Fraction Calculator

The formula

gcd

Greatest common divisor, found by the Euclidean algorithm — repeatedly replace the pair with the smaller number and the remainder until the remainder is zero.

Input format

Either a whole number such as 7, or a fraction such as 3/4 or -5/8. A negative sign goes on the numerator; the denominator must be a positive whole number and cannot be zero.

Mixed numbers

Shown in the output when the result is larger than one, but not accepted as input. Convert first: 2 3/4 is (2×4 + 3)/4 = 11/4.

Worked example

Adding 3/4 and 5/6.

• Common denominator: 4 × 6 = 24
• Numerator: 3×6 + 5×4 = 18 + 20 = 38
• Raw result: 38/24
• gcd(38, 24) = 2 → 19/12
• As a mixed number: 1 whole and 7/12

1 7/12, or about 1.583333.

Why the answer is exact and the decimal is not

Fractions carry the answer without loss: 19/12 is precisely the sum, whereas 1.583333 is a truncation of a repeating decimal. This matters wherever the value gets used again — thirds, sixths, sevenths and ninths never terminate in decimal, so a decimal answer accumulates error every time it is fed into another calculation. The decimal shown alongside the fraction is there to make the size easy to judge, not to be used as the result.

The cross-multiplication shortcut

Multiplying the two denominators always produces a valid common denominator, which is what makes the method above work for any pair without hunting for a lowest common multiple. It sometimes overshoots — adding 1/4 and 1/6 gives 10/24 rather than 5/12 — but the simplification step at the end fixes that, so the answer is always in lowest terms regardless of the route taken.

Division is multiplication by the reciprocal

Dividing by 2/3 is identical to multiplying by 3/2, which is why the division formula above swaps the second fraction. It is also the reason a numerator of zero in the second fraction is rejected: flipping it would put zero in the denominator, and dividing by zero has no answer.

Where negatives sit

Put the sign on the numerator — write -3/4 rather than 3/-4. The calculator computes the sign of the result from the signs of both parts and always presents it on the numerator, so a result that ought to be negative reads cleanly instead of hiding a minus in the denominator.

Common questions

How do I add fractions with different denominators?

Cross-multiply: multiply each numerator by the other denominator, add the two products for the new numerator, and multiply the denominators for the new one. Then divide both by their greatest common divisor to simplify. For 3/4 + 5/6 that gives 38/24, which reduces to 19/12.

Can I enter a mixed number like 2 1/2?

Not directly — the input takes whole numbers or simple fractions. Convert first by multiplying the whole part by the denominator and adding the numerator: 2 1/2 becomes 5/2. Results larger than one are displayed as mixed numbers even though they cannot be entered that way.

Why does the calculator refuse a denominator of zero?

Because a fraction with a zero denominator has no value. The same rule blocks dividing by a fraction whose numerator is zero, since dividing by it means flipping it and putting zero underneath.

Does it always simplify the answer?

Yes. Every result is divided by the greatest common divisor of its numerator and denominator before it is shown, so 38/24 is never displayed — you get 19/12. That happens regardless of which operation produced it.