Question

Eight employees of an organization have been rated on a scale of 1 to 50 for their performance. All ratings are integers. The overall average rating of the eight employees is 30. While the five employees with the highest ratings average 38, the five employees with the lowest ratings average 25.

Which of the following, about the ratings obtained by the eight employees, is DEFINITELY FALSE?

Accepted Answer

D. Let ratings in increasing order be $a_1 \le a_2 \le a_3 \le a_4 \le a_5 \le a_6 \le a_7 \le a_8$. Sum of all 8 = $8 	imes 30 = 240$. Sum of top 5 ($a_4 + a_5 + a_6 + a_7 + a_8$) = $5 	imes 38 = 190$. Sum of bottom 5 ($a_1 + a_2 + a_3 + a_4 + a_5$) = $5 	imes 25 = 125$. From overall sum: $a_1 + a_2 + a_3 = 240 - 190 = 50$. $a_6 + a_7 + a_8 = 240 - 125 = 115$. $a_4 + a_5 = 190 - 115 = 75$. If highest rating $a_8 = 40$: Then $a_6 + a_7 = 115 - 40 = 75$. Since $a_5 \ge a_4$ and $a_4 + a_5 = 75$, max possible $a_4 \le 37$. Since $a_6 \le a_7$, and $a_6 + a_7 = 75$, min possible $a_6 \ge 38$. This requires $a_5 \le a_6$, which gives $a_5 = 37.5$, impossible since ratings are integers. Hence, $a_8$ cannot be 40. Option D is DEFINITELY FALSE.

Employee Ratings Definitely False Analysis

Explanation

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