Simplifying fractions: how to simplify a fraction step by step

What does simplifying fractions mean?

Simplifying a fraction means finding the equivalent fraction with the smallest possible numbers. ⁶⁄₁₅ and ²⁄₅ are worth exactly the same, but ²⁄₅ is simpler to read, compare and work with.

The shaded parts are the same length. Simplifying doesn't change the value of a fraction, only the way it's written. It's sometimes called reducing a fraction, or cancelling, but the amount never gets smaller.

A fraction is in its simplest form when the only number that divides into both the numerator and the denominator is 1. ²⁄₅ is in its simplest form, because nothing except 1 goes into both 2 and 5.

How to simplify fractions

To simplify a fraction, divide the numerator and the denominator by the same number.

1. Find a common factor. A factor is a number that divides in exactly. A common factor goes into both the numerator and the denominator.
2. Divide both numbers by it. Top and bottom, by the same number.
3. Check. If the new numbers still share a factor other than 1, divide again.

The quickest way is to use the highest common factor: the biggest number that goes into both. Then one step is enough.

Worked example 1: simplify six-fifteenths

The factors of 6 are 1, 2, 3 and 6. The factors of 15 are 1, 3, 5 and 15. The only common factor other than 1 is 3.

Divide both by 3: 6 ÷ 3 = 2, and 15 ÷ 3 = 5.

So ⁶⁄₁₅ = ²⁄₅. The only common factor of 2 and 5 is 1, so this is the simplest form.

Worked example 2: simplify four-twelfths, two ways

4 and 12 have more than one common factor: 2 and 4.

In one step, with the highest common factor. The highest common factor is 4. 4 ÷ 4 = 1 and 12 ÷ 4 = 3, so ⁴⁄₁₂ = ¹⁄₃.

In two steps. If your child spots 2 first, that's fine. Halving both gives ⁴⁄₁₂ = ²⁄₆. 2 and 6 still share a factor of 2, so halve again: ²⁄₆ = ¹⁄₃.

Both ways end at the same answer. Government guidance for teachers says children should learn to use the highest common factor, but that simplifying in more than one step is also right, as long as they check at the end.

Worked example 3: simplify eighteen-twenty-fourths

Look for the highest common factor of 18 and 24. Both are in the 6 times table: 6 × 3 = 18 and 6 × 4 = 24. Nothing bigger goes into both, so the highest common factor is 6.

18 ÷ 6 = 3, and 24 ÷ 6 = 4. So ¹⁸⁄₂₄ = ³⁄₄.

Simplifying fractions bigger than 1

Fractions bigger than 1 simplify in the same way. Government guidance uses this example: ⁹⁄₁₂ + ¹¹⁄₁₂ = ²⁰⁄₁₂.

Simplify first. 20 and 12 have a highest common factor of 4, so ²⁰⁄₁₂ = ⁵⁄₃. Then, as a mixed number, ⁵⁄₃ = 1 ²⁄₃.

Or make a mixed number first. ²⁰⁄₁₂ = 1 ⁸⁄₁₂. Then simplify the fraction part: ⁸⁄₁₂ = ²⁄₃, giving 1 ²⁄₃ again.

Tips for finding common factors

Good times tables knowledge makes this much faster, because factors are just times table facts read backwards.

• Both numbers even? Then 2 is a common factor. Halve both.
• Both end in 0 or 5? Then 5 is a common factor.
• Is the numerator a factor of the denominator? If it goes into the denominator exactly, the fraction simplifies to a unit fraction. ³⁄₉ = ¹⁄₃ and ⁷⁄₂₈ = ¹⁄₄.
• Not sure? List the factors of the smaller number and test each one, biggest first.

The times table grid is handy for spotting which tables two numbers share.

Simplifying fractions in Year 5 and Year 6

Simplifying fractions is in the Year 6 national curriculum: children use common factors to simplify fractions, and give the answers to fraction multiplications in their simplest form. They meet the idea earlier. The Year 4 notes suggest spotting that ⁶⁄₉ can be written as ²⁄₃, and in Year 5 children learn to find common factors of two numbers and to write equivalent fractions.

In Year 6, simplifying also helps with adding and subtracting fractions, multiplying fractions and comparing fractions. See the Year 5 and Year 6 guides for the rest of each year.

How to simplify fractions in algebra

Algebraic fractions come at secondary school, in the GCSE years, not at primary. The idea is exactly the same: divide the top and the bottom by a common factor. For example, 6a/9 simplifies to 2a/3, because 6 and 9 share a factor of 3. If your child is in primary school, the number skills on this page are the ones they need.

Common mistakes to watch for

• Dividing only one number. ⁴⁄₈ divided only on the bottom becomes ⁴⁄₂, which is a different amount. Always divide top and bottom by the same number.
• Stopping too soon. ¹²⁄₁₈ halved is ⁶⁄₉, but 6 and 9 still share 3. The simplest form is ²⁄₃.
• Subtracting instead of dividing. Taking 2 from the top and bottom of ⁴⁄₆ gives ²⁄₄, which isn't equivalent. Simplifying is always dividing.
• Thinking the fraction got smaller. ¹⁄₃ uses smaller numbers than ⁴⁄₁₂, but it's the same amount.

How to help at home

• Practise factors. Ask "what goes into 12?" and "what goes into both 12 and 18?"
• Ask "can it be simpler?" After any fraction answer, check together whether the top and bottom share a factor.
• Use the inverse. Multiply back to check: ²⁄₅ with top and bottom multiplied by 3 should give ⁶⁄₁₅ again.
• Keep the times tables sharp. Spotting common factors is mostly table knowledge.

Simplifying fractions questions to try

Write each fraction in its simplest form.

1. ⁵⁄₁₀
2. ⁶⁄₈
3. ⁹⁄₁₂
4. ¹⁰⁄₂₅
5. ¹⁶⁄₂₄
6. ¹⁴⁄₄, as a mixed number

Answers

1. ¹⁄₂ (divide both by 5)
2. ³⁄₄ (divide both by 2)
3. ³⁄₄ (divide both by 3)
4. ²⁄₅ (divide both by 5)
5. ²⁄₃ (divide both by 8, the highest common factor)
6. 3 ¹⁄₂: ¹⁴⁄₄ = ⁷⁄₂, which is 3 wholes and a half

For everything else children learn in these years, see the KS2 maths guide.