Multiplying Fractions

What you will learn

• Multiply two fractions.
• Explain why the rule works using an area model.
• Simplify, and multiply a fraction by a whole number.

Before you start: you should be able to simplify fractions and find a fraction of an amount. See Equivalent Fractions and Simplifying and Finding a Fraction of an Amount.

The idea

“Of” often means multiply. A half of a third is $\frac{1}{2} \times \frac{1}{3}$. Draw a square. Cut it into 3 columns and shade one column for the third. Cut it into 2 rows and mark one row for the half. They overlap in 1 of 6 small rectangles.

The rule

Multiply the tops and multiply the bottoms:

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

The bottoms count how many small rectangles the square is cut into. The tops count how many of them are in the overlap.

A part of a part

Take three quarters of a bar, then take half of that. You are left with three eighths of the bar.

Whole numbers

Write a whole number over $1$. So $3 \times \frac{2}{5} = \frac{3}{1} \times \frac{2}{5} = \frac{6}{5}$. It is three lots of two fifths.

Simplify early

You can divide a top and a bottom by a common factor before multiplying. This keeps the numbers small.

Worked examples

Example 1: Multiply and simplify

Work out $\frac{2}{5} \times \frac{3}{4}$.

1. Tops: $2 \times 3 = 6$. Bottoms: $5 \times 4 = 20$.
2. $\frac{6}{20}$ simplifies by dividing both by $2$.

Answer: $\frac{3}{10}$

Example 2: Simplify first

Work out $\frac{4}{9} \times \frac{3}{8}$.

1. $4$ and $8$ both divide by $4$, giving $1$ and $2$.
2. $3$ and $9$ both divide by $3$, giving $1$ and $3$.
3. Now multiply: $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$.

Answer: $\frac{1}{6}$

Example 3: Mixed number

Work out $1\frac{1}{2} \times \frac{2}{3}$.

1. Write $1\frac{1}{2}$ as $\frac{3}{2}$.
2. $\frac{3}{2} \times \frac{2}{3}$: the $3$s cancel and the $2$s cancel.
3. The answer is $\frac{1}{1} = 1$.

Answer: $1$

Tip: always change mixed numbers to improper fractions before multiplying.

Why it matters

Scaling a recipe, finding a share of a share and working out probabilities of two events together all use fraction multiplication. In algebra, the same rule multiplies terms such as $\frac{x}{2} \times \frac{y}{3}$.