How to Simplify Fractions
A fraction is in its simplest (lowest) form when the numerator and denominator share no common factor other than 1. Simplifying divides both by their Greatest Common Divisor (GCD).
Step-by-Step Method
Simplify 18/24:
Step 1 — Find GCD(18, 24)
18 = 2 × 3² | 24 = 2³ × 3 → GCD = 6
Step 2 — Divide both by GCD
18 ÷ 6 = 3 | 24 ÷ 6 = 4
Result: 3/4Finding GCD with the Euclidean Algorithm
GCD(48, 36):
48 = 1 × 36 + 12
36 = 3 × 12 + 0
GCD = 12
48/36 simplified = 4/3 (improper → 1⅓ as mixed number)Improper Fractions and Mixed Numbers
Convert 22/8 to simplest mixed number:
Step 1 — Simplify: GCD(22,8)=2 → 11/4
Step 2 — Divide: 11 ÷ 4 = 2 remainder 3
Result: 2¾Quick Divisibility Checks
- Both even → divide by 2 first
- Both end in 0 or 5 → divide by 5
- Both digit-sums divisible by 3 → divide by 3
- If no quick factor found → use prime factorization
Simplify fractions instantly: Free Simplify Fraction Calculator
Step-by-Step Simplification
To simplify a fraction: (1) Find the GCD of the numerator and denominator. (2) Divide both by the GCD. Example: 36/48. GCD(36,48): 36 = 2²×3², 48 = 2⁴×3. GCD = 2²×3 = 12. So 36/48 = 3/4. Verify: 3 and 4 share no common factors — the fraction is fully simplified. A fraction is in lowest terms when GCD(numerator, denominator) = 1.
Finding GCD: Two Methods
- Prime factorisation: Factorise both numbers and take the product of common prime factors (lowest powers). Clear but slow for large numbers.
- Euclidean algorithm: Repeatedly apply GCD(a,b) = GCD(b, a mod b) until remainder is 0. GCD(48,36): GCD(48,36) → GCD(36,12) → GCD(12,0) = 12. Fast even for large numbers.
Frequently Asked Questions
What does it mean for a fraction to be in lowest terms?
A fraction a/b is in lowest terms (fully simplified, reduced) when GCD(a,b) = 1 — the numerator and denominator share no common factors except 1. Equivalently, no integer greater than 1 divides both evenly. 7/8 is in lowest terms (7 is prime, 8 = 2³, no common factors). 6/9 is not in lowest terms (both divisible by 3); reduced: 2/3.
Can I simplify a fraction where the numerator is larger than the denominator?
Yes — improper fractions (numerator ≥ denominator) simplify the same way. 18/12: GCD(18,12) = 6, so 18/12 = 3/2. You can also convert to a mixed number: 3/2 = 1½. Whether to leave as an improper fraction or convert to a mixed number depends on context — improper fractions are usually preferred in algebra; mixed numbers in everyday measurement.
Does simplifying a fraction change its value?
No. Simplification only changes the representation, not the value. 36/48 = 3/4 = 0.75 exactly. Multiplying or dividing both numerator and denominator by the same non-zero number (equivalent fractions) preserves value. This is the same as multiplying by 1 in the form k/k.