The Maximum Mark in an Examination is 100 and the Minimum is 0 GMAT Problem Solving

Question - 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?

1. 11
2. 46
3. 99
4. 54
5. 100

‘The maximum mark in an examination is 100 and the minimum is 0’ is the topic for GMAT Quantitative Reasoning. GMAT quantitative reasoning section checks the ability of the candidate to solve mathematical, quantitative problems and interpret graphic data. This section of the GMAT Quant comprises 31 questions which needs to be completed in 62 minutes. In GMAT Problem Solving candidates logical and analytical reasoning skills. In this section, candidates indicate the best five answer choices. 

Solution and Explanation 
There is one approach to solve the problem statement.
The average mark of seven students such that no two of them have scored the same marks is 88.
This means: (the sum of all seven scores)/7 = 88
If we multiply both sides by 7, we get: the sum of all seven scores = 616
The median score is 92
So if we write all seven scores in ascending order, the middlemost score will be 92 as follows: _, _, _, 92, _, _, _
What is the maximum possible difference between the highest and the least mark obtained by these seven students?
In order to maximize the range, we must maximize the biggest number and minimize the smallest number.
Let's start by maximizing the biggest number.
Since the maximum score is 100, our resulting set looks like this: _, _, _, 92, _, _, 100
At this point we want to minimize the smallest value.
Since the sum of all seven scores = 616, we want to maximize the value of every score except the smallest score.
When we do this we get: _, 90, 91, 92, 98, 99, 100
The sum of the 6 biggest numbers = 90 + 91 + 92 + 98 + 99 + 100 = 570
So, the last remaining number (the smallest number) = 616 - 570 = 46
So the seven scores are as follows: 46, 90, 91, 92, 98, 99, 100
Range = biggest - smallest = 100 - 46 = 54
Hence, the correct answer is D

Correct Answer: D

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