Simplify Fractions Calculator

A fraction is in simplest form (also called lowest terms) when the numerator and denominator share no common factor except $1$.

Reducing does not change the value. Dividing top and bottom by the same number is dividing by $1$:

$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$

$\frac{18}{24}$ and $\frac{3}{4}$ are the same amount — both are $0.75$. The pieces are just bigger in $\frac{3}{4}$.

This is why simplifying is safe anywhere in a calculation, and why answers are marked wrong for not being reduced: $\frac{6}{8}$ is a correct value written in an unfinished form.

A fast first pass uses the divisibility tests: both even means divide by $2$; both digit sums divisible by $3$ means divide by $3$; both ending in $0$ or $5$ means divide by $5$. Running those three in turn reduces most fractions you will meet without any factor lists at all.

1. List the factors of each number and find the largest they share — the greatest common factor.
   • $18$: $1, 2, 3, 6, 9, 18$
   • $24$: $1, 2, 3, 4, 6, 8, 12, 24$
   • Largest shared factor: $6$.
2. Divide the numerator by the GCF. $18 \div 6 = 3$.
3. Divide the denominator by the same GCF. $24 \div 6 = 4$.
4. Answer: $\frac{3}{4}$.

You do not have to find the greatest factor on the first try. Chipping away works too: $\frac{45}{60} = \frac{9}{12} = \frac{3}{4}$, dividing by $5$ then by $3$. It takes more rounds but lands in the same place — provided you keep going until nothing divides both.

For awkward numbers, break both into primes and cancel what matches:

$\frac{84}{126} = \frac{2^2 \times 3 \times 7}{2 \times 3^2 \times 7} = \frac{2}{3}$

The shared part is $2 \times 3 \times 7 = 42$, which is the GCF. Prime factorisation finds it without any guessing.

Where people go wrong:

• Stopping early. $\frac{18}{24} \to \frac{9}{12}$ is progress, not an answer; $9$ and $12$ still share a $3$.
• Cancelling digits instead of factors. $\frac{16}{64}$ is not $\frac{1}{4}$ because "the sixes cancel" — you may only divide by whole factors, and here the GCF genuinely is $16$.
• Subtracting instead of dividing. $\frac{18}{24}$ is not $\frac{12}{18}$; simplifying is division, never subtraction.
• Improper fractions still reduce. $\frac{150}{45} = \frac{10}{3} = 3\frac{1}{3}$.

One shortcut worth knowing: if the numerator divides evenly into the denominator, the numerator is the GCF. $\frac{7}{21}$ reduces straight to $\frac{1}{3}$ in a single step, and $\frac{9}{9}$ reduces to $1$.