UKMT - Grey Kangaroo - 2023 - 25

25

A rugby team scored 24 points, 17 points and 25 points in the seventh, eighth and ninth games of their season. Their mean points-per-game was higher after 9 games than it was after their first 6 games. What is the smallest number of points that they could score in their 10th game for their mean number of points-per-game to exceed 22?

A 22 B 23 C 24 D 25 E 26

Answer

SOLUTIOn

Let X be the total number of points scored after 6 games and let y be the number of points scored in the 10th game. Since the mean score after 9 games is greater than the mean score after 6 games, we have X/6 < (X+24+17+25)/9 and hence 3X < 6 × 66 and so X < 132. Similarly, since the mean after 10 games is greater than 22, (X+66+y)/10 > 22 and hence X + 66 + y > 220 and so X + y > 154. Therefore X ≤ 131 and X + y ≥ 155. Hence the smallest number of points they could score in the 10th game is 155 − 131 = 24.

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