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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]  08 Oct 2010, 13:58
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Question Stats:

54% (02:10) correct 46% (01:04) wrong based on 86 sessions
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 14 Jun 2013, 04:12, edited 2 times in total.
Edited the question
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Re: 15 homes in town X [#permalink]  08 Oct 2010, 14:01
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Expert's post
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.

More at: gmatprep-2010-statistics-92909.html

Hope it's clear.
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Re: 15 homes in town X [#permalink]  24 Dec 2010, 00:35
why 3rd is wrong ? because if the median is 130000 , atleast a few elements shoulce be below it

pls help
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Re: 15 homes in town X [#permalink]  24 Dec 2010, 00:59
Expert's post
hirendhanak wrote:
why 3rd is wrong ? because if the median is 130000 , atleast a few elements shoulce be below it

pls help

Not necessarily.

If a set has odd number of terms the median of a set is the middle number when arranged in ascending or descending order;
If a set has even number of terms the median of a set is the average of the two middle terms when arranged in ascending or descending order.

So, for a set with odd number of elements the median is just the middle number: it's not necessary for any number of terms in a set to be more or less than the median. Consider the set {1, 1, 1,} --> median=1 or the set {1, 1, 2} --> median=1 no term is less than 1...

So if we take: $$x_1=x_2=x_3=x_4=x_5=x_6=x_7=x_8=130=median$$, and $$x_9=x_{10}=x_{11}=x_{12}=x_{13}=x_{14}=x_{15}\approx{173}$$ we can see that III is wrong with this scenario, so it's not always true.

Hope it's clear.
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Re: Mean / Median Question from GMAT Prep Test 1 [#permalink]  11 Mar 2011, 17:40
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Median means 7 prices were above the median and 7 below the median.

I. At least one of the homes was sold for more than $165,000. Total sales = 150K * 15 = 2250K Lets say all the seven homes above the median were sold for 165K Hence 165K * 7 + 130K * 8 = 1159K + 1040K = 2199K - Not high enough. Hence one home MUST have been above 165K A, D and E left. II. At least one of the homes was sold for more than$130,000 and less than $150,000. This is not true you can have the mean 150K even when 7 numbers are below 130K and 7 numbers are above 150K. I will not solve this scenario since this is intuitive. III. At least one of the homes was sold for less than$130,000.

This is not true since we can have 8 numbers at 130K and 7 numbers above 130K and have the mean 150K. I will not solve this scenario since this is intuitive.

D and E out. A remains.

CDM770234 wrote:
Can someone help me solve the following GMAT Prep Test 1 question:

23. Last month 15 homes were sold in town X. The average (Arithmetic Mean) price 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 & III

I do not agree/understand the official answer!
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Re: 15 homes in town X [#permalink]  17 Mar 2011, 08:49
Thanks, gmat1220!

I saw that this question was already posted with detailed analysis. Unfortunately, my search did not find it.

Thanks for responding in spite of that, gmat1220!
Re: 15 homes in town X   [#permalink] 17 Mar 2011, 08:49
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