Calculating with fractions - BBC Bitesize (2025)

If you were to add \(\frac{1}{2}\) and \(\frac{1}{3}\), it is hard to picture what the answer would be.

Rewriting the fractions with a common bottom number, or denominator (in this case, \({6}\)), makes it easier.

Remember, you can only add and subtract fractions when the bottom numbers, or denominators, are the same.

So, to add or subtract fractions:

1. Change the fractions so they have the same denominator.
2. Add or subtract the top numbers, or numerators.

Example

\(\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\)

\(\frac{7}{10} - \frac{2}{5} = \frac{7}{10} - \frac{4}{10} = \frac{3}{10}\)

Mixed numbers

To add or subtract mixed numbers, it is usually easiest to change them to improper fractions first and then change the answer back into a mixed number (if needed).

Have a go

Have a go

Multiplying fractions

\(\frac{1}{2}\) of \(\frac{1}{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)

\(\frac{2}{3}\) of \(\frac{4}{5} = \frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)

Multiply the numerators to find the new numerator, multiply the denominators to find the new denominator, then simplify where necessary.

Dividing fractions

When you divide \({10}\) by \({2}\), you are working out how many \({2}\)s there are in \({10}\).

\(10 \div 2 = 5\), so there are five \({2}\)s in \({10}\).

In a similar way, when dividing \({2}\) by \(\frac{1}{2}\), you are working out how many \(\frac{1}{2}\)s there are in \({2}\).

There are four \(\frac{1}{2}\)s in \({2}\), so:

\(2 \div \frac{1}{2} = 4\)

If you divide \(1 \frac{1}{2}\) by \(\frac{1}{4}\) you are working out how many \(\frac{1}{4}\)s there are in \(1 \frac{1}{2}\).

There are six \(\frac{1}{4}\)s in \(1 \frac{1}{2}\), so:

\(1\frac{1}{2} \div \frac{1}{4} = 6\)

Do you see a pattern?

Let's write out those calculations a different way.

• \(2 \div \frac{1}{2} = 4\) and \(2 \times 2 = 4\), so \(2 \div \frac{1}{2}\) is the same as \(2 \times 2\)

• \(1\frac{1}{2} \div \frac{1}{4} = \frac{3}{2} \div \frac{1}{4} = 6\), so \(\frac{3}{2} \div \frac{1}{4}\) is the same as \(\frac{3}{2} \times 4 = \frac{12}{2} = 6\)

So, '\(\div\frac{1}{2}\)' is the same as ‘\(\times 2\)’.

And '\(\div\frac{1}{4}\)' is the same as ‘\(\times 4\)’.

To divide fractions, turn the second fraction upside down, then multiply.

Have a go

Have a go

To find a fraction of a quantity, divide by the number on the bottom of the fraction (denominator) and multiply by the number on the top (numerator).

Example

A teacher has a box containing 54 pencils and gives out \(\frac{2}{3}\) of them to his class.

1. How many pencils has he given out?

2. How many pencils are left in the box?

Divide by the (denominator) bottom number (3) to find one third.

\(54 ÷ 3 = 18\)

Now multiply by the (numerator) top number (2) to find two thirds.

\({18} \times {2} = 36\)

The teacher gives out 36 pencils.

How many pencils are left in the box?

\(54 – 36 = 18\) pencils are left.

How to work out a fraction of an amount

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