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| My National Review Online "Diary" column for June 2005 included the following brainteaser.
Bowling Problem —————— A person is bowling and has scores of 140, 85, 125, and 150. In his fifth round his score is within 3 of the average score for the first 5 rounds. How many possible medians are there for the set of 5 scores?
Hareendra Yalamanchili, to whom I owe the problem, says: "It is from the MathCounts competition for middle schoolers. You get about 30 seconds to solve it.” ———————————————
Solution Average of the five rounds is (140 + 85 + 125 + 150 + X) / 5, where X is the score for the 5th round. This is equal to 100 + X / 5. So I know that 97 + X / 5 <= X <= 103 + X / 5. Subtracting X / 5 all through: 97 <= 4X / 5 <= 103. Multiplying by 5 / 4 all through: 121¼ <= X <= 128¾. Since a bowling score must be a whole number, the only possibilities are X = 122, 123, 124, 125, 126, 127, or 128. Laying out the five-round scores in these seven cases, in ascending score order for each, I have: 85, 122, 125, 140, 150 85, 123, 125, 140, 150 85, 124, 125, 140, 150 85, 125, 125, 140, 150 85, 125, 126, 140, 150 85, 125, 127, 140, 150 85, 125, 128, 140, 150 The medians in the seven cases are 125, 125, 125, 125, 126, 127, 128. So there are four possible medians. Answer: 4 |
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