Problem Solving

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.

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.

Average Sale price = 150,000
Total Cost = 150,000 x 15 = 2250,000


Statement 1:
Consider that all the homes untill median were sold for 130k then ==> 130,000x8 = 1,040,000
Lets consider that all the homes after median were sold for 165k ==> 1,155,000
Adding = 1040,000 + 1155,000 ==> 2195,000 ... HENCE there must be a HOME sold for more than 165k
AGREED

Statement 2:
We can have a house which was sold for say 140k and then the price of other house will be must increased
NOT AGREED

Statement 3:
We say in Statement 1 that we can have all the homes sold for 130k.. also this statemnet has no upper binding on limit.
NOT AGREED

"I Only"