Fraction Calculator exact and simplified
Enter two fractions and an operation. You get the answer in lowest terms, as a mixed number, and as a decimal.
Nothing Is Rounded
The whole calculation stays in whole numbers until the very last step.
Zero Denominators Are Undefined
A fraction needs a non-zero denominator, and you cannot divide by zero.
How do you calculate with fractions?
At a Glance
A fraction is one integer divided by another. Multiplying and dividing are easy — multiply straight across, or flip the second fraction and multiply. Adding and subtracting need a common denominator first, because you can only combine parts of the same size.
Each operation has its own rule, and all four keep the numbers as integers:
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/bcSimplification uses the Euclidean algorithm to find the greatest common divisor of the numerator and denominator, then divides both by it. For 38 and 24 the greatest common divisor is 2, so 38/24 reduces to 19/12.
Keeping everything in integers matters. A decimal calculator turns 1/3 into 0.3333333333333333, and 1/3 + 1/6 then comes out as 0.4999999999999999 rather than exactly 1/2. This page never converts to a decimal until the answer is already known exactly.
These are the values the calculator loads by default.
Find a common denominator
4 × 6 = 24. It does not have to be the lowest common denominator — simplifying at the end fixes that.
Rewrite both fractions
3/4 becomes 18/24, and 5/6 becomes 20/24.
Add the numerators
18 + 20 = 38, so the answer is 38/24.
Simplify
The greatest common divisor of 38 and 24 is 2, giving 19/12.
Express it three ways
19 ÷ 12 = 1 remainder 7, so it is 1 7/12 as a mixed number and 1.583333 as a decimal.
The arithmetic here is exact, so the limitations are all about what a fraction can represent.
An improper fraction is not wrong. 19/12 and 1 7/12 are the same number. Improper fractions are easier to keep calculating with; mixed numbers are easier to picture. School answers usually want the mixed form, algebra usually wants the improper one.
The decimal may be the approximation. 19/12 is exact; 1.583333 is the same value rounded to six places. Where a fraction has a repeating decimal expansion, the fraction is the precise form and the decimal is not.
Zero denominators are rejected rather than guessed at. A denominator of zero has no value, and dividing by a fraction whose numerator is zero is division by zero. Both return no result.
Very large inputs can exceed exact integer arithmetic. Numerators and denominators are limited to a million here; multiplying two fractions near that bound produces intermediate values beyond the range where every integer is exactly representable.
To go the other way, use the Decimal to Fraction Converter. For the greatest common divisor on its own, use the GCF and LCM Calculator.