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Inside Collection (Textbook):

Textbook by: Denny Burzynski, Wade Ellis. E-mail the authors

Division of Whole Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to divide whole numbers. By the end of the module students should be able to be able to divide a whole number by a single or multiple digit divisor and interpret a calculator statement that a division results in a remainder.

Section Overview

• Division with a Single Digit Divisor
• Division with a Multiple Digit Divisor
• Division with a Remainder
• Calculators

Division with a Single Digit Divisor

Our experience with multiplication of whole numbers allows us to perform such divisions as 75÷575÷5 size 12{"75" div 5} {}. We perform the division by performing the corresponding multiplication, 5×Q=755×Q=75 size 12{5 times Q="75"} {}. Each division we considered in (Reference) had a one-digit quotient. Now we will consider divisions in which the quotient may consist of two or more digits. For example, 75÷575÷5 size 12{"75" div 5} {}.

Let's examine the division 75÷575÷5 size 12{"75" div 5} {}. We are asked to determine how many 5's are contained in 75. We'll approach the problem in the following way.

1. Make an educated guess based on experience with multiplication.
2. Find how close the estimate is by multiplying the estimate by 5.
3. If the product obtained in step 2 is less than 75, find out how much less by subtracting it from 75.
4. If the product obtained in step 2 is greater than 75, decrease the estimate until the product is less than 75. Decreasing the estimate makes sense because we do not wish to exceed 75.

We can suggest from this discussion that the process of division consists of

The Four Steps in Division

1. an educated guess
2. a multiplication
3. a subtraction
4. bringing down the next digit (if necessary)

The educated guess can be made by determining how many times the divisor is contained in the dividend by using only one or two digits of the dividend.

Sample Set A

Example 1

Find 75÷575÷5 size 12{"75" div 5} {}.

5 75 5 75 Rewrite the problem using a division bracket.

10 5 75 10 5 75
Make an educated guess by noting that one 5 is contained in 75 at most 10 times.
Since 7 is the tens digit, we estimate that 5 goes into 75 at most 10 times.

10 5 75 - 50 ̲ 25 10 5 75 - 50 ̲ 25
Now determine how close the estimate is.
10 fives is 10×5=5010×5=50 size 12{"10" times 5="50"} {}. Subtract 50 from 75.
Estimate the number of 5's in 25.
There are exactly 5 fives in 25.

5 10 10 fives + 5 fives = 15 fives. There are 15 fives contained in 75. 5 75 - 50 ̲ 25 - 25 ̲ 0 5 10 10 fives + 5 fives = 15 fives. There are 15 fives contained in 75. 5 75 - 50 ̲ 25 - 25 ̲ 0

Check:

Thus, 75÷5=1575÷5=15 size 12{"75" div 5="15"} {}.

The notation in this division can be shortened by writing.

15 5 75 - 5↓ ̲ 25 - 25 ̲ 0 15 5 75 - 5↓ ̲ 25 - 25 ̲ 0
Divide: 5 goes into 7 at most 1 time. Multiply: 1 × 5 = 5. Write 5 below 7. Subtract: 7 - 5 = 2. Bring down the 5. Divide: 5 goes into 7 at most 1 time. Multiply: 1 × 5 = 5. Write 5 below 7. Subtract: 7 - 5 = 2. Bring down the 5. Divide: 5 goes into 25 exactly 5 times. Multiply: 5 × 5 = 25. Write 25 below 25. Subtract: 25 - 25 = 0. Divide: 5 goes into 25 exactly 5 times. Multiply: 5 × 5 = 25. Write 25 below 25. Subtract: 25 - 25 = 0.

Example 2

Find 4,944÷84,944÷8 size 12{4,"944" div 8} {}.

8 4944 8 4944
Rewrite the problem using a division bracket.

600 8 4944 - 4800 ̲ 144 600 8 4944 - 4800 ̲ 144
8 goes into 49 at most 6 times, and 9 is in the hundreds column. We'll guess 600.
Then, 8×600=48008×600=4800 size 12{8 times "600"="4800"} {}.

10 600 8 4944 - 4800 ̲ 144 -   80 ̲ 64 10 600 8 4944 - 4800 ̲ 144 -   80 ̲ 64
8 goes into 14 at most 1 time, and 4 is in the tens column. We'll guess 10.

