I'm happy to help with this.
First of all, I believe this is the complete text of the question, with the answer choices.
Sets F and G contain 20 positive integers, respectively. If the average of the numbers in F is greater than the greatest number of G, which of the following must be true?
I. The average of F is greater than that of G.
II. The median of F is greater than that of G.
III. The range of G is greater than that of F.
A. I only
B. II only
C. I and II
D. I,II, and III
E. II and III
I don't know the source, and I am little suspicious of the question because of the grammatical mistake in its phrasing --- the word "respectively" is used incorrect, creating ambiguity. The real GMAT, of course, will have flawless grammar on each and every math question.
Be that as it may. How do we solve this?
I'm going to assume that what they meant to say is --- Set F and Set G each contain 20 numbers. Minor variations from what will not affect the solution.
We are told: the average of the numbers in F is greater than the greatest number of G. Right away, that tells us that there are at least a few numbers in F that are much bigger than any number in G. The biggest number in F&G combined would have to be in F. Notice, this
doesn't mean the smallest number in F&G combined would be in G --- we can't conclude that.
I. The average of F is greater than that of G.
Well, the average of any set must be bigger than the minimum of that set and smaller than the maximum of that set. The average won't necessary be symmetrically located between min and max, but it definitely must be smaller than the max and bigger than the min. From this we know (max of G) > (average of G).
Well, the question tells us (average of F) > (max of G). Combining the inequalities, we know (average of F) > (max of G) > (average of G), and by transitivity of inequality, we know that (average of F) > (average of G). Statement I is
true.
II. The median of F is greater than that of G.
This is not necessarily true. The median really only tells us what's happening right at the middle of the list, and gives us no information about outliers at either end. For example, Set F could be eleven 1's and nine 10's ==> meanF = 5.05, and medianF = 1, while G could be twenty 3's ==> meanG = medianG = 3. That's a simple example where (average F) > (max of G) but (median G > median F). Thus, Statement II is
not necessarily true.
III. The range of G is greater than that of F.
Range = max - min. Well, in the example set I gave above --- Set F = eleven 1's and nine 10's, and Set G = twenty 3's ---- Set F has a range of 9, and Set G has a range of 0. So the range of G doesn't have to be greater. Thus, Statement III is
not necessarily true.
Only Statement I must be true. That gives an answer of
A.
Does that make sense? Please let me know if you have any questions?
Mike
