Homes sold- Median and Mean

josh80 wrote:Last month, 15 homes were sold in Town X. The average 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 $13,000 but 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 only
E) I and III only

OA A

The key word in this question is MUST. As in, which of the following MUST be true. So, if it's possible that one scenario is not true, we can eliminate it.

So, let's looks at one possible scenario and see which answer choices we can eliminate.

Aside: To make things simpler, let's divide all of the prices by 1000.

First, we'll use a nice rule that says: sum of all values = (mean)(number of values)
So, the sum of all 15 prices = ($150)(15) = $2250.

If the median is $130, then the middlemost value is $130

So, one possible scenario is:
130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 430

Aside: To find the last value (430), I took the sum of all 15 numbers (2250) and subtracted (14)(130)

Notice that this scenario tells us that II and III need not be true (since our scenario does not conform to either one).

Since answer choices B, C, D and E all include either II or III, we can eliminate them.

This leaves us with A, which must be the correct answer.

Cheers,
Brent