# 6.6 Division of decimals  (Page 2/3)

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$\begin{array}{ccc}\frac{\text{432}}{\text{100}}\cdot \text{10}=\frac{\text{432}}{\underset{\text{10}}{\overline{)100}}}\cdot \frac{\stackrel{1}{\overline{)10}}}{1}& =& \frac{\text{432}\cdot 1}{\text{10}\cdot 1}=\frac{\text{432}}{\text{10}}\hfill \\ & =& \text{43}\frac{2}{\text{10}}\hfill \\ & =& 43.2\hfill \end{array}$

We have converted the division $4\text{.}\text{32}÷1\text{.}8$ into the division $\text{43}\text{.}2÷\text{18}$ , that is,

$1.8\overline{)4.32}\to 18\overline{)43.2}$

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.

$\text{32}\text{.}\text{66}÷7\text{.}1$

$7.1\overline{)32.66}$ • 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, $\text{32}\text{.}\text{66}÷7\text{.}1=4\text{.}6$ .

Check: $\text{32}\text{.}\text{66}÷7\text{.}1=4\text{.}6$ if $4\text{.}6×7\text{.}1=\text{32}\text{.}\text{66}$

$1\text{.}\text{0773}÷0\text{.}\text{513}$ • 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, $1\text{.}\text{0773}÷0\text{.}\text{513}=2\text{.}1$ .

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

$\text{12}÷0\text{.}\text{00032}$

$0.00032\overline{)12.00000}$

• 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.

Checking assures us that $\text{12}÷0\text{.}\text{00032}=\text{37},\text{500}$ .

## Practice set b

Find the decimal representation of each quotient.

$9\text{.}\text{176}÷3\text{.}1$

2.96

$5\text{.}\text{0838}÷1\text{.}\text{11}$

4.58

$\text{16}÷0\text{.}\text{0004}$

40,000

$8,\text{162}\text{.}\text{41}÷\text{10}$

816.241

$8,\text{162}\text{.}\text{41}÷\text{100}$

81.6241

$8,\text{162}\text{.}\text{41}÷1,\text{000}$

8.16241

$8,\text{162}\text{.}\text{41}÷\text{10},\text{000}$

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

$\underset{\text{places}}{\underset{\text{5 decimal}}{\underbrace{0.12345}}}×\underset{\text{places}}{\underset{\text{4 decimal}}{\underbrace{0.4567}}}$

produces $5+4=9$ 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.

$\text{12}\text{.}\text{596}÷4\text{.}7$

 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.

$0\text{.}\text{5696376}÷0\text{.}\text{00123}$

 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.

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scanning tunneling microscope
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