8 10 600 8 4944 - 4800 ̲ 144 -   80 ̲ 64 - 64 ̲ 0 8 10 600 8 4944 - 4800 ̲ 144 -   80 ̲ 64 - 64 ̲ 0

8 goes into 64 exactly 8 times.
600 eights + 10 eights + 8 eights = 618 eights.

Check:

Thus, 4,944÷8=6184,944÷8=618 size 12{4,"944" div 8="618"} {}.

As in the first problem, the notation in this division can be shortened by eliminating the subtraction signs and the zeros in each educated guess.

Divide: 8 goes into 49 at most 6 times. Multiply: 6 × 8 = 48. Write 48 below 49. Subtract: 49 - 48 = 1. Bring down the 4. Divide: 8 goes into 49 at most 6 times. Multiply: 6 × 8 = 48. Write 48 below 49. Subtract: 49 - 48 = 1. Bring down the 4. Divide: 8 goes into 14 at most 1 time. Multiply: 1 × 8 = 8. Write 8 below 14. Subtract: 14 - 8 = 6. Bring down the 4. Divide: 8 goes into 14 at most 1 time. Multiply: 1 × 8 = 8. Write 8 below 14. Subtract: 14 - 8 = 6. Bring down the 4. Divide: 8 goes into 64 exactly 8 times. Multiply: 8 × 8 = 64. Write 64 below 64. Subtract: 64 - 64 = 0. Divide: 8 goes into 64 exactly 8 times. Multiply: 8 × 8 = 64. Write 64 below 64. Subtract: 64 - 64 = 0.

Note:

Not all divisions end in zero. We will examine such divisions in a subsequent subsection.

Practice Set A

Perform the following divisions.

Exercise 1

126÷7126÷7 size 12{"126" div 7} {}

18

Exercise 2

324÷4324÷4 size 12{"324" div 4} {}

81

Exercise 3

2,559÷32,559÷3 size 12{2,"559" div 3} {}

853

Exercise 4

5,645÷55,645÷5 size 12{5,"645" div 5} {}

1,129

Exercise 5

757,125÷9757,125÷9 size 12{"757","125" div 9} {}

84,125

Division with a Multiple Digit Divisor

The process of division also works when the divisor consists of two or more digits. We now make educated guesses using the first digit of the divisor and one or two digits of the dividend.

Sample Set B

Example 3

Find 2,232÷362,232÷36 size 12{2,"232" div "36"} {}.

36 2232 36 2232

Use the first digit of the divisor and the first two digits of the dividend to make the educated guess.

3 goes into 22 at most 7 times.

Try 7: 7×36=2527×36=252 size 12{7 times "36"="252"} {} which is greater than 223. Reduce the estimate.

Try 6: 6×36=2166×36=216 size 12{6 times "36"="216"} {} which is less than 223.

Multiply: 6 × 36 = 216. Write 216 below 223. Subtract: 223 - 216 = 7. Bring down the 2. Multiply: 6 × 36 = 216. Write 216 below 223. Subtract: 223 - 216 = 7. Bring down the 2.

Divide 3 into 7 to estimate the number of times 36 goes into 72. The 3 goes into 7 at most 2 times.

Try 2: 2×36=722×36=72 size 12{2 times "36"="72"} {}.

Check:

Thus, 2,232÷36=622,232÷36=62 size 12{2,"232" div "36"="62"} {}.

Example 4

Find 2,417,228÷8022,417,228÷802 size 12{2,"417","228" div "802"} {}.

802 2417228 802 2417228

First, the educated guess: 24÷8=324÷8=3 size 12{"24" div 8=3} {}. Then 3×802=24063×802=2406 size 12{3 times "802"="2406"} {}, which is less than 2417. Use 3 as the guess. Since 3×802=24063×802=2406 size 12{3 times "802"="2406"} {}, and 2406 has four digits, place the 3 above the fourth digit of the dividend.

Subtract: 2417 - 2406 = 11.
Bring down the 2.

The divisor 802 goes into 112 at most 0 times. Use 0.

Multiply: 0 × 802 = 0. Subtract: 112 - 0 = 112. Bring down the 2. Multiply: 0 × 802 = 0. Subtract: 112 - 0 = 112. Bring down the 2.

The 8 goes into 11 at most 1 time, and 1×802=8021×802=802 size 12{1 times "802"="802"} {}, which is less than 1122. Try 1.

