Reduced Fraction — Definition, Formula & Examples

A reduced fraction is a fraction where the numerator and denominator have been divided by their greatest common factor so that no number other than 1 divides evenly into both.

A fraction $\frac{a}{b}$ is said to be reduced (or in lowest terms) if $\gcd(a, b) = 1$ , meaning the greatest common divisor of the numerator $a$ and the denominator $b$ is 1.

How It Works

To reduce a fraction, find the greatest common factor (GCF) of the numerator and denominator, then divide both by that number. For example, the GCF of 12 and 18 is 6, so

\frac{12}{18}

becomes

\frac{2}{3}

. If the GCF is 1, the fraction is already reduced. Every fraction has exactly one reduced form.

Worked Example

Problem: Reduce the fraction 24/36 to lowest terms.

Find the GCF: List the factors of 24 (1, 2, 3, 4, 6, 8, 12, 24) and 36 (1, 2, 3, 4, 6, 9, 12, 18, 36). The greatest common factor is 12.

\gcd(24, 36) = 12

Divide both parts by the GCF: Divide the numerator and denominator each by 12.

\frac{24 \div 12}{36 \div 12} = \frac{2}{3}

Answer:

\frac{24}{36}

reduced to lowest terms is

\frac{2}{3}

Why It Matters

Reducing fractions makes them easier to compare, add, and subtract. Teachers in pre-algebra and algebra courses typically require answers in reduced form, and standardized tests mark unreduced fractions as incomplete.

Common Mistakes

Mistake: Dividing by a common factor that is not the greatest common factor, then stopping too early.

Correction: After dividing, check whether the new numerator and denominator still share a factor. Repeat until no common factor remains, or find the GCF first to reduce in one step.