Simplest Form of a Fraction — Definition, Formula & Examples

Simplest form of a fraction is when the numerator and denominator have no common factor other than 1. You cannot reduce the fraction any further.

A fraction $\frac{a}{b}$ is in simplest form (also called lowest terms) when $\gcd(a, b) = 1$ , meaning the greatest common factor of the numerator and denominator is 1.

Key Formula

\frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)}

Where:

$a$ = The numerator of the fraction
$b$ = The denominator of the fraction
$\gcd(a,b)$ = The greatest common factor of a and b

How It Works

To simplify a fraction, find the greatest common factor (GCF) of the numerator and denominator. Then divide both the numerator and the denominator by that GCF. The result is the fraction in simplest form. For example, if the GCF is 4, you divide both parts by 4. If the GCF is already 1, the fraction is already in simplest form.

Worked Example

Problem: Write the fraction 12/18 in simplest form.

Find the GCF: List the factors of 12 (1, 2, 3, 4, 6, 12) and 18 (1, 2, 3, 6, 9, 18). The greatest common factor is 6.

\gcd(12, 18) = 6

Divide both parts by the GCF: Divide the numerator and denominator each by 6.

\frac{12 \div 6}{18 \div 6} = \frac{2}{3}

Check: The only common factor of 2 and 3 is 1, so the fraction is fully simplified.

\gcd(2, 3) = 1

Answer:

\frac{12}{18}

in simplest form is

\frac{2}{3}

Why It Matters

Teachers expect answers in simplest form on nearly every fractions assignment and test. Simplifying also makes fractions easier to compare — it is much quicker to see that

\frac{2}{3}

is larger than

\frac{1}{2}

than to compare

\frac{12}{18}

with

\frac{9}{18}

Common Mistakes

Mistake: Dividing the numerator and denominator by a common factor that is not the greatest common factor, then stopping.

Correction: If you divide by a smaller factor (say 2 instead of 6), the fraction is not fully reduced yet. Keep simplifying until no common factor remains, or find the GCF first to do it in one step.