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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to divide decimals. By the end of the module students should understand the method used for dividing decimals, be able to divide a decimal number by a nonzero whole number and by another, nonzero, decimal number and be able to simplify a division of a decimal by a power of 10.

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   

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 ÷ 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

Thus, 4 . 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.

Long division. 196.8 divided by 6. 6 goes into 19 3 times, with a remainder of 1. Bring the 6 down. 6 goes into 16 twice, with a remainder of 4. Bring the 8 down, and the decimal place with it. 6 goes into 48 8 times, with a remainder of zero. The quotient is 32.8 Notice that the decimal points appear in the same column.

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.

114.1 ÷ 7 = 7

16.3 7 114.1 7      ̲ 44    42    ̲ 2.1 2.1 ̲ 0

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

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

16.3 4 2           7 ̲ 114.1 True.

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

Long division. 0.02068 divided by 4. 4 goes into 20 5 times, with no remainder. 4 goes into 6 once, with a remainder of 2. Bring down the 8. 4 goes into 28 7 times, with a remainder of zero. The quotient is 0.00517.
Place zeros in the tenths and hundredths positions. (See Step 3.)

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

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Practice set a

Find the following quotients.

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

61.5

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

1.884

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

0.0741

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

0.000062

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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 4 32 100 ÷ 1 8 10 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

The divisor is 18 10 size 12{ { {"18"} over {"10"} } } {} . We can convert 18 10 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 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 18 10 size 12{ { {"18"} over {"10"} } } {} to a whole number by multiplying it by 10, we must also multiply the numerator 432 100 size 12{ { {"432"} over {"100"} } } {} by 10.

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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