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# 6.6 Decimals: Division of Decimals

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

Note: You are viewing an old version of this document. The latest version is available here.

## Section Overview

• The Logic Behind the Method
• A Method of Dividing a Decimal By a Nonzero Whole Number
• A Method of Dividing a Decimal by a Nonzero Decimal
• Dividing Decimals by Powers of 10

## The Logic Behind the Method

As we have done with addition, subtraction, and multiplication of decimals, we will study a method of division of decimals by converting them to fractions, then we will make a general rule.

We will proceed by using this example: Divide 196.8 by 6.

32    6 196.8 18     ̲ 16   12   ̲ 4   32    6 196.8 18     ̲ 16   12   ̲ 4

We have, up to this point, divided 196.8 by 6 and have gotten a quotient of 32 with a remainder of 4. If we follow our intuition and bring down the .8, we have the division 4.8÷64.8÷6 size 12{4 "." 8 div 6} {}.

4 . 8 ÷ 6 = 4 8 10 ÷ 6 = 48 10 ÷ 6 1 = 48 8 10 1 6 1 = 8 10 4 . 8 ÷ 6 = 4 8 10 ÷ 6 = 48 10 ÷ 6 1 = 48 8 10 1 6 1 = 8 10

Thus, 4.8÷6=.84.8÷6=.8 size 12{4 "." 8 div 6= "." 8} {}.

Now, our intuition and experience with division direct us to place the .8 immedi­ately to the right of 32.

From these observations, we suggest the following method of division.

## A Method of Dividing a Decimal by a Nonzero Whole Number

### Method of Dividing a Decimal by a Nonzero Whole Number

To divide a decimal by a nonzero whole number:

1. Write a decimal point above the division line and directly over the decimal point of the dividend.
2. Proceed to divide as if both numbers were whole numbers.
3. If, in the quotient, the first nonzero digit occurs to the right of the decimal point, but not in the tenths position, place a zero in each position between the decimal point and the first nonzero digit of the quotient.

### Sample Set A

Find the decimal representations of the following quotients.

#### Example 1

114.1÷7=7114.1÷7=7

16.3 7 114.1 7     ̲ 44    42    ̲ 2.1 2.1 ̲ 0 16.3 7 114.1 7     ̲ 44    42    ̲ 2.1 2.1 ̲ 0

Thus, 114.1÷7=16.3114.1÷7=16.3 size 12{"114" "." 1 div 7="16" "." 3} {}.

Check: If 114.1÷7=16.3114.1÷7=16.3 size 12{"114" "." 1 div 7="16" "." 3} {}, then 716.3716.3 size 12{7 cdot "16" "." 3} {} should equal 114.1.

16.3 4 2           7 ̲ 114.1 True. 16.3 4 2           7 ̲ 114.1 True.

#### Example 2

0.02068÷40.02068÷4 size 12{0 "." "02068" div 4} {}

Place zeros in the tenths and hundredths positions. (See Step 3.)

Thus, 0.02068÷4=0.005170.02068÷4=0.00517 size 12{0 "." "02068" div 4=0 "." "00517"} {}.

### Practice Set A

Find the following quotients.

#### Exercise 1

184.5÷3184.5÷3 size 12{"184" "." 5 div 3} {}

61.5

#### Exercise 2

16.956÷916.956÷9 size 12{"16" "." "956" div 9} {}

1.884

#### Exercise 3

0.2964÷40.2964÷4 size 12{0 "." "2964" div 4} {}

0.0741

#### Exercise 4

0.000496÷80.000496÷8 size 12{0 "." "000496" div 8} {}

0.000062

## A Method of Dividing a Decimal By a Nonzero Decimal

Now that we can divide decimals by nonzero whole numbers, we are in a position to divide decimals by a nonzero decimal. We will do so by converting a division by a decimal into a division by a whole number, a process with which we are already familiar. We'll illustrate the method using this example: Divide 4.32 by 1.8.

