Simplify Fractions Calculator

How to use this calculator

Enter a numerator and denominator — the whole-number box is optional, for mixed-number input like 2 6/8. The calculator finds the greatest common factor (GCF) of the numerator and denominator, divides both by it, and shows the fraction in lowest terms. If the reduced fraction is improper (numerator ≥ denominator), it's also converted to a mixed number.

Negative fractions, zero, and whole-number results (like 24/8 = 3) all work — the step-by-step breakdown below the answer shows exactly how each case was handled.

How to simplify a fraction — the GCF method

Every simplification follows the same two-step recipe:

1. Find the GCF (greatest common factor, also called GCD) of the numerator and denominator — the largest number that divides both evenly.
2. Divide numerator and denominator by the GCF. The result is the same value, written with smaller numbers.

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

For 8/12: the GCF of 8 and 12 is 4, so 8 ÷ 4 = 2 and 12 ÷ 4 = 3 — the simplified fraction is $\frac{2}{3}$. If the GCF is 1 (like 3/4), the fraction is already in simplest form and nothing changes.

Why dividing by the GCF doesn't change the value

Dividing numerator and denominator by the same nonzero number is the same as multiplying by $\frac{1}{1}$ (in disguise) — it changes how the value is written, not the value itself. $\frac{8}{12}$ and $\frac{2}{3}$ both equal exactly 0.666…, because $\frac{8}{12} = \frac{2 \times 4}{3 \times 4} = \frac{2}{3} \times \frac{4}{4} = \frac{2}{3} \times 1$.

That's also why the GCF — the largest shared factor — is the right number to divide by: any smaller shared factor leaves the fraction only partly reduced, and any factor that isn't shared by both numbers would change the value.

Improper fractions and mixed numbers

If the numerator is greater than or equal to the denominator after reducing, the fraction is improper — its value is 1 or more. Long division splits it into a whole part and a proper-fraction remainder:

$\frac{a}{b} = q\,\frac{r}{b}, \quad a \div b = q \text{ remainder } r$

Worked examples

Example 1 — 8/12

GCF(8, 12) = 4. 8 ÷ 4 = 2, 12 ÷ 4 = 3. Answer: $\frac{2}{3}$ (already proper — no mixed number needed).

Example 2 — 45/10

GCF(45, 10) = 5. 45 ÷ 5 = 9, 10 ÷ 5 = 2 → $\frac{9}{2}$. Since 9 ≥ 2, convert: 9 ÷ 2 = 4 remainder 1. Answer: $\frac{9}{2} = 4\frac{1}{2}$.

Example 3 — mixed number 2 6/8

Convert to improper first: $(2 \times 8) + 6 = 22$, so $\frac{22}{8}$. GCF(22, 8) = 2 → $\frac{11}{4}$. Since 11 ≥ 4: 11 ÷ 4 = 2 remainder 3. Answer: $\frac{11}{4} = 2\frac{3}{4}$.

Example 4 — -8/12

GCF(8, 12) = 4 (the GCF uses the absolute value). -8 ÷ 4 = -2, 12 ÷ 4 = 3. Answer: $-\frac{2}{3}$.

Common fractions reference

Precision note

Every computation uses arbitrary-precision integers (BigInt), not floating-point numbers — so the GCF and the reduced fraction are always exact, no matter how large the numerator or denominator is.