Five distinct positive integers represent monthly retention counts, and they sum to 60. What is the greatest possible value of their median?
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Answer & Explanation
Correct answer
C
To make the median (the 3rd of five sorted distinct positive integers) as large as possible, push the other four values to be as small as the constraints allow. The two values below the median should be the smallest distinct positive integers, 1 and 2. The two values above the median must still be distinct and larger than it, so the smallest they can be is its closest neighbors, median + 1 and median + 2.
Let the median be m. Then 1 + 2 + m + (m + 1) + (m + 2) = 3m + 6 = 60, so m = 18, realized by {1, 2, 18, 19, 20}.
The key move is keeping the two values above the median as small as possible (m + 1 and m + 2) rather than letting them grow, because every unit added to them is taken from the sum the median could otherwise use. Checking m = 19 forces a minimum total of 1 + 2 + 19 + 20 + 21 = 63, which exceeds 60, so 18 is the maximum.
There is no shortcut that skips rebuilding the five-integer set: finding the maximum median means constructing this exact minimal arrangement.