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# Last month 15 homes were sold in Town X. The avg. sale price

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Last month 15 homes were sold in Town X. The avg. sale price [#permalink]  31 Oct 2008, 08:52
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Last month 15 homes were sold in Town X. The avg. 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
[Reveal] Spoiler: OA

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Last edited by Bunuel on 25 Jan 2012, 02:17, edited 1 time in total.
Edited the question and added the OA
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Re: PS : Sale Price of home [#permalink]  31 Oct 2008, 09:11
I am with A.

If we consider all higher values than median as 165000, they would offset only 105000 of the deficit that less than equal to median values have created. Considering all values to the left of median equal to 130000, we require 160000 to reach the mean from 7 values to the right.

Thus one of them must be > 165000
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Re: PS : Sale Price of home [#permalink]  31 Oct 2008, 10:57
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linau1982 wrote:
IMO E
1 is true, because average is 150K and median is 130K, therefore there must have been some bigger sales
3 is true, because if median is 130K there must have been some houses sold for less than that

Think of the prices laid out as follows:

__ __ __ __ __ __ __ $130,000 __ __ __ __ __ __ __$130,000 is the median, but we also know that $150,000 is the average, so obviously there must be some homes priced higher than$130k. Now, the higher priced the homes that precede the median of $130k, the less the homes priced higher than$130k have to be to create a mean of $150k. So let's fill out the scenario of the highest priced homes left of the$130k:

$130k$130k $130k$130k $130k$130k $130k$130k __ __ __ __ __ __ __

Now, the remaining seven must be high enough to create an average of $150k. Some quick calculations will show that even in this scenario, even if every home priced above the median were$170k, the mean would still be less than $150k. Based on this, we know that I MUST be true. Eliminate any choice that doesn't list I. B and C are out. We can also quickly deduce that II is not necessarily true since we just showed that all the prices above the median could be greater than$170k, or one could be $131k, and the rest much higher- you get the idea. The point is that we don't know, so II is out, and any of the 5 choices that list II. Eliminate D. We are down to A and E. III is out because from the example above, we know that all of the homes that precede the median-priced home could be$130k or less, but it is not true that one HAS to be less than $130k. Eliminate E. A is your answer. By the way, the reason I immediately jumped to the example of all of the prices preceding$130k equaling $130k instead of playing around with ranges, is that questions like this almost invariably hinge on your ability to quickly recognize that average/median questions like this involve a sort of tug-of-war between the ends of the average. You should think of the layout of prices as a see-saw. The median and mean are fixed in this case, so if you lessen one end, the other end must be raised, and vice versa. In this example, option I is asserting that at least one price can't be less than$165k, so you should quickly recognize that you need to set up a case to disprove this by making the prices as close to the median as possible on both ends.
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Re: PS : Sale Price of home [#permalink]  31 Oct 2008, 12:54
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amitdgr wrote:
Source : GMATPrep

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

i will go with A..

15 houses sold so the median is the 8th house..

worst case all houses upto 8th are 130K, which means

sum/15=150 or sum=2250K, 130K*8=1040K or 1210K left for the remaining 7 houses if all the 7 most expensive houses were 165K, then 7*165K=1050 but we still have 1210K-1050K leftover which means definitely the 15th house is over 165K..

I just porved II and III wrong..so..

A it is
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Re: Statistics from GMATPrep [#permalink]  11 Oct 2009, 20:04
Statement II is not necessarily true.
Since Mean is 150 and median is 130, you could have houses either side of the average (but not between 130 and 150).
for e.g, 120/180, 110/190, 100/200, 90/210, 80/220, 70/230, 60/260 and the median 130.

Statement III is not true because you could have 8 houses @130 and still have the average be 150.

So by POE, answer is A.
I would start with trying to prove which of the options can be proved to be untrue.

It's quiet possible to have a house @140k. Say 140/160, that would give 150 as the mean. It's possible, but not mandatory, hence St. II is out.
Re: Statistics from GMATPrep   [#permalink] 11 Oct 2009, 20:04
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