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

3 perfect scores of 10 There can't be 5x10 because the mode (the commonest score) is 7. Hence if there are 5x10 there must be 6 x7, but the median (the middle score) is 8 and this would make at least 12 scores which is impossible. There can't be 4x10. If there were 4 x10 there would be 5x7 (7=mode), 1x8(median) + either another 8 or 9. Assuming the best case scenario (2 x8). The total of the 11 scores = 91. They should add up to 88 (mean = 8, (11x8=88) If there are 3x10, there must be at least 4x7(mode =7), 3more 8+(median = 8), and 1 x less than 7. 3x10, 3x8, 4x7 and 1x6 fits the bill. Mean= (30+24+28+6)/11 = 88/11 = 8 OK Median is the sixth number in the series = 8 OK The mode is the commonest number. Mode = 7 OK

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