Calculating with Fractions

Method — Adding and Subtracting

1. Look at the denominators. If they match, the fractions already count the same-sized parts.

2. If they differ, choose a common denominator. This is a number both denominators divide into exactly.

3. Write equivalent fractions. Multiply the top and bottom of each fraction by the same number. This changes the pieces we count, but keeps each fraction’s value.

4. Add or subtract the numerators. Keep the common denominator.

5. Simplify the answer. Divide the top and bottom by a shared factor until no shared factor greater than 1 remains. A factor is a whole number that divides exactly.

How do I choose a common denominator?

List a few multiples of each denominator. A multiple is a number in its times table.

For 3: 3, 6, 9, 12, …

For 4: 4, 8, 12, …

The first positive number in both lists is 12. It is the lowest common multiple, so it gives us the smallest common denominator.

Multiplying the two denominators also gives a suitable common denominator. It may be larger than necessary. For 4 and 6, 24 works, but 12 keeps the numbers smaller.

Worked Examples

Example 1 — Add with the same denominator

Work out $\frac{1}{5}$ + $\frac{2}{5}$.

Both fractions count fifths. Add the numerators:

1 + 2 = 3

$\frac{1}{5}$ + $\frac{2}{5}$ = $\frac{3}{5}$

The denominator stays 5. The answer is already in its simplest form.

Example 2 — Subtract and simplify

Work out $\frac{5}{8}$ − $\frac{1}{8}$.

Both fractions count eighths. Subtract the numerators:

5 − 1 = 4

$\frac{5}{8}$ − $\frac{1}{8}$ = $\frac{4}{8}$

Divide the top and bottom of the answer by 4:

4 ÷ 4 = 1

8 ÷ 4 = 2

$\frac{4}{8}$ = $\frac{1}{2}$

Example 3 — Add with different denominators

Work out $\frac{2}{3}$ + $\frac{1}{4}$.

Use twelfths, because 12 is in both the 3 and 4 times tables.

For thirds, multiply the top and bottom by 4:

$\frac{2}{3}$ = $\frac{8}{12}$

For quarters, multiply the top and bottom by 3:

$\frac{1}{4}$ = $\frac{3}{12}$

Now all the parts are twelfths. Add their counts:

$\frac{8}{12}$ + $\frac{3}{12}$ = $\frac{11}{12}$

$\frac{2}{3}$ + $\frac{1}{4}$ = $\frac{11}{12}$

The answer is already in its simplest form.

Example 4 — Subtract with different denominators

Work out $\frac{3}{4}$ − $\frac{1}{6}$.

The smallest common denominator is 12. Multiply the top and bottom of the first fraction by 3:

$\frac{3}{4}$ = $\frac{9}{12}$

Multiply the top and bottom of the second fraction by 2:

$\frac{1}{6}$ = $\frac{2}{12}$

Subtract the numerators:

$\frac{9}{12}$ − $\frac{2}{12}$ = $\frac{7}{12}$

$\frac{3}{4}$ − $\frac{1}{6}$ = $\frac{7}{12}$

Example 5 — Multiply fractions

Work out $\frac{2}{3}$ × $\frac{4}{5}$. The symbol × means “multiply”. Here it asks for two thirds of four fifths.

The rectangle is split into 3 rows and 5 columns: 15 equal parts. Four columns show four fifths. Take two of those three rows.

Eight of the fifteen parts are dark blue. This shows why we multiply the numerators and multiply the denominators:

2 × 4 = 8

3 × 5 = 15

$\frac{2}{3}$ × $\frac{4}{5}$ = $\frac{8}{15}$

Simplifying before or after multiplying

Now consider $\frac{3}{4}$ × $\frac{2}{9}$. Multiplying first gives:

$\frac{3}{4}$ × $\frac{2}{9}$ = $\frac{6}{36}$

Divide the numerator and denominator by 6:

$\frac{6}{36}$ = $\frac{1}{6}$

You can do those divisions earlier to keep the numbers small. This is called cancelling.

Divide the numerator 3 and denominator 9 by 3:

$\frac{3}{4}$ × $\frac{2}{9}$ = $\frac{1}{4}$ × $\frac{2}{3}$

Then divide the numerator 2 and denominator 4 by 2:

$\frac{1}{4}$ × $\frac{2}{3}$ = $\frac{1}{2}$ × $\frac{1}{3}$

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

Both methods divide the overall numerator and denominator by the same factors. They give the same answer. Cancel only factors in a multiplication; do not cancel across an addition or subtraction sign.