Subtract 1122-802 = 320 1122-802=320
Bring down the 8.

8 goes into 32 at most 4 times.

4×802=32084×802=3208 size 12{4 times "802"="3208"} {}.

Use 4.

Check:

Thus, 2,417,228÷802=3,0142,417,228÷802=3,014 size 12{2,"417","228" div "802"=3,"014"} {}.

Practice Set B

Perform the following divisions.

Exercise 6

1,376÷321,376÷32 size 12{1,"376" div "32"} {}

43

Exercise 7

6,160÷556,160÷55 size 12{6,"160" div "55"} {}

112

Exercise 8

18,605÷6118,605÷61 size 12{"18","605" div "61"} {}

305

Exercise 9

144,768÷48144,768÷48 size 12{"144","764" div "48"} {}

3,016

Division with a Remainder

We might wonder how many times 4 is contained in 10. Repeated subtraction yields

10 -   4 ̲ 6 - 4 ̲ 2 10 -   4 ̲ 6 - 4 ̲ 2

Since the remainder is less than 4, we stop the subtraction. Thus, 4 goes into 10 two times with 2 remaining. We can write this as a division as follows.

2 4 10 -   8 ̲ 2 2 4 10 -   8 ̲ 2
Divide: 4 goes into 10 at most 2 times. Multiply: 2 × 4 = 8. Write 8 below 0. Subtract: 10 - 8 = 2. Divide: 4 goes into 10 at most 2 times. Multiply: 2 × 4 = 8. Write 8 below 0. Subtract: 10 - 8 = 2.

Since 4 does not divide into 2 (the remainder is less than the divisor) and there are no digits to bring down to continue the process, we are done. We write

2R2 4    10 - 8 ̲ 2 2R2 4    10 - 8 ̲ 2 or 10÷4=2R22 with remainder 210÷4=2R22 with remainder 2

Sample Set C

Example 5

Find 85÷385÷3 size 12{"85" div 3} {}.

Divide: 3 goes into 8 at most 2 times. Multiply: 2 × 3 = 6. Write 6 below 8. Subtract: 8 - 6 = 2. Bring down the 5. Divide: 3 goes into 8 at most 2 times. Multiply: 2 × 3 = 6. Write 6 below 8. Subtract: 8 - 6 = 2. Bring down the 5. Divide: 3 goes into 25 at most 8 times. Multiply: 3 × 8 = 24. Write 24 below 25. Subtract: 25 - 24 = 1. Divide: 3 goes into 25 at most 8 times. Multiply: 3 × 8 = 24. Write 24 below 25. Subtract: 25 - 24 = 1.

There are no more digits to bring down to continue the process. We are done. One is the remainder.

Check: Multiply 28 and 3, then add 1.

28 ×   3 ̲ 84 +   1 ̲ 85 28 ×   3 ̲ 84 +   1 ̲ 85

Thus, 85÷3=28R185÷3=28R1 size 12{"85" div 3="28"R1} {}.

Example 6

Find 726÷23726÷23 size 12{"726" div "23"} {}.

Check: Multiply 31 by 23, then add 13.

Thus, 726÷23=31R13726÷23=31R13 size 12{"726" div "23"="31"`R"13"} {}.

Practice Set C

Perform the following divisions.

Exercise 10

75÷475÷4 size 12{"75" div 4} {}

18 R3

Exercise 11

346÷8346÷8 size 12{"346" div 8} {}

43 R2

Exercise 12

489÷21489÷21 size 12{"489" div "21"} {}

23 R6

Exercise 13

5,016÷825,016÷82 size 12{5,"016" div "82"} {}

61 R14

Exercise 14

41,196÷6741,196÷67 size 12{"41","196" div "67"} {}

614 R58

Calculators

The calculator can be useful for finding quotients with single and multiple digit divisors. If, however, the division should result in a remainder, the calculator is unable to provide us with the particular value of the remainder. Also, some calculators (most nonscientific) are unable to perform divisions in which one of the numbers has more than eight digits.

Sample Set D

Use a calculator to perform each division.

