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Consider the product of 3.2 and 1.46. Changing each decimal to a fraction, we have
$\begin{array}{ccc}(3\text{.}2)(1\text{.}\text{46})& =& 3\frac{2}{\text{10}}\cdot 1\frac{\text{46}}{\text{100}}\hfill \\ & =& \frac{\text{32}}{\text{10}}\cdot \frac{\text{146}}{\text{100}}\hfill \\ & =& \frac{\text{32}\cdot \text{146}}{\text{10}\cdot \text{100}}\hfill \\ & =& \frac{\text{4672}}{\text{1000}}\hfill \\ & =& 4\frac{\text{672}}{\text{1000}}\hfill \\ & =& \text{four and six hundred seventy-two thousandths}\\ & =& \text{4}\text{.}\text{672}\hfill \end{array}$
Thus, $(3\text{.}2)(1\text{.}\text{46})=\text{4}\text{.}\text{672}$ .
Notice that the factor
$\left(\begin{array}{}\text{3.2 has 1 decimal place,}\\ \text{1.46 has 2 decimal places,}\\ \text{and the product}\\ \text{4.672 has 3 decimal places.}\end{array}\right\}1+2=3$
Using this observation, we can suggest that the sum of the number of decimal places in the factors equals the number of decimal places in the product.
Find the following products.
$6\text{.}\text{5}\cdot \text{4}\text{.}3$
Thus, $6\text{.}5\cdot 4\text{.}3=\text{27}\text{.}\text{95}$ .
$\text{23}\text{.}4\cdot 1\text{.}\text{96}$
Thus, $\text{23}\text{.}4\cdot 1\text{.}\text{96}=\text{45}\text{.}\text{864}$ .
Find the product of 0.251 and 0.00113 and round to three decimal places.
Now, rounding to three decimal places, we get
Find the following products.
$1\text{.}\text{054}\cdot \text{0}\text{.}\text{16}$
0.16864
$0\text{.}\text{00031}\cdot \text{0}\text{.}\text{002}$
0.00000062
Find the product of 2.33 and 4.01 and round to one decimal place.
9.3
Calculators can be used to find products of decimal numbers. However, a calculator that has only an eight-digit display may not be able to handle numbers or products that result in more than eight digits. But there are plenty of inexpensive ($50 - $75) calculators with more than eight-digit displays.
Find the following products, if possible, using a calculator.
$2\text{.}\text{58}\cdot \text{8}\text{.}\text{61}$
Display Reads | ||
Type | 2.58 | 2.58 |
Press | × | 2.58 |
Type | 8.61 | 8.61 |
Press | = | 22.2138 |
The product is 22.2138.
$0\text{.}\text{006}\cdot \text{0}\text{.}\text{0042}$
Display Reads | ||
Type | .006 | .006 |
Press | × | .006 |
Type | .0042 | 0.0042 |
Press | = | 0.0000252 |
We know that there will be seven decimal places in the product (since $\text{3}+\text{4}=\text{7}$ ). Since the display shows 7 decimal places, we can assume the product is correct. Thus, the product is 0.0000252.
$0\text{.}\text{0026}\cdot \text{0}\text{.}\text{11976}$
Since we expect $\text{4}+\text{5}=\text{9}$ decimal places in the product, we know that an eight-digit display calculator will not be able to provide us with the exact value. To obtain the exact value, we must use "hand technology." Suppose, however, that we agree to round off this product to three decimal places. We then need only four decimal places on the display.
Display Reads | ||
Type | .0026 | .0026 |
Press | × | .0026 |
Type | .11976 | 0.11976 |
Press | = | 0.0003114 |
Rounding 0.0003114 to three decimal places we get 0.000. Thus, $0\text{.}\text{0026}\cdot \text{0}\text{.}\text{11976}=\text{0}\text{.}\text{000}$ to three decimal places.
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