A set of 15 different integers has median of 25 and a range

[BTGmoderatorDC]

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

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

OA D

Source: GMAT Prep

[fskilnik@GMATH]

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

A. 32
B. 37
C. 40
D. 43
E. 50
Source: GMAT Prep

WLOG (without loss of generality) we may assume that:
$$a = {x_1} < {x_2} < \ldots < {x_7} < {x_8} = 25 < {x_9} < \ldots < {x_{14}} < {x_{15}} = a + 25\,\,\,\,\,{\text{ints}}$$
Considering this powerful structure, the problem is trivialized:
$$?\,\, = \,\,\left( {a + 25} \right)\,\,\max \,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,a\,\,\max $$
$$a\,\,\max \,\,\,\,\, \Leftrightarrow \,\,\,\,\left( {{x_7},{x_6},{x_5},{x_4},{x_3},{x_2},{x_1} = a} \right) = \left( {24,23,22,21,20,19,18} \right)\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a = 18\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a + 25 = 43$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

[[email protected]]

Hi All,

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.

Final Answer: D

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

[Scott@TargetTestPrep]

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

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

OA D

Source: GMAT Prep

We are given that there are 15 different integers in a set with a median of 25 and a range of 25. We must determine the greatest possible integer that could be in the set. To determine this integer, we need to first determine the greatest possible value of the least integer from the set.

Since there are 15 total integers in the set, there are 7 integers before the median and 7 integers after the median if we list them in order. We must also keep in mind that each integer is different. We want the smallest integer in the set to be as large as possible. Thus, the first 8 integers, including the median, are the following:

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

Since the range of this set is 25, the greatest number in this set must be 25 more than the smallest integer in the set, and thus the largest number in the set is 18 + 25 = 43.

Answer: D

[Brent@GMATPrepNow]

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

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

OA D

Source: GMAT Prep

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 43

Answer: D