6.5: Multiplication of Decimals (2024)

Table of Contents

The Logic Behind the Method The Method of Multiplying Decimals Calculators Multiplying Decimals by Powers of 10 Multiplication in Terms of “Of” Exercises FAQs References

The Logic Behind the Method

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

$\begin{array} {rcl} {(3.2)(1.46)} & = & {3 \dfrac{2}{10} \cdot 1 \dfrac{46}{100}} \\ {} & = & {\dfrac{32}{10} \cdot \dfrac{146}{100}} \\ {} & = & {\dfrac{32 \cdot 146}{10 \cdot 100}} \\ {} & = & {\dfrac{4672}{1000}} \\ {} & = & {4 \dfrac{672}{1000}} \\ {} & = & {\text{four and six hundred seventy-two thousandths}} \\ {} & = & {4.672} \end{array}$

Thus, $(3.2)(1.46) = 4.672$

Notice that the factor

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

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.

The Method of Multiplying Decimals

Method of Multiplying Decimals

To multiply decimals:

Multiply the numbers as if they were whole numbers.
Find the sum of the number of decimal places in the factors.
The number of decimal places in the product is the sum found in step 2.

Sample Set A

Find the following products.

$6.5 \cdot 4.3$

Solution

Thus, $6.5 \cdot 4.3 = 27.95.$

Sample Set A

$23.4 \cdot 1.96$

Solution

Thus, $23.4 \cdot 1.96 = 45.864.$

Sample Set A

Find the product of 0.251 and 0.00113 and round to three decimal places.

Solution

Now, rounding to three decimal places, we get

Practice Set A

Find the following products.

$5.3 \cdot 8.6$

Answer: 45.58

Practice Set A

$2.12 \cdot 4.9$

Answer: 10.388

Practice Set A

$1.054 \cdot 0.16$

Answer: 0.16864

Practice Set A

$0.00031 \cdot 0.002$

Answer: 0.00000062

Practice Set A

Find the product of 2.33 and 4.01 and round to one decimal place.

Answer: 9.3

Practice Set A

$10 \cdot 5.394$

Answer: 53.94

Practice Set A

$100 \cdot 5.394$

Answer: 539.4

Practice Set A

$1000 \cdot 5.394$

Answer: 5,394

Practice Set A

$10,000 \cdot 5.394$

Answer: 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.

$2.58 \cdot 8.61$

Solution

		Display Reads
Type	2.58	2.58
Press	$\times$	2.58
Type	8.61	8.61
Press	=	22.2138

The product is 22.2138.

Sample Set B

$0.006 \cdot 0.0042$

Solution

		Display Reads
Type	.006	.006
Press	$\times$	.006
Type	.0042	0.0042
Press	=	0.0000252

We know that there will be seven decimal places in the product (since $3 + 4 = 7$). Since the display shows 7 decimal places, we can assume the product is correct. Thus, the product is 0.0000252.

Sample Set B

$0.0026 \cdot 0.11976$

Solution

Since we expect $4 + 5 = 9$ 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	$\times$	.0026
Type	.11976	0.11976
Press	=	0.0003114

Rounding 0.0003114 to three decimal places we get 0.000. Thus, $0.0026 \cdot 0.11976 = 0.000$ to three decimal places.

Practice Set B

Use a calculator to find each product. If the calculator will not provide the exact product, round the result to four decimal places.

$5.126 \cdot 4.08$

Multiplying Decimals by Powers of 10

There is an interesting feature of multiplying decimals by powers of 10. Consider the following multiplications.

Multiplication	Number of Zeros in the Power of 10	Number of Positions the Decimal Point Has Been Moved to the Right
$10 \cdot 8.315274 = 83.15274$	1	1
$100 \cdot 8.315274 = 831.5274$	2	2
$1,000 \cdot 8.315274 = 8,315.274$	3	3
$10,000 \cdot 8.315274 = 83,152.74$	4	4

Multiplying a Decimal by a Power of 10
To multiply a decimal by a power of 10, move the decimal place to the right of its current position as many places as there are zeros in the power of 10. Add zeros if necessary.