# 6.5 Multiplication of decimals

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to multiply decimals. By the end of the module students should understand the method used for multiplying decimals, be able to multiply decimals, be able to simplify a multiplication of a decimal by a power of 10 and understand how to use the word "of" in multiplication.

## Section overview

• The Logic Behind the Method
• The Method of Multiplying Decimals
• Calculators
• Multiplying Decimals By Powers of 10
• Multiplication in Terms of “Of”

## The logic behind the method

Consider the product of 3.2 and 1.46. Changing each decimal to a fraction, we have

$\begin{array}{ccc}\left(3\text{.}2\right)\left(1\text{.}\text{46}\right)& =& 3\frac{2}{\text{10}}\cdot 1\frac{\text{46}}{\text{100}}\hfill \\ & =& \frac{\text{32}}{\text{10}}\cdot \frac{\text{146}}{\text{100}}\hfill \\ & =& \frac{\text{32}\cdot \text{146}}{\text{10}\cdot \text{100}}\hfill \\ & =& \frac{\text{4672}}{\text{1000}}\hfill \\ & =& 4\frac{\text{672}}{\text{1000}}\hfill \\ & =& \text{four and six hundred seventy-two thousandths}\\ & =& \text{4}\text{.}\text{672}\hfill \end{array}$

Thus, $\left(3\text{.}2\right)\left(1\text{.}\text{46}\right)=\text{4}\text{.}\text{672}$ .

Notice that the factor

$\left(\begin{array}{}\text{3.2 has 1 decimal place,}\\ \text{1.46 has 2 decimal places,}\\ \text{and the product}\\ \text{4.672 has 3 decimal places.}\end{array}}1+2=3$

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. ## Method of multiplying decimals

To multiply decimals,
1. Multiply the numbers as if they were whole numbers.
2. Find the sum of the number of decimal places in the factors.
3. The number of decimal places in the product is the sum found in step 2.

## Sample set a

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 ## Practice set a

Find the following products.

$5\text{.}3\cdot \text{8}\text{.}6$

45.58

$2\text{.}\text{12}\cdot \text{4}\text{.}9$

10.388

$1\text{.}\text{054}\cdot \text{0}\text{.}\text{16}$

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

$\text{10}\cdot \text{5}\text{.}\text{394}$

53.94

$\text{100}\cdot \text{5}\text{.}\text{394}$

539.4

$\text{1000}\cdot \text{5}\text{.}\text{394}$

5,394

$\text{10,000}\cdot \text{5}\text{.}\text{394}$

59,340

## Calculators

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.

## Sample set b

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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