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Average and Median analysis

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Average and Median analysis

by amirhakimi » Wed Nov 06, 2013 5:09 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 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 $130,000 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 is A

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by Brent@GMATPrepNow » Wed Nov 06, 2013 6:17 am
amirhakimi wrote: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 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 $130,000 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 is A
Difficulty-wise, I'd place this question around 650.


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,
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by GMATGuruNY » Wed Nov 06, 2013 7:58 am
Last month 15 homes were sold in town X. The average sale price of the homes were $150,000 & the median sale price was $130,000. Which of the following statements MUST be true?

1) At least one of the homes was sold for more than $165,000
2) At least one of the homes was sold for more than $130,000 & less than $150,000.
3) 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
Try to prove that the answer choices DON'T have to be true.
To make the math easier, divide all of the given numbers by 1000.

The sum of the 15 prices = 15*150 = 2250.

The following solution is similar to Brent's, with one suggestion:
Before testing cases, look for a statement that seems easy to disprove.

If it's possible to have a sum of 2250 without including a price below 130, we can eliminate any answer choice that includes III.
Since the median price is 130, it's possible that 14 of the homes were sold for 130 each.
The price of the 15th home would then be 2250 - 14*130 = 430, yielding the following list of prices:

130, 130....130, median=130, 130...130, 430.

Eliminate any answer choice that includes III, since the list above does not include a price below 130.
Eliminate C and E.

Eliminate any remaining answer choice that includes II, since the list above does not include a price between 130 and 150.
Eliminate B and D.

The correct answer is A.
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