Fraction Calculator: Add, Subtract, Multiply & Divide

Summary

A fraction represents a part of a whole, written as one integer (the numerator) over another (the denominator). Fractions are fundamental to arithmetic and appear throughout mathematics, cooking, engineering, and finance. This calculator performs addition, subtraction, multiplication, and division on two fractions, automatically simplifying the result using the greatest common divisor (GCD) and converting to mixed numbers when appropriate.

How it works

Each arithmetic operation on fractions follows a specific rule:

1. Addition and subtraction require a common denominator. The calculator finds the least common denominator (LCD), converts both fractions, then adds or subtracts the numerators. The general formula works by cross-multiplying, which always yields a common denominator (though not always the least one — the result is simplified afterward).
2. Multiplication is the simplest operation: multiply the numerators together and the denominators together, then simplify.
3. Division inverts the second fraction (reciprocal) and multiplies. This is the classic “keep, change, flip” rule.

After computing the raw result, the calculator divides both numerator and denominator by their GCD to produce the fully simplified fraction. If the numerator is larger than the denominator, it also converts to a mixed number (e.g., 17/12 becomes 1 5/12).

Finding the GCD

The greatest common divisor is computed using the Euclidean algorithm: repeatedly divide the larger number by the smaller and take the remainder, until the remainder is zero. The last non-zero remainder is the GCD. For example, GCD(8, 12): 12 mod 8 = 4, 8 mod 4 = 0, so GCD = 4.

The formulas

Simplification

After any operation, divide numerator and denominator by their GCD:

Worked example

Inputs explained

• First fraction (a/b) — the numerator and denominator of the first fraction. Both must be integers; the denominator cannot be zero.
• Operation — select addition (+), subtraction (-), multiplication (*), or division (/).
• Second fraction (c/d) — the numerator and denominator of the second fraction. For division, the numerator cannot be zero (division by zero).
• Mixed numbers — if you enter a mixed number (e.g., 1 2/3), the calculator converts it to an improper fraction first (5/3) before performing the operation.

Outputs explained

• Result (improper fraction) — the fully simplified result as a single fraction. For example, 17/12 rather than 34/24.
• Result (mixed number) — if the numerator exceeds the denominator, the result is also shown as a mixed number (e.g., 1 5/12).
• Decimal equivalent — the fraction expressed as a decimal, rounded to a reasonable number of decimal places.
• Step-by-step breakdown — the intermediate calculations showing how the common denominator was found and how simplification was applied.

Assumptions & limitations

• Integer inputs only — the calculator works with integer numerators and denominators. Decimal inputs are not supported directly; convert 0.75 to 3/4 first.
• No complex fractions — nested fractions (fractions within fractions) must be simplified to a single fraction before entry.
• Overflow for very large numbers — while JavaScript can handle very large integers, fractions with numerators or denominators exceeding 2^53 may lose precision due to floating-point representation.
• Single operation at a time — for chain calculations (e.g., 1/2 + 1/3 + 1/4), perform them sequentially, feeding the result of one operation into the next.
• Negative fractions — negative signs are normalized to the numerator. A fraction like 3/(-4) is displayed as -3/4.