Multiple choice

In a certain game, each of $5$ players received a score between $0$ and $100$, inclusive. If their average (arithmetic mean) score was $80$, what is the greatest possible number of the $5$ players who could have received a score of $50$?

  1. None

  2. One

  3. Two

  4. Three
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total sum of 5 scores = 5 * 80 = 400. To maximize the number of 50s, minimize the other scores. Let x be the number of 50s. The remaining (5-x) scores must be as high as possible (100). 50x + 100(5-x) >= 400. 50x + 500 - 100x >= 400, so 100 >= 50x, x <= 2. The maximum is 2.

AI explanation

The total score for the 5 players is 5 times 80, which equals 400. To maximize the number of players scoring 50, assume the remaining players score the maximum possible value of 100. If three players scored 50, the maximum total score would be 3 times 50 plus 2 times 100, which equals 350; since this is less than 400, three players is impossible. If two players scored 50, the total of the other three could be 300, meaning they can all score 100, making two the maximum possible number of players with a score of 50.

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