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Last month 15 homes were sold in Town X. The average

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Manager
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Last month 15 homes were sold in Town X. The average [#permalink] New post 06 Oct 2007, 17:13
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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.

(a) I only
(b) II only
(c) III only
(d) I and II
(e) I and III
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 [#permalink] New post 06 Oct 2007, 21:45
This is certainly a good question.

The average sale price of the homes = $150,000
No of homes = 15
=> Total sale price = 150000*15= 2250000

Now median is 130000.

Let's assume the case when home1 , home2 ......home8 are all 130k homes.
(This is the extreme case when median will remain 130k and other 7 homes will be of highest price.)

So total cost of 8 houses = 130k*8 = 1040000

Cost of remaining 7 houses = 2250000 - 1040000 = 1210000
Average = 1210000/7 ~ 172k
So At least one of the homes was sold for more than $165,000.

Hence A.

I would appreciate if somebody can give any faster way of solving this problem.

- Brajesh
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 [#permalink] New post 07 Oct 2007, 03:25
OA is A.

Like Brajesh, if anyone knows a quicker/shorter way of solving it, please let us know.

Thanks Brajesh!

A.
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 [#permalink] New post 07 Oct 2007, 11:00
If 8 are at 130 they need 20 each to get to 150. 20*8=160
160/7(remaining homes)=app. 23

150+23=173

The reamining 7 homes must average 173
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 [#permalink] New post 07 Oct 2007, 12:08
JDMBA - simply superb solution. Love it.

Thanks for the shortcut.

A.
  [#permalink] 07 Oct 2007, 12:08
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