Let's look at this problem as 432100÷1810432100÷1810 size 12{4 { {"32"} over {"100"} } div 1 { {8} over {"10"} } } {}.

4 32 100 ÷ 1 8 10 = 4 32 100 1 8 10 = 432 100 18 10 4 32 100 ÷ 1 8 10 = 4 32 100 1 8 10 = 432 100 18 10

The divisor is 18101810 size 12{ { {"18"} over {"10"} } } {}. We can convert 18101810 size 12{ { {"18"} over {"10"} } } {} into a whole number if we multiply it by 10.

18 10 10 = 18 10 1 10 1 1 = 18 18 10 10 = 18 10 1 10 1 1 = 18 size 12{ { {"18"} over {"10"} } cdot "10"= { {"18"} over { { { {1}} { {0}}} cSub { size 8{1} } } } cdot { { { { {1}} { {0}}} cSup { size 8{1} } } over {1} } ="18"} {}

But, we know from our experience with fractions, that if we multiply the denomina­tor of a fraction by a nonzero whole number, we must multiply the numerator by that same nonzero whole number. Thus, when converting 18101810 size 12{ { {"18"} over {"10"} } } {} to a whole number by multiplying it by 10, we must also multiply the numerator 432100432100 size 12{ { {"432"} over {"100"} } } {} by 10.

432 100 10 = 432 100 10 10 1 1 = 432 1 10 1 = 432 10 = 43 2 10 = 43.2 432 100 10 = 432 100 10 10 1 1 = 432 1 10 1 = 432 10 = 43 2 10 = 43.2

We have converted the division 4.32÷1.84.32÷1.8 size 12{4 "." "32" div 1 "." 8} {} into the division 43.2÷1843.2÷18 size 12{"43" "." 2 div "18"} {}, that is,

1.8 4.32 18 43.2 1.84.321843.2

Notice what has occurred.

If we "move" the decimal point of the divisor one digit to the right, we must also "move" the decimal point of the dividend one place to the right. The word "move" actually indicates the process of multiplication by a power of 10.

### Method of Dividing a Decimal by a Decimal Number

To divide a decimal by a nonzero decimal,

1. Convert the divisor to a whole number by moving the decimal point to the position immediately to the right of the divisor's last digit.
2. Move the decimal point of the dividend to the right the same number of digits it was moved in the divisor.
3. Set the decimal point in the quotient by placing a decimal point directly above the newly located decimal point in the dividend.
4. Divide as usual.

### Sample Set B

Find the following quotients.

#### Example 3

32.66÷7.132.66÷7.1 size 12{"32" "." "66" div 7 "." 1} {}

7.1 32.667.132.66

• The divisor has one decimal place.
• Move the decimal point of both the divisor and the dividend 1 place to the right.
• Set the decimal point.
• Divide as usual.

Thus, 32.66÷7.1=4.632.66÷7.1=4.6 size 12{"32" "." "66" div 7 "." 1=4 "." 6} {}.

Check: 32.66÷7.1=4.632.66÷7.1=4.6 size 12{"32" "." "66" div 7 "." 1=4 "." 6} {} if 4.6×7.1=32.664.6×7.1=32.66 size 12{4 "." 6 times 7 "." 1="32" "." "66"} {}

4.6 × 7.1 ̲ 46 322   ̲ 32.66 True. 4.6 × 7.1 ̲ 46 322   ̲ 32.66 True.

#### Example 4

1.0773÷0.5131.0773÷0.513 size 12{1 "." "0773" div 0 "." "513"} {}

• The divisor has 3 decimal places.
• Move the decimal point of both the divisor and the dividend 3 places to the right.
• Set the decimal place and divide.

Thus, 1.0773÷0.513=2.11.0773÷0.513=2.1 size 12{1 "." "0773" div 0 "." "513"=2 "." 1} {}.

Checking by multiplying 2.1 and 0.513 will convince us that we have obtained the correct result. (Try it.)

