Official Solution:


Statements (1) and (2) combined are insufficient. Consider:

\(\{14, 18, 24, 24\}\). The median is 21;

\(\{18, 18, 22, 22\}\). The median is 20.


Answer: E
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hello Bunuel,

This explanation seems a bit confusing to me .
The question says that the mean is "20".
Then it should include "20" in set S,
But the examples you took do not include 20.
can you please explain each statement more clearly?

The mean of {1, 2} is 1.5, which is NOT in the set. So, the mean of a set is not necessarily a member of the set.
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(1) In set S there are as many numbers larger than 20 as there are numbers smaller than 20

=> Dont know the number of numbers smaller and larger 20 => Cannot know the middle point of the number set => Insufficient

(2) All numbers in set S are even integers => Just know there are two Median for this number set => Insufficient

(1)+(2): Also insufficient

=> Answer: E

Initially got confused by statement
(1) In set SS there are as many numbers larger than 20 as there are numbers smaller than 20.

Was under the impression that number 20 has to in the set..
for example : {-30, -25, 20, 25, 30}, here there are two numbers are below 20 and two numbers above 20.. Mean = 20, Median = 20
from Bunuel's example set : {14, 18, 24, 24}, here again two numbers are below 20 and two above 20 .. Mean = 20, Median = 21

Clearly Statement (1) is Not Sufficient !!!

Can we apply a simple logic here, that to find the Median we shud have no of terms and also an assurance that the set is in AP ??

I think this is a high-quality question and the explanation isn't clear enough, please elaborate.

Check more solutions here: https://gmatclub.com/forum/if-the-mean- ... 34780.html

Hope it helps.
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Great question . I fell for the trap that 20 will be a part of the set and hence marked statement 1 as sufficient. Now I realised my mistake.Thanks

I think this is a high-quality question and the explanation isn't clear enough, please elaborate. This explanation doesn't take you through why each part is wrong. It is insufficient.

I think it is a nice deceptive question. After reading S1, i assumed that 20 will be part of the set and hence will be the median. However, the statement implies that you can have elements greater than or smaller than 20.

We can easily create a set with an average of 20

20,20,20,20, here the median=20 from this, we can also easily create a new set with an average of 20 (YES)

5,7,30,38, here the median is 37/2 (NO).

From this, we can also easily create a set with all even numbers:

6,8,28,38, here the median is 18 (NO) or

4,10,30,36, here the median is 20 (YES)
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Hello Sir,

I am still not able to arrive how both statements together are insufficient.....Can you please explain me with a simple example sir

You might find the following discussion helpful: https://gmatclub.com/forum/if-the-mean- ... 34780.html
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I guess the hard part of this problem is to come up with two instances to show the median may or may not be 20

Bunuel has these two sets as the solutions, but how did he come up with such numbers? Lets try to find out...

{14,18,24,24}. The median is 21;
{18,18,22,22}. The median is 20.

I guess he started with an easy set, {18,18,22,22}. Mean=20, median =20, easy!
Now our goal is to get to a set which has a mean=20 but median is NOT 20.

Step I: I want to space the last two elements a bit further off,
{18,18,24,24}, here the sum=84 (and the goal is to keep the sum=80, so that the mean is 20)

Step II: I want to compensate the extra 4 in the sum in the first two elements
{16,16,24,24}, now my mean =20, but my median is also 20

Step III: I want to finally change the second term, such that the median is NOT 20 anymore
{14,18,24,24}, so now my mean is still 20, but my median is 21

I think this is how Bunuel was thinking when he came up with these numbers. Of course we can make more such cases.

Answer: E