Question: If the mean of set S does not exceed the mean of the subset of set S, which of the following must be true about the set S?

1. Set S contains only one element
2. All the elements in set S are equal
3. The median of set S equals the mean of set S

1. None of the three qualities is necessary
2. II only
3. III only
4. II and III only
5. I, II, III

“If the mean of set S does not exceed mean of the subset of set S, which of the following must be true about the set S?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”.
To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer:

Approach Solution (1):

“The mean of set S does not exceed mean of any subset of set S” --- set S can be:

A: S = {x} – S contains only one element
B: S = {X, X, …} – S contains more than one element and all elements are equal
What is that? Because if set S contains two (or more) different elements, then we can always consider the subset with the smallest number and the mean of this subset will be less than mean of entire set
Example: S = {3, 5} mean of S = 4. Pick the subset with smallest numbers = {3} mean of s’ = 3 --- 3 < 4
Now let’s consider the statements:

I: Set S contains only one element- not always true, we can have scenario B too (S = {X, X, …});
II: The elements in set S are equal- true for both A and B scenarios, hence always true;
III: The median of set S equals the mean of set S- true for both the statements A and B , hence always true

So the statements II and III are always true.

Correct option: D

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