Adding and subtracting fractions — Mathematics, 11–13
Adding and subtracting fractions when one denominator is a multiple of the other, with mental strategies and estimates.
Same-size parts first
You can only add or subtract parts of the same size. If the denominators are equal, add or subtract the numerators and keep the denominator: 3/8 + 2/8 = 5/8. If they are different but one is a multiple of the other, rewrite the fraction with the smaller denominator as an equivalent one with the larger denominator, then do the same: 1/2 + 1/4 = 2/4 + 1/4 = 3/4.
Why it works, and doing it in your head
The denominator tells the size of the part and the numerator tells how many there are: adding halves to quarters would be adding things of different sizes. Writing everything in quarters, you count equal parts. The properties of addition still hold: you can swap and group, as in 1/4 + 1/2 + 3/4 = (1/4 + 3/4) + 1/2 = 1 + 1/2. And reference fractions help: 1/2 + 1/2 = 1 and 1 − 1/4 = 3/4.
Calculating 1/2 + 1/6 and 7/8 − 1/4
Calculate 1/2 + 1/6 and 7/8 − 1/4. For 1/2 + 1/6: 6 is a multiple of 2 (2 × 3 = 6), so 1/2 = 3/6, and 3/6 + 1/6 = 4/6. For 7/8 − 1/4: 8 is a multiple of 4 (4 × 2 = 8), so 1/4 = 2/8, and 7/8 − 2/8 = 5/8. Notice that the denominator does not change when you add or subtract: only the numerator does.
Adding tops and bottoms
Many people calculate 1/2 + 1/4 by adding the numerators and the denominators: 1 + 1 = 2 and 2 + 4 = 6, so 2/6. It feels logical, because you “added everything”. But 2/6 equals 1/3, which is less than 1/2, and a sum cannot be smaller than one of its parts. Estimating first helps: a half plus a quarter is more than a half, so the right result is 3/4.
Sharing a snack, estimating a sum
In a pack of biscuits, you eat 1/4 and your brother eats 1/2: together they ate 1/4 + 1/2 = 3/4, and 1/4 is left. To estimate 5/8 + 1/2: 5/8 is a little more than 1/2, so the sum is a little more than 1. The exact value confirms it: 5/8 + 4/8 = 9/8 = 1 + 1/8. Estimating before calculating catches mistakes such as adding tops and bottoms.
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