Adding and Subtracting Fractions

You can only add or subtract fractions when the parts are the same size, which means the denominators must match. 2 fifths + 1 fifth = 3 fifths, in the same way that 2 apples + 1 apple = 3 apples. The denominator is the name of the parts, so it stays the same; only the numerators are added or subtracted.

When the denominators are different, you first rename the fractions as equivalent fractions with a common denominator. Then add or subtract the numerators, and finally simplify.

Any common multiple of the denominators will work, but the lowest common multiple (LCM) keeps the numbers small. To find it, list the multiples of each denominator until one appears in both lists.

For 5/6 − 1/4: multiples of 6 are 6, 12, 18…; multiples of 4 are 4, 8, 12…, so the LCM is 12. Multiplying the denominators together (6 × 4 = 24) also works, but you then have to simplify more at the end: 20/24 − 6/24 = 14/24, which simplifies to 7/12, the same answer.

When the denominators share no common factor, their product is the LCM. For 2/5 + 3/4, the LCM of 5 and 4 is 20: 2/5 = 8/20 and 3/4 = 15/20, so the total is 23/20, or 1 3/20.

Method 1: improper fractions. Turn each mixed number into an improper fraction, then add or subtract as usual. This always works, and it is the method shown in example 3 on the infographic: 3 1/4 − 1 2/3 = 13/4 − 5/3 = 39/12 − 20/12 = 19/12 = 1 7/12.

Method 2: wholes and parts separately. Add or subtract the whole numbers, then the fractions. For 2 1/3 + 1 1/2: the wholes give 3, and 1/3 + 1/2 = 2/6 + 3/6 = 5/6, so the answer is 3 5/6.

Method 2 needs care with subtraction. In 3 1/4 − 1 2/3, the fraction parts become 3/12 − 8/12, and 3/12 is too small. Exchange one whole for 12/12: 3 3/12 becomes 2 15/12. Now 2 15/12 − 1 8/12 = 1 7/12. Both methods give the same answer.

Always finish by simplifying (divide top and bottom by their highest common factor) and, if the answer is an improper fraction, decide whether to write it as a mixed number. Ask what form the question wants; both 4/3 and 1 1/3 are correct.

A quick estimate catches most slips. 5/6 lies between 3/4 and 1, so 5/6 − 1/4 must lie between 1/2 and 3/4; 7/12 is just over a half, so it fits. For mixed numbers, decimals help: 3 1/4 − 1 2/3 is about 3.25 − 1.67 = 1.58, and 1 7/12 is about 1.58 too.

Adding and subtracting fractions comes up whenever amounts are measured in parts: combining 3/4 of an hour and 1/2 an hour of homework (1 1/4 hours), working out how much of a pizza is left after two people have eaten 1/3 and 1/4 of it (5/12 left), or adding lengths on a plan. The same skills are needed later in algebra, where fractions contain letters instead of numbers but follow exactly the same rules.