Multiplying fractions

Multiplying fractions is one of the easiest operations in mathematics , easier than adding.

Idea

To multiply $\dfrac{a}{b} \times \dfrac{c}{d}$, multiply numerators together and denominators together: $\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d}.$

For example, $\tfrac{2}{3} \times \tfrac{5}{7} = \tfrac{10}{21}$.

Simplify-before-multiplying trick. If the numerator of one fraction shares a common factor with the denominator of the other, cancel them first. This keeps numbers small.

Example: $\tfrac{2}{3} \times \tfrac{9}{8}$. Cancel $2$ in $\tfrac{2}{8}$ to get $\tfrac{1}{4}$. Cancel $3$ in $\tfrac{9}{3}$ to get $\tfrac{3}{1}$. So $\tfrac{1}{1} \times \tfrac{3}{4} = \tfrac{3}{4}$.

Geometric meaning. $\tfrac{1}{2} \times \tfrac{1}{3}$ asks: what is half of a third? Draw a rectangle; shade $\tfrac{1}{3}$ (one strip out of three). Now shade half of that strip , you get $\tfrac{1}{6}$ of the whole rectangle. So $\tfrac{1}{2} \times \tfrac{1}{3} = \tfrac{1}{6}$.

Multiplying a whole number by a fraction. Treat the whole number as a fraction over $1$. So $5 \times \tfrac{2}{3} = \tfrac{5}{1} \times \tfrac{2}{3} = \tfrac{10}{3} = 3\tfrac{1}{3}$.

Mixed numbers. Convert to improper first, then multiply, then convert back. Example: $1\tfrac{1}{2} \times 2\tfrac{1}{3} = \tfrac{3}{2} \times \tfrac{7}{3} = \tfrac{21}{6} = \tfrac{7}{2} = 3\tfrac{1}{2}$.

Worked examples

Example 1. Compute $\tfrac{3}{4} \times \tfrac{2}{5}$.

$\dfrac{3 \times 2}{4 \times 5} = \dfrac{6}{20} = \dfrac{3}{10}$. Or simplify first: $\tfrac{3}{4} \times \tfrac{2}{5}$, cancel $2$ in $\tfrac{2}{4}$: $\tfrac{3}{2} \times \tfrac{1}{5} = \tfrac{3}{10}$.

Example 2. Compute $\tfrac{5}{6} \times \tfrac{12}{25}$.

Cancel $5$: $\tfrac{1}{6} \times \tfrac{12}{5}$. Cancel $6$ and $12$ (factor $6$): $\tfrac{1}{1} \times \tfrac{2}{5} = \tfrac{2}{5}$.

Example 3. Compute $6 \times \tfrac{2}{3}$.

$\tfrac{6}{1} \times \tfrac{2}{3} = \tfrac{12}{3} = 4$.

Example 4. Compute $2\tfrac{1}{2} \times 1\tfrac{1}{5}$.

Improper: $\tfrac{5}{2} \times \tfrac{6}{5}$. Cancel $5$: $\tfrac{1}{2} \times \tfrac{6}{1} = \tfrac{6}{2} = 3$.

Try it yourself

1. Compute $\tfrac{1}{2} \times \tfrac{1}{3}$.
2. Compute $\tfrac{3}{5} \times \tfrac{10}{9}$ (simplify before multiplying).
3. Compute $4 \times \tfrac{3}{8}$.
4. Compute $\tfrac{7}{8} \times \tfrac{4}{21}$.
5. Compute $1\tfrac{1}{4} \times 2\tfrac{2}{5}$.
6. Find $\tfrac{2}{3}$ of $\tfrac{9}{10}$.
7. Compute $\tfrac{5}{12} \times \tfrac{6}{25}$.
8. Multiply three fractions: $\tfrac{1}{2} \times \tfrac{2}{3} \times \tfrac{3}{4}$.

Activity

Take a square piece of paper. Fold it in half , that's $\tfrac{1}{2}$. Now fold the half into thirds , each tiny piece is $\tfrac{1}{2} \times \tfrac{1}{3} = \tfrac{1}{6}$. Count the small rectangles in the whole square: six. Multiplication of fractions, visualised.