Fraction Calculator
Add, subtract, multiply, divide fractions and mixed numbers. Shows simplified result, mixed form, decimal, and step-by-step.
Result
Adding and subtracting fractions
You can't add 1/2 + 1/3 directly because they have different denominators (different-sized pieces). You have to convert both to the same-sized pieces first by finding a common denominator — typically the least common multiple (LCM) of the two denominators.
For 1/2 + 1/3: LCM(2, 3) = 6. Convert: 1/2 = 3/6, 1/3 = 2/6. Now add: 3/6 + 2/6 = 5/6. Same approach for subtraction. The calculator handles the LCM math and shows the steps.
Multiplying and dividing fractions
Easier than addition. Multiplying: multiply numerators, multiply denominators. 2/3 × 3/4 = 6/12 = 1/2.
Dividing: multiply by the reciprocal. 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9. The reciprocal of a/b is b/a (flip it). This is why “divide by 2” equals “multiply by 1/2.”
Simplifying with the GCD
A fraction is “simplified” when numerator and denominator share no common factor besides 1. To simplify: find the GCD (greatest common divisor) of numerator and denominator, divide both by it. The Euclidean algorithm finds GCD quickly: GCD(a, b) = GCD(b, a mod b), repeat until b = 0.
Example: simplify 18/24. GCD(18, 24) = 6. 18 ÷ 6 = 3. 24 ÷ 6 = 4. Result: 3/4.
Mixed numbers vs improper fractions
7/4 (improper) and 1 3/4 (mixed) are the same value. For arithmetic, improper fractions are easier — convert mixed to improper first: 1 3/4 = (1 × 4 + 3) / 4 = 7/4. After computing, convert back to mixed if it's the more natural form (cooking, construction, music).
Decimal vs fraction precision
Some fractions have terminating decimals (1/4 = 0.25), others have repeating decimals that you can never write down exactly (1/3 = 0.333..., 1/7 = 0.142857142857...). For exact arithmetic, fractions win. For comparison and easy mental math, decimals usually win.
In computing: 0.1 + 0.2 ≠ 0.3 in IEEE 754 floating point, due to binary representation issues. Fractional libraries (BigInt-based) avoid this for precise calculations. Standard JavaScript Number is fine for everyday math but watch for accumulating rounding errors.
Real-life fraction problems
- Cooking: doubling 3/4 cup recipes (= 1 1/2 cups), halving 1/3 cup (= 1/6 cup).
- Construction: cutting a 7 1/2 foot board in thirds (= 2 1/2 feet each).
- Music: time signatures, note values (quarter notes, eighth notes, dotted notes).
- Probability: 1 in 6 odds = 1/6 ≈ 16.67%.
- Maps and scales: 1/24,000 scale = 1 inch represents 24,000 inches (2,000 feet).
- Stock prices (historical): prices used to be quoted in eighths and sixteenths before decimalization in 2001.
For other math tools: Percentage Calculator for relative changes, Unit Converter for measurement conversions, and Statistics Calculator for descriptive stats.