Problem Solving

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 $130,000 and less than 150,000
III. At least one of the homes was sold for less than 130,000.

I only
II only
III only
I and II
I and III

Thanks!

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.
If it's possible to have a sum of 2250 without including a price below 130, we can eliminate 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.

Please note that by examining what we were trying to disprove, we were able to determine the correct answer by testing only one set of prices.

GMATGuruNY wrote:

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.
If it's possible to have a sum of 2250 without including a price below 130, we can eliminate 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.

Please note that by examining what we were trying to disprove, we were able to determine the correct answer by testing only one set of prices.

I understand your approach and it works well, but why did you try to disprove 2 first? I still don't understand why statement 1 must be true.....

alex.gellatly wrote: I still don't understand why statement 1 must be true.....

Divide all the costs by 1000 for simplicty
Average cost of 15 houses = $150
Median = $130


Statement 1:At least one of the homes was sold for more than $165,000

Lets say no house was sold for more than $165,000
Highest mean possible = (130+130+130+130+130+130+130+130+165+165+165+165+165+165+165)/15 = 146.33 < 150

To increase the value of mean, cost of atleast 1 house must be increased.
Cost of homes sold for $130 cannot be increased, as that would change the median.
Hence, atleast one of the homes must cost more than $165.
Statement 1 is correct.