Multiplying fractions

The friendliest of the four

No matching, no rewriting. Multiply straight across, then simplify.

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

Common denominators belong to addition and subtraction, where the parts must be the same size. Multiplying makes new parts, so nothing needs matching first.

Why both pairs multiply

Draw a square as one whole. Shade $\tfrac{3}{4}$ of it going down, then $\tfrac{2}{3}$ of it going across. The overlap is the answer.

The cuts run in two directions at once, so the square holds $3 \times 4 = 12$ cells — that is the new denominator. The overlap covers $2 \times 3 = 6$ of them — that is the new numerator.

The rule is a count of cells, not a convention. The area model shows it directly.

The product can be smaller

Multiplying by less than $1$ shrinks the number, because “of” takes a part rather than repeating.

Multiply by	The answer
more than $1$	grows
exactly $1$	unchanged
less than $1$	shrinks

Knowing which way the answer should move catches most slips before the arithmetic does.

Worked examples

Work out $\tfrac{3}{5} \times \tfrac{2}{7}$.

$$\frac{3 \times 2}{5 \times 7} = \frac{6}{35}$$

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

Cancel before multiplying: $4$ and $8$ share $4$; $3$ and $9$ share $3$.

$$\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$$

Whole and mixed numbers

Write a whole number over $1$. Convert a mixed number to an improper fraction first — a mixed number is a sum, and its parts cannot be multiplied separately.

$$2\tfrac{1}{2} \times \frac{2}{3} = \frac{5}{2} \times \frac{2}{3} = \frac{10}{6} = \frac{5}{3}$$