Simplify Fraction
Reduce any fraction to its lowest terms using the Greatest Common Divisor (GCD).
How to Simplify a Fraction
Simplifying (or reducing) a fraction means writing it in its lowest terms — where the numerator and denominator have no common factors other than 1. The process uses the Greatest Common Divisor (GCD):
- Find the GCD of the numerator and denominator.
- Divide both the numerator and denominator by the GCD.
- The result is the simplified fraction.
Worked Example
Simplify 36/48:
- List factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- List factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- GCD(36, 48) = 12
- 36 ÷ 12 = 3 and 48 ÷ 12 = 4
- Result: 36/48 = 3/4
The Euclidean Algorithm for GCD
For large numbers, the Euclidean algorithm is the most efficient method:
Example: GCD(48, 18)
- 48 = 2 × 18 + 12 → GCD(48,18) = GCD(18,12)
- 18 = 1 × 12 + 6 → GCD(18,12) = GCD(12,6)
- 12 = 2 × 6 + 0 → GCD(12,6) = 6
So GCD(48,18) = 6 and 48/18 = 8/3.
Common Uses
- Math homework: Simplifying answers to their required form.
- Cooking: Scaling recipe fractions — 6/8 cup becomes 3/4 cup.
- Measurements: Converting 12/16 inch to 3/4 inch on a ruler.
- Probability: Expressing 15/20 probability as 3/4.
- Algebra: Simplifying rational expressions and algebraic fractions.
Common Mistakes to Avoid
- Only dividing one part: You must divide both numerator AND denominator by the GCD.
- Stopping too early: 12/18 → 6/9 is not fully simplified (both still divisible by 3). Always check GCD of the result.
- Negative fractions: -6/9 simplifies to -2/3. The negative sign belongs to the fraction, not just the numerator.
Frequently Asked Questions
How do you simplify a fraction?
Find the GCD of numerator and denominator, then divide both by it. For 18/24: GCD(18,24) = 6, so 18/24 = 3/4.
What is the Greatest Common Divisor (GCD)?
The GCD is the largest number that divides both the numerator and denominator exactly. GCD(12,18) = 6 because 6 is the largest factor shared by both.
When is a fraction fully simplified?
A fraction is in lowest terms when GCD(numerator, denominator) = 1 — they share no common factors. For example, 3/4 is simplified since GCD(3,4) = 1.
How does the Euclidean algorithm find the GCD?
Repeatedly divide: GCD(48,18) → GCD(18,12) → GCD(12,6) → GCD(6,0) = 6. The algorithm stops when the remainder is 0.
Is the Simplify Fraction tool free?
Yes, completely free. No registration, no limits, no hidden fees.