CAT Practice : Averages, Ratios, Mixtures

Question that is logical-reasoning based rather than algebraic. Given the average of a subset of an overall set, and maximum and minimum values overall, how does one find the average of the overall set.

Averages - maximum possible value

Q.2: The average score in an examination of 10 students of a class is 60. If the scores of the top five students are not considered, the average score of the remaining students falls by 5. The pass mark was 40 and the maximum mark was 100. It is also known that none of the students failed. If each of the top five scorers had distinct integral scores, the maximum possible score of the topper is......
1. 99
2. 100
3. 87
4. 95

• Correct Answer
Choice A. 99

Explanatory Answer

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Detailed Solution

10 students have scored 600 marks amongst them, and no one is allowed to score lesser than 40 or higher than 100. The idea now is to maximize what the highest scorer gets.

The 5 least scores have an average of 55, which means that they have scored 55 x 5 = 275 marks amongst them. This leaves 325 marks to be shared amongst the top 5 students. Lets call them a, b, c, d and e. Now, in order to maximize what the top scorer “e” gets, all the others have to get the least possible scores (and at the same time, they should also get distinct integers.)

The least possible score of the top 5 should be at least equal to the highest of the bottom 5. Now we want to make sure that the highest of the bottom 5 is the least possible. This can be done by making all scores equal to 55. If some scores are less than 55, some other scores have to be higher than 55 to compensate and make the average 55. Thus the highest score is the least only when the range is 0.

So now, we have the lowest value that the top 5 can score, which is 55. The others have to get distinct integer scores, and as few marks as possible, so that “e” gets the maximum.
So, 55 + 56 + 57 + 58 + e = 325
e = 99 marks.
Answer choice (A)

Correct Answer: 99

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