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From 11 positive integer scores on a 10-point quiz, the mean is 8, the median is 8, and the mode is 7. Find the maximum number of perfect scores possible on this test.

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  1. total points = 11*8 = 88
    So, with median 8, the scores are
    x x x x x 8 x x x x x
    If the mode is 7, then for max scores of 10, we want the fewest under 8:
    x x x 7 7 8 x x x x x
    Now we cannot have duplicate scores below 7 (only one mode), so
    1 2 3 7 7 8 x x x x x
    They have to add up to 88, so since the bottom 6 scores add to 28, that leaves 60 points for the top 5 scores. If they were all 10's, that would only be 50 points, so we need another 10 points among the low scores. If we add 3 to the bottom scores, that is only 9 more, so let's have a 3rd 7 score, giving us
    4 5 7 7 7 8 10 10 10 10 10

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