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The percentage of households with an annual income of more [#permalink]

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20 Aug 2004, 09:57

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The percentage of households with an annual income of more than $40,000 is higher in Merton County than in any other county. However, the percentage of households with an annual income of $60,000 or more is higher in Sommer County.

If the statements above are true, which of the following must also be true?

(A) The percentage of households with an annual income of $80,000 is higher in Sommer County than in Merton County. (B) Merton County has the second highest percentage of households with an annual income of $60,000 or more. (C) Some households in Merton County have an annual income between $40,000 and $60,000. (D) The number of households with an annual income of more than $40,000 is greater in Merton County than in Sommer County. (E) Average annual household income is higher in Sommer County than in Merton County.

If everyone in Merlon who made over 40,000 also made over 60,000, then Merlon would have the highest %> 60. Hence at least 1 person in Merlon must make between 40 and 60.

The percentage of households with an annual income of more than $40,000 is higher in Merton county than in any other county. However, the percentage of households with an annual income of $60,000 or more is highest in Sommer County.

If the statements above are true, which of the following can properly be concluded on the basis of them?

A) No household in Merton county has an annual income of $60,000 or more. - Household in Merton may have one or 2 or less people with 60K income. This is unknown. Hence - Out.

B) Some households in Merton county have an annual income between $40,000 and $60,000 - May be that some of the merton household have income more than 60K and none of them between 40k and 60K. This is not stated, and we cannot assume. - Hence OUT.

C) THe number of households with an annual income of more than $40,000 is greater in Merton than in Sommer County - This is clearly stated in the main statement. This is the correct one. Unless until stated or can be referred clearly, we cannot assume..

D) Average annual household income is higher in Sommer than in Merton County - We do not have numbers, hence we cannot know about the average. Hence OUT

E) The percentage of households with an annual income of $80,000 is higher in Sommer than in Merton County- Same case as D. Hence OUT.
_________________

Carsen: The Stimilus says that the percentage of household is higher in Merton county but it says nothing about the "the number".....We just know about the percentages but nothing about absolute numbers.

B) Some households in Merton county have an annual income between $40,000 and $60,000 - May be that some of the merton household have income more than 60K and none of them between 40k and 60K. This is not stated, and we cannot assume. - Hence OUT.

I disagree. We know that the percentage in Merlon of >40K is greater than the percentage in Sommer of 40K.

HENCE, if no one made between 40K and 60K in Merlon, the greatest percentage of >60K would be MERLON and not summer, therefore, it is necessary that atleast 1 person make between 40K and 60K in Merlon.

I agree with SigEpUCI.....I also chose "B" as my Final Answer.

After a while i will post the official answer (the reason i posted this question is coz i was shocked when i saw the OA). I dont understand how it could be the Official Answer.

Going by general fundamentals on avgs's we know that when it comes to weighted avg's the avg inclines itself towards that number whose weight is more. So the avg income in Somer county is more that in Merton county.

Let me elaborate with some numbers.

Merton county - majortiy if >40k income (Say 45k). Lets assume 100 people and 60 people with >40k income. The remaining would be less than 40k (say 35k).

Avg = (60*45 + 40*35)/100 = 41k (closer to 40k than 35k)

For Sommer country - going by the same logic and similar no.s, we have

The stimulus reads 'The percentage of households with an annual income of more than $40,000 is higher in Merton county than in any other county. However, the percentage of households with an annual income of $60,000 or more is highest in Sommer County'

The above info is given already. Why do we need to presume the numbers you have outlined..
Summer county, 1 person > 60000, 100 people at 1000 yearly income.
1 person is wrong, it should be atleast 51%. Others can be 1000 perfect.

Merlot, 0 person > 60000, 100 peaple at 40000. Again 51% should be astleast 41k 9or any number just above 40k)

[summer average = 1100
merlot average = 40000] ..this will change. [/i]

The stimulus reads 'The percentage of households with an annual income of more than $40,000 is higher in Merton county than in any other county. However, the percentage of households with an annual income of $60,000 or more is highest in Sommer County'

The above info is given already. Why do we need to presume the numbers you have outlined.. Summer county, 1 person > 60000, 100 people at 1000 yearly income. 1 person is wrong, it should be atleast 51%. Others can be 1000 perfect.

Merlot, 0 person > 60000, 100 peaple at 40000. Again 51% should be astleast 41k 9or any number just above 40k)

[summer average = 1100 merlot average = 40000] ..this will change. [/i]

In order to disapprove a conclusion of "True for all X", one just needs to show a single case when the conclusion is not False. I assigned arbitrary numbers to show that answer D does not hold true for every case.

The percentage of households with an annual income of more than $40,000is higher in Merton county than in any other county. However, the percentage of households with an annual income of $60,000 or moreis highest in Sommer County

Let me shoot down C.
[THe number of households with an annual income of more than $40,000 is greater in Merton than in Sommer County - This is clearly stated in the main statement. This is the correct one. Unless until stated or can be referred clearly, we cannot assume..]

