Fraction Calculator
Add, subtract, multiply and divide fractions with full step-by-step workings
// History
How to Do Fraction Arithmetic
Fractions represent parts of a whole. The top number (numerator) tells how many parts you have; the bottom number (denominator) tells how many parts make up a whole. All fraction operations follow a consistent set of rules.
The Four Operations
Mixed Numbers
A mixed number has a whole part and a fraction (e.g. 2¾). To convert to an improper fraction: multiply the whole by the denominator and add the numerator. 2¾ = (2×4 + 3)/4 = 11/4. This calculator handles mixed numbers automatically.
Finding the GCD
The Greatest Common Divisor (GCD) — also called the Highest Common Factor (HCF) — is the largest number that divides both the numerator and denominator exactly. It is used to simplify fractions to their lowest terms. This calculator uses the Euclidean algorithm to find the GCD.
Fractions: The Rules and Why They Work
Built and verified by Andrius R. · Updated June 2026
Fraction arithmetic comes down to four operations and one idea: you can only add or subtract pieces that are the same size. Everything else follows.
Adding and subtracting: find a common denominator
Thirds and quarters are different-sized pieces, so convert both to a size that fits each: the least common denominator of 3 and 4 is 12.
2/3 = 8/12 and 3/4 = 9/12 → 8/12 + 9/12 = 17/12 = 1 5/12.
Any common denominator works (e.g. 24); the least one just keeps numbers small. Subtraction is identical with a minus sign.
Multiplying: straight across
Multiply numerators, multiply denominators: 2/3 × 3/4 = 6/12 = 1/2. No common denominator needed. Pro move — cancel before multiplying: the 3s cancel immediately, leaving 2/4 = 1/2 with no big numbers ever appearing. Why it works: "2/3 of 3/4" means taking two-thirds of a three-quarter piece, and scaling shrinks both dimensions at once.
Dividing: multiply by the reciprocal
"Keep, change, flip": 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9. The logic isn't magic — division asks "how many 3/4s fit into 2/3?", and dividing by a number is the same as multiplying by its inverse, exactly as dividing by 2 equals multiplying by ½.
Simplifying: divide out the GCD
To reduce 24/36, find the greatest common divisor of 24 and 36 — which is 12 — and divide both: 2/3. If hunting the GCD feels slow, divide by any obvious common factor repeatedly (24/36 → 12/18 → 6/9 → 2/3); you reach the same place. Our LCM & GCD calculator handles the lookup instantly.
Mixed numbers and improper fractions
To compute with mixed numbers, convert first: 2 1/3 = (2×3 + 1)/3 = 7/3. To convert back, divide: 17/12 → 17 ÷ 12 = 1 remainder 5 → 1 5/12. Calculations are far less error-prone in improper form; mixed form is for presenting the answer.
Fractions ↔ decimals ↔ percentages
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.3% |
| 1/4 | 0.25 | 25% |
| 1/8 | 0.125 | 12.5% |
| 2/3 | 0.666… | 66.7% |
| 3/4 | 0.75 | 75% |
Fraction → decimal is just division (3 ÷ 4 = 0.75). Decimal → fraction: put the digits over the matching power of ten and simplify (0.75 = 75/100 = 3/4). A useful fact: a fraction in lowest terms gives a terminating decimal only when its denominator's prime factors are exclusively 2s and 5s — which is exactly why 1/3 repeats forever and 1/8 doesn't.
The two classic mistakes
- Adding straight across: 1/2 + 1/3 ≠ 2/5. (It's 5/6 — and 2/5 is actually smaller than 1/2, so the error fails a basic sanity check.)
- Forgetting that dividing by a fraction makes things bigger: 6 ÷ 1/2 = 12, not 3. "How many halves are in six?" — twelve.
From the Blog
// Reciprocal Trick
To divide by a fraction, flip it (find its reciprocal) and multiply. ½ ÷ ¼ = ½ × 4/1 = 2.
// Simplify First
When multiplying fractions, simplify diagonally before multiplying — it keeps numbers small. 4/9 × 3/8 → 1/3 × 1/2 = 1/6.
// LCD not just CD
Use the Least Common Denominator (not just any common denominator) to keep numbers manageable when adding or subtracting.
// Mixed Numbers
Always convert mixed numbers to improper fractions before doing arithmetic, then convert back at the end.