Fraction Calculator

All four operations, with the working shown — including why division by a fraction means multiplying by its reciprocal.

The four operations

Multiplication is the easy one — straight across, top times top and bottom times bottom. Addition and subtraction need a common denominator first, because you cannot add quarters to sixths any more than you can add metres to pounds.

This calculator uses the product of the denominators as the common denominator and then simplifies. Using the lowest common multiple instead would keep the intermediate numbers smaller, but the final simplified answer is identical either way.

Why dividing means multiplying by the reciprocal

The rule is taught as “keep, change, flip” and rarely explained. The reason is that dividing by a number is the same as multiplying by its multiplicative inverse — the value that multiplies with it to give 1.

The reciprocal of 2/3 is 3/2, because 2/3 × 3/2 = 6/6 = 1. So dividing by 2/3 is multiplying by 3/2. Concretely: “how many halves are in 3?” is 3 ÷ 1/2, and the answer is 6 — which is 3 × 2/1. Dividing by a number smaller than one makes things bigger, and that is why.

Simplifying and mixed numbers

A fraction is in lowest terms when the numerator and denominator share no common factor above 1. To get there, divide both by their greatest common divisor, found efficiently with the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is zero.

For 18/24: GCD(18, 24) = 6, so the fraction reduces to 3/4.

An improper fraction — numerator larger than denominator — converts to a mixed number by dividing: 19/5 is 3 remainder 4, so 3 4/5. Improper fractions are generally easier to compute with, which is why they are preferred in algebra, while mixed numbers are easier to picture, which is why they are preferred in cooking and carpentry.