could be the median?

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jain2016: Master | Next Rank: 500 Posts; Posts: 313; Joined: Tue Oct 13, 2015 7:01 am; Thanked: 2 times

by jain2016 » Tue Apr 26, 2016 10:37 am

Given distinct positive integers 1, 11, 3, x, 2, and 9, which of the following could be the median?

A) 3

B) 5

C) 7

D) 8

E) 9

OAB

Hi Experts ,

Please explain.

Many thanks in advance.

SJ

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Source: Beat The GMAT — Problem Solving | Tue Apr 26, 2016 10:37 am

DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 amFollowed by: 128 members; Location: Boston, MA; Thanked: 1153 times; GMAT Score:770

by DavidG@VeritasPrep » Tue Apr 26, 2016 11:06 am

jain2016 wrote:Given distinct positive integers 1, 11, 3, x, 2, and 9, which of the following could be the median?

A) 3

B) 5

C) 7

D) 8

E) 9

OAB

Hi Experts ,

Please explain.

Many thanks in advance.

SJ

If there are six elements in a set, the median will be the average of the third and fourth term.

Our set: 1, 2, 3, 9, 11 and x

Because the elements in the set are distinct and positive integers, the smallest possible value for x is 4. If 3<x<9, the third and fourth term will be 3 and x, giving us a median of (3+x)/2. If x is greater than 9, the third and fourth term will be 3 and 9, and their average will be (3+9)/2 = 6. Not an answer, so we know that 3<x<9.

Now let's test the answers.

A) 3; This would only work if x = 3, but we're dealing with distinct integers and the set already contains a 3, so this is out.
B) 5; If the median is 5, then (3+x)/2 = 5. and x = 7. Looks like we have our answer.

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Wed Apr 27, 2016 2:37 pm

Let's start by arranging the set:

{1, 2, 3, 9, 11}

We know the numbers are distinct positive integers, so x must be GREATER than 3. (It can't be 1, 2, or 3, because then it wouldn't be distinct, and it can't be < 1, or it wouldn't be positive.)

From there, we could try a few x's.

If x = 4, the median = (3+4)/2 = 3.5
If x = 5, the median = (3+5)/2 = 4
If x = 6, the median = (3+6)/2 = 4.5
If x = 7, the median = (3+7)/2 = 5

Touchdown! 5 works, so we're set.

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OptimusPrep: Master | Next Rank: 500 Posts; Posts: 410; Joined: Fri Mar 13, 2015 3:36 amFollowed by: 8 members; Location: Worldwide; Thanked: 120 times; GMAT Score:770

by OptimusPrep » Wed Apr 27, 2016 8:28 pm

jain2016 wrote:Given distinct positive integers 1, 11, 3, x, 2, and 9, which of the following could be the median?

A) 3

B) 5

C) 7

D) 8

E) 9

OAB

Hi Experts ,

Please explain.

Many thanks in advance.

SJ

Since we are talking about the median, the first thing that we need to do is to arrange the numbers in ascending order

{1, 2, 3, 9, 11}
If the number of elements in a set are even, then the median is the average of the middle two terms if the series is in ascending/descending order
If the number of elements in a set are odd, then the median is the middle term is the series if in ascending/descending order

Now in the question at hand, we are given that the numbers are all distinct and positive. Hence the number has to >=4
If x = 4, series becomes {1, 2, 3, 4, 9, 11}
Median = 3+4/2 = 3.5

If x = 5, series becomes {1, 2, 3, 5, 9, 11}
Median = 3+5/2 = 5

If x = 6, series becomes {1, 2, 3, 6, 9, 11}
Median = 3+6/2 = 4.5

If x = 7, series becomes {1, 2, 3, 7, 9, 11}
Median = 3+7/2 = 5
Hence the Correct Option: B

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