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.

Method of Dividing a Decimal by a Nonzero Whole Number

To divide a decimal by a nonzero whole number:

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 1.84.32→1843.2

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,

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.

In problems 4 and 5 of Section 8, 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  ̲ 0  816.241 10 8162.410 80            ̲ 16          10          ̲ 62        60        ̲ 24      20      ̲ 41    40    ̲ 10  10  ̲ 0 

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.

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

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

For the following 9 problems, find each solution.

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.