Fraction Calculator

Fraction arithmetic: one rule for adding, a different one for multiplying

Multiplying fractions is the easy case — multiply across the top, multiply across the bottom, done. Adding is the one people get wrong, because you cannot add the numerators until the denominators match. This calculator does both, reduces the answer, and shows it as a mixed number and a decimal.

How it works

• Adds, subtracts, multiplies and divides two fractions, giving the answer in lowest terms.
• Shows the same result three ways: as a fraction, as a mixed number, and as a decimal.
• Converts a decimal back into the simplest fraction that represents it.

a/b + c/d = (ad + cb) / bd
a/b - c/d = (ad - cb) / bd
a/b x c/d = ac / bd
a/b / c/d = ad / bc      (multiply by the reciprocal)

then divide both parts by their greatest common divisor

Worked example

Adding 1/2 and 1/3, then reducing.

1. common denominator 2 x 3 = 6
2. first fraction 1/2 = 3/6
3. second fraction 1/3 = 2/6
4. add the numerators 3 + 2 = 5
5. result 5/6
6. gcd(5, 6) = 1 already in lowest terms

Five sixths. Note that the intuitive wrong answer — adding tops and bottoms to get 2/5 — is smaller than either starting fraction, which is a quick way to catch the mistake without redoing the arithmetic.

Reading the result

• Dividing by a fraction is multiplying by its reciprocal, which is why dividing by 1/2 makes a number bigger. “How many halves fit into this?” is the question division actually asks, and the answer is twice as many as there are wholes.
• Reducing means dividing both parts by their greatest common divisor. It changes how the fraction is written, never its value, so 6/8 and 3/4 are the same number — one is just the form you would write down.
• Decimal to fraction uses a continued-fraction expansion rather than multiplying by a power of ten. Typing 0.333333333 gives 1/3, not 333333333/1000000000, because the smallest denominator within tolerance is almost always the answer somebody wants.

Common questions

Why do I need a common denominator to add but not to multiply?

Because a fraction is a count of parts, and you can only add counts of the same-sized part. Two thirds plus one quarter is not three of anything until both are rewritten in twelfths. Multiplication asks a different question — a fraction of a fraction — and scaling both dimensions needs no shared unit.

What is a mixed number for?

Readability. 7/3 and 2 1/3 are the same value, but the mixed form tells you immediately that it is a bit over two. Improper fractions are easier to compute with, which is why this tool shows both.