Fraction Calculator

How to use the fraction calculator

Four operations, four different rules, and only one of them works the way people expect. Multiplication is the easy case: multiply across the top, multiply across the bottom, reduce. Division is that with the second fraction turned upside down. Addition and subtraction are where the work is, because they need a common denominator first.

The rule that gets broken most often is that only the numerators are added. Three quarters plus one quarter is four quarters, not four eighths — the denominator names the size of the pieces, and adding two fractions does not make the pieces smaller. Once the denominators match, the bottom number is simply carried through unchanged.

This calculator does every step in whole numbers, which sounds like an implementation detail and is not. A third has no exact decimal form, so a tool that converts to decimals first and adds afterwards can return 0.9999999999999998 where the answer is exactly 1. Working in numerators and denominators throughout removes that failure entirely.

Common denominators can be found two ways. Multiplying the denominators together always works and needs no thought, but it can produce large numbers that then reduce a long way. Using their least common multiple keeps the arithmetic small: for sixths and eighths, 24 is the least common multiple while multiplying gives 48, which then halves back down.

Reducing, and why the greatest common factor is the right divisor

A fraction is in lowest terms when its numerator and denominator share no factor above one. Getting there in a single step means dividing both by their greatest common factor. Dividing by anything smaller works but leaves more to do: 12/18 divided by 2 is 6/9, which still reduces to 2/3.

There is a shortcut worth knowing for multiplication: you can cancel diagonally before multiplying. In 2/3 × 3/4 the 3 on top and the 3 underneath cancel, leaving 2/1 × 1/4 = 2/4 = 1/2. Same answer, smaller numbers, and much less chance of an arithmetic slip on the way.

The words, and one that is quietly being retired

A proper fraction has a numerator smaller than its denominator. An improper fraction does not — 7/3 and 4/4 both qualify. A mixed number pairs a whole number with a proper fraction, so 7/3 becomes 2 1/3. All three terms are standard school vocabulary in England and the United States alike.

The word "improper" is the one under quiet pressure. California's mathematics framework refers to "so-called improper fractions" and points out that the label wrongly implies something needs fixing. Nothing does: 7/3 is a perfectly good number, and for most arithmetic it is the more convenient form. Where a plainer phrase helps, "a fraction greater than one" says it without the judgement.

The formula, worked line by line

Four formulas, and a fifth for turning a mixed number into something you can compute with. Each is written below in the form that generalises, rather than as a worked example, because the worked examples are in the table above.

It is worth holding on to because the wrong rule is seductive: adding across the top and across the bottom is what multiplication does, and it looks like consistency. It gives 2/8 for a half plus a quarter, which is a quarter — smaller than either of the things being added.

a/b + c/d = (ad + cb) / bd
a/b − c/d = (ad − cb) / bd
a/b × c/d = ac / bd
a/b ÷ c/d = ad / bc
w n/d = (|w| × d + n) / d, with the sign of w

Why decimals are the wrong intermediate

A fraction converts to a terminating decimal only when its denominator, once reduced, has no prime factors besides two and five. A third, a sixth, a seventh and a ninth all fail that test and become infinite repeating decimals, which a computer must cut short. Every cut introduces an error that then propagates through the rest of the calculation.

Fractions that do and do not terminate

That is why this page keeps everything in integers until the last moment. The decimal at the bottom of the results is produced once, from the final reduced fraction, and never fed back into anything.

Questions people ask