How to Simplify Fractions: The GCD Method Explained

A fraction like 24/36 and a fraction like 2/3 represent exactly the same value, but one is far easier to work with. Simplifying (or "reducing") a fraction means rewriting it with the smallest possible numerator and denominator while keeping its value unchanged — and the reliable way to do that is with the greatest common divisor.

What "simplifying" a fraction actually means

A fraction is in its simplest form when the numerator and denominator share no common factors other than 1. Multiplying or dividing both the top and bottom of a fraction by the same number never changes its value — it just changes what the fraction looks like. Simplifying takes advantage of this by dividing both numbers by the largest number that evenly divides both.

The GCD method, step by step

The greatest common divisor (GCD), also called the greatest common factor, is the largest number that divides evenly into both the numerator and denominator. Once you find it, dividing both parts of the fraction by the GCD gives you the simplest form in a single step:

For 24/36: list the factors of 24 (1, 2, 3, 4, 6, 8, 12, 24) and the factors of 36 (1, 2, 3, 4, 6, 9, 12, 18, 36). The largest number appearing in both lists is 12, so dividing both parts by 12 gives 24÷12 / 36÷12 = 2/3 — already fully simplified, since 2 and 3 share no common factors besides 1.

A faster way to find the GCD: the Euclidean algorithm

Listing out every factor works fine for small numbers, but becomes tedious for larger ones. The Euclidean algorithm finds the GCD much faster by repeatedly applying one simple rule: divide the larger number by the smaller number, and replace the larger number with the remainder. Repeat until the remainder is 0 — the last non-zero remainder is the GCD.

For example, finding the GCD of 84 and 126: 126 ÷ 84 leaves a remainder of 42. Then 84 ÷ 42 leaves a remainder of 0. Since the last non-zero remainder was 42, the GCD of 84 and 126 is 42 — and 84/126 simplifies to 2/3.

Adding, subtracting, multiplying, and dividing fractions

Simplification also matters for combining fractions. Multiplying fractions is direct — multiply numerators together and denominators together, then simplify the result:

Adding or subtracting fractions requires a common denominator first — typically the least common multiple of the two denominators — before the numerators can be combined. Dividing fractions is multiplication by the reciprocal: flip the second fraction, then multiply. In every case, simplifying the final result (using the GCD method above) keeps the answer in its cleanest, most usable form.