Fractions of an Amount — KS3 Maths Explained

To find a fraction of an amount, divide the amount by the denominator (bottom number), then multiply by the numerator (top number). This two-step method works for any fraction and is tested in every KS3 maths paper from Year 7 to Year 9.

What does "fraction of an amount" mean?

When a question asks for a fraction of an amount — for example, three-fifths of 40 — it is asking: "If you split 40 into 5 equal parts, how much do 3 of those parts make?"

The rule:

Fraction of an amount = (amount ÷ denominator) × numerator

This is sometimes written as: divide by the bottom, multiply by the top.

Step-by-step method

Follow two clear steps for every fraction-of-an-amount question:

Divide the amount by the denominator to find one equal part (the unit fraction).

Multiply that result by the numerator to find the required number of parts.

Worked example 1 — three-fifths of 40

Step 1: 40 ÷ 5 = 8 (one-fifth of 40)

Step 2: 8 × 3 = 24 (three-fifths of 40)

Answer: 24

Worked example 2 — two-sevenths of 63

Step 1: 63 ÷ 7 = 9 (one-seventh of 63)

Step 2: 9 × 2 = 18 (two-sevenths of 63)

Answer: 18

Worked example 3 — five-eighths of 96

Step 1: 96 ÷ 8 = 12 (one-eighth of 96)

Step 2: 12 × 5 = 60 (five-eighths of 96)

Answer: 60

Finding fractions of an amount with a calculator

With a calculator the process is the same, but you can also multiply first and then divide — the order does not matter.

Fraction of amount = amount × numerator ÷ denominator

Example: four-ninths of 279

279 × 4 ÷ 9 = 1116 ÷ 9 = 124

Answer: 124

This alternative order can avoid awkward decimals appearing mid-calculation on a non-calculator paper — try dividing first, and if the result is not a whole number, multiply first instead.

Unit fractions vs non-unit fractions

Type | Definition | Example
Unit fraction | Numerator is 1 | 1/5 of 60 = 12
Non-unit fraction | Numerator > 1 | 3/5 of 60 = 36

For a unit fraction, you only need step 1 — divide by the denominator. For a non-unit fraction, both steps are needed.

Fractions of an amount with mixed numbers

Sometimes a question involves a mixed number such as 2¾ of an amount. Convert the mixed number to an improper fraction first.

Example: 2¾ of 36

Convert: 2¾ = (2 × 4 + 3)/4 = 11/4

Step 1: 36 ÷ 4 = 9

Step 2: 9 × 11 = 99

Answer: 99

Fractions of amounts in context (word problems)

Many KS3 and GCSE exam questions embed fraction-of-an-amount calculations inside a real-life context. The method is identical — identify the whole amount and the fraction, then apply the two-step rule.

Worked example 4 — a real-life context

A school has 840 students. Three-sevenths travel by bus. How many students travel by bus?

Step 1: 840 ÷ 7 = 120

Step 2: 120 × 3 = 360

Answer: 360 students travel by bus.

Worked example 5 — finding the remainder

In the same school, five-twelfths play a sport. How many students do NOT play a sport?

Step 1: 840 ÷ 12 = 70

Step 2: 70 × 5 = 350 (students who play a sport)

Students who do NOT play: 840 − 350 = 490

Answer: 490 students do not play a sport.

The national curriculum and fractions

The DfE's KS3 mathematics programme of study requires pupils to use the four operations (including division) applied to fractions, and to solve problems involving fractions in a variety of contexts. The statutory national curriculum for Key Stage 3 and 4 identifies fractions of an amount as a core skill that underpins ratio, proportion, and percentage work across Years 7 to 9.

Common mistakes to avoid

Mistake | Error example | Correct approach
Dividing by the numerator, not the denominator | 3/5 of 40: dividing by 3 first | Always divide by the denominator (bottom) first
Forgetting the multiplication step | Stopping at 40 ÷ 5 = 8 and writing 8 as the answer | 8 is one-fifth; multiply by 3 to get three-fifths = 24
Mixing up the fraction and the amount | Writing 5 ÷ 40 instead of 40 ÷ 5 | Divide the whole amount by the denominator
Not converting a mixed number first | Using 2¾ directly in a calculation without converting | Convert to an improper fraction (11/4) before dividing

Frequently asked questions

How do you find a fraction of an amount without a calculator?

Divide the amount by the denominator first to find the value of one equal part. Then multiply that result by the numerator. As long as the amount divides evenly by the denominator, the whole calculation can be done in your head or on paper. If it does not divide evenly, try multiplying the amount by the numerator first and then dividing by the denominator.

What is the difference between a unit fraction and a non-unit fraction?

A unit fraction has 1 as its numerator — for example, one-third or one-quarter. To find a unit fraction of an amount, you only divide by the denominator. A non-unit fraction has a numerator greater than 1 — for example, three-quarters. You divide by the denominator and then multiply by the numerator to find the required number of equal parts.

Why does dividing by the denominator and multiplying by the numerator work?

The denominator tells you how many equal parts the whole is split into. Dividing the amount by the denominator gives one equal part. The numerator tells you how many of those parts you need. Multiplying by the numerator gives the total of those parts. This is simply the definition of what a fraction represents.

Can I multiply first and then divide instead?

Yes. Multiplication is commutative, so (amount × numerator) ÷ denominator gives the same result as (amount ÷ denominator) × numerator. On a calculator, this makes no difference. Without a calculator, dividing first is usually easier because it tends to produce a smaller, whole-number intermediate answer — but choose whichever order avoids awkward decimals.

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