How to multiply fractions (and mixed numbers)

Look for a number on top and a number on the bottom (in either fraction) that share a factor:

(6 and 15 share 3; 14 and 7 share 7.) Multiplying first gives $\tfrac{84}{105}$, which simplifies to the same $\tfrac{4}{5}$, but with more work.

Convert to improper fractions first:

Multiplying the whole numbers and fractions separately ($1 \times 2$ and $\tfrac{1}{2} \times \tfrac{2}{3}$) gives the wrong answer.

• $\tfrac{3}{8} \times 32 = \tfrac{3 \times 32}{8} = 12$ (or $32 \div 8 \times 3$).
• "$\tfrac{2}{5}$ of 45" means $\tfrac{2}{5} \times 45 = 18$.
• A fraction of a fraction: $\tfrac{1}{2}$ of $\tfrac{3}{4}$ is $\tfrac{3}{8}$.

$\tfrac{2}{3} \times \tfrac{4}{5}$ means "two-thirds of four-fifths". Draw a rectangle split into 5 columns, shade 4; then split it into 3 rows and take 2. The overlap is 8 small rectangles out of 15: $\tfrac{8}{15}$.

• Finding a common denominator (needed for adding, not multiplying).
• Not converting mixed numbers.
• Cancelling two numbers that are both on top.
• Forgetting to simplify the answer.

(a) $\tfrac{3}{4} \times \tfrac{2}{9}$, (b) $\tfrac{5}{6} \times 18$, (c) $2\tfrac{1}{4} \times 1\tfrac{1}{3}$. Answers: $\tfrac{1}{6}$, 15, 3. Check with the fraction calculator; see also how to divide fractions.

Multiply all the tops and all the bottoms, cancelling where you can:

Cancelling first is even quicker: the 2s and 3s cancel, leaving $\tfrac{1}{4}$.

A school has 840 pupils. $\tfrac{3}{7}$ of them walk to school. How many walk?

$840 \div 7 = 120$, and $120 \times 3 = 360$ pupils.

Of those who walk, $\tfrac{1}{4}$ are in Year 7. How many Year 7 pupils walk? $360 \div 4 = 90$. Or in one step: $\tfrac{1}{4} \times \tfrac{3}{7} = \tfrac{3}{28}$ of the school, and $\tfrac{3}{28} \times 840 = 90$.

"Of" in a question is your signal to multiply.

Letters cancel just like numbers, as long as they're factors:

Here $x^2 \div x = x$ and $2 \div 6 = \tfrac{1}{3}$. This comes up in GCSE Higher algebraic fraction questions.

A quick estimate catches most mistakes. $\tfrac{2}{3} \times \tfrac{4}{5}$ means "two-thirds of something a bit less than 1", so the answer must be a bit less than $\tfrac{2}{3}$. $\tfrac{8}{15} \approx 0.53$, which fits. If you'd added instead and got $\tfrac{6}{8}$, the estimate would warn you. You can also check with the fraction calculator, which shows the working.