How this calculator works
This calculator handles the two things people most often need to do with fractions: combine two of them with addition, subtraction, multiplication, or division, or reduce a single fraction to its simplest form. Every result is shown three ways — as a simplified fraction, as a mixed number when the value is greater than one whole, and as a decimal — so you can use whichever form fits the problem you're solving.
Fraction arithmetic only works cleanly once both fractions share a common denominator (for addition and subtraction) or once you multiply straight across (for multiplication and division). This calculator does that arithmetic for you and always reduces the answer, the same way a textbook answer key would.
The formulas
Add: a/b + c/d = (ad + bc) / bdSubtract: a/b − c/d = (ad − bc) / bdMultiply: a/b × c/d = ac / bdDivide: a/b ÷ c/d = ad / bcSimplify: divide both a and b by gcd(a, b)The greatest common divisor (gcd) is found with Euclid's algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is zero.
Worked example: 1/2 + 1/3
- Cross-multiply the numerators and denominators: (1 × 3) + (1 × 2) = 5, and the new denominator is 2 × 3 = 6.
- The unreduced sum is 5/6.
- gcd(5, 6) = 1, so 5/6 is already in lowest terms — it's the final answer.
- As a decimal, 5/6 ≈ 0.8333.
Frequently asked questions
Why do I need a common denominator to add fractions?
A fraction's denominator tells you the size of the pieces being counted. You can only add or subtract counts of the same-size piece, so both fractions are rewritten over a shared denominator (bd) before combining the numerators.
Why does dividing by a fraction flip it upside down?
Dividing by a number is the same as multiplying by its reciprocal. Dividing by c/d is multiplying by d/c, which is why a/b ÷ c/d becomes a/b × d/c = ad/bc.
What does it mean when the answer is a mixed number?
A mixed number (like 1 5/12) is used when the result is an improper fraction — the numerator is larger than the denominator, meaning the value is more than one whole. 17/12 and 1 5/12 are the same amount written two different ways.
Do the inputs have to be whole numbers?
Yes. This calculator expects integer numerators and denominators, since that's how fractions are conventionally defined. To work with a decimal like 2.5, convert it to a fraction first (2.5 = 5/2).
What happens if I divide by zero?
Dividing by a fraction whose numerator is zero is undefined, the same as dividing by zero in ordinary arithmetic, so the calculator won't return a result for that case.