Four operations, two of them easy
Multiplying and dividing fractions is straightforward. Adding and subtracting them is the part that trips people up, and it is worth being clear about why: addition needs the two fractions to be counting the same size of piece, and multiplication does not.
If you have 2/3 of a pizza and 1/4 of a pizza, you cannot add 2 and 1 to get 3 of anything, because thirds and quarters are different sizes. You have to re-cut both into twelfths first. Multiplication has no such requirement — 2/3 of 1/4 is a straightforward scaling, and the numerators and denominators multiply independently.
That asymmetry is the whole difficulty, and every rule below follows from it.
Adding and subtracting
Rewrite both fractions over a common denominator, add or subtract the numerators, keep the denominator, then simplify.
a/b + c/d = (ad + cb) / bd
Multiplying the two denominators always produces a workable common denominator. It is not always the lowest one, which means more simplifying at the end, but it never fails. The lowest common denominator is the least common multiple of b and d — which is bd divided by their greatest common divisor.
Worked example — 2/3 + 1/4
- Common denominator: 3 × 4 = 12
- 2/3 becomes 8/12 (multiply top and bottom by 4)
- 1/4 becomes 3/12 (multiply top and bottom by 3)
- Add the numerators: 8 + 3 = 11
- Result: 11/12
Here 12 is already the lowest common denominator, and 11/12 is already in lowest terms.
Worked example — 5/6 − 3/8
- Product of denominators: 48. Lowest common denominator: 24, since 6 and 8 share a factor of 2.
- Using 24: 5/6 = 20/24, and 3/8 = 9/24
- Subtract: 20 − 9 = 11
- Result: 11/24
Had you used 48 instead, you would have got 40/48 − 18/48 = 22/48, which simplifies to the same 11/24. Both routes are correct; the lowest common denominator just gets there with smaller numbers.
Multiplying
Multiply the numerators, multiply the denominators, simplify.
a/b × c/d = ac / bd
That is the entire rule. No common denominator, no rewriting.
Worked example — 2/3 × 3/5
- Numerators: 2 × 3 = 6
- Denominators: 3 × 5 = 15
- 6/15 shares a factor of 3, so it reduces to 2/5
You can cancel before multiplying, which keeps the numbers small: the 3 in the first numerator and the 3 in the second denominator cancel, leaving 2/1 × 1/5 = 2/5 directly. This is called cross-cancelling, and it is worth the habit when the numbers are large.
Dividing
Flip the second fraction and multiply.
a/b ÷ c/d = a/b × d/c = ad / bc
The flip is not a trick to memorise. Dividing by a number means multiplying by its reciprocal — dividing by 2 is multiplying by 1/2 — and the reciprocal of c/d is d/c. The rule is the definition of division applied to a fraction.
Worked example — 1/2 ÷ 3/4
- Reciprocal of 3/4 is 4/3
- 1/2 × 4/3 = 4/6
- Simplify: 2/3
A useful sanity check: dividing by a fraction smaller than 1 makes the result bigger. Here 2/3 is larger than 1/2, which is right, because 3/4 fits into 1/2 less than once.
Mixed numbers
A mixed number such as 2 3/4 combines a whole part and a fraction part. It is easier to visualise, which is why recipes and tape measures use it, but harder to compute with.
Converting to an improper fraction: multiply the whole number by the denominator, add the numerator, keep the denominator. So 2 3/4 = (2 × 4 + 3) / 4 = 11/4.
Converting back: divide the numerator by the denominator. The quotient is the whole part and the remainder is the new numerator. 11 ÷ 4 is 2 remainder 3, giving 2 3/4.
The trap is negatives. In a mixed number, the sign applies to the whole quantity, not just the whole part. So −2 1/2 means −(2 + 1/2) = −5/2, not −2 + 1/2 = −3/2. This calculator applies the sign to the whole quantity, which is the standard convention, and it is worth being explicit because reading it the other way flips answers.
Simplifying, and Euclid's algorithm
A fraction is in lowest terms when the numerator and denominator share no common factor other than 1. To get there, divide both by their greatest common divisor.
Finding the GCD by inspection works for small numbers. For anything larger, Euclid's algorithm — over two thousand years old and still the method every computer uses — is faster than factorising:
Take 1,071 and 462.
- 1,071 ÷ 462 = 2 remainder 147
- 462 ÷ 147 = 3 remainder 21
- 147 ÷ 21 = 7 remainder 0
- The last non-zero remainder, 21, is the GCD
So 1071/462 reduces to 51/22. Trying to spot that by factorising both numbers would take considerably longer.
Fractions, decimals, and why one of them is exact
Every fraction can be written as a decimal by dividing the numerator by the denominator. Whether that decimal terminates is entirely determined by the denominator once the fraction is in lowest terms.
A decimal terminates if and only if the denominator's only prime factors are 2 and 5 — the primes that divide 10.
- 1/8: denominator 2³, terminates at 0.125
- 3/20: denominator 2² × 5, terminates at 0.15
- 1/3: denominator 3, repeats as 0.333…
- 1/7: denominator 7, repeats with a six-digit cycle, 0.142857142857…
There is nothing special about thirds. They repeat in base ten because 3 does not divide 10. In base three, 1/3 would be written exactly as 0.1, and it would be 1/2 that repeated.
This is also the reason computers get decimal arithmetic subtly wrong. Binary floating point, defined by IEEE 754, can represent a fraction exactly only when its denominator is a power of two. Ten is not a power of two, so 0.1 cannot be stored exactly, and 0.1 + 0.2 comes out as 0.30000000000000004 in most languages. Fraction arithmetic sidesteps this entirely by keeping whole-number numerators and denominators throughout — which is why this calculator computes on integers and converts to a decimal only at the very end, for display.
Where the arithmetic goes wrong
Adding denominators. 1/2 + 1/3 is not 2/5. This is the single most common fraction error, and the way to see it is wrong is that 2/5 is smaller than 1/2 — adding a positive quantity made the answer smaller.
Forgetting to simplify. 4/8 is not wrong, but it is not finished. Most marking schemes and most readers expect lowest terms.
Cancelling across an addition. You can cancel a common factor between a numerator and a denominator in a product. You cannot cancel terms across a sum: in (a + b)/b, the b values do not cancel to give a.
Mishandling negative mixed numbers. Covered above, and worth checking whenever a negative sign appears next to a whole part.
Assuming a bigger denominator means a bigger fraction. 1/100 is smaller than 1/2. The denominator counts how many pieces the whole was cut into, so more pieces means smaller pieces.
What this calculator does not do
It handles two fractions and one operation at a time. Chains of operations with precedence, algebraic fractions with variables in them, and continued fractions are all outside its scope.
It also works with integers only, so a decimal typed into a numerator box is rejected rather than silently rounded. If you want to turn a decimal into a fraction, multiply by a power of ten to clear the point — 0.375 is 375/1000 — and simplify from there; 375 and 1000 share a factor of 125, giving 3/8.
The conventions here follow the standard school curricula linked below, both of which set out the expected methods explicitly, and the floating-point discussion follows IEEE 754-2019.