This GMAT Sample Math question is a descriptive statistics question focusing on Arithmetic Mean & Median. A GMAT hard math question. Helps understand the concept of how to maximize a parameter. An interesting GMAT 700 level problem solving practice question.
Question 8: Positive integers from 1 to 45, inclusive are placed in 5 groups of 9 each. What is the highest possible average of the medians of these 5 groups?
We are dividing the numbers into sets of 9. So, we will have 5 sets.
The median of each set is the middle number when the 9 numbers are arranged in ascending order.
So, the median is the 5th number when the numbers in a set are written in ascending order.
Therefore, we will have 4 numbers that are greater than the median in each set.
We need to maximize the median in each group in order to maximize the average of all the medians.
The highest possible median is 41 as there should be 4 numbers greater than the median in a group of 9.
So, if we have a group that has a, b, c, d, 41, 42, 43, 44, 45, the median will be 41.
In this set, it is essential not to expend any more high values on a, b, c, or d as these do not affect the median.
For example, the median of a group that comprises 1, 2, 3, 4, 41, 42, 43, 44, 45 will be 41.
The next group can be 5, 6, 7, 8, 36, 37, 38, 39, 40. The median will be 36.
The first 4 numbers in any set is not of much consequence. Our focus is on the numbers upward of the median in any set.
Extrapolating the findings in the two sets listed above, to maximize medians in all the 5 groups, the medians of the 5 groups will have to be 21, 26, 31, 36, and 41.
The average of the highest possible medians will be the average of these 5 numbers = 31.
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