Median question from GMAT Prep

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Median question from GMAT Prep

by prakashchandra » Tue Nov 05, 2013 3:06 pm
A set of 15 different integers has a median of 25 and a range of 25.What is the greatest possible integer that could be in the set?

A. 32
B. 37
C. 40
D. 43
E. 50

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by Brent@GMATPrepNow » Tue Nov 05, 2013 3:08 pm
prakashchandra wrote:A set of 15 different integers has a median of 25 and a range of 25.What is the greatest possible integer that could be in the set?

A. 32
B. 37
C. 40
D. 43
E. 50
Let's tackle this one step at a time.

First, we have 15 different integers.
We can let these 15 spaces represent the 15 numbers written in ascending order: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

If the median is 25, we can add this as the middle value: _ _ _ _ _ _ _ 25 _ _ _ _ _ _ _
Notice that 7 of the remaining numbers must be greater than 25 and the other 7 remaining number must be less than 25.

Since, we are told that the range is 25, we know that the greatest number minus the smallest number = 25

Now notice two things:
1) Once we know the value of the smallest number, the value of the greatest number is fixed.
For example, if the smallest number were 10, then the greatest number would have to be 35 in order to have a range of 25
Similarly, if the smallest number were 12, then the greatest number would have to be 37 in order to have a range of 25

2) If we want to maximize the value of the greatest number, we need to maximize the value of the smallest number.

So, how do we maximize the value of the smallest number in the set?
To do this, we must maximize each of the 7 numbers that are less than the median of 25.

Since the 15 numbers are all different, the largest values we can assign to the numbers less than the median of 25 are as follows:
18 19 20 21 22 23 24 25 _ _ _ _ _ _ _ (this maximizes the value of the smallest number)

If 18 is the maximum value we can assign to the smallest number, and if the range of the 15 numbers is 25, then greatest number must equal 43 (since 43 - 18 = 25)

So, the numbers are as follows: 18 19 20 21 22 23 24 25 _ _ _ _ _ _ 43 (the missing numbers don't really matter here)

This means the answer is [spoiler]43 = D[/spoiler]

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by GMATGuruNY » Tue Nov 05, 2013 5:55 pm
A set of 15 different integers has a median of 25 and a range of 25. What is the greatest possible integer that could be in this set?
32
37
40
43
50
Range = biggest - smallest.
Thus:
25 = biggest - smallest.
Smallest = biggest - 25.

An alternate approach is to PLUG IN THE ANSWERS, which represent the greatest possible integer that could be in the set.
Since we need the greatest possible integer that could be in the set, we should start with the GREATEST answer choice.

Answer choice E: 50
Smallest = 50-25 = 25.
Since all the integers must be different, the smallest integer cannot be equal to the median.
Eliminate E.

Answer choice D: 43
Smallest = 43-25 = 18.
Thus, the 7 integers below the median of 25 could be {18,19,20,21,22,23,24}.
The 6 integers between 25 (the median) and 43 (the biggest) could be any 6 different integers between 25 and 43.
This works.

The correct answer is D.
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