Finding Fractions of a Number

What "a fraction of a number" means

When someone says "I ate half of the cookies" or "a third of the class walk to school", they are finding a fraction of a number. It is one of the most useful skills in everyday maths — sharing food, working out discounts, splitting a bill or reading a recipe all use it.

A fraction shows a part of a whole. The bottom number, the denominator, tells you how many equal pieces the whole is split into. The top number, the numerator, tells you how many of those pieces you take. If you would like a refresher on what fractions are, read our Introduction to Fractions lesson first.

So finding 1/4 of 20 means: split 20 into 4 equal groups, then take one of those groups.

The simplest case: a unit fraction

A unit fraction has a numerator of 1, like 1/2, 1/3, 1/5 or 1/10. To find a unit fraction of a number, you only need to divide.

Rule: To find a unit fraction of a number, divide the number by the denominator.

Example 1 — Find 1/4 of 20.

1. The denominator is 4, so split 20 into 4 equal groups.
2. Divide: 20 ÷ 4 = 5.
3. So 1/4 of 20 = 5.

You can picture this as 20 sweets shared fairly between 4 children — each gets 5.

Example 2 — Find 1/3 of 18.

1. The denominator is 3, so divide 18 into 3 equal groups.
2. 18 ÷ 3 = 6.
3. So 1/3 of 18 = 6.

When the numerator is more than 1

Most fractions, like 2/3 or 3/5, have a numerator bigger than 1. There is just one extra step: after dividing, you multiply.

Rule: Divide by the denominator to find one part. Then multiply by the numerator to find how many parts you want.

Example 3 — Find 2/3 of 12.

1. Divide by the denominator: 12 ÷ 3 = 4. That is one third.
2. Multiply by the numerator: 4 × 2 = 8. That is two thirds.
3. So 2/3 of 12 = 8.

Example 4 — Find 3/4 of 16.

1. Divide by the denominator: 16 ÷ 4 = 4 (one quarter).
2. Multiply by the numerator: 4 × 3 = 12 (three quarters).
3. So 3/4 of 16 = 12.

A neat check: the leftover quarter is also 4, and 12 + 4 = 16, the whole. Your parts should always add back up to the whole.

A handy reference table

Here is a table showing the two steps — divide, then multiply — for several fractions.

Find	Divide (number ÷ denominator)	Multiply (× numerator)	Answer
1/2 of 10	10 ÷ 2 = 5	5 × 1	5
1/5 of 25	25 ÷ 5 = 5	5 × 1	5
2/5 of 25	25 ÷ 5 = 5	5 × 2	10
3/4 of 8	8 ÷ 4 = 2	2 × 3	6
2/3 of 15	15 ÷ 3 = 5	5 × 2	10
5/6 of 12	12 ÷ 6 = 2	2 × 5	10

Notice the pattern: the denominator decides how big each part is, and the numerator decides how many parts you collect.

Why this works

Why do we divide first? Think back to the meaning of the denominator. In 2/3, the 3 says the whole is cut into three equal parts. You cannot take "two of the parts" until you know how big one part is — and the only way to find that is to share the number into 3 equal groups, which is division.

Then the numerator simply counts up the parts you want. 2/3 means "take 2 of those 3 equal parts", so you multiply the size of one part by 2. This is exactly why finding a fraction of a number is the same as multiplying: 2/3 of 12 gives the same answer as 2/3 × 12.

It also explains a useful safety check. If the fraction is less than 1 (the numerator is smaller than the denominator), your answer must be smaller than the number you started with. If you ever find 2/3 of 12 and get something bigger than 12, you know a mistake has crept in.

A practice activity

You only need a handful of small objects — buttons, coins, dried pasta or counters.

1. Count out 12 objects.
2. To find 1/3, split them into 3 equal piles. Each pile is one third. Count one pile: 4.
3. To find 2/3, take 2 of those piles: 4 + 4 = 8.
4. Now try 3/4 of 12. First make 4 equal piles (3 in each), then take 3 piles: 3 + 3 + 3 = 9.

Then solve these on paper (answers below): (a) 1/2 of 14, (b) 1/5 of 30, (c) 2/3 of 9, (d) 3/4 of 20, (e) 2/5 of 20.

Answers: (a) 7, (b) 6, (c) 6, (d) 15, (e) 8.

Where this leads

Finding fractions of a number is the foundation for working with Equivalent Fractions and later for percentages, since a percentage is just a fraction out of 100. Practise with real-life sharing — a pizza, a bag of grapes, your pocket money — and the idea will quickly feel natural.