3.5: Multiplying decimals

To multiply two decimal numbers together, write each decimal as a fraction with a whole number as the numerator and a product of 10s as the denominator. For example,

• 4.6 x 0.27
• = (46 / 10) x (27 / 100)    [whole number numerators, product of 10s denominators]
• = (46 x 27) / (10 x 100)    [use 3.3: Multiplying fractions to multiply numerators and denominators]
• = 1242 / 1000    [use 3.1: Multiplying integers to calculate 46 x 27]
• = 1.242    [dividing by 1000 is equivalent to moving the decimal point three places]

This justifies why the standard procedure for multiplying decimals works:

• Multiply the numbers without decimal points (46 x 27 = 1242).
• Count the total number of digits to the right of the decimal point in each number (one in 4.6 + two in 0.27 = three in total).
• Place the decimal point that many places (three) from the end (1.242).

Here’s another example,

• 0.35 x 2.6
• = (35 / 100) x (26 / 10)
• = (35 x 26) / (100 x 10)
• = 910 / 1000
• = 0.91

In this example, the decimal point in the answer (0.91) looks like it is only two places from the end. However, we can see that it is three places from the end if we write the answer as 0.910. This is why it can be clearer to solve decimal multiplication problems by multiplying fractions rather than applying the standard procedure. Deciding where the decimal point goes in the standard procedure can lead to errors when there are zeros at the end of the whole-number product.

The video below works through some examples of multiplying decimals.