Simplifying a fraction using the GCD

What does it mean to simplify a fraction?

Simplifying a fraction means finding an equivalent fraction with a smaller numerator and denominator. For example, 6/8 = 3/4 (we have divided both the numerator and the denominator by 2).

The GCD method

To simplify a fraction a/b as much as possible in a single step:

1. Calculate the GCD(a, b).
2. Divide both the numerator AND the denominator by this GCD.

Example: simplify 18/24

• GCD(18, 24) = 6
• 18 / 6 = 3 and 24 / 6 = 4
• Therefore, 18/24 = 3/4

This fraction 3/4 is said to be irreducible: it cannot be simplified any further, as GCD(3, 4) = 1.

Another example, in several steps

Simplify 60/84:

• If you don’t think of the GCD straight away, you can divide step by step: 60/84 = 30/42 (divided by 2) = 15/21 (divided by 2 again) = 5/7 (divided by 3)
• By calculating GCD(60, 84) = 12 directly, we get 60/12 = 5 and 84/12 = 7, so 60/84 = 5/7 straight away.

Summary table

Original fraction	GCD(numerator, denominator)	Simplified fraction
18/24	6	3/4
60/84	12	5/7
21/35	7	3/5

Common pitfall

A common mistake is to divide only the numerator or only the denominator: you must ALWAYS divide both by the same number, otherwise the fraction changes in value.