The HCF (highest common factor) is the largest whole number that divides exactly into two or more numbers. For 12 and 18 it is 6, because 6 is the biggest number that divides both with no remainder.

Why divide by the HCF instead of any common factor?

Dividing by any common factor simplifies a fraction, but dividing by the HCF reaches the simplest form in a single step. Using a smaller factor still works, but you have to repeat the process.

Does simplifying change the value of a fraction?

No. Simplifying divides the top and bottom by the same number, which is the same as dividing by 1. The fraction looks different but represents exactly the same amount.

Simplifying Fractions Using the HCF | Math

What "simplest form" means

Two fractions can describe the same amount but look very different. 6/8 and 3/4 are equal, yet 3/4 is tidier. A fraction is in its simplest form (also called lowest terms) when the only number that divides both the top and the bottom is 1.

Simplifying does not change the value — it just writes the fraction in the cleanest possible way, which makes it easier to compare, add and understand.

The slow way (and why we want faster)

You can simplify by dividing the top and bottom by any common factor, then repeating:

24/36 → divide by 2 → 12/18
12/18 → divide by 2 → 6/9
6/9 → divide by 3 → 2/3 ✅

That works, but it took three steps. There is a one-step method using the HCF.

Finding the HCF

The highest common factor (HCF) is the largest whole number that divides both numbers exactly.

Method — list the factors:

To find the HCF of 12 and 18:

Number	Factors
12	1, 2, 3, 4, 6, 12
18	1, 2, 3, 6, 9, 18

The shared factors are 1, 2, 3 and 6. The highest is 6, so HCF(12, 18) = 6.

For bigger numbers, you can instead use prime factors: 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. Multiply the primes they share — one 2 and one 3 — to get 2 × 3 = 6.

Simplifying in one step

Once you know the HCF, divide both the numerator and denominator by it.

Worked example 1: simplify 12/18

HCF of 12 and 18 = 6.
Divide top and bottom by 6: 12 ÷ 6 = 2, 18 ÷ 6 = 3.
Answer: 2/3. Check: HCF(2, 3) = 1, so it is fully simplified. ✅

Worked example 2: simplify 24/36

HCF of 24 and 36 = 12.
24 ÷ 12 = 2, 36 ÷ 12 = 3.
Answer: 2/3 in a single step.

Worked example 3: simplify 45/60

HCF of 45 and 60 = 15.
45 ÷ 15 = 3, 60 ÷ 15 = 4.
Answer: 3/4.

Checking you are finished

After simplifying, ask: "Is there any number bigger than 1 that divides both the new top and bottom?"

2/3 → factors of 2 are 1, 2; factors of 3 are 1, 3 → only 1 shared → done.
6/9 → both divide by 3 → not done, simplify again to 2/3.

If you accidentally used a common factor that was not the HCF, you can simplify again — you will still reach the same simplest form.

A handy table of HCFs

Fraction	HCF of top and bottom	Simplest form
8/12	4	2/3
9/15	3	3/5
16/24	8	2/3
35/49	7	5/7
7/10	1	7/10 (already simplest)

Try it yourself: the HCF race

Write down five "messy" fractions, such as 10/25, 18/24, 14/21, 30/45 and 8/20. For each one:

List the factors of the top and bottom.
Circle the HCF.
Divide both by it and write the simplest form.

Time yourself, then check no fraction can be simplified further.

Why this matters

Simplifying with the HCF is faster, cleaner and the standard way answers are written in exams. It also sharpens the factor skills you will reuse in factors and multiples and connects to the equal-value idea in equivalent fractions.

Simplifying Fractions Using the HCF

Key takeaways