The GRE maths sample question given below is a problem solving question in statistics. This sample question is a GRE hard math question.
Question 1: The maximum mark in an examination is 100 and the minimum is 0. The average mark of seven students such that no two of them have scored the same marks is 88. If the median score is 92 and all the marks are integers, what is the maximum possible difference between the highest and the least mark obtained by these seven students?
Let the marks of seven students in ascending order be S1, S2, S3, S4, S5, S6, and S7.
Data Point 1: Average mark of seven students is 88.
Hence, sum of the marks of the seven students = 7 × 88 = 616.
Data Point 2: The median is 92.
Therefore, mark scored by S4 is 92.
Data Point 3: No two of them have scored the same marks. All marks are distinct.
Data Point 4: Maximum mark in the exam is 100 and the minimum mark is 0.
So, S7 cannot exceed 100 and S1 cannot be lesser than 0.
The difference between the highest and the lowest mark is maximized when the highest mark is maimized and the lowest mark is minmized. i.e., maximize S7 and minimize S1.
Rather easy. Theoretically, the highest mark cannot be more than 100.
So, Max S7 = 100
S1 is minimized when the remaining 4 marks S2, S3, S5, and S6 are also maximized.
Why only these 4 marks?
S7 is already maximized. S4 is 92 and cannot be changed. S1 has to be minimized. That leaves us with these 4 marks.
Maximum values for S5 and S6 are 98 and 99 respectively.
We know S4 = 92
So, Min S1 + Max S2 + Max S3 + 92 + 98 + 99 + 100 = 616
Or Min S1 + Max S2 + Max S3 = 227
How to maximize S2 and S3?
S2 and S3 should be lesser than S4 = 92.
Hence, max S3 = 91 and max S2 = 90.
So, Min S1 + 90 + 91 = 227
⇒ Min S1 = 46
Max S7 = 100; Min S1 = 46
Maximum possible difference between highest and lowest mark, S7 - S1 = 100 - 46 = 54.
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