Fraction Calculator

Fraction Math Made Simple

Fractions represent part of a whole, written as a numerator (top number) over a denominator (bottom number). While the concept is simple, the arithmetic rules for adding, subtracting, multiplying, and dividing fractions are different from regular number arithmetic, and mixing up the rules is one of the most common math mistakes. This guide covers all six operations this calculator handles, with the exact steps for each.

Adding and Subtracting Fractions

Unlike multiplication, you can't add or subtract fractions directly unless they share the same denominator. The trick is to find a common denominator first, which is done by cross-multiplying.

Example: 1/2 + 1/3. Common denominator via cross-multiplication: (1×3 + 1×2) ÷ (2×3) = (3+2) ÷ 6 = 5/6.

Example: 3/4 − 1/6. (3×6 − 1×4) ÷ (4×6) = (18−4) ÷ 24 = 14/24, which simplifies to 7/12.

Multiplying Fractions

Multiplication is the simplest fraction operation: multiply the numerators together, then multiply the denominators together. No common denominator needed.

Example: 2/3 × 3/5 = (2×3) ÷ (3×5) = 6/15, which simplifies to 2/5.

Dividing Fractions

Dividing fractions uses a trick called "keep, change, flip": keep the first fraction, change division to multiplication, and flip (invert) the second fraction.

Example: 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2. This makes intuitive sense too, there are 2 quarters in a half.

Simplifying a Fraction

Simplifying means reducing a fraction to its lowest terms by dividing both numerator and denominator by their greatest common divisor (GCD).

Example: Simplify 18/24. The GCD of 18 and 24 is 6. 18 ÷ 6 = 3, and 24 ÷ 6 = 4. Simplified: 3/4.

A simplified fraction represents the exact same value as the original but is easier to read, compare, and work with in further calculations.

Converting a Fraction to a Decimal

Converting is the most direct operation of all: just divide the numerator by the denominator.

Example: 5/8 as a decimal: 5 ÷ 8 = 0.625.

This is useful whenever you need to compare a fraction to a decimal or percentage figure, plug it into a formula that expects decimal input, or just get an intuitive sense of a fraction's size.

Why the Common Denominator Rule Exists

The reason addition and subtraction need a common denominator, while multiplication doesn't, comes down to what each operation actually means. Adding 1/2 and 1/3 only makes sense once both are expressed in the same "unit size" (sixths, in this case), similar to how you can't directly add "3 apples and 2 oranges" without first converting to a common category like "pieces of fruit." Multiplication, by contrast, is really asking "what's a third of a half," which doesn't require matching units at all, it's just scaling one fraction by another.