How to add, subtract, multiply and divide fractions

Steps

1. To add or subtract, cross-multiply: a/b + c/d = (ad + bc) ÷ bd. For 1/2 + 1/3 that is 5/6.
2. To multiply, multiply the numerators and the denominators: 2/3 × 3/5 = 6/15.
3. To divide, multiply by the reciprocal: 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2.
4. Simplify by dividing the top and bottom by their greatest common divisor: 6/15 becomes 2/5 and 4/2 becomes 2.
5. If the top is bigger than the bottom, write it as a mixed number. Or use the fraction calculator.

Worked examples

Subtract: 3/4 − 1/6 = (3 × 6 − 1 × 4) ÷ (4 × 6) = 14/24, which simplifies to 7/12. Multiply: 2/3 × 3/5 = 6/15 = 2/5. Divide: 1/2 ÷ 1/4 = 2.

Add with a result above one: 7/8 + 3/4 = (7 × 4 + 3 × 8) ÷ (8 × 4) = 52/32, which simplifies to 13/8. As a mixed number that is 1 5/8, and as a decimal 1.625.

Why a common denominator is needed

You can only add or subtract pieces of the same size. Halves and thirds are different sizes, so rewrite both as sixths first: 1/2 = 3/6 and 1/3 = 2/6, and then 3/6 + 2/6 = 5/6. Cross-multiplying does this in one step, at the price of a larger denominator that you may then need to simplify.

Common slips to avoid

Do not add the denominators: 1/2 + 1/3 is not 2/5. Do not forget to flip only the second fraction when dividing. And always check whether the answer can be simplified. A fraction is in its simplest form when the top and bottom share no divisor except 1.