#### Example 5

12÷0.0003212÷0.00032 size 12{"12" div 0 "." "00032"} {}

0.00032 12.000000.0003212.00000

• The divisor has 5 decimal places.
• Move the decimal point of both the divisor and the dividend 5 places to the right. We will need to add 5 zeros to 12.
• Set the decimal place and divide.

This is now the same as the division of whole numbers.

37500. 32 1200000. 96         ̲ 240     224     ̲ 160    160    ̲ 000 37500. 32 1200000. 96         ̲ 240     224     ̲ 160    160    ̲ 000

Checking assures us that 12÷0.00032=37,50012÷0.00032=37,500 size 12{"12" div 0 "." "00032"="37","500"} {}.

### Practice Set B

Find the decimal representation of each quotient.

#### Exercise 5

9.176÷3.19.176÷3.1 size 12{9 "." "176" div 3 "." 1} {}

2.96

#### Exercise 6

5.0838÷1.115.0838÷1.11 size 12{5 "." "0838" div 1 "." "11"} {}

4.58

#### Exercise 7

16÷0.000416÷0.0004 size 12{"16" div 0 "." "0004"} {}

40,000

#### Exercise 8

8,162.41÷108,162.41÷10 size 12{8,"162" "." "41" div "10"} {}

816.241

#### Exercise 9

8,162.41÷1008,162.41÷100 size 12{8,"162" "." "41" div "100"} {}

81.6241

#### Exercise 10

8,162.41÷1,0008,162.41÷1,000 size 12{8,"162" "." "41" div 1,"000"} {}

8.16241

#### Exercise 11

8,162.41÷10,0008,162.41÷10,000 size 12{8,"162" "." "41" div "10","000"} {}

0.816241

## Calculators

Calculators can be useful for finding quotients of decimal numbers. As we have seen with the other calculator operations, we can sometimes expect only approximate results. We are alerted to approximate results when the calculator display is filled with digits. We know it is possible that the operation may produce more digits than the calculator has the ability to show. For example, the multiplication

0.12345 5 decimal places × 0.4567 4 decimal places 0.12345 5 decimal places × 0.4567 4 decimal places

produces 5+4=95+4=9 size 12{5+4=9} {} decimal places. An eight-digit display calculator only has the ability to show eight digits, and an approximation results. The way to recognize a possible approximation is illustrated in problem 3 of the next sample set.

### Sample Set C

Find each quotient using a calculator. If the result is an approximation, round to five decimal places.

#### Example 6

12.596÷4.712.596÷4.7 size 12{"12" "." "596" div 4 "." 7} {}

 Display Reads Type 12.596 12.596 Press ÷ 12.596 Type 4.7 4.7 Press = 2.68

Since the display is not filled, we expect this to be an accurate result.

#### Example 7

0.5696376÷0.001230.5696376÷0.00123 size 12{0 "." "5696376" div 0 "." "00123"} {}

 Display Reads Type .5696376 0.5696376 Press ÷ 0.5696376 Type .00123 0.00123 Press = 463.12

Since the display is not filled, we expect this result to be accurate.

#### Example 8

0.8215199÷4.1130.8215199÷4.113 size 12{0 "." "8215199" div 4 "." "113"} {}

 Display Reads Type .8215199 0.8215199 Press ÷ 0.8215199 Type 4.113 4.113 Press = 0.1997373

There are EIGHT DIGITS — DISPLAY FILLED! BE AWARE OF POSSIBLE APPROXI­MATIONS.

We can check for a possible approximation in the following way. Since the division 4 1234 123 can be checked by multiplying 4 and 3, we can check our division by performing the multiplication

4.113 3 decimal places × 0.1997373 7 decimal places 4.113 3 decimal places × 0.1997373 7 decimal places

This multiplication produces 3+7=103+7=10 size 12{3+7="10"} {} decimal digits. But our suspected quotient contains only 8 decimal digits. We conclude that the answer is an approximation. Then, rounding to five decimal places, we get 0.19974.

