For each ordered pair of positive integers (x, y) satisfying 3x + 5y = 47, consider the product xy. What is the median of all such products?
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Answer & Explanation
Correct answer
C
This item fuses a Diophantine search with a statistics question, and the sting is that the median of the products is not the product of the middle solution. First find every solution: x = (47 − 5y)/3 must be a positive integer, so 47 − 5y must be a positive multiple of 3. Test y = 1: x = 42/3 = 14. Test y = 4: x = 27/3 = 9. Test y = 7: x = 12/3 = 4. The remaining y values up to 9 leave 47 − 5y not divisible by 3, and y at 10 or more makes x nonpositive. So the solutions are (14, 1), (9, 4), and (4, 7), with products 14, 36, and 28.
Here is the trap: the tempting move is to grab the middle solution, (9, 4), and call its product the median, but that product, 36, is actually the largest of the three, which is exactly choice D. Sort the products first: 14, 28, 36. The middle value is 28. Choice A is the smallest product, choice B is the mean, 26, not the median, and choice E just repeats the 47 from the equation. Sort the products, then take the middle: the median is 28, choice C.