A list of number of 25 integers has a median of 50 and a range of 50.

Question

A list of number of 25 integers has a median of 50 and a range of 50. What is the greatest possible integer that could be in this set?

A. 62
B. 68
C. 75
D. 88
E. 100

Bunuel · Accepted Answer

Consider 25 numbers in ascending order to be  x_1 ,  x_2 ,  x_3 , ...,  x_{25} .

The median of a set with odd number of elements is the middle number (when arranged in ascending or descending order), so the median of given set is  x_{13}=50 ;

The range of a set is the difference between the largest and the smallest numbers of a set, so the range of given set is  50=x_{25}-x_{1}  -->  x_{25}=50+x_{1} ;

We want to maximize  x_{25} , hence we need to maximize  x_{1} . The maximum value of  x_{1}  is the median, so 50, hence the maximum value of  x_{25}  is  x_{25}=50+50=100 .

The set could be {50, 50, 50, ..., 50, 100}

Answer: E.

Now,  in order the answer to be 88, as given in the OA , the question should state that "A set of 25  different  integers has a median of 50..."

In this case since all integers must be distinct then the maximum value of  x_{1}  will be  median-12=50-12=38  and thus the maximum value of  x_{25}  is  x_{25}=38+50=88 .

The set could be { 38 , 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,  50 , 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61,  88 }

Answer: D.

Hope it's clear.

HKHR · Answer

Shouldn't the question actually be "25 different positive integers" then?

Bunuel · Answer

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

hdwnkr · Answer

The range 50 stops us from taking numbers non-consecutive numbers. Right?

ENGRTOMBA2018 · Answer

Not necessarily. One possible answer could have been 38,38,38... (9 more times),50,50,50,... (9 more times), 88. This way your set has a median value of 50 and range is 50 as well. In this case the elements of the set are not consecutive integers. No where in the question is it stated that the integers have to be different!

for questions like these, it is always better to use the options provided as follows:

You know that the median of 25 integers (or the 13th term) has to be 50 and that the range has to be 50.

Now start with option (E). Can the max. be 100? Lets check. If the max. is 100, then the minimum has to be 50 (for range to be 50) and we can then have a set as : 50,50...(22 more times), 100. Yes, this is a possible answer and thus is the correct answer. You dont have to spend any more time on this. But I agree with Bunuel that for the OA to be 88, there needs to additional information that can negate 100 as the maximum possible integer value.

shashankism · Answer

Set of 25 integers 
median = 50 
Range = 50

So, Mid number i.e. 13 th number is 50 
{x1, x2, ......., x12, 50, x14, x15, ...... x25}
For the integer to be greatest and range to be 50 , we will have to maximize the lowest integer.

This is possible by keeping all the numbers on the left hand side of x13 = 50 as 50.
So set becomes {50,50,......., 50,50, x14, x15, ...... x25}

Now greatest integer can go upto 50+50 = 100.

Answer E

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

A list of number of 25 integers has a median of 50 and a range of 50.

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

A list of number of 25 integers has a median of 50 and a range of 50.

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards