Simplify Fractions Calculator
Reduce a fraction to lowest terms by dividing numerator and denominator by their greatest common divisor (GCD). Returns simplified fraction or mixed number.
Last updated: September 2026
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About this calculator
A fraction is in lowest terms (or simplest form) when its numerator and denominator share no common factor greater than 1. To simplify, compute GCD(numerator, denominator) via the Euclidean algorithm — repeatedly replace (a, b) with (b, a mod b) until b = 0; the remaining a is the GCD. Then divide both numerator and denominator by the GCD.
Euclidean algorithm example: GCD(18, 24). 18 mod 24 = 18. Swap: (24, 18). 24 mod 18 = 6. Swap: (18, 6). 18 mod 6 = 0. GCD = 6. So 18/24 = 3/4.
Simplification is a canonical form: every equivalent fraction has exactly one lowest-terms representative, which makes them easy to compare (are 6/8 and 9/12 equal? Both simplify to 3/4, so yes). Negative fractions are normalized so the denominator is positive: −3/−4 becomes 3/4; 3/−4 becomes −3/4.
If the numerator is a multiple of the denominator, the fraction is an integer (e.g. 15/5 = 3). If the numerator's absolute value exceeds the denominator, it is an improper fraction that can be written as a mixed number (e.g. 7/3 = 2 ⅓).
How to use
Example — simplify 18/24. GCD(18, 24) = 6. 18 ÷ 6 = 3, 24 ÷ 6 = 4. Simplified: 3/4. Example — 84/126 as a mixed number. GCD = 42. 84/42 = 2, 126/42 = 3. Simplified 2/3. Because 2 < 3, no mixed form is needed. If we had 126/42 instead, it would simplify to 3/1 = 3, an integer.
Frequently asked questions
How does the calculator find the GCD?
It uses the Euclidean algorithm, which is the fastest classical method: repeatedly replace (a, b) with (b, a mod b) until the second number is zero. The remaining first number is the GCD. This works in O(log(min(a, b))) steps and is exact for any integer inputs the calculator can handle.
What if the denominator is zero?
Division by zero is undefined. The calculator returns "undefined" for that case rather than a numerical result. In mathematics, x/0 is not equal to any real number — the expression is meaningless.
Why normalize the sign to a positive denominator?
By convention, fractions are written with a positive denominator. −3/4 is standard; 3/−4 is equivalent but non-standard. The calculator moves any denominator negative sign onto the numerator so the result reads normally.
Can I simplify a fraction with decimals?
Not directly — the GCD is only defined for integers. To simplify 0.75/1.5, first multiply both by 100 (or the smallest power of 10 that clears the decimals): 75/150. Then GCD(75, 150) = 75, giving 1/2. Or use a decimal-to-fraction converter to get the exact rational form first.