Adding and Subtracting Fractions

If the denominators are the same, add the numerators and keep the denominator: 1/5 + 2/5 = 3/5. If the denominators are different, change the fractions so they match first, then add the numerators. Simplify the result if you can, and turn an improper fraction into a mixed number when the question asks for one.

Find the lowest common multiple of the denominators and expand both fractions to it, because you can only add parts of the same size. For 2/3 + 4/9, the multiples of 3 and 9 first meet at 9, so 2/3 becomes 6/9 and 6/9 + 4/9 = 10/9, which is 1 1/9. For 1/4 + 1/2, 1/2 is 2/4, so the answer is 3/4.

Subtracting works like adding. With the same denominator, subtract the numerators and keep the denominator. With different denominators, make them the same first. For 5/6 − 1/4, list the multiples of 6 and 4: the smallest shared one is 12. So 5/6 = 10/12 and 1/4 = 3/12, and 10/12 − 3/12 = 7/12.

A proper fraction has a numerator smaller than its denominator, so it is worth less than 1, like 1/3. An improper fraction has a numerator greater than or equal to its denominator, so it is worth 1 or more, like 8/5 or 7/7. A mixed number is a whole number with a proper fraction, like 2 3/5, which means 2 + 3/5.

Divide the numerator by the denominator: the whole-number answer is the whole part and the remainder is the new numerator, over the same denominator. 125 ÷ 6 is 20 remainder 5, so 125/6 = 20 5/6. To go back, multiply the whole number by the denominator and add the numerator: 20 × 6 + 5 = 125, so 20 5/6 = 125/6.

The same two rules, but GCSE questions often use mixed numbers, so convert them to improper fractions first. For 6 1/2 − 2 3/4: 6 1/2 = 13/2 = 26/4 and 2 3/4 = 11/4, so 26/4 − 11/4 = 15/4, which is 3 3/4. Give the answer as a mixed number in its simplest form unless the question says otherwise.