Fraction Calculator (Add, Subtract, Multiply, Divide, Simplify)
Add, subtract, multiply, divide, and simplify fractions. Returns lowest-terms result + decimal + mixed number representation.
Result
5/6
Decimal: 0.833333
Calculation
1/2 + 1/3 = 5/6Related calculators
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Fractions are the bedrock of arithmetic — and the operation many adults forget how to do by hand. The four basic operations (add, subtract, multiply, divide) all follow simple rules but require different setups: addition and subtraction need a common denominator; multiplication is straight numerator-times-numerator over denominator-times-denominator; division flips the second fraction. This calculator handles all four plus simplification to lowest terms via the Euclidean algorithm for GCD.
Every result comes back in three forms: simplified fraction (e.g. 6/8 → 3/4), decimal (0.75), and mixed number when the result is improper (7/3 → 2 1/3). Useful for: cooking (1/3 cup × 2 batches), construction (3/4 inch + 1/8 inch = 7/8 inch), school homework, and financial ratios where exact rational arithmetic matters more than decimal approximations.
The math behind fraction operations
Addition / subtraction: find a common denominator (the simplest is product of denominators), convert both fractions, then add/subtract numerators. 1/2 + 1/3 → (3 + 2)/6 = 5/6. The result is then simplified to lowest terms.
Multiplication: numerator × numerator over denominator × denominator. 2/3 × 3/4 = 6/12 = 1/2. Simplification often happens automatically.
Division: flip the second fraction (reciprocal) and multiply. (3/4) ÷ (1/2) = (3/4) × (2/1) = 6/4 = 3/2 = 1 1/2.
Simplification: divide numerator and denominator by their GCD (greatest common divisor), found via the Euclidean algorithm. 8/12: GCD(8,12)=4, simplified = 2/3. Sign convention: negative is always on the numerator (3/-4 → -3/4).
Practical examples
Adding fractions with different denominators
Setup: 1/2 + 1/3 — needs a common denominator.
LCM of 2 and 3 = 6. (3/6) + (2/6) = **5/6**
Takeaway: When denominators differ, find LCM (least common multiple) for the common denominator.
Multiplying with auto-simplification
Setup: 2/3 × 3/4 — does the result simplify?
(2 × 3) / (3 × 4) = 6/12 = **1/2** (GCD = 6)
Takeaway: After multiplying, divide by GCD to get lowest terms.
Dividing improper fractions to mixed number
Setup: 7/2 ÷ 3/4 = ?
(7/2) × (4/3) = 28/6 = 14/3 = **4 2/3** as mixed number
Takeaway: Improper results convert to mixed numbers for easier reading.
Simplifying a high-GCD fraction
Setup: 48/64 — what is the simplest form?
GCD(48, 64) = 16 → 48/64 = **3/4**
Takeaway: Euclidean algorithm: gcd(48,64) = gcd(16,48) = gcd(16,32) = gcd(16,16) = 16.
Frequently asked questions
What is GCD and why does it matter?▾
Greatest Common Divisor — the largest integer that divides both numerator and denominator. Dividing by GCD reduces the fraction to lowest terms (e.g. 8/12 ÷ 4 = 2/3). Computed via the Euclidean algorithm: gcd(a,b) = gcd(b, a mod b).
Why does 1/2 ÷ 0/5 return an error?▾
Division by zero is undefined. The fraction 0/5 equals zero; dividing by zero has no answer in standard arithmetic.
Can I enter mixed numbers?▾
Convert to improper fraction first: 2 1/3 = (2×3 + 1)/3 = 7/3. The calculator only accepts numerator/denominator pairs. After computing, the result block shows mixed number form when applicable.
Why does 3/-4 become -3/4?▾
Canonical sign convention: negatives belong on the numerator. 3/-4, -3/4, and -(3/4) all represent the same value; we standardise to numerator-negative.
When are fractions better than decimals?▾
When the result is repeating: 1/3 = 0.333... is exactly 1/3 as a fraction but inexact as decimal. Engineering tolerances, recipes scaled by integer factors, and probability problems often prefer exact fractions over rounded decimals.
Sources & references
Cross-check every number in this calculator against the primary sources below.
- ReferenceKhan Academy — Fractions
- AcademicWolfram MathWorld — Euclidean algorithm