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]

Cheers,
Brent

Hi anksm22,

This question can be solved by TESTing THE ANSWERS.

This prompt provides a number of math facts to work with:

1) 15 DIFFERENT integers
2) The median (meaning the 8th number) is 25.....since the 15 numbers are all DIFFERENT, there are 7 numbers that are less than 25 and 7 other numbers that are greater than 25.
3) The range is 25....so the largest number - smallest number = 25

We're asked for the GREATEST POSSIBLE integer in the set of 15 numbers.

Since we have 15 DIFFERENT integers, 7 of the numbers are less than the median (25) and the range = 25, the greatest number CANNOT = 50 (since the smallest number + 25 = largest number).

Let's try 43

Largest = 43?
Median = 25

You COULD have 7 numbers greater than 25 (including 43).

Range = 25
43 - 25 = 18 = Smallest value?

With 18 as the minimum, you COULD have 18, 19, 20, 21, 22, 23, 24 --> 7 numbers that are smaller than 25.

This PROVES that 43 is the GREATEST possible value of the set.

Final Answer: D

GMAT assassins aren't born, they're made,
Rich