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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 $4\text{.}\text{32}\xf71\text{.}8$ into the division $\text{43}\text{.}2\xf7\text{18}$ , 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.
Find the following quotients.
$\text{32}\text{.}\text{66}\xf77\text{.}1$
$7.1\overline{)32.66}$
Thus, $\text{32}\text{.}\text{66}\xf77\text{.}1=4\text{.}6$ .
Check: $\text{32}\text{.}\text{66}\xf77\text{.}1=4\text{.}6$ if $4\text{.}6\times 7\text{.}1=\text{32}\text{.}\text{66}$
$\begin{array}{cc}\hfill 4.6& \\ \hfill \underline{\times 7.1}& \\ \hfill 46& \\ \hfill \underline{322\text{}}& \\ \hfill 32.66& \text{True.}\hfill \end{array}$
$1\text{.}\text{0773}\xf70\text{.}\text{513}$
Thus, $1\text{.}\text{0773}\xf70\text{.}\text{513}=2\text{.}1$ .
Checking by multiplying 2.1 and 0.513 will convince us that we have obtained the correct result. (Try it.)
$\text{12}\xf70\text{.}\text{00032}$
$0.00032\overline{)12.00000}$
This is now the same as the division of whole numbers.
$\begin{array}{c}\hfill 37500.\\ \hfill 32\overline{)1200000.}\\ \hfill \underline{96}\\ \hfill 240\\ \hfill \underline{224}\\ \hfill 160\\ \hfill \underline{160}\\ \hfill 000\end{array}$
Checking assures us that $\text{12}\xf70\text{.}\text{00032}=\text{37},\text{500}$ .
Find the decimal representation of each quotient.
$\mathrm{8,}\text{162}\text{.}\text{41}\xf7\text{10}$
816.241
$\mathrm{8,}\text{162}\text{.}\text{41}\xf7\text{100}$
81.6241
$\mathrm{8,}\text{162}\text{.}\text{41}\xf7\mathrm{1,}\text{000}$
8.16241
$\mathrm{8,}\text{162}\text{.}\text{41}\xf7\text{10},\text{000}$
0.816241
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
$\underset{\text{places}}{\underset{\text{5 decimal}}{\underbrace{0.12345}}}\times \underset{\text{places}}{\underset{\text{4 decimal}}{\underbrace{0.4567}}}$
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.
Find each quotient using a calculator. If the result is an approximation, round to five decimal places.
$\text{12}\text{.}\text{596}\xf74\text{.}7$
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.
$0\text{.}\text{5696376}\xf70\text{.}\text{00123}$
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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