Question 15.
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?
Question Explanation
It is given that the average of the first five highest-rated employees is 38. So, the sum of the ratings of the top 5 highest-rated employees is 38 * 5 = 190.
It is given that the average of the first five lowest-rated employees is 25. So, the sum of the ratings of the top 5 lowest-rated employees is 25 * 5 = 125.
The overall average rating of all the employees is given as 30.
So, the sum of the ratings of all the employees is given as 30 * 8 = 240.
The sum of the 3 highest rated employees' ratings can be obtained by subtracting the sum of the 5 lowest rated players' ratings from the overall rating, which is 240 - 125 = 115.
The sum of the 3 lowest rated employees' ratings can be obtained by subtracting the sum of the 5 highest rated players' ratings from the overall rating, which is 240 - 190 = 50.
So, the sum of the 4th and 5th highest-rated employees is 190 - 115 = 75.
Now let us look at the options to eliminate the wrong option,
Option A) It is given that the second-highest rating is 38. It is not an incorrect option because there is a possible case of the second highest being 38 and satisfying all the above conditions.
The first 5, in that case, can be 39, 38, 38, 38, 37, which satisfies the above conditions.
Option B) Same as the above case, we can have the 4th and 5th ratings to be 38 and 37, and in that case, the median of the ratings of the employees becomes . So, this is not an incorrect option.
Option C) We know that the sum of the lowest three ratings is 50, and in that case, there are possibilities of the lowest rating being 1 and the sum of the three is 50. For example, 37, 12, 1 is one of the cases. Hence, this is not an incorrect option.
Option D) The highest rating cannot be 40 because if it is 40, then the sum of the 2nd and 3rd becomes 75. In that case, the 3rd rated person's rating has to be less than or equal to 37, and we know that the 4th person's rating must be greater than or equal to 38. So, if the third person rating is 37 or less, then there is no possibility as the 4th person rating must be less than 3rd person rating. So, option D is incorrect.
Option E) As explained in option C, the set of the last three rated employees can be 37, 12 and 1, and in this case, we can see that the third lowest player's rating is 37. So, E is not an incorrect option.
Hence, the correct answer is option D.
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