Fraction Calculator
Free fraction calculator. Simplify any fraction to its lowest terms using GCD, or add, subtract, multiply, and divide two fractions. Instantly get the simplified fraction, decimal equivalent, and mixed number form.
Simplify fractions and perform addition, subtraction, multiplication, and division with exact results.
A fraction calculator lets you simplify fractions to their lowest terms or perform exact arithmetic — addition, subtraction, multiplication, and division — on two fractions. Results appear as a simplified fraction, a decimal, and a mixed number, so you can work with whichever form suits your needs.
What Is a Fraction?
A fraction represents a part of a whole. It consists of two integers:
- Numerator (top number) — how many parts you have.
- Denominator (bottom number) — how many equal parts the whole is divided into.
For example, 3/4 means 3 out of 4 equal parts.
Types of Fractions
| Type | Description | Example |
|---|---|---|
| Proper | Numerator < denominator | 3/4 |
| Improper | Numerator ≥ denominator | 7/4 |
| Mixed number | Whole + proper fraction | 1¾ |
| Equivalent | Different form, same value | 2/4 = 1/2 |
How to Simplify Fractions
Simplifying (reducing) a fraction means dividing both numerator and denominator by their Greatest Common Divisor (GCD).
Steps:
- Find GCD(numerator, denominator) using the Euclidean algorithm.
- Divide both by the GCD.
Example: Simplify 18/24
- GCD(18, 24) = 6
- 18 ÷ 6 = 3; 24 ÷ 6 = 4
- Result: 3/4
How to Add and Subtract Fractions
Same Denominator
Simply add or subtract the numerators: a/c ± b/c = (a ± b)/c
Example: 3/8 + 1/8 = 4/8 = 1/2
Different Denominators
Use the formula: a/b + c/d = (a×d + c×b) / (b×d), then simplify.
Example: 1/2 + 1/3
- (1×3 + 1×2) / (2×3) = 5/6
- GCD(5, 6) = 1 → already simplified
- Result: 5/6
How to Multiply and Divide Fractions
Multiplication
a/b × c/d = (a×c) / (b×d)
Example: 2/3 × 3/4 = 6/12 = 1/2
Division
a/b ÷ c/d = (a×d) / (b×c) — multiply by the reciprocal.
Example: 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 (or 1½)
Worked Examples
Example 1 — Adding Unlike Fractions
5/6 + 7/8 = ?
Using the formula:
- Numerator: 5×8 + 7×6 = 40 + 42 = 82
- Denominator: 6×8 = 48
- GCD(82, 48) = 2
- Result: 41/24 = 1 and 17/24 ≈ 1.708
Example 2 — Dividing Fractions (Recipe Scaling)
A recipe calls for 3/4 cup of flour, but you want to make half the recipe.
3/4 ÷ 2 = 3/4 × 1/2 = 3/8 cup of flour.
Example 3 — Simplifying a Large Fraction
Simplify 360/480:
- GCD(360, 480) = 120
- 360 ÷ 120 = 3; 480 ÷ 120 = 4
- Result: 3/4
Common Mistakes
- Adding denominators directly. 1/2 + 1/3 ≠ 2/5. Always find a common denominator first.
- Forgetting to simplify after arithmetic. Always check if the result can be reduced.
- Dividing instead of multiplying by the reciprocal. To divide by a fraction, flip the second fraction and multiply.
- Ignoring the sign. Negative fractions: (-3)/4 = 3/(-4) = -3/4. The denominator should always be positive in the simplified form.
Frequently Asked Questions
What is the GCD and why does it matter? The Greatest Common Divisor (GCD) is the largest integer that divides both numbers without remainder. Dividing numerator and denominator by the GCD gives the simplest form of a fraction. The Euclidean algorithm finds it efficiently: GCD(a, b) = GCD(b, a mod b) repeated until b = 0.
What is the difference between an improper fraction and a mixed number? They represent the same value in different forms. 7/4 is an improper fraction (numerator ≥ denominator). 1¾ is the mixed number form (1 whole + 3/4). This calculator shows both.
Can fractions have negative denominators? Mathematically yes, but by convention the denominator is always written as positive. A negative fraction is shown with the minus sign on the numerator: -3/4.
How do I compare two fractions? Convert both to the same denominator (or to decimals) and compare. For example, which is larger, 5/8 or 3/5? 5/8 = 0.625, 3/5 = 0.600, so 5/8 is larger.