Problem Solving

Hi J_shreyans,

When it comes to maximizing or minimizing a value in a group of numbers, you have to think about what the other numbers would need to be to accomplish your goal.

Here, we have a group of 15 DISTINCT (meaning DIFFERENT) integers with a median of 25 and a RANGE of 25. That range will dictate how large the largest value can be.

With a median of 25, we know that 7 numbers are LESS than 25 and 7 numbers are GREATER than 25:

_ _ _ _ _ _ _ 25 _ _ _ _ _ _ _

To maximize the largest value, we need to maximize the smallest value. Here's how we can do it:

18 19 20 21 22 23 24 25 _ _ _ _ _ _ _

With 18 as the smallest value, and a range of 25, the largest value would be 43.

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

Here's the complete question:

A set of 15 different integer has a median of 25 & a range of 25. What is the greatest possible integer that could be in this 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