Mental calculation with fractions — Mathematics, 11–13
Add and multiply fractions faster by swapping and grouping terms, without always reducing to a common denominator.
Same properties, now with fractions
The commutative and associative properties also hold with fractions. So when adding you can first join the terms that already share a denominator and complete each other: 1/3 + 2/7 + 2/3 = (1/3 + 2/3) + 2/7 = 1 + 2/7 = 9/7. When multiplying you look for pairs that give 1 or that simplify: 3/4 × 4/3 = 1 and 2/5 × 7 × 5/2 = (2/5 × 5/2) × 7 = 7. You do not have to reduce everything to one denominator.
Shorter than a common denominator
A common denominator always works, but with several denominators it produces big numbers and easy slips. The properties give a shortcut when the numbers allow it: you spot fractions that complete a whole or cancel out, and the calculation fits in your head. It is also easier to explain: “1/6 and 5/6 make a whole” says more than a pile of steps with 42 in the denominator.
A sum and a product without common denominators
Sum: 5/6 + 2/7 + 1/6 + 5/7. Swapping the order: (5/6 + 1/6) + (2/7 + 5/7) = 6/6 + 7/7 = 1 + 1 = 2. With the common denominator 42 it would be 35/42 + 12/42 + 7/42 + 30/42 = 84/42 = 2: correct, but longer. Product: 2/3 × 9 × 5/4 × 8. Pairing each fraction with the number that simplifies it, (2/3 × 9) × (5/4 × 8) = 6 × 10 = 60. Multiplying everything in one go gives 720/12 = 60.
The pair is not the same
The reasonable mistake is to think the pair that “cancels out” is the same in addition and in multiplication. In multiplication, 3/4 × 4/3 = 1. But in addition, 3/4 + 4/3 = 9/12 + 16/12 = 25/12, which is not 1. In addition, the pair that completes a whole is the one that adds up to 1: 3/4 + 1/4 = 1. Before grouping, ask yourself: am I adding or multiplying? To add, look for parts that complete a whole; to multiply, look for inverses and numbers that simplify.
Times, measures and parts of parts
The properties help with adding times and measures: 3/4 h + 1/2 h + 1/4 h = (3/4 + 1/4) + 1/2 = 1 + 1/2 = 3/2 h, that is 1 h 30 min. They also help to take a fraction of a fraction: 2/3 of 3/4 of 12 is (2/3 × 3/4) × 12 = 1/2 × 12 = 6. Look first for pairs that complete or simplify, and only then reach for the common denominator.
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