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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, 14:58
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66% (02:24) correct 33% (01:30) wrong based on 1 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
[Reveal] Spoiler: OA

Last edited by Bunuel on 02 Mar 2012, 13:14, edited 1 time in total.
Edited the question
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Re: 15 homes in town X [#permalink]  08 Oct 2010, 15:01
1
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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

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, 01:35
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, 01:59
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, 18:40
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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, 09:49
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, 09:49
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