Last month 15 homes were sold in Town X. The avg. sale price of the homes was $150,000 and the median sale price was $130,000. Which of the following statements must be true?

I. At least one of the homes was sold for more than $165,000.
II. At least one of the homes was sold for more than $130,0000 and less than $150,000
III. At least one of the homes was sold for less than $130,000.

A. I only
B. II only
C. III only
D. I and II
E. I and III
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Think of the prices laid out as follows:

__ __ __ __ __ __ __ $130,000 __ __ __ __ __ __ __

$130,000 is the median, but we also know that $150,000 is the average, so obviously there must be some homes priced higher than $130k. Now, the higher priced the homes that precede the median of $130k, the less the homes priced higher than $130k have to be to create a mean of $150k. So let's fill out the scenario of the highest priced homes left of the $130k:

$130k $130k $130k $130k $130k $130k $130k $130k __ __ __ __ __ __ __

Now, the remaining seven must be high enough to create an average of $150k. Some quick calculations will show that even in this scenario, even if every home priced above the median were $170k, the mean would still be less than $150k.

Based on this, we know that I MUST be true. Eliminate any choice that doesn't list I. B and C are out.

We can also quickly deduce that II is not necessarily true since we just showed that all the prices above the median could be greater than $170k, or one could be $131k, and the rest much higher- you get the idea. The point is that we don't know, so II is out, and any of the 5 choices that list II. Eliminate D.

We are down to A and E.

III is out because from the example above, we know that all of the homes that precede the median-priced home could be $130k or less, but it is not true that one HAS to be less than $130k. Eliminate E. A is your answer.

By the way, the reason I immediately jumped to the example of all of the prices preceding $130k equaling $130k instead of playing around with ranges, is that questions like this almost invariably hinge on your ability to quickly recognize that average/median questions like this involve a sort of tug-of-war between the ends of the average. You should think of the layout of prices as a see-saw. The median and mean are fixed in this case, so if you lessen one end, the other end must be raised, and vice versa. In this example, option I is asserting that at least one price can't be less than $165k, so you should quickly recognize that you need to set up a case to disprove this by making the prices as close to the median as possible on both ends.

It's quiet possible to have a house @140k. Say 140/160, that would give 150 as the mean. It's possible, but not mandatory, hence St. II is out.