How to multiply and divide fractions

Multiplying fractions and dividing fractions use closely related steps. We can multiply straight across, and we can turn division into multiplication by a reciprocal. Let’s work through both operations and simplify the results.

Multiply the numerators and the denominators

The product of the numerators becomes the new numerator. The product of the denominators becomes the new denominator.

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

The denominators $b$ and $d$ must be nonzero. For example,

\frac{3}{4} \times \frac{1}{7} = \frac{3 \times 1}{4 \times 7} = \frac{3}{28}

Example: Multiply two fractions

\begin{array}{l} \frac{2}{3} \times \frac{4}{11} \\ = \frac{2 \times 4}{3 \times 11} = \frac{8}{33} \end{array}

The numerator and denominator have no common factor greater than $1$ , so the answer is already in lowest terms.

Divide by multiplying by the reciprocal

The reciprocal of a nonzero fraction $\frac{c}{d}$ is $\frac{d}{c}$ . To divide, keep the first fraction and multiply it by the reciprocal of the second fraction. Division requires a nonzero divisor.

\frac{3}{4} \div \frac{1}{7} = \frac{3}{4} \times \frac{7}{1} = \frac{21}{4}

Example: Divide and simplify

\begin{array}{l} \frac{2}{3} \div \frac{4}{11} = \frac{2}{3} \times \frac{11}{4} \\ = \frac{22}{12} = \frac{2 \cdot 11}{2 \cdot 6} \\ = \frac{11}{6} \end{array}

The fraction $\frac{11}{6}$ is an improper fraction because its numerator is larger than its denominator. It is a valid answer in lowest terms.

Multiply a fraction by a whole number

Write the whole number with denominator $1$ . Then use the same multiplication rule.

\frac{3}{4} \times 7 = \frac{3}{4} \times \frac{7}{1} = \frac{21}{4}