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

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08 Oct 2010, 14:58

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Difficulty:

55% (hard)

Question Stats:

52% (02:16) correct
48% (01:05) wrong based on 128 sessions

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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

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

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.

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.

Re: Mean / Median Question from GMAT Prep Test 1 [#permalink]

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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.

The answer choices are:

A. I only B. II only C. III only D. I and II E. I & III

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