Example 7

328 ÷ 8 328 ÷ 8 size 12{"328" div 8} {}

 Type 328 Press ÷ Type 8 Press =

Example 8

53 , 136 ÷ 82 53 , 136 ÷ 82 size 12{"53","136" div "82"} {}

 Type 53136 Press ÷ Type 82 Press =

Example 9

730 , 019 , 001 ÷ 326 730 , 019 , 001 ÷ 326 size 12{"730","019","001" div "326"} {}

We first try to enter 730,019,001 but find that we can only enter 73001900. If our calculator has only an eight-digit display (as most nonscientific calculators do), we will be unable to use the calculator to perform this division.

Example 10

3727 ÷ 49 3727 ÷ 49 size 12{"3727" div "49"} {}

 Type 3727 Press ÷ Type 49 Press =

This number is an example of a decimal number (see (Reference)). When a decimal number results in a calculator division, we can conclude that the division produces a remainder.

Practice Set D

Use a calculator to perform each division.

Exercise 15

3,330÷743,330÷74 size 12{3,"330" div "74"} {}

45

Exercise 16

63,365÷11563,365÷115 size 12{"63","365" div "115"} {}

551

Exercise 17

21,996,385,287÷5321,996,385,287÷53 size 12{"21","996","385","287" div "53"} {}

Solution

Since the dividend has more than eight digits, this division cannot be performed on most nonscientific calculators. On others, the answer is 415,026,137.4

Exercise 18

4,558÷674,558÷67 size 12{4,"558" div "57"} {}

Solution

This division results in 68.02985075, a decimal number, and therefore, we cannot, at this time, find the value of the remainder. Later, we will discuss decimal numbers.

Exercises

For the following problems, perform the divisions.

The first 38 problems can be checked with a calculator by multiplying the divisor and quotient then adding the remainder.

