M12-14

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

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.

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 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.

I think that this is a high-quality question.

I am sharing my approach.

Mean=20, Median=?

From S1: \(Numbers_{>20}=Numbers_{<20}\)
Let's pick numbers in such a way that their mean is 20.
18 is 2 less than 20, 22 is 2 more than 20.
Case-I: S=\(\{18, 18, 22, 22\}\)
Mean=20, Median=20
14 & 18 are 6 & 2 less than 20 respectively i.e. -8.
24 is 4 more than 20. i.e. 4+4=8
Case-II: S=\(\{14, 18, 24, 24\}\)
Mean=20, Median=21
Insufficient

From S2: All numbers are even
Again, we can consider the cases we had constructed in S1.
Case-I: S=\(\{18, 18, 22, 22\}\)
Mean=20, Median=20
Case-II: S=\(\{14, 18, 24, 24\}\)
Mean=20, Median=21
Insufficient

S1 & S2:
No new information.
Insufficient.

I think this is a high-quality question and I agree with explanation.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.