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2005 AMC 10B Problem 19

Problem 19 of 25HarderAlgebra

On a certain math exam, 10%10\% of the students got 7070 points, 25%25\% got 8080 points, 20%20\% got 8585 points, 15%15\% got 9090 points, and the rest got 9595 points. What is the difference between the mean and the median score on this exam?

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Solution

The percentage scoring 9595 is 100−10−25−20−15=30.100 - 10 - 25 - 20 - 15 = 30. The mean is 0.10(70)+0.25(80)+0.20(85)+0.15(90)+0.30(95)=86. \begin{aligned} &0.10(70) + 0.25(80) \\ &\quad {}+ 0.20(85) + 0.15(90) \\ &\quad {}+ 0.30(95) = 86. \end{aligned} Since 35%35\% scored below 8585 and 35%35\% scored above 85,85, the middle student scored 85,85, so the median is 85.85. The difference is 86−85=1.86 - 85 = 1. Thus, B is the correct answer.

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Concepts: mean · median (data) · percentage

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.