Fractions of amounts: divide by the bottom, times by the top

How to find a fraction of an amount

To find a fraction of an amount, divide by the denominator, then multiply by the numerator. The denominator tells you how many equal parts to split the amount into. The numerator tells you how many of those parts you want.

Here's three quarters of 24. Split 24 into 4 equal parts, then take 3 of them:

So ³⁄₄ of 24 = 18. Children often learn this as "divide by the bottom, times by the top". The rhyme is handy, but it's worth your child knowing why it works: three quarters is just 3 one-quarters.

How to find a fraction of a quantity: start with unit fractions

A unit fraction has 1 on top, like a half, a third or a quarter. To find a unit fraction of a quantity, you only need the division step. A quarter of 20 is 20 shared into 4 equal parts:

Government guidance for teachers puts it in these words: "To find one-fifth of 15, we divide 15 into 5 equal parts." This is why the times tables matter so much for fractions. If your child knows that 4 × 5 = 20, they know that a quarter of 20 is 5 straight away. In Year 3, the guidance keeps to halves, quarters, eighths, fifths and tenths, because those match the 2, 4, 8, 5 and 10 times tables children know by then.

How to get a fraction of a number when the top isn't 1

When the numerator is bigger than 1, find one part first, then multiply. Here's a worked example: what is two thirds of 15?

1. Find one third: 15 ÷ 3 = 5.
2. You want two thirds, so take 2 of those parts: 2 × 5 = 10.

So ²⁄₃ of 15 = 10. A bar model shows it clearly:

A quick check: two thirds is more than half, and 10 is more than half of 15. If the answer came out smaller than half, something has gone wrong.

Bigger numbers: using written methods

By the end of Year 5, government guidance expects children to find fractions of amounts beyond the times tables, using written methods. Its example is four ninths of 3,411.

First find one ninth with short division (the bus stop method):

1. 9 doesn't go into 3, so write 0 above (some schools leave this space blank) and carry the 3 to the next digit.
2. The 3 thousands carried make 34 hundreds. How many 9s in 34? 3, because 3 × 9 = 27, with 7 left over. Write 3 above and carry the 7 to the next digit.
3. The 7 hundreds carried make 71 tens. How many 9s in 71? 7, because 7 × 9 = 63, with 8 left over. Write 7 above and carry the 8 to the next digit.
4. The 8 tens carried make 81 ones. 81 ÷ 9 = 9. Write 9 above.

Then multiply by 4 with short multiplication:

1. Ones: 9 × 4 = 36. Write 6 in the ones column and carry 3 into the tens column.
2. Tens: 7 × 4 = 28, plus the 3 carried makes 31. Write 1 in the tens column and carry 3 into the hundreds column.
3. Hundreds: 3 × 4 = 12, plus the 3 carried makes 15. Write 15.

So four ninths of 3,411 is 1,516. The steps are exactly the same as for two thirds of 15. Only the numbers have grown.

How do you find a fraction of something that isn't a number?

The same idea works for shapes, lengths, money and time. A fraction of a shape is the shaded share of equal parts, so this bar shows three quarters:

The parts must be equal. A shape cut into 4 pieces of different sizes isn't showing quarters, even if one piece is shaded.

For measures, use the number: a third of 60 minutes is 60 ÷ 3 = 20 minutes, and a fifth of £35 is 35 ÷ 5 = 7 pounds.

How to find a fraction of a percentage

A fraction of a percentage works just like a fraction of any other number: divide by the denominator, multiply by the numerator. A half of 30% is 15%, and three quarters of 40% is 30%, because 40 ÷ 4 = 10 and 3 × 10 = 30. Percentages start in Year 5, so this comes up in the upper years of primary school.

Working backwards to the whole

In Year 6, children also work the other way: they know the fraction and the part, and find the whole. The national curriculum's example: if a quarter of a length is 36 cm, then the whole length is 36 × 4 = 144 cm.

Government guidance gives a harder one. Miss Reeves has given out five sixths of her tangerines and has 30 left. How many did she start with? The 30 left over are the last sixth, so one sixth is 30, and the whole is 6 × 30 = 180 tangerines. Drawing a bar in 6 equal parts makes this much easier to see.

Fractions of amounts year by year

Year	What the national curriculum asks for
Year 1	Find a half and a quarter of an object, shape or quantity
Year 2	Find a third, a quarter, two quarters and three quarters of a length, shape, set of objects or quantity, and write sums like "half of 6 = 3"
Year 3	Find unit fractions and simple non-unit fractions of a set of objects, and link unit fractions to dividing
Year 4	Solve problems with harder fractions of quantities, including non-unit fractions where the answer is a whole number
Year 5	Find fractions of numbers and quantities, and see multiplying by a fraction as finding a fraction of an amount
Year 6	Work back from a unit fraction to find the whole amount

Common mistakes to watch for

• Dividing by the numerator. For two thirds of 15, some children work out 15 ÷ 2. The bottom number says how many equal parts.
• Stopping halfway. For three quarters of 24, the answer is 18, not 6. The 6 is only one quarter.
• Unequal parts. Shading 1 piece of a shape cut into 4 uneven pieces isn't a quarter.
• Missing the "how many are left" step. "16 apples, a quarter are eaten, how many are left?" needs 16 − 4 = 12, not just a quarter of 16.

How to help at home

• Share real things. Splitting 12 grapes equally between 3 people, then asking what a third is, is the whole idea in one snack.
• Ask for the next fraction. Once your child knows a quarter of 20 is 5, ask for three quarters.
• Use money and time. Half an hour, a quarter of an hour, and a fifth of £10 are all fractions of amounts.
• Keep the division facts sharp. Fractions of amounts are quick when the times tables are secure. Our printable times table grid helps for checking.

For what fractions mean in the first place, see fractions explained. Finding a fraction of an amount also leads into multiplying fractions in Year 6.

Fractions of amounts questions to try

1. What is a quarter of 28?
2. What is three fifths of 35?
3. What is two thirds of 18?
4. Rohan has £32 and spends a quarter of it on a toy. How much does he spend?
5. A 15 litre tin of paint is two thirds used. How many litres have been used?
6. Four fifths of the runners have finished a race. 92 have finished. How many runners are there altogether?

Answers

1. 7 (28 ÷ 4 = 7)
2. 21 (one fifth is 7, and 3 × 7 = 21)
3. 12 (one third is 6, and 2 × 6 = 12)
4. £8 (32 ÷ 4 = 8)
5. 10 litres (one third is 5, and 2 × 5 = 10)
6. 115 (one fifth is 92 ÷ 4 = 23, and the whole is 5 × 23 = 115)