Multiplying Fractions, Decimals and Percentages

1. A whole number times a fraction: read × as "times"

You already know that 3 × 4 means 3 groups of 4, or 4 + 4 + 4 = 12. The same idea works when the second number is a fraction. When a whole number comes first, we read the × sign as "times". So $3\times 2\frac{2}{3}$ means *3 times $2\frac{2}{3}$*, that is, 3 groups of $2\frac{2}{3}$.

We could add: $2\frac{2}{3}+2\frac{2}{3}+2\frac{2}{3}$. The whole numbers give 6 and the thirds give $\frac{6}{3}=2$, so the total is 8. Repeated addition works, but it becomes slow when the whole number is big, like 15 × $\frac{2}{3}$. So we use a shorter method.

1. 1Change every number into a common fraction. A whole number goes over 1 ($3=\frac{3}{1}$) and a mixed number becomes an improper fraction ($2\frac{2}{3}=\frac{8}{3}$).
2. 2Multiply all the numerators together and all the denominators together: $\frac{3}{1}\times \frac{8}{3}=\frac{24}{3}$.
3. 3Simplify the result: $\frac{24}{3}=8$. If the answer is improper, you may write it as a mixed number.

▶ Notice that three equal jumps of $2\frac{2}{3}$ land exactly on 8 — the same answer as the short method.

2. A fraction times a number or a fraction: read × as "of"

When the fraction comes first, we read × as "of". So $\frac{2}{3}\times 5$ means *$\frac{2}{3}$ of 5*, and $\frac{2}{3}\times \frac{1}{2}$ means *$\frac{2}{3}$ of a half*. The method is exactly the same: write everything as common fractions, multiply the tops, multiply the bottoms, then simplify.

For example, $\frac{2}{3}\times 5=\frac{2}{3}\times \frac{5}{1}=\frac{10}{3}=3\frac{1}{3}$. And $\frac{2}{3}\times \frac{1}{2}=\frac{2\times 1}{3\times 2}=\frac{2}{6}=\frac{1}{3}$. The picture below shows why the denominators are multiplied: cutting thirds into halves makes sixths.

▶ Watch the overlap: two of the six small pieces are covered by both colours, so $\frac{2}{3}$ of $\frac{1}{2}$ is $\frac{2}{6}=\frac{1}{3}$.

3. Multiplying decimal fractions

A decimal is just a fraction with a denominator of 10, 100 or 1,000, so the same idea applies. For example $0.3\times 0.4=\frac{3}{10}\times \frac{4}{10}=\frac{12}{100}=0.12$. Notice that tenths × tenths gives hundredths: the denominators multiply, so the number of decimal places adds up.

1. 1Ignore the decimal points and multiply the numbers as whole numbers.
2. 2Count the total number of decimal places in the numbers you multiplied.
3. 3Put the decimal point in the answer so that it has that same number of decimal places (count from the right, adding zeros in front if needed).

4. Finding a percentage of a quantity

Per cent means out of 100, so a percentage is a fraction with denominator 100. To find 28% of GH₵40, we multiply: $\frac{28}{100}\times 40=\frac{1120}{100}=11.2$. So the discount on Akosua's bag is GH₵11.20, not GH₵28. Kwame forgot that 28% means 28 out of every 100, and the bag costs only GH₵40.

▶ 28 of the 100 small squares are shaded. That is why 28% can be written as $\frac{28}{100}$ or 0.28 before you multiply.

You can use the fraction form or the decimal form, whichever is easier: 28% of 40 = 0.28 × 40 = 11.2. Percentages with fractions or decimals in them work the same way. Write $3\frac{1}{2}\%$ as 3.5% = 0.035, and 0.2% as 0.002.

Worked examples

Try each one yourself first, then open the solution and compare every step.