Jose drove 279 miles (mi) on 18 gallons of gas. Estimate his mileage.(Hint: Find the number of miles per gallon.)

92. Construction. A contractor can build a house in 27 days. Estimate how many housescan be built in 265 days.

93. Inheritances. Twelve people are to share equally in an estate totaling $26,875.Estimate how much money each person will receive.

94. Business. There is $365 left in the budget to purchase pens. If each box of pens costs$18, estimate the number of boxes of pens that can be ordered.

95. Monthly payments. Tara purchased a used car for $1850 by paying $275 down andthe rest in equal monthly payments over a period of 18 months. Estimate the amountof her monthly payments.

96. Consumer purchases. Art has $275 to spend on shirts. If the cost of a shirt is $23,estimate the number of shirts that Art can buy.

97. You are going to recarpet your living room. You have budgeted $1500 for the carpetand installation.

(a) Determine how much carpet you will need to do the job. Draw a sketch tosupport your measurements.

(b) What is the highest price per square yard you can pay and still stay withinbudget?

(c) Go to a local store and determine the total cost of doing the job for three differentgrades of carpet. Be sure to include padding, labor costs, and any other expenses.

(d) What considerations (other than cost) would affect your decision about what typeof carpet to install?

(e) Write a brief paragraph indicating your final decision, and give supportingreasons.

98. Division is the inverse operation of multiplication. Many daily activities haveinverses. For each of the following activities, state the inverse activity:

(a) Spending money

(b) Going to sleep

(c) Turning down the volume on your CD player

(d) Getting dressed

ANSWERS

85.

86.

87.

88.

89.

90.

91.

92.

93.

94.

95.

96.

97.

98.

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99. If you have no money in your pocket and want to divide it equally among your fourfriends, how much does each person get? Use this situation to explain division ofzero by a nonzero number.

100. Explain the difference between division by zero and division of zero by a naturalnumber.

101. Division is not associative. For example, 8 � 4 � 2 will produce different resultsif 8 is divided by 4 and then divided by 2 or if 8 is divided by the result of 4 � 2.In the following, place parentheses in the proper place so that the expression istrue.

(a) 16 � 8 � 2 � 4 (b) 16 � 8 � 2 � 1

(c) 125 � 25 � 5 � 1 (d) 125 � 25 � 5 � 25

(e) Is there any situation in which the order of how the operation of division isperformed produces the same result? Give an example.

102. Division is not commutative. For example, 15 � 5 � 5 � 15. What must be true ofthe numbers a and b if a � b � b � a?

103. Your class goes to a local amusement park. A ride can carry 15 passengers in eachcycle.

(a) If a new cycle starts every 5 min, how many cycles does the ride make everyhour?

(b) How many passengers can ride every hour?

(c) How long would it take all the students in your class to complete the ride?

Answers1. Divisor, dividend, quotient 3. 4 5. 7 7. 9 pages 9. 700 km11. 40 miles/hour 13. 744 hours/month 15. 6 17. 7 19. 821. 8 r3 23. 7 r2 25. 7 r1 27. 0 29. Undefined 31. 033. 16 r3 35. 54 37. 36 r5 39. 392 41. 679 r6 43. 2769 r145. 1087 r3 47. 1830 r1 49. 4473 r2 51. 18 r28 53. 23 r555. 52 r27 57. 257 r10 59. 188 r6 61. 305 r7 63. 4 r1965. 7 boxes 67. 9 bars 69. 5 pictures 71. 138 tickets 73. $14575. 46 calls 77. 32 min 79. $458 81. 20 83. 5085. 100 87. 30 89. 20 91. 15 mi/gal 93. $2700 95. $8097. 99. 101. 103.

ANSWERS

99.

100.

101.

102.

103.

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Using a Scientific Calculatorto Divide

Of course, division is easily done by using your calculator. However, as we will see, somespecial things come up when we use a calculator to divide. First let’s outline the steps ofdivision as it is done on a calculator.

Divide

1. Enter the dividend. 2380

2. Press the divide key.

3. Enter the divisor. 35

4. Press the equals key. The desired quotient is now in

your display.

The display shows

We mentioned some of the difficulties related to division with 0 earlier. Let’s experimenton the calculator.

68

�

�

35B2380.

87

Example 1

Using a Scientific Calculator to Divide

To find 0 � 5, we use this sequence:

0 5

Display

There is no problem with this. Zero divided by any whole number other than 0 is just 0.

0

��

We’ve seen what happens when dividing zero by another number, but what happenswhen we try to divide by zero? More importantly to this section, how does the calculatorhandle division by zero? Example 2 illustrates this concept.


What is the result when you use your calculator to perform the following operation?

0 � 17


Another special problem comes up when a remainder is involved in a division problem.

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Example 3

NOTE We will say more aboutthis later. For now, just beaware that the calculator willnot give you a remainder in theform we have been using in thischapter.

�

QuotientRemainder

7 is the whole-number part of thequotient as before.

0.7105263 is the decimal form of theremainder, 27, as a fraction of 38.

Example 4

Do you see that the calculator hasdone the division as the first step?


In a previous section, we divided 293 by 38 and got 7 with remainder 27.

293 38 7.7105263��

The calculator can also help you combine division with other operations.


To find 18 � 2 � 3, use this sequence:

18 2 3

Display 12

��



458 � 36

Example 2

NOTE You may find that youmust “clear” your calculatorafter trying this.


To find 5 � 0, we use this sequence:

5 0

Display

If we try this sequence, the calculator gives us an error! Do you see why? Division by 0 isnot allowed. Try this on your calculator to see how this error is indicated.

Error

��



17 � 0

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Example 5


To find 6 � 3 � 2, use this sequence:

6 3 2

Display 4

��


Use your calculator to compute.

15 � 5 � 7


Use your calculator to compute.

18 � 6 � 5

C H E C K Y O U R S E L F A N S W E R S

1. 0 2. Error message 3. 12.72222 4. 10 5. 15


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Name

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ANSWERS

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2.

3.

4.

5.

6.

7.

8.

9.

10.

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12.

13.

14.

15.

16.

Calculator Exercises

Use your calculator to perform the indicated operations.

1. 5940 � 45 2. 2808 � 36

3. 36,182 � 79 4. 36,232 � 56

5. 583,467 � 129 6. 464,184 � 189

7. 6 � 9 � 3 8. 18 � 6 � 3

9. 24 � 6 � 4 10. 32 � 8 � 4

11. 4368 � 56 � 726 � 33 12. 1176 � 42 � 1572 � 524

13. 3 � 8 � 8 � 8 � 12 14. 5 � 6 � 6 � 18

15. (18 � 87) � 15 16. (89 � 14) � 25

Answers1. 132 3. 458 5. 4523 7. 9 9. 16 11. 100 13. 12815. 7

90

Documents

1.6 Division - McGraw Hill Higher Education trombones 4 19 trombones $55 11 $5 When one denominate number is divided by another, the result will get the units of the dividend over