Fraction Simplifier
Reduce a fraction to its simplest form by dividing the numerator and denominator by their greatest common factor.
Calculator
- GCF reduction
- Sign normalization
- Lowest-terms proof
- Handles zero numerator
Result & calculation receipt
- Method
- Divide numerator and denominator by their greatest common factor
- Formula
n/d = (n ÷ g) / (d ÷ g), g = gcd(|n|, |d|)- Substitution
- 18/24 = 18÷6 / 24÷6 = 3/4
- Steps
- gcd(|18|, |24|) = 6
- 18 ÷ 6 = 3, 24 ÷ 6 = 4
- Denominator already positive.
- Simplest form: 3/4
- Verification
- gcd(|3|, 4) = 1 = 1, so the fraction is fully reduced.
- Exact result
- 3/4
- Decimal result
- 0.75
- Engine
- math-solvers v1.4.0
How this calculator works
The calculator finds the greatest common factor of the numerator and denominator with the Euclidean algorithm, divides both by it, and moves any negative sign to the numerator so the denominator stays positive. It then confirms the reduced fraction cannot simplify further by checking that its numerator and denominator are coprime.
Worked example
18/24 has a greatest common factor of 6, so dividing both parts by 6 gives 3/4. The check confirms gcd(3, 4) = 1, so 3/4 is fully reduced. A fraction like 5/7, whose parts share no factor, is already in lowest terms and is returned unchanged with that note.
Key insight
Reducing a fraction never changes its value — 18/24 and 3/4 are the same number — only its form. Doing it in one step with the greatest common factor is faster and less error-prone than repeatedly cancelling small common factors.
Frequently asked questions
What does simplifying a fraction mean?
It means writing the same value with the smallest possible whole-number numerator and denominator, by dividing both by every factor they share. 18/24 and 3/4 are equal; 3/4 is the simplified form.
How is the fraction reduced in one step?
By dividing the numerator and denominator by their greatest common factor. Since the GCF contains all shared factors, one division puts the fraction in lowest terms.
How do I know a fraction is fully reduced?
When the greatest common factor of its numerator and denominator is 1 — that is, they are coprime. The calculator confirms this for the result.
What happens to a negative fraction?
The sign is placed on the numerator and the denominator is kept positive, so −6/8 becomes −3/4. The value is unchanged.
What if the numerator is zero?
0 over any non-zero denominator is 0, returned as 0. A zero denominator is undefined and returns an error.
Methodology & verification
The fraction is reduced by dividing the numerator and denominator by their greatest common factor (Euclidean algorithm), with the sign normalized onto the numerator so the denominator stays positive. Inputs are validated as integers with a non-zero denominator. The receipt confirms the reduced pair is coprime (gcd = 1), proving the fraction cannot simplify further, and reports the decimal value alongside the exact fraction.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.