2015/16 AHSMC Part 1 Problem 10

Problem 10 of 16

Each person has two legs. Some are sitting on three-legged stools while the others are sitting on four-legged chairs such that all the stools and chairs are occupied. If the total number of legs is $39$ , how many people are there?

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Solution

Let the number of stools be $m$ and the number of chairs be $n$ . Then $5m + 6n = 39$ . Hence $m$ is a multiple of $3$ but not a multiple of $6$ . Moreover, $5m < 39$ so that $m \le 7$ . It follows that $m = 3$ and $n = (39 - 3 \times 5) \div 6 = 4$ , so that the number of people is $3+4=7$ . The answer is (C).

University of Alberta

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