### Practice Set C

Find each quotient using a calculator. If the result is an approximation, round to four decimal places.

#### Exercise 12

42.49778÷14.26142.49778÷14.261 size 12{"42" "." "49778" div "14" "." "261"} {}

2.98

#### Exercise 13

0.001455÷0.2910.001455÷0.291 size 12{0 "." "001455" div 0 "." "291"} {}

0.005

#### Exercise 14

7.459085÷2.11927.459085÷2.1192 size 12{7 "." "459085" div 2 "." "1192"} {}

##### Solution

3.5197645 is an approximate result. Rounding to four decimal places, we get 3.5198

## Dividing Decimals By Powers of 10

In problems 4 and 5 of Practice Set B, we found the decimal representations of 8,162.41÷108,162.41÷10 size 12{8,"162" "." "41" div "10"} {} and 8,162.41÷1008,162.41÷100 size 12{8,"162" "." "41" div "100"} {}. Let's look at each of these again and then, from these observations, make a general statement regarding division of a decimal num­ber by a power of 10.

816.241 10 8162.410 80          ̲ 16        10        ̲ 62       60       ̲ 24     20     ̲ 41   40   ̲ 10  10  ̲ 816.241 10 8162.410 80          ̲ 16        10        ̲ 62       60       ̲ 24     20     ̲ 41   40   ̲ 10  10  ̲

Thus, 8,162.41÷10=816.2418,162.41÷10=816.241 size 12{8,"162" "." "41" div "10"="816" "." "241"} {}.

Notice that the divisor 10 is composed of one 0 and that the quotient 816.241 can be obtained from the dividend 8,162.41 by moving the decimal point one place to the left.

81.6241 100 8162.4100 800          ̲ 162         100         ̲ 62 4      60 0      ̲ 2 41    2 00    ̲ 410  400  ̲ 100 100 ̲ 0 81.6241 100 8162.4100 800          ̲ 162         100         ̲ 62 4      60 0      ̲ 2 41    2 00    ̲ 410  400  ̲ 100 100 ̲ 0

Thus, 8,162.41÷100=81.62418,162.41÷100=81.6241 size 12{8,"162" "." "41" div "100"="81" "." "6241"} {}.

Notice that the divisor 100 is composed of two 0's and that the quotient 81.6241 can be obtained from the dividend by moving the decimal point two places to the left.

Using these observations, we can suggest the following method for dividing decimal numbers by powers of 10.

### Dividing a Decimal Fraction by a Power of 10

To divide a decimal fraction by a power of 10, move the decimal point of the decimal fraction to the left as many places as there are zeros in the power of 10. Add zeros if necessary.

### Sample Set D

Find each quotient.

#### Example 9

9,248.6÷1009,248.6÷100 size 12{9,"248" "." 6 div "100"} {}
Since there are 2 zeros in this power of 10, we move the decimal point 2 places to the left.

#### Example 10

3.28÷10,0003.28÷10,000 size 12{3 "." "28" div "10","000"} {}

Since there are 4 zeros in this power of 10, we move the decimal point 4 places to the left. To do so, we need to add three zeros.

### Practice Set D

Find the decimal representation of each quotient.

#### Exercise 15

182.5÷10182.5÷10 size 12{"182" "." 5 div "10"} {}

18.25

#### Exercise 16

182.5÷100182.5÷100 size 12{"182" "." 5 div "100"} {}

1.825

#### Exercise 17

182.5÷1,000182.5÷1,000 size 12{"182" "." 5 div 1,"000"} {}

0.1825

#### Exercise 18

182.5÷10,000182.5÷10,000 size 12{"182" "." 5 div "10","000"} {}

0.01825

#### Exercise 19

646.18÷100646.18÷100 size 12{"646" "." "18" div "100"} {}

6.4618

#### Exercise 20

21.926÷1,00021.926÷1,000 size 12{"21" "." "926" div 1,"000"} {}

0.021926

## Exercises

For the following 30 problems, find the decimal representation of each quotient. Use a calculator to check each result.

