Fraction Calculator
Add, subtract, multiply or divide fractions. Result is automatically simplified.
Fraction Operation Rules
| Operation | Rule | Example |
|---|---|---|
| Addition | Common denominator first | 1/4 + 2/3 = 3/12 + 8/12 = 11/12 |
| Subtraction | Common denominator first | 3/4 − 1/3 = 9/12 − 4/12 = 5/12 |
| Multiplication | Top × top, bottom × bottom | 2/3 × 3/5 = 6/15 = 2/5 |
| Division | Multiply by reciprocal | 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6 |
The Wikipedia article on fractions covers historical notation and additional operations like comparisons, conversions, and decimals.
The "1/2 + 1/3" classic mistake
Naive arithmetic: 1/2 + 1/3 = (1+1)/(2+3) = 2/5. Wrong.
You can't simply add numerators and denominators because the fractions are measuring different-sized pieces. 1/2 is one piece out of a pie cut in 2; 1/3 is one piece out of a pie cut in 3 — those pieces aren't the same size. To add them, you need a common denominator (6 works: 6 = 2×3):
1/2 = 3/6 and 1/3 = 2/6 → 3/6 + 2/6 = 5/6
5/6 ≈ 0.833, which is much closer to "halfway plus a third" than the wrong answer of 2/5 = 0.4.
Simplifying — How and Why
To simplify (reduce to "lowest terms"), divide both numerator and denominator by their greatest common factor (GCF). Example: 12/18 → both divisible by 6 → 2/3. The calculator above simplifies automatically using the Euclidean algorithm for GCF.
Why simplify? Mathematically, 12/18 and 2/3 represent identical values. But 2/3 is easier to read, compare to other fractions, and use in further calculations. Most textbooks and tests expect simplified answers.
Improper Fractions vs Mixed Numbers
- Proper fraction: numerator < denominator. Example: 3/4
- Improper fraction: numerator ≥ denominator. Example: 7/3
- Mixed number: whole number + proper fraction. Example: 2⅓ (same value as 7/3)
Mixed numbers are easier to visualize ("two and a third pizzas"). Improper fractions are easier for arithmetic. Convert between them: improper-to-mixed = divide and keep the remainder; mixed-to-improper = multiply whole × denominator, add numerator.
Common Mistakes
- Adding/subtracting without common denominators — the #1 fraction mistake.
- Cross-multiplying when you should regular-multiply — cross-multiplication is for solving proportions, not for products.
- Forgetting to flip the second fraction in division — division means "multiply by reciprocal."
- Not simplifying the final answer — usually expected in school answers.