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

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:

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

The correct answer is D.

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?

Given 15 different integers, lets say

n1, n2, n3, n4, n5, n6, n7, n8, n9, n10, n11, n12, n13, n14, n15

Also given median is 25 i.e. n8 = 25

n1, n2, n3, n4, n5, n6, n7, 25, n9, n10, n11, n12, n13, n14, n15

As each integer is different we need to find the maximum values for all those numbers before the median.

the maximum value n7 can have is one less then the median i.e. 24, similarly n6 will be one less than 24 i.e. 23 ... using this process the values for all number before the median would be..

18, 19, 20, 21, 22, 23, 24, 25, n9, n10, n11, n12, n13, n14, n15

Also given the range is 25 i.e. n15 - n1 (18) = 25
The maximum value n15 can have is 25 + n1 (18) = 43

50 can not be the highest number.

for the range to be 25, (50 - least number) = 25, i.e least number is 25
But it's given that 25 is median and since each number is different, there must be 7 smaller numbers than 25

8 different numbers from 1st to 8th number gives lowest number = 25-8= 18
hence max number = 18+25 = 43.

Try to right down 25 in the middle as a median and 7 numbers to the left and 7 nubers to the right.
You will see clearly that the minimum possible least number is 18 to the left of 25.
Hence 18+25 -->43

18 19 20 21 22 23 24 25 ....... --> the least possible

Similar question to practice:


Discussed here: a-set-of-25-different-integers-has-a-median-of-50-and-a-129345.html

Variation of this question: a-set-of-25-integers-has-a-median-of-50-and-a-range-of-128463.html
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The largest in the set minus the smallest in the set should be 25
All the integers are different

Use the options. If 50 is the largest and the range is 25, the smallest is 25, which could have been if the integers were allowed to be same in the set.


Next option to check is 43. 43 - 25 = 18. Can 18 be a part of the set and at the same time there are no repetitions and 25 is the median? Yes

18, 19, 20, 21, 22, 23, 25, 6 more numbers, 43

Hence D
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Hi hdwnkr,

Excellent use of TESTing THE ANSWERS! While that approach isn't nearly as useful overall as TESTing VALUES or the frequent 'math' approaches that will always be an option, it WILL be applicable at least a handful of times on Test Day. Having that approach in your skill-set will help you to quickly (and correctly) solve those few questions and move on with confidence.

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

Thanks a lot for the encouragement. Much appreciated! Helps when I am just days away from the D Day
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Haven't you made the assumption that the integers in the set are consecutive integers whereas the question does not specify that they are consecutive?

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. 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
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Hi Mitty,

Since the question asks for the LARGEST POSSIBLE integer that could be in the set of numbers, we have to tailor our work around a certain 'math idea' - since we have a range of 25, to get the largest possible integer, we need the smallest integer in the set to be as big as possible. Working 'backwards' from the median - and keeping in mind that all of the values are distinct - the numbers BELOW the median would have to be consecutive. In that way, we could make the smallest number as big as possible.

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

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


RELATED VIDEO

The constraints are median (which is easy to handle - just that the 8th integer will be 25) and range (which is 25). The important thing is that all integers are different. For the range to be constant and then have the greatest possible integer, the smallest integer should be as large as possible.

Since all integers are distinct, the smallest integer should be 7 less than 25 i.e. 18 (so 18, 19, 20, 21, 22, 23, 24, 25 are the first 8 integers)

For the range to be 25, the greatest integer should be 18+25 = 43

Answer (D)
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I think the easiest way to approach this problem is process elimination
Let's look at answer D which is 43
25 is the 8th number in the sequence

43 - x = 25
x = 43 - 25
x = 18

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

Therefore, the answer is D

Hi,

I have a question here, it is mentioned that the median is 25 or we can say the 8th element is 25 and the range is 25 but it's nowhere mentioned that the numbers are consecutive. Then how did you assume them to be consecutive integers?

If I take 37 as the greatest number, then the first element must be 12 and it can still satisfy 12,15,17,19,21,23,24,25

25 can be the median in this case too..or is it a strict rule to assume the numbers in consecutive order? Please explain

As per my knowledge, for the range to be constant, and to find the greatest integer, the smallest integer must be as large as possible.. Here 18 can be the largest..so we have to take 43..

Hi Pratyaksh2791,

This question asks us to find the greatest possible number that could be in this set. Since 25 is the MEDIAN of the group of 15 INTEGERS, we know that 7 integers are greater than 25 and 7 integers are less than 25. We're told that the largest integer is exactly 25 more than the smallest integer, so to maximize the biggest value, we also have to maximize the smallest value. Since we're restricted to INTEGERS, the only way to get that maximum result is if the 7 integers that are less than 25 are CONSECUTIVE integers:

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

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

So does that mean that this set does not contain any -ve integers? I am assuming it can not but why?

Hi Rocket7,

The question tells us that the RANGE of the numbers is 25... and asks us to find the largest POSSIBLE integer that could be in the set. If you look at the five answer choices - and then subtract 25 from each, you'll see that the SMALLEST number that would correspond to each of those answers would be a positive integer. Thus, the set CANNOT contain any negative numbers.

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