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Five students took a test scored from 0 to 100, and each student scored at least 60. The mean of the five scores is 76. Did the highest-scoring student score above 90?
(1) The median score is 74.
(2) Each of the four lowest scores is at least 73.
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Correct answer
B
The five scores lie in [60, 100] with mean 76, so they sum to 380. Ordering s₁ ≤ s₂ ≤ s₃ ≤ s₄ ≤ s₅, the question is whether s₅ > 90. The key tool is that s₅ = 380 − (s₁ + s₂ + s₃ + s₄): the top score is large exactly when the other four are small. So to find s₅'s maximum, minimize the other four (push them to their floors); to find s₅'s minimum, maximize the other four (push them up to where the ordering allows). Doing the direction wrong is the classic slip.
Statement (1): the median is s₃ = 74. S₅ maximum, minimize the others: put s₁, s₂ at the 60 floor and s₃ = 74 (the median), leaving 380 − 60 − 60 − 74 = 186 for s₄ + s₅ with s₄ ≤ s₅ ≤ 100. Taking s₅ = 100 leaves s₄ = 86, giving the valid list (60, 60, 74, 86, 100), sum 380, median 74. So s₅ can be 100 (answer yes). S₅ minimum, maximize the others: push s₁ = s₂ = s₃ = 74 (the most they can be, since they are ≤ the median) and split the remaining 380 − 222 = 158 between s₄ and s₅ with s₄ ≤ s₅. That gives s₄ = s₅ = 79, so s₅ can be as low as 79 (answer no).
The answer straddles 90, so statement (1) alone is not sufficient.
Statement (2): each of the four lowest scores is at least 73, so s₁ + s₂ + s₃ + s₄ ≥ 4 × 73 = 292. Therefore s₅ = 380 − (s₁ + s₂ + s₃ + s₄) ≤ 380 − 292 = 88. The highest score can be at most 88, which is below 90: a determinate no. (The bound is tight: 73, 73, 73, 73, 88 sums to 380, so 88 is actually reached.) Sufficient.
The primary distractor is '(1) alone', solvers treat the median as the informative statement and dismiss 'each of the four lowest is at least 73' as a vague floor, choosing (A). But that floor is surgical: a floor on the four lowest scores is a ceiling on the highest (the sum is fixed at 380), and it caps s₅ at 88. Statement (1), the median, actually straddles once you run both optimization directions. Choosing (C) assumes both are needed; (2) alone resolves. Choosing (D) over-credits the straddling median as also sufficient.
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