Fraction calculator

Working with fractions means finding a common denominator, cross-multiplying and simplifying at the end. This calculator does all three steps and returns the answer in the three useful forms: lowest terms, decimal and mixed number.

Negative numerators and denominators are accepted.

For addition and subtraction everything is brought to a common denominator by cross multiplication. For multiplication, numerators multiply together and denominators multiply together. For division, the second fraction is inverted and then multiplied.

The final step is always dividing numerator and denominator by their greatest common divisor, which gives the fraction in lowest terms.

Adding needs a common denominator, multiplying does not

This is the asymmetry that confuses most. To add you must equalise denominators first; to multiply you simply multiply numerators together and denominators together, directly. And dividing is multiplying by the inverted fraction. That is why multiplying fractions is easier than adding them, the opposite of whole numbers.

Not every fraction has an exact decimal

A third is 0.3333 without end. A fraction gives an exact decimal only if its simplified denominator factors into twos and fives alone, because those are the factors of ten. A seventh, a third or a sixth produce infinite repeating patterns, and there the fraction is more precise than any truncated decimal.

Why the denominator cannot be zero

Division by zero is undefined. If three divided by zero gave some number, that number multiplied by zero would have to return three, and anything times zero is zero. It is not a technical limitation: the operation simply has no meaning.

Where they still beat decimals

In cooking, in music, in construction with inch measurements and in any calculation where exactness matters. A third of a cup is exact; 0.33 cups is an approximation that compounds error when scaling a recipe.