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median / range problem

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by Brent@GMATPrepNow » Fri Apr 19, 2013 7:18 am
vishal_2804 wrote:A set of 15 different integers has a median of 25 and 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
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by Anju@Gurome » Fri Apr 19, 2013 7:18 am
vishal_2804 wrote:A set of 15 different integers has a median of 25 and range of 25. what is the greatest possible integer that could be in this set?
Median of a set of 15 different integers will be the 8th integer of the series when arranged according to their values.

Now, Largest - Smallest = Range ---> Largest = (Smallest + Range)
As the range is fixed, we can maximize largest number by maximizing the smallest number.

Maximum possible value of the smallest integer in the set is (25 - 7) = 18, as all the terms are different and 25 is the 8th term.

Hence, greatest possible integer in the set = (18 + Range) = (18 + 25) = 43

The correct answer is D.
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by Gurpreet singh » Wed Jun 22, 2016 5:53 am
Hi Anju,

Is this a rule

"Maximum possible value of the smallest integer in the set is (25 - 7)"

Can we also use it to find the maximum possible value if median is given?
Regards

Anju@Gurome wrote:
vishal_2804 wrote:A set of 15 different integers has a median of 25 and range of 25. what is the greatest possible integer that could be in this set?
Median of a set of 15 different integers will be the 8th integer of the series when arranged according to their values.

Now, Largest - Smallest = Range ---> Largest = (Smallest + Range)
As the range is fixed, we can maximize largest number by maximizing the smallest number.

Maximum possible value of the smallest integer in the set is (25 - 7) = 18, as all the terms are different and 25 is the 8th term.

Hence, greatest possible integer in the set = (18 + Range) = (18 + 25) = 43

The correct answer is D.
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by Gurpreet singh » Wed Jun 22, 2016 5:57 am
Hi Brent,
please make me understand this

"If we want to maximize the value of the greatest number, we need to maximize the value of the smallest number"

In a set should not the smallest nos be min possible so that the greatest no is maximum?
Regards
Gurpreet

Brent@GMATPrepNow wrote:
vishal_2804 wrote:A set of 15 different integers has a median of 25 and 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
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by Brent@GMATPrepNow » Wed Jun 22, 2016 7:59 am
Gurpreet singh wrote:Hi Brent,
please make me understand this

"If we want to maximize the value of the greatest number, we need to maximize the value of the smallest number"

In a set should not the smallest nos be min possible so that the greatest no is maximum?
Regards
Gurpreet
In this question we're told that range = 25
Range = max # - min #
So, for example, if the minimum value = 1, then the maximum value = 26
Likewise, if the minimum value = 40, then the maximum value = 65
Or if the minimum value = -20, then the maximum value = 5
As you can see, the bigger the minimum value, the bigger the maximum value.

Our goal is to get the biggest maximimum value possible.
To do so, we must make the minimum value as big as possible too.

Does that help?

Cheers,
Brent
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by [email protected] » Wed Jun 22, 2016 9:08 am
Hi Gurpreet singh,

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.

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by Gurpreet singh » Thu Jun 23, 2016 2:53 am
Thank you Brent & Rich
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by Matt@VeritasPrep » Thu May 11, 2017 8:22 pm
Gurpreet singh wrote:he value of the greatest number, we need to maximize the value of the smallest number"

In a set should not the smallest nos be min possible so that the greatest no is maximum?
Regards
Gurpreet
Typically yes, but remember that here the greatest number can be written as a function of the smallest number:

Greatest = Smallest + 25

So you want the Smallest to be as BIG as possible to make the Greatest also as big as possible.
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by Scott@TargetTestPrep » Thu Jun 01, 2017 4:07 pm
vishal_2804 wrote:A set of 15 different integers has a median of 25 and 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

We are given that there are 15 different integers in a data set, and its median is 25 and its range is 25. We need to determine the greatest possible integer that could be in the set.

Since there are 15 different integers and the median is 25, we know there are 7 integers less than 25 and 7 integers greater than 25. We can list the integers (using blanks to substitute unknown integers) as follows:

__, __, __, __, __, __, __, 25, __, __, __, __, __, __, __

Since we need to find the greatest possible integer that could be in the set, we also need to find the largest possible value of the smallest integer that could be in the set.

Since the integers are different, we can use consecutive integers for the first half of the data set leading up to the median. Let's list the integers (again using blanks to substitute unknown integers) as follows:

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

As we can see, the largest possible value of the smallest integer that could be in the set is 18. Since the range is 25, the largest possible value of the greatest integer that could be in the set is 18 + 25 = 43.

Answer: D

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