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

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Last month 15 homes were sold in Town X. The average [#permalink]

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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 $130,0000 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

OPEN DISCUSSION OF THIS QUESTION IS HERE: last-month-15-homes-were-sold-in-town-x-the-average-128477.html

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[Reveal] Spoiler: OA

Last edited by Bunuel on 02 Mar 2012, 13:20, edited 2 times in total.
Edited the question and added the OA. Topic is locked.

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Re: Homes Sold PS Q [#permalink]

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New post 12 May 2009, 20:42
No of Households = 15
Median household = 15 +1 /2 = 8
Median sales price = 13k and Average sale = 15k

Total sales = 15 * 15K = 225k

1) at least one of the homes sold was sold for greater than 165,000.

since Median sales price is 13 k. Let us keep the first 8 house at that price.
13K * 8 = 104K Total sales price of remaining houses = 225K - 104K = 121K
Clearly buy putting all the remaining houses at 17K = 17K * 7 = 119K. Still will not reach 121K.
So i put some at 17 and some 18 = 17K * 5 + 18K * 2 = 121K.

So 1 is true.

2 ) at least one of the homes sold was sold for greater than 130,000 and less than 150,000.

From above u can see houses do have to lie b/w 13K and 15K. Not True.

3) at least one of the homes sold was sold for less than 130,000.

Can be or we can place at the houses at 13 K as we did above. Not true

Ans A

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Re: Homes Sold PS Q [#permalink]

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New post 02 Mar 2012, 12:29
:oops: still not clear to me.
Can someone please explain??

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Re: Homes Sold PS Q [#permalink]

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New post 02 Mar 2012, 13:19
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Babzsn84 wrote:
:oops: still not clear to me.
Can someone please explain??


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 $130,0000 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

Note that we are asked which MUST be true.

Given: \({x_1}+{x_2}+...+{x_7}+{x_8=130}+{x_9}+...+{x_{15}}=15*150=2250\)

Let's start with the first one and try to make it false.

I. At least one of the homes was sold for more than $165,000.

Worst case scenario, when \(x_{15}\) has the least value (trying to make it less than 165), would be when \(x_1=x_2=x_3=x_4=x_5=x_6=x_7=x_8=130=max\), and \(x_9=x_{10}=x_{11}=x_{12}=x_{13}=x_{14}=x_{15}=min\):
\(8*130+7x=2250\) --> \(x_{min}\approx{173}\).

So we got that I is always true: At least one of the homes was sold for more than $165,000 (as for the worst case scenario we got that least value of \(x_{15}>165\)).

But if we take the scenario which we considered: \(x_1=x_2=x_3=x_4=x_5=x_6=x_7=x_8=130\), and \(x_9=x_{10}=x_{11}=x_{12}=x_{13}=x_{14}=x_{15}\approx{173}\) we can see that II and III with this scenario are false. So II and III are not always true.

Answer: A (I only).

In case of any question please post it here: last-month-15-homes-were-sold-in-town-x-the-average-128477.html

Hope it helps.
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Re: Homes Sold PS Q   [#permalink] 02 Mar 2012, 13:19
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