Last month \(15\) homes were sold in Town \(X.\) The average (arithmetic mean) sale price of the homes was \(\$150,000\)

Vincen wrote: ↑

Fri Jun 18, 2021 7:38 am

Last month \(15\) homes were sold in Town \(X.\) The average (arithmetic mean) 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

Answer: A

Source: GMAT Prep

The key word in this question is MUST
So, if it's possible to create a scenario in which the statement is not true, we can eliminate it.

So, let's create a 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, 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 statements II and III need not be true.
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