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 problem combines a search for integer solutions with a median calculation, and the sting is that the median product is not the product of the middle solution pair. Start by finding every solution to 3x + 5y = 47 in positive integers. Solve for x: x = (47 − 5y)/3, so 47 − 5y must be a positive multiple of 3. Testing y = 1 gives x = 42/3 = 14; y = 4 gives x = 27/3 = 9; y = 7 gives x = 12/3 = 4. Every other value of y up to 9 leaves 47 − 5y not divisible by 3, and y ≥ 10 makes x zero or negative. The three solutions are (14, 1), (9, 4), and (4, 7), giving products 14, 36, and 28.
To find the median, sort the products first: 14, 28, 36. The middle value is 28, which is choice C. The trap is to seize the middle solution pair, (9, 4), and treat its product as the median, but that product, 36, is actually the largest of the three, not the middle one. Sorting is what the median demands, and only after sorting does 28 emerge as the center.