Page 132 - ArithBook5thEd ~ BCC

4.7 Multiplication of general decimals

We already know how to multiply and divide decimals when one of them is a power of 10. If neither
factor is a power of ten, there is more to the process than just moving the decimal point. Consider the
following example.
Example 163. Multiply 0.3 × 0.07.

Solution. Writing this as a product of ordinary fractions, we get

3 7 3 × 7 21
0.3 × 0.07 = × = = .
10 100 10 × 100 1000
3
The last fraction can be written 21 ÷ 10 ,or 21 × 10 −3 which, from the results of the last section, is
equal to 0.021. Thus,
0.3 × 0.07 = 0.021.

Multiplication of two decimals always involves a whole number multiplication for the numerator
(3 × 7 = 21 in the example) and a multiplication of powers of 10 for thedenominator (10 × 100 =
1
3
2
10 × 10 =1000 = 10 in the example). By looking back at the previous section, it is easy to see the
product of two powers of 10 is itself a power of 10. Which power of 10? The rule is simple:
k
n
10 × 10 =10 (n+k)

This implies that the number of decimal places in the product of two (or more) decimals is the sum of
the numbers of decimal places in the factors. Thus any set of decimals can be multiplied by following
a two step procedure:

To multiply two or more decimals:

• Multiply the decimals as if they were whole numbers, ignoringthe decimal
points (this gives the numerator of the decimal fraction);
• Add the number of decimal places in each factor (this gives thedenominator
of the decimal fraction by specifying the number of decimal places).

Example 164. Find the product 21.02 × 0.004.
Solution. Temporarily ignoring the decimal points, we multiply 2102 × 4= 8408. Now 21.02 has two
decimal places, and 0.004 has three. So the product will have 2+ 3 = 5 decimal places. In other words,

5
21.02 × 0.004 = 8408 ÷ 10 =0.08408.

Example 165. Find the product of 12, 0.3, and 0.004.

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