### Exercise 21

4.8÷34.8÷3 size 12{4 "." 8÷3} {}

1.6

### Exercise 22

16.8÷816.8÷8 size 12{"16" "." 8÷8} {}

### Exercise 23

18.5÷518.5÷5 size 12{"18" "." 5÷5} {}

3.7

### Exercise 24

12.33÷312.33÷3 size 12{"12" "." "33"÷3} {}

### Exercise 25

54.36÷954.36÷9 size 12{"54" "." "36"÷9} {}

6.04

### Exercise 26

73.56÷1273.56÷12 size 12{"73" "." "56"÷"12"} {}

### Exercise 27

159.46÷17159.46÷17 size 12{"159" "." "46"÷"17"} {}

9.38

### Exercise 28

12.16÷6412.16÷64 size 12{"12" "." "16"÷"64"} {}

### Exercise 29

37.26÷8137.26÷81 size 12{"37" "." "26"÷"81"} {}

0.46

### Exercise 30

439.35÷435439.35÷435 size 12{"439" "." "35"÷"435"} {}

### Exercise 31

36.98÷4.336.98÷4.3 size 12{"36" "." "98"÷4 "." 3} {}

8.6

### Exercise 32

46.41÷9.146.41÷9.1 size 12{"46" "." "41"÷9 "." 1} {}

### Exercise 33

3.6÷1.53.6÷1.5 size 12{3 "." 6÷1 "." 5} {}

2.4

### Exercise 34

0.68÷1.70.68÷1.7 size 12{0 "." "68"÷1 "." 7} {}

### Exercise 35

50.301÷8.150.301÷8.1 size 12{"50" "." "301"÷8 "." 1} {}

6.21

### Exercise 36

2.832÷0.42.832÷0.4 size 12{2 "." "832"÷0 "." 4} {}

### Exercise 37

4.7524÷2.184.7524÷2.18 size 12{4 "." "7524"÷2 "." "18"} {}

2.18

### Exercise 38

16.2409÷4.0316.2409÷4.03 size 12{"16" "." "2409"÷4 "." "03"} {}

### Exercise 39

1.002001÷1.0011.002001÷1.001 size 12{1 "." "002001"÷1 "." "001"} {}

1.001

### Exercise 40

25.050025÷5.00525.050025÷5.005 size 12{"25" "." "050025"÷5 "." "005"} {}

### Exercise 41

12.4÷3.112.4÷3.1 size 12{"12" "." 4÷3 "." 1} {}

4

### Exercise 42

0.48÷0.080.48÷0.08 size 12{0 "." "48"÷0 "." "08"} {}

### Exercise 43

30.24÷2.1630.24÷2.16 size 12{"30" "." "24"÷2 "." "16"} {}

14

### Exercise 44

48.87÷0.8748.87÷0.87 size 12{"48" "." "87"÷0 "." "87"} {}

### Exercise 45

12.321÷0.11112.321÷0.111 size 12{"12" "." "321"÷0 "." "111"} {}

111

### Exercise 46

64,351.006÷1064,351.006÷10 size 12{"64","351" "." "006"÷"10"} {}

### Exercise 47

64,351.006÷10064,351.006÷100 size 12{"64","351" "." "006"÷"100"} {}

643.51006

### Exercise 48

64,351.006÷1,00064,351.006÷1,000 size 12{"64","351" "." "006"÷1,"000"} {}

### Exercise 49

64,351.006÷1,000,00064,351.006÷1,000,000 size 12{"64","351" "." "006"÷1,"000","000"} {}

0.064351006

### Exercise 50

0.43÷1000.43÷100 size 12{0 "." "43"÷"100"} {}

For the following 5 problems, find each quotient. Round to the specified position. A calculator may be used.

