Problem 7 · 1995 AJHSME
Medium
Fractions, Decimals & Percents
complement-fraction
At Clover View Junior High, half of the students go home on the school bus, one fourth go home by automobile, and one tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home?
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Answer: B — 3/20.
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Hint 1 of 2
Everyone goes home SOME way, so the four groups together make 1 whole. The walkers are simply 'everything that's left over' — that's a subtraction, not a fourth fraction to compute directly.
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Hint 2 of 2
Add the three known fractions over a common denominator, then take them away from 1. Denominators 2, 4, 10 all fit into 20.
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Approach: the walkers are the leftover — subtract the rest from the whole
- The whole student body is 1. Since walkers are 'the rest,' find them by taking the three known groups away from 1 — much cleaner than trying to figure walkers out on their own.
- Over the common denominator 20: bus + car + bike = 12 + 14 + 110 = 1020 + 520 + 220 = 1720.
- Walkers = 1 − 1720 = 3/20.
- You'll see it again: whenever the parts must fill a whole, the unknown 'remaining' part is fastest found as (whole − everything else). This is complementary counting with fractions.
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