A class with 2N students took a quiz, on which the possible scores were
0, 1, . . . , 10. Each score appeared at least once.
The average score for this class was exactly 7.4.
Show that the class can be divided into two groups of N members each, such that that the average score for each group was exactly 7.4.
(In reply to
An example, not a proof by Larry)
Another observation, not sure if it's important, but N must be a multiple of 5, and in fact must be at least 15, in order to meet the conditions that every score appears once and the average ends in 0.4. So the example given of 30 students is the smallest case possible.
Larry's example also can help illustrate the strategy I outlined. In his case, I'd initially split them into two groups as follows:
10 9
9 9
9 9
9 9
9 9
9 9
9 9
9 9
9 8
8 8
8 8
7 6
5 4
3 2
1 0
The sum of the larger group is 114 and the sum of the smaller group is 108, so we just need to swap two students with a difference of 3 to make them equal - 9 and 6, 7 and 4, 5 and 2 or 3 and 0 all would yield solutions, in addition to the one Larry provided. I claim (without proof yet) that this will always be possible.
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Posted by tomarken
on 2024-02-24 14:38:23 |
re: An example, not a proof
