Problem 14 · 2020 Math Kangaroo
Medium
Counting & Probability
Fractions, Decimals & Percents
sum-constraint
In a class, every student either only swims, or only dances, or does both. Three eighths of the students in the class swim. Exactly five students do both — that is, they swim and dance. At least how many students are in the class?
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Answer: A — 16
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Hint 1 of 2
Three eighths of the class swim, so the class size must be a multiple of 8.
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Hint 2 of 2
The five who do both must fit inside the swimmers; find the smallest multiple of 8 that allows that.
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Approach: make the swimmer count work out
- Swimmers = 3/8 of the class, so the total must be a multiple of 8.
- The 5 students who do both are among the swimmers, so swimmers ≥ 5, meaning 3/8 of the total ≥ 5.
- The smallest multiple of 8 giving at least 5 swimmers is 16 (3/8 of 16 = 6 swimmers, which holds the 5).
- So at least 16 students, choice A.
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