Fraction Calculator
Exact fraction arithmetic with mixed numbers, simplification and decimal conversion. Results are computed with whole-number arithmetic, so there is no rounding error.
How to type fractions
Type a fraction with a slash (3/4), a mixed number with a space between the whole part and the fraction (2 1/3), a whole number (5) or a decimal (0.375). A minus sign applies to the whole mixed number: -1 1/2 means −3/2.
The rules behind each operation
a/b + c/d = (a×d + c×b) / (b×d)
a/b − c/d = (a×d − c×b) / (b×d)
a/b × c/d = (a×c) / (b×d)
a/b ÷ c/d = (a×d) / (b×c)
After each operation the result is simplified by dividing the top and bottom by their greatest common divisor (GCD).
Worked example
1 1/2 + 2/3: first convert the mixed number, 1 1/2 = 3/2. Then 3/2 + 2/3 = (3×3 + 2×2) / (2×3) = 13/6. As a mixed number that is 2 1/6, and as a decimal 2.1666…, written 2.1(6) because the 6 repeats forever.
Simplifying fractions
To simplify 42/56, find the largest number that divides both: GCD(42, 56) = 14. Then 42 ÷ 14 = 3 and 56 ÷ 14 = 4, so 42/56 = 3/4. The calculator finds the GCD with Euclid's algorithm, which works for numbers of any size.
Repeating decimals
A simplified fraction has a terminating decimal only when its denominator has no prime factors other than 2 and 5. So 3/8 = 0.375 ends, but 1/3 = 0.333… and 1/7 = 0.142857142857… repeat forever. The calculator marks the repeating block with parentheses: 1/7 = 0.(142857).
Common questions
How do I turn a decimal into a fraction?
Write the decimal over a power of ten, then simplify: 0.375 = 375/1000 = 3/8. Enter the decimal in the "simplify or convert" box to do this automatically.
Why is dividing by a fraction the same as multiplying by its reciprocal?
Dividing by c/d asks how many c/d fit into a number. Multiplying by d/c gives the same answer: 1 ÷ 1/4 = 1 × 4 = 4, because four quarters fit into one.
What happens with a zero denominator?
A fraction with 0 on the bottom is undefined, and so is dividing by a fraction equal to zero. The calculator tells you instead of showing a misleading result.