Multiplying and Dividing Fractions

Why this matters

Recipes, scale drawings and measurements are full of fractions: ¾ of a 164½-litre tank, a plank 4¼ m long cut into pieces of ⅝ m. Multiplying and dividing them quickly and without a calculator is a core Year 9 skill.

Why dividing means multiplying by the reciprocal

You already know that $9 \div 3 = 9 \times \frac{1}{3}$ and $8 \div 2 = 8 \times \frac{1}{2}$ . Dividing by any number is multiplying by one over that number — its reciprocal.

Number	Reciprocal	Check
3	$\frac{1}{3}$	$3 \times \frac{1}{3} = 1$
$\frac{1}{2}$	2	$\frac{1}{2} \times 2 = 1$
$\frac{3}{4}$	$\frac{4}{3}$	$\frac{3}{4} \times \frac{4}{3} = 1$
$1\frac{3}{8} = \frac{11}{8}$	$\frac{8}{11}$	$\frac{11}{8} \times \frac{8}{11} = 1$

“Of” means multiply

$\frac{7}{10}$ of $8\frac{1}{3}$ m is $\frac{7}{10} \times \frac{25}{3} = \frac{175}{30} = 5\frac{5}{6}$ m.

The mistake to avoid

When dividing, only the second fraction is flipped. This working is wrong:

$3\tfrac{2}{3} \div 1\tfrac{1}{11} = \tfrac{11}{3} \div \tfrac{12}{11} \;\;\xrightarrow{\ \text{wrong!}\ }\;\; \tfrac{3}{11} \times \tfrac{11}{12}$

The first fraction was flipped as well. Correctly: $\frac{11}{3} \times \frac{11}{12} = \frac{121}{36} = 3\frac{13}{36}$ .

Sense check. Dividing by a mixed number (bigger than 1) must give a smaller answer. Dividing by a proper fraction (less than 1) must give a bigger one.

Key rules and formulas

Multiplying

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

Multiply the tops and multiply the bottoms. Cancel any numerator with any denominator first to keep the numbers small.

Mixed numbers become improper first

$1 \frac{1}{2} = \frac{3}{2}, 4 \frac{2}{5} = \frac{22}{5}$

Never multiply the whole numbers and the fractions separately:

1 \frac{1}{2} \times 1 \frac{1}{2}

is not

1 \frac{1}{4}

Dividing: flip and multiply

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

Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of

\frac{3}{4}

\frac{4}{3}

, because

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

Bigger or smaller?

Multiplying by a number greater than 1 makes things bigger; by a proper fraction, smaller. Dividing works the other way round.

Worked examples

Worked example 1

Multiplying mixed numbers

Calculate

1 \frac{1}{15} \times \frac{3}{8} \times 10

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Squaring a mixed number

Calculate

(1 \frac{4}{5})^{2}

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Dividing mixed numbers

Calculate

2 \frac{1}{4} \div 1 \frac{3}{8}

Try it yourself first, then reveal the solution one step at a time.

Multiplying and Dividing Fractions

In this lesson you will learn to…

Why this matters

Why dividing means multiplying by the reciprocal

“Of” means multiply

The mistake to avoid

Key rules and formulas

Multiplying

Mixed numbers become improper first

Dividing: flip and multiply

Bigger or smaller?

Worked examples

Multiplying mixed numbers

Squaring a mixed number

Dividing mixed numbers

Practice: multiplying and dividing fractions