Formula
To simplify a fraction: find the GCD (Greatest Common Divisor) of the numerator and denominator, then divide both by the GCD. The result is the fraction in its lowest terms.
How to use this calculator
Enter the numerator and denominator of any fraction. The calculator finds the GCD, divides both parts, and shows the simplified result as a fraction, mixed number (where applicable), decimal, and percentage — along with a three-step GCD breakdown.
Simplifying fractions formula
A fraction is in its simplest form (lowest terms) when the numerator and denominator share no common factor other than 1. The reduction formula is:
$$\frac{a}{b} = \frac{a \div \gcd(a, b)}{b \div \gcd(a, b)}$$
GCD (Greatest Common Divisor) is also written as GCF (Greatest Common Factor) or HCF (Highest Common Factor) — all refer to the same value: the largest integer that divides both numbers exactly.
Finding the GCD
Three methods exist, each suited to different situations:
Method 1: Euclidean algorithm (fastest)
Repeatedly divide the larger number by the smaller and take the remainder. Stop when the remainder is 0 — the last non-zero remainder is the GCD:
$$\gcd(48, 36): \quad 48 = 1 \times 36 + 12 \implies \gcd(36, 12) \implies 36 = 3 \times 12 + 0 \implies \gcd = 12$$
This method works efficiently on any pair of integers, regardless of size.
Method 2: Prime factorisation
Write both numbers as products of prime factors. The GCD is the product of all shared prime factors. For 18 and 24: 18 = 2 × 3² and 24 = 2³ × 3. Shared factors: 2¹ × 3¹ = 6. GCD = 6.
Method 3: Listing factors
List all factors of both numbers and find the largest common one. Practical only for small numbers. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Largest common: 6.
Worked examples
Example 1: straightforward reduction
$$\frac{18}{24}: \quad \gcd(18, 24) = 6 \implies \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$$
GCD(18, 24) = 6. Divide: 18 ÷ 6 = 3, 24 ÷ 6 = 4. Result: 3/4. Check: GCD(3, 4) = 1 ✓
Example 2: different path, same result
$$\frac{36}{48}: \quad \gcd(36, 48) = 12 \implies \frac{36 \div 12}{48 \div 12} = \frac{3}{4}$$
GCD(36, 48) = 12. Both 18/24 and 36/48 simplify to 3/4 — they are equivalent fractions at different scales.
| Fraction | GCD | Simplified | Decimal |
|---|---|---|---|
| 2/4 | 2 | 1/2 | 0.5 |
| 6/9 | 3 | 2/3 | 0.6667 |
| 8/12 | 4 | 2/3 | 0.6667 |
| 15/25 | 5 | 3/5 | 0.6 |
| 14/21 | 7 | 2/3 | 0.6667 |
| 24/36 | 12 | 2/3 | 0.6667 |
| 45/60 | 15 | 3/4 | 0.75 |
| 100/250 | 50 | 2/5 | 0.4 |
Already in simplest form
If the GCD of numerator and denominator equals 1, the fraction cannot be reduced further — it is already in its simplest form:
$$\frac{7}{13}: \quad \gcd(7, 13) = 1 \implies \text{already in simplest form}$$
7 and 13 are both prime and share no common factor. GCD = 1, so 7/13 is already simplified. Any fraction where both numerator and denominator are prime numbers (and unequal) is automatically in lowest terms.
Improper fractions and mixed numbers
An improper fraction (numerator ≥ denominator) is simplified exactly like a proper fraction. The simplified result may still be improper:
$$\frac{22}{8}: \quad \gcd(22, 8) = 2 \implies \frac{22 \div 2}{8 \div 2} = \frac{11}{4} = 2\frac{3}{4}$$
22/8 → GCD(22, 8) = 2 → 11/4. As a mixed number: 11 ÷ 4 = 2 remainder 3, so 2 3/4.
Note: simplifying and converting to a mixed number are two separate operations. Simplifying reduces the fraction; converting to a mixed number changes its representation. This calculator performs both.
Negative fractions
For negative fractions, the GCD is applied to the absolute values. The sign is preserved in the numerator:
$$\frac{-12}{16}: \quad \gcd(12, 16) = 4 \implies \frac{-12 \div 4}{16 \div 4} = \frac{-3}{4}$$
Convention: the negative sign is placed on the numerator, not the denominator. −3/4 is standard; 3/−4 is avoided in final answers.
Common mistakes
Dividing by a common factor that is not the GCD
Any common factor reduces the fraction, but only dividing by the GCD reduces it in one step to its simplest form. Dividing 18/24 by 2 gives 9/12 — correct but not fully simplified. Dividing by GCD(18, 24) = 6 reaches 3/4 directly.
Thinking that simplification changes the value
Simplifying a fraction does not change its value — it changes its representation. 18/24, 9/12, 6/8, 3/4 are all the same number (0.75). Simplest form is the most compact representation.
Stopping too early
A common error is to divide by a factor that is not the GCD, resulting in a fraction that can still be reduced. Always verify the result: if GCD(new numerator, new denominator) = 1, the simplification is complete.
Simplifying across addition or subtraction
You cannot "cancel" numerators and denominators across a + or − sign. For example, (2 + 3)/(2 + 5) ≠ (1 + 3)/(1 + 5). Simplification applies only to multiplication and division of fractions, not to their components separately.
Frequently asked questions
How do you simplify a fraction?
Find the GCD of the numerator and denominator, then divide both by it. The result is the fraction in its lowest terms.
What does simplest form mean?
A fraction is in simplest form when its numerator and denominator share no common factor other than 1 — their GCD is 1.
What is the GCD?
The Greatest Common Divisor is the largest integer that divides both numbers exactly. Dividing both parts of a fraction by their GCD reduces it to simplest form in one step.
How do you find the GCD?
The Euclidean algorithm is the fastest method: divide the larger by the smaller, take the remainder, and repeat until the remainder is 0. The last non-zero remainder is the GCD.
Can an improper fraction be simplified?
Yes — apply the same GCD method. The simplified result may remain improper, or it may convert to a whole number or mixed number.
What if the GCD is 1?
The fraction is already in simplest form and cannot be reduced further. This applies to all fractions where numerator and denominator are coprime.