Exercise 19

52÷452÷4 size 12{"52" div 4} {}

13

Exercise 20

776÷8776÷8 size 12{"776" div 8} {}

Exercise 21

603÷9603÷9 size 12{"603" div 9} {}

67

Exercise 22

240÷8240÷8 size 12{"240" div 8} {}

Exercise 23

208÷4208÷4 size 12{"208" div 4} {}

52

Exercise 24

576÷6576÷6 size 12{"576" div 6} {}

Exercise 25

21÷721÷7 size 12{"21" div 7} {}

3

Exercise 26

0÷00÷0 size 12{0 div 0} {}

Exercise 27

140÷2140÷2 size 12{"140" div 2} {}

70

Exercise 28

528÷8528÷8 size 12{"528" div 8} {}

Exercise 29

244÷4244÷4 size 12{"244" div 4} {}

61

Exercise 30

0÷70÷7 size 12{0 div 7} {}

Exercise 31

177÷3177÷3 size 12{"177" div 3} {}

59

Exercise 32

96÷896÷8 size 12{"96" div 8} {}

Exercise 33

67÷167÷1 size 12{"67" div 1} {}

67

Exercise 34

896÷56896÷56 size 12{"896" div "56"} {}

Exercise 35

1,044÷121,044÷12 size 12{1,"044" div "12"} {}

87

Exercise 36

988÷19988÷19 size 12{"988" div "19"} {}

Exercise 37

5,238÷975,238÷97 size 12{5,"238" div "97"} {}

54

Exercise 38

2,530÷552,530÷55 size 12{2,"530" div "55"} {}

Exercise 39

4,264÷824,264÷82 size 12{4,"264" div "82"} {}

52

Exercise 40

637÷13637÷13 size 12{"637" div "13"} {}

Exercise 41

3,420÷903,420÷90 size 12{3,"420" div "90"} {}

38

Exercise 42

5,655÷875,655÷87 size 12{5,"655" div "87"} {}

Exercise 43

2,115÷472,115÷47 size 12{2,"115" div "47"} {}

45

Exercise 44

9,328÷229,328÷22 size 12{9,"328" div "22"} {}

Exercise 45

55,167÷7155,167÷71 size 12{"55","167" div "71"} {}

777

Exercise 46

68,356÷9268,356÷92 size 12{"68","356" div "92"} {}

Exercise 47

27,702÷8127,702÷81 size 12{"27","702" div "81"} {}

342

Exercise 48

6,510÷316,510÷31 size 12{6,"510" div "31"} {}

Exercise 49

60,536÷9460,536÷94 size 12{"60","536" div "94"} {}

644

Exercise 50

31,844÷3831,844÷38 size 12{"31","844" div "38"} {}

Exercise 51

23,985÷4523,985÷45 size 12{"23","985" div "45"} {}

533

Exercise 52

60,606÷7460,606÷74 size 12{"60","606" div "74"} {}

Exercise 53

2,975,400÷2852,975,400÷285 size 12{2,"975","400" div "285"} {}

10,440

Exercise 54

1,389,660÷7951,389,660÷795 size 12{1,"389","660" div "795"} {}

Exercise 55

7,162,060÷8797,162,060÷879 size 12{7,"162","060" div "879"} {}

Solution

8,147 remainder 847

Exercise 56

7,561,060÷9097,561,060÷909 size 12{7,"561","060" div "909"} {}

Exercise 57

38÷938÷9 size 12{"38" div 9} {}

4 remainder 2

Exercise 58

97÷497÷4 size 12{"97" div 4} {}

Exercise 59

199÷3199÷3 size 12{"199" div 3} {}

66 remainder 1

Exercise 60

573÷6573÷6 size 12{"573" div 6} {}

Exercise 61

10,701÷1310,701÷13 size 12{"10","701" div "13"} {}

823 remainder 2

Exercise 62

13,521÷5313,521÷53 size 12{"13","521" div "53"} {}

Exercise 63

3,628÷903,628÷90 size 12{3,"628" div "90"} {}

40 remainder 28

Exercise 64

10,592÷4310,592÷43 size 12{"10","592" div "43"} {}

Exercise 65

19,965÷3019,965÷30 size 12{"19","965" div "30"} {}

665 remainder 15

Exercise 66

8,320÷218,320÷21 size 12{8,"320" div "21"} {}

Exercise 67

61,282÷6461,282÷64 size 12{"61","282" div "64"} {}

957 remainder 34

Exercise 68

1,030÷281,030÷28 size 12{1,"030" div "28"} {}

Exercise 69

7,319÷117,319÷11 size 12{7,"319" div "11"} {}

665 remainder 4

Exercise 70

3,628÷903,628÷90 size 12{3,"628" div "90"} {}

Exercise 71

35,279÷7735,279÷77 size 12{"35","279" div "77"} {}

458 remainder 13

Exercise 72

52,196÷5552,196÷55 size 12{"52","196" div "55"} {}

Exercise 73

67,751÷6867,751÷68 size 12{"67","751" div "68"} {}

Solution

996 remainder 23

For the following 5 problems, use a calculator to find the quo­tients.

Exercise 74

4,346÷534,346÷53 size 12{4,"346" div "53"} {}

Exercise 75

3,234÷773,234÷77 size 12{3,"234" div "77"} {}

42

Exercise 76

6,771÷376,771÷37 size 12{6,"771" div "37"} {}

Exercise 77

4,272,320÷5204,272,320÷520 size 12{4,"272","320" div "520"} {}

8,216

Exercise 78

7,558,110÷6517,558,110÷651 size 12{7,"558","110" div "651"} {}

Exercise 83

A certain brand of refrigerator has an automatic ice cube maker that makes 336 ice cubes in one day. If the ice machine makes ice cubes at a constant rate, how many ice cubes does it make each hour?

Solution

14 cubes per hour

Exercise 84

A beer manufacturer bottles 52,380 ounces of beer each hour. If each bottle contains the same number of ounces of beer, and the manufacturer fills 4,365 bottles per hour, how many ounces of beer does each bottle contain?

Exercise 85

A computer program consists of 68,112 bits. 68,112 bits equals 8,514 bytes. How many bits in one byte?

Solution

8 bits in each byte

Exercise 86

A 26-story building in San Francisco has a total of 416 offices. If each floor has the same number of offices, how many floors does this building have?

Exercise 87

A college has 67 classrooms and a total of 2,546 desks. How many desks are in each classroom if each classroom has the same number of desks?

38

Exercises for Review

Exercise 88

((Reference)) What is the value of 4 in the number 124,621?

Exercise 89

((Reference)) Round 604,092 to the nearest hundred thousand.

600,000

Exercise 90

((Reference)) What whole number is the additive identity?

Exercise 91

((Reference)) Find the product. 6,256×1006,256×100 size 12{6,"256" times "100"} {}.

625,600

Exercise 92

((Reference)) Find the quotient. 0÷110÷11 size 12{0 div "11"} {}.

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