I12-10

Question

A university conducted an entrance test for a research program, with ten candidates participating. Each candidate was assigned a distinct integer score based on their performance.
 
The top candidate scored 90 points. The difference in score between the top candidate and the seventh-ranked candidate was 25, while the difference in score between the fourth-ranked and tenth-ranked candidates was 6.
 
Select for  Median  the median score of the ten participants, and for  Range  the range of the scores of the ten participants. Make only two selections, one in each column.

Bunuel · Answer

Official Solution:

Given that the difference in score between the top candidate and the seventh-ranked candidate was 25, the 7th candidate scored 90 - 25 = 65 points.
 
From the 4th to the 10th candidate, inclusive, there are 7 people: (4th, 5th, 6th, 7th, 8th, 9th, 10th). Since the range of their scores is 6 and all scores are distinct integers, their scores must be consecutive integers. With the 7th candidate scoring 65, the scores are: 4th = 68, 5th = 67, 6th = 66, 7th = 65, 8th = 64, 9th = 63, and 10th = 62.
 
Therefore, the median score would be the average of the two middle scores, 5th and 6th, so 66.5, and the range would be the difference between the highest and lowest scores, so the difference between the top, 90, and the 10th, 62, so 28.

Correct answer:

Median  "66.5"

Range  "28"

Bunuel · Answer

I have revised the question, solution, and formatting by adding more details to enhance clarity.

Dereno · Answer

Another nice question in this set

Vijendra.D · Answer

This is a great question that’s helpful for learning.

anushree01 · Answer

The greatest catch here is this :

From the 4th to the 10th candidate, inclusive, there are 7 people: (4th, 5th, 6th, 7th, 8th, 9th, 10th). Since the range of their scores is 6 and all scores are distinct integers, their scores must be consecutive integers.

If you catch this in time, its a cakewalk then

dhruv23320 · Answer

I like the solution - it’s helpful.

agrasan · Answer

I like the solution - it’s helpful.

Vijendra.D · Answer

1st term == X-6.,,,, 2nd term = ___
3rd term ==___.,,,,, 4th term == 65 
5th term == ___.,,,,,6th term == ___
7th term == X.,,,,,,, 8th term ==___
9th term == ___.,,,,,10th term == 90.

Median == (5th + 6th)term/2

4th term = 65 -----> 5th and 6th term >65 
So median could be 66 or 66.5

If median == 66 ----> 5th term = 65 and 6th term = 67 ----> Not Possible
If median == 66.5 ----> 5th term = 66 and 6th term = 67 ----> Possible

DimitrisChatzi · Answer

I did not quite understand the solution. Could you please explain why we have to assume because the integers are disctict they also have to be consecutive?

Bunuel · Answer

Because there are 7 distinct integer scores (4th through 10th) but their range is only 6.

For 7 distinct integers, the smallest possible range happens when they are consecutive. If they were not consecutive, at least one “gap” of 1 or more would appear, and that would force the max minus min to be greater than 6.

So “distinct integers” does not automatically mean consecutive, but “7 distinct integers with range 6” forces consecutiveness.

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