Simplify Fractions Calculator
Reduce any fraction to lowest terms in one step. Enter a whole-number numerator and denominator — negatives and improper fractions welcome — and get the simplified fraction, the GCD that was divided out, the mixed-number form, and the decimal.
Example: with Numerator (top) 24 · Denominator (bottom) 36 → Simplified fraction: 2/3.
- Greatest common divisor12 — divided top and bottom by 12
- As a mixed number2/3 (already a proper fraction)
- As a decimal≈ 0.666667 (repeating)
Computed by the calculator below using its default values. Change any input to see your own numbers.
Divide numerator and denominator by their greatest common divisor — found with Euclid's algorithm — and the fraction is in lowest terms in a single step: 24/36 ÷ 12/12 = 2/3.
Lowest terms in one step, not five
A fraction is in lowest terms when its top and bottom share no factor bigger than 1. You can get there by repeatedly dividing out small common factors — 24/36 becomes 12/18, then 6/9, then 2/3 — but the direct route is to divide both numbers by their greatest common divisor once. This calculator finds the GCD with Euclid's algorithm, an exact integer method that never touches floating-point, so 2/3 comes out as exactly 2/3.
Simplifying never changes the value, only the notation: 24/36 and 2/3 name the same point on the number line because you divided top and bottom by the same amount. That is also why the decimal forms match (both are 0.666...), and why answers in lowest terms are the convention on tests — every equivalent fraction reduces to exactly one such form.
How it’s calculated
GCD computed by Euclid's algorithm (repeated remainders, exact integer arithmetic); simplified fraction = (|n| ÷ GCD) / (|d| ÷ GCD) with the sign carried on the numerator. Mixed number splits an improper result into whole part |n|÷|d| (floored) and remainder. The decimal is exact when the reduced denominator contains only factors 2 and 5 (shown to exactly the needed places); otherwise it repeats and is shown to 6 places.
Inputs must be whole numbers — for a fraction with decimals in it (like 2.5/4), scale both parts by 10 first (25/40) and simplify that.
Two routes from 24/36 to lowest terms
| Step | Divide both by | Result |
|---|---|---|
| Start | 24/36 | |
| Small factor | 2 | 12/18 |
| Small factor again | 2 | 6/9 |
| Small factor again | 3 | 2/3 |
| Or one step | GCD = 12 | 2/3 |
Computed by dividing numerator and denominator by common factors; the GCD (12 = 2 × 2 × 3) collapses the ladder into one division.
Common mistakes
- Dividing only the numerator (or only the denominator) — both must be divided by the same number or the value changes.
- Stopping too early: 24/36 reduced by 2 gives 12/18, which still hides a common factor of 6.
- Dropping a negative sign during the reduction — the simplified fraction keeps the original's sign.
- Canceling across addition, like turning (2+3)/(2+5) into 3/5 — cancellation only works on factors, not terms.
Frequently asked questions
How do you simplify a fraction?
Divide the numerator and denominator by their greatest common divisor. For 24/36 the GCD is 12, so 24 ÷ 12 = 2 and 36 ÷ 12 = 3, giving 2/3 in a single step.
What is the GCD and how is it found?
The greatest common divisor is the largest whole number dividing both values evenly. Euclid's algorithm finds it fast: replace the larger number with the remainder of the division and repeat — gcd(36, 24) → gcd(24, 12) → gcd(12, 0) = 12.
How do I know a fraction is fully simplified?
When the GCD of the top and bottom is 1 (the numbers are coprime). 6/9 is not done because both divide by 3; 2/3 is done because no whole number above 1 divides both 2 and 3.
What is the difference between an improper fraction and a mixed number?
An improper fraction has a top at least as large as its bottom (48/36 → 4/3); a mixed number writes the same value as a whole part plus a proper fraction (1 1/3). Both are correct — mixed numbers just read more naturally for measurements.
Why does the decimal sometimes say repeating?
A fraction terminates only when its reduced denominator has no prime factors other than 2 and 5. 5/8 terminates at 0.625 because 8 = 2³, but 2/3 repeats forever (0.666...) because of the 3.