Example 6 — Divide by a fraction

First think about 2 ÷ $\frac{1}{2}$. The symbol ÷ means “divide”. This asks: how many halves fit into two wholes?

There are four halves altogether.

2 ÷ $\frac{1}{2}$ = 4

Counting halves doubles the number of wholes. In this case, dividing by one half has the same effect as multiplying by 2.

The reciprocal of a non-zero fraction is made by swapping its numerator and denominator. A fraction multiplied by its reciprocal equals 1.

For example, the reciprocal of $\frac{4}{5}$ is $\frac{5}{4}$:

$\frac{4}{5}$ × $\frac{5}{4}$ = 1

Dividing by a fraction is equivalent to multiplying by its reciprocal. Why? Multiplying both numbers in a division by the same non-zero number leaves the answer unchanged. Multiplying by the reciprocal makes the divisor 1.

The divisor is the number you are dividing by. Once it becomes 1, only the first number multiplied by the reciprocal is left.

Now work out $\frac{2}{3}$ ÷ $\frac{4}{5}$.

Keep the first fraction. Multiply it by the reciprocal of the second:

$\frac{2}{3}$ ÷ $\frac{4}{5}$ = $\frac{2}{3}$ × $\frac{5}{4}$

Multiply the numerators and denominators:

$\frac{2}{3}$ × $\frac{5}{4}$ = $\frac{10}{12}$

Divide the top and bottom by 2:

$\frac{10}{12}$ = $\frac{5}{6}$

We cannot divide by zero, and zero has no reciprocal.

Example 7 — Work with mixed numbers

A mixed number combines a whole number and a fraction. For example, 1 $\frac{1}{2}$ means 1 + $\frac{1}{2}$, not 1 × $\frac{1}{2}$.

Before multiplying or dividing, write the whole amount as one fraction.

One whole is two halves. Add the extra half:

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

This is an improper fraction: its numerator is at least as large as its denominator. It represents one whole or more.

For any positive mixed number, multiply the whole number by the denominator, then add the numerator. Keep the denominator.

For 1 $\frac{1}{2}$: 1 × 2 + 1 = 3, so the numerator is 3.

Now work out 1 $\frac{1}{2}$ × $\frac{2}{3}$.

1 $\frac{1}{2}$ × $\frac{2}{3}$ = $\frac{3}{2}$ × $\frac{2}{3}$

$\frac{3}{2}$ × $\frac{2}{3}$ = $\frac{6}{6}$

$\frac{6}{6}$ = 1

To turn an improper fraction back into a mixed number, count the whole groups of the denominator.

For $\frac{7}{3}$, seven thirds contain two full groups of three thirds, with one third left.

$\frac{7}{3}$ = 2 $\frac{1}{3}$

Example 8 — A GCSE-style measuring problem

A jug contains 2 $\frac{1}{4}$ litres of water. You pour out $\frac{3}{4}$ of a litre. How much water remains?

Convert the mixed number into quarters:

2 × 4 + 1 = 9

2 $\frac{1}{4}$ = $\frac{9}{4}$

Subtract the amount poured out:

$\frac{9}{4}$ − $\frac{3}{4}$ = $\frac{6}{4}$

Simplify by dividing the numerator and denominator by 2:

$\frac{6}{4}$ = $\frac{3}{2}$

Three halves is one whole and one half.

1 $\frac{1}{2}$ litres remain.

Example 9 — challenge for both tiers: combine the steps

Work out (1 $\frac{1}{2}$ − $\frac{2}{3}$) ÷ $\frac{5}{6}$.

This combines the methods already learnt. Work inside the brackets first.

Convert the mixed number to an improper fraction:

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

Use sixths to subtract:

$\frac{3}{2}$ − $\frac{2}{3}$ = $\frac{9}{6}$ − $\frac{4}{6}$

$\frac{9}{6}$ − $\frac{4}{6}$ = $\frac{5}{6}$

Now do the division outside the brackets:

$\frac{5}{6}$ ÷ $\frac{5}{6}$ = $\frac{5}{6}$ × $\frac{6}{5}$

$\frac{5}{6}$ × $\frac{6}{5}$ = $\frac{30}{30}$

$\frac{30}{30}$ = 1

This makes sense: one amount divided by the same non-zero amount is 1.