### Exercise 51

11.2944÷6.2411.2944÷6.24 size 12{"11" "." "2944"÷6 "." "24"} {}

 Actual Quotient Tenths Hundredths Thousandths

#### Solution

 Actual Quotient Tenths Hundredths Thousandths 1.81 1.8 1.81 1.810

### Exercise 52

45.32931÷9.0145.32931÷9.01 size 12{"45" "." "32931"÷9 "." "01"} {}

 Actual Quotient Tenths Hundredths Thousandths

### Exercise 53

3.18186÷0.663.18186÷0.66 size 12{3 "." "18186"÷0 "." "66"} {}

 Actual Quotient Tenths Hundredths Thousandths

#### Solution

 Actual Quotient Tenths Hundredths Thousandths 4.821 4.8 4.82 4.821

### Exercise 54

4.3636÷44.3636÷4 size 12{4 "." "3636"÷4} {}

 Actual Quotient Tenths Hundredths Thousandths

### Exercise 55

0.00006318÷0.0180.00006318÷0.018 size 12{0 "." "00006318"÷0 "." "018"} {}

 Actual Quotient Tenths Hundredths Thousandths

#### Solution

 Actual Quotient Tenths Hundredths Thousandths 0.00351 0.0 0.00 0.004

For the following 9 problems, find each solution.

### Exercise 56

Divide the product of 7.4 and 4.1 by 2.6.

### Exercise 57

Divide the product of 11.01 and 0.003 by 2.56 and round to two decimal places.

0.01

### Exercise 58

Divide the difference of the products of 2.1 and 9.3, and 4.6 and 0.8 by 0.07 and round to one decimal place.

### Exercise 59

A ring costing $567.08 is to be paid off in equal monthly payments of$46.84. In how many months will the ring be paid off?

12.11 months

### Exercise 63

A woman notices that on slow speed her video cassette recorder runs through 296.80 tape units in 10 minutes and at fast speed through 1098.16 tape units. How many times faster is fast speed than slow speed?

3.7

### Exercise 64

A class of 34 first semester business law students pay a total of \$1,354.90, disregarding sales tax, for their law textbooks. What is the cost of each book?

### Calculator Problems

For the following problems, use calculator to find the quotients. If the result is approximate (see Sample Set C Example 8) round the result to three decimal places.

### Exercise 65

3.8994÷2.013.8994÷2.01 size 12{3 "." "8994"÷2 "." "01"} {}

1.94

### Exercise 66

0.067444÷0.0520.067444÷0.052 size 12{0 "." "067444"÷0 "." "052"} {}

### Exercise 67

14,115.628÷484.7414,115.628÷484.74 size 12{"14","115" "." "628"÷"484" "." "74"} {}

29.120

### Exercise 68

219,709.36÷9941.6219,709.36÷9941.6 size 12{"219","709" "." "36"÷"9941" "." 6} {}

### Exercise 69

0.0852092÷0.492710.0852092÷0.49271 size 12{0 "." "0852092"÷0 "." "49271"} {}

0.173

### Exercise 70

2.4858225÷1.116112.4858225÷1.11611 size 12{2 "." "4858225"÷1 "." "11611"} {}

### Exercise 71

0.123432÷0.11110.123432÷0.1111 size 12{0 "." "123432"÷0 "." "1111"} {}

1.111

### Exercise 72

2.102838÷1.03052.102838÷1.0305 size 12{2 "." "102838"÷1 "." "0305"} {}

### Exercises for Review

#### Exercise 73

((Reference)) Convert 478478 size 12{4 { {7} over {8} } } {} to an improper fraction.

398398

#### Exercise 74

((Reference)) 2727 size 12{ { {2} over {7} } } {} of what number is 4545 size 12{ { {4} over {5} } } {}?

#### Exercise 75

((Reference)) Find the sum. 415+710+35415+710+35 size 12{ { {4} over {"15"} } + { {7} over {"10"} } + { {3} over {5} } } {}.

##### Solution

47304730 or 1173011730

#### Exercise 76

((Reference)) Round 0.01628 to the nearest ten-thousandths.

#### Exercise 77

((Reference)) Find the product (2.06)(1.39)

2.8634

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