This is definately not the case.

Houses in Merlot = 10, all of them make >40000 and all < 60000= hence 100%

House in Summer = 100, 20 > 40000 and all make > 60000 = hence 20% > 40000, BUT more numbers.

So C (just like D) can not be concluded. To be a valid answer, it must hold true for every possible assignment that obeys the rules in the premise.

Proof by contradiction for B. (HERE IT IS)

Assume B is not true.

We know that the %>40 in Merlot is greater than the %>40 in Sumner.

if B is not true, than everyone in Merlot makes >60. in that case, the first premise that Merlot has the highest proportion over 40 really means, Merlot has the highest proportion over 60. This directly contradicts the second premise. Hence B can not be false, and must be true.

If someone would like to assign numbers to the problem in an attempt to prove B wrong, I would love to see it.

If the OA is not B, then the OA is wrong. Whats the source of this question?

The percentage of households with an annual income of more than $40,000is higher in Merton county than in any other county. However, the percentage of households with an annual income of $60,000 or moreis highest in Sommer County

The percentages are relative to one another.

If the percent of people >40 in Merton is 2%, and in every other county (including Sommer) is 1%, the premises are still valid.

This is getting even better. SigEpUCI, dont mistake me, I am stretching this argument so as to get it right - not in the sense of an argument. Ok, here is my defense,

1. B is a right assumption - need not be a conclusion.
2. Going by your argument, if people in Merton county can have a 60K earning - then people in Sommer country can be assumed to have a number like 100K.
3. I just read the 2%, 1% stuff you had replied on - I think the stimulus perhaps have been wrongly read. My understanding is that the stimulus says that majority of people in Merton county has incomes >than 40k and that the majority of people in Sommer county has incomes >than 60k.

The more this stimulus is creating discussion - the ambiguity of the stimulus is only getting stronger. So, if not anything, I agree that the question is not the GMAT quality.

This is getting even better. SigEpUCI, dont mistake me, I am stretching this argument so as to get it right - not in the sense of an argument. Ok, here is my defense,

1. B is a right assumption - need not be a conclusion. 2. Going by your argument, if people in Merton county can have a 60K earning - then people in Sommer country can be assumed to have a number like 100K. 3. I just read the 2%, 1% stuff you had replied on - I think the stimulus perhaps have been wrongly read. My understanding is that the stimulus says that majority of people in Merton county has incomes >than 40k and that the majority of people in Sommer county has incomes >than 60k.

The more this stimulus is creating discussion - the ambiguity of the stimulus is only getting stronger. So, if not anything, I agree that the question is not the GMAT quality.

Yeah, agree with that. I'm not trying to get into an arugment here, only a friendly debate. However, I don't agree that B is an assumption, i think B can absoluty be concluded from the premises. If you could provide a situation where the premises are both true, and B is not true, i would be very impressed!

Lets assume for a second that M county has 40% population above 40,000 a year.

We know that S county has less than 40% population above 40,000 by the stimulus. We therefore also now that whatever percent of the population that makes greater than 60,000 is also less than 40% (since those who make greater than 60,000 are a subset of those who make greater than 40,000).

Now, if no one in M county made between 40000 and 60000, that would mean 40% of the population made more than 60000. (since 40% made more that 40000, and no one made between 40000 and 60000.

Since we know that that percent of people in S county who made more than 60000 is necessarily less than 40%, we arrive at the conclusion that
M is higher than S in terms of percentage making greater than 60000. This contradicts the second premise. Hence B must be true.

Some people must make between 40000 and 60000 in M, so that S (who lacks the biggest percentage in the 40000+ range, can gain the highest percentage in the 60000+ range.

Personally i think the OA should be "B". "D" in my mind is wrong for all the reasons that SigEpUIC explained.

We know nothing about the either "distribution of the salary (no. of people with a certain salary range" or the "the # of people in a particular county....Therefore all theories on "avg. salaries" or "the # of people with a certain salary is wrong. Because for any statistics that are calculated to prove that C, D are correct, i think there is another set of # that will disprove C,D.

I'm just struggling with the OA....When i first solved this question i didnt give "E' even a thought.

C is directly stated from the premise 1. It doesnt seem to involve the next premise, which is not right for any particular conclusion.

In case of B, premise 1 states that the % of households above $40000 is more in M county, and % of household is above $60000 is more in S county.

Lets say x % in county M and y% in county S are the corresponding figures we are talking here.

since y is above $60000, it has to be above $40000 too. assuming that there is no household in M county is between 40000-60000, we would end up having x% above $60000, and which means that x > y, but we know for sure this is not so from premise 2. so there has to some number which is reduced from x, lets say z, which exists in between 40000-60000.

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