How to Simplify Fractions: KS3 Maths Step-by-Step Guide

To simplify a fraction, divide both the numerator and the denominator by their highest common factor (HCF). The result is an equivalent fraction in its lowest terms — it represents the same value but with the smallest possible whole numbers.

What does simplifying a fraction mean?

A fraction is in its simplest form (or lowest terms) when the numerator and denominator share no common factor other than 1. For example, 6/8 is not in its simplest form because both 6 and 8 can be divided by 2. Dividing both by 2 gives 3/4, which is in its simplest form.

Simplifying does not change the value of the fraction — 6/8 and 3/4 are equal. It simply makes the fraction easier to work with in calculations and is expected by KS3 and GCSE mark schemes.

What is the highest common factor?

The highest common factor (HCF) of two numbers is the largest number that divides exactly into both. To find it:

List all the factors of each number.

Identify the factors that appear in both lists.

Select the largest one.

Example: find the HCF of 12 and 18

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18

Common factors: 1, 2, 3, 6. The HCF is 6.

How to simplify a fraction: step-by-step

Step 1 — Find the HCF of the numerator and denominator

Step 2 — Divide both the numerator and denominator by the HCF

Step 3 — Check the result: the new numerator and denominator should share no common factor other than 1

Worked example 1

Simplify 10/15.

Step 1 — Factors of 10: 1, 2, 5, 10. Factors of 15: 1, 3, 5, 15. HCF = 5.

Step 2 — Divide both by 5:
10 ÷ 5 = 2 and 15 ÷ 5 = 3

Step 3 — Result: 2/3

Do 2 and 3 share a common factor other than 1? No. The fraction is fully simplified.

Answer: 2/3

Worked example 2

Simplify 24/36.

Step 1 — Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. HCF = 12.

Step 2 — Divide both by 12:
24 ÷ 12 = 2 and 36 ÷ 12 = 3

Step 3 — Result: 2/3

Answer: 2/3

Worked example 3

Simplify 45/60.

Step 1: Factors of 45: 1, 3, 5, 9, 15, 45. Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. HCF = 15.

Step 2: 45 ÷ 15 = 3, 60 ÷ 15 = 4

Answer: 3/4

Simplifying in stages (dividing by smaller factors)

If you cannot spot the HCF immediately, you can simplify in stages by dividing by a smaller common factor more than once. You will reach the same answer, but it takes more steps.

Worked example 4 (staged approach)

Simplify 48/72.

Spot that both are even, so divide by 2: 48/72 → 24/36
Both still even: 24/36 → 12/18
Both still even: 12/18 → 6/9
Both divisible by 3: 6/9 → 2/3

Answer: 2/3 (same result as finding HCF = 24 directly)

The staged approach always works, but finding the HCF in one step is quicker.

Simplifying fractions that contain large numbers

When the numbers are large, use prime factorisation to find the HCF reliably.

Worked example 5

Simplify 84/126.

Prime factorisation of 84: 84 = 2 × 2 × 3 × 7
Prime factorisation of 126: 126 = 2 × 3 × 3 × 7

Common prime factors: 2, 3, and 7.
HCF = 2 × 3 × 7 = 42.

84 ÷ 42 = 2, 126 ÷ 42 = 3

Answer: 2/3

Simplifying improper fractions and mixed numbers

The same method applies to improper fractions (where the numerator is greater than the denominator).

Worked example 6

Simplify 18/12.

HCF of 18 and 12: factors of 18 are 1, 2, 3, 6, 9, 18; factors of 12 are 1, 2, 3, 4, 6, 12. HCF = 6.

18 ÷ 6 = 3, 12 ÷ 6 = 2 → 3/2

This can also be written as the mixed number 1 1/2.

Answer: 3/2 (or 1 1/2)

Summary table

Fraction | HCF | Simplified form
6/8 | 2 | 3/4
10/15 | 5 | 2/3
24/36 | 12 | 2/3
45/60 | 15 | 3/4
84/126 | 42 | 2/3
18/12 | 6 | 3/2

Common mistakes to avoid

Mistake 1 — Dividing only the numerator or only the denominator.
You must divide both by the same number to keep the fraction equivalent.

Mistake 2 — Stopping too early.
If you divide by a common factor that is not the HCF, you will produce a fraction that can be simplified further. Always check whether the new numerator and denominator still share a common factor.

Mistake 3 — Confusing simplify with solve.
Simplifying does not change the value. 3/4 and 6/8 are the same amount.

Mistake 4 — Forgetting to simplify the final answer.
KS3 and GCSE mark schemes require answers in their simplest form. Always check as your last step.

How simplifying fractions fits the KS3 national curriculum

The Department for Education's KS3 mathematics programme of study requires pupils to "use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factors [and] lowest common multiples." Simplifying fractions draws on all of these skills. Working with equivalent fractions and lowest terms is a core number skill set out in the statutory national curriculum for Key Stage 3 and 4, underpinning later work on ratio, proportion, and algebraic fractions at GCSE.

Frequently asked questions

What is the quickest way to simplify a fraction at KS3?

The quickest method is to find the HCF of the numerator and denominator in one step and divide both by it. To find the HCF efficiently, list factors of the smaller number first (there are fewer of them) and check which also divide the larger number. The biggest one that does is the HCF.

How do I know whether a fraction is already in its simplest form?

A fraction is in its simplest form when 1 is the only number that divides exactly into both the numerator and the denominator. To check, test whether both numbers share a common factor of 2, 3, 5, or 7 — these catch most cases at KS3. If none of those divide both evenly, the fraction is almost certainly in its simplest form.

Can a fraction with a prime numerator always be simplified?

Not always, but it is simple to check. A prime number has only two factors: 1 and itself. So a fraction with a prime numerator can only be simplified if the denominator is a multiple of that prime. For example, 7/21 simplifies to 1/3 because 21 = 3 × 7. But 7/15 is already in its simplest form because 15 is not divisible by 7.

Is simplifying the same as cancelling?

Yes — "cancelling" is an informal term for the same process. You divide numerator and denominator by the same factor, effectively "cancelling" it out. The formal term used in KS3 and GCSE mark schemes is "simplify" or "express in lowest terms."

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