Fraction Calculator (add, subtract, multiply, divide)

Fraction arithmetic

A fraction is a division waiting to happen: the numerator on top divided by the denominator underneath. Working with two of them follows four rules, and this calculator applies them and reduces the answer to lowest terms.

a/b + c/d = (ad + bc) / bd  ·  a/b − c/d = (ad − bc) / bd

a/b × c/d = ac / bd  ·  a/b ÷ c/d = ad / bc

Why adding needs a common denominator

You can only add pieces that are the same size, so halves and thirds have to be rewritten as sixths before they can combine. Multiplying by bd does that for both fractions at once — hence the (ad + bc)/bd pattern. Multiplication needs no such step, which is why it's the easier operation, and dividing is just multiplying by the second fraction flipped upside down.

Reducing to lowest terms

Once you have an answer, divide the top and bottom by their greatest common divisor. So 4/6 has a GCD of 2 and reduces to 2/3 — the same value written more simply. The GCD and LCM calculator finds that divisor for any pair of numbers.

Worked example

For 1/2 + 1/3: the common denominator is 6, so it becomes 3/6 + 2/6 = 5/6, which is already in lowest terms — about 0.833, or 83.3%. For 1/2 ÷ 1/4, flip the second and multiply: 1/2 × 4/1 = 4/2 = 2. Dividing by a fraction smaller than one makes the result bigger, which surprises people but follows straight from the rule.

Exact, which is the point

Unlike most of the calculators here, this one has no approximation in it at all. Fractions are held as integers and reduced by the greatest common divisor, so a third plus a sixth returns exactly one half rather than 0.49999999999999994. The only practical limit is the size of the integers involved: very large numerators and denominators can exceed what a JavaScript number represents exactly, which is far beyond anything arising from ordinary arithmetic.