### Multiplication of decimals

A very common student error is to write:

0.3 × 0.2 = 0.6

Clearly, that is to be avoided and suggests that multiplying one decimal by another is best done by converting the decimals to fractions. The fractions can be multiplied in the usual way and converted back to decimals.

For example:

\begin{align}0.6×0.4&=\dfrac{6}{10}×\dfrac{4}{10}\\\\ &=\dfrac{24}{100}\\\\ &=0.24\end{align}

There is another commonly used method for multiplying decimals. Students seem to not carry out this method well. It was a pre-calculator method that was used widely.

The steps are as follows:

Ignore the decimal points and multiply the factors as if they were whole numbers.
Insert a decimal point in the product so that the total number of decimal places is the same on both sides of the equation.

For example:

2.5 × 0.06

25 × 6 = 150

#### Step 2:

2.5 × 0.06 = 0.150 = 0.15

#### Example 1

Multiply 0.03 × 0.18 using both methods as described above.

#### Solution

\begin{align}0.03×0.18&=\dfrac{3}{100}×\dfrac{18}{100}\\\\ &=\dfrac{54}{10000}\\\\ &=0.0054\end{align}
\begin{align}0.03 × 0.18 = 0.0054\hspace{4mm}(3 × 18 = 54.\ \text{There are 4 decimal places}.)\end{align}

### Summary

• Multiply decimals by converting each decimal to a fraction, multiplying the fractions (without cancelling) and converting the result back to a decimal.

Or

• Follow the procedure of:
1. Ignore the decimal point and multiply as if the factors were whole numbers.
2. Place the decimal point so that the number of decimal places in the answer is equal to the total number of decimal places in the factors.