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 Post subject: GMAT Statistics
PostPosted: Mon May 13, 2013 9:24 am 
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Four barrels of water have an average capacity of 8 gallons. What is the maximum possible capacity of the smallest barrel if the median is 10 gallons?

A. 1
B. 2
C. 3
D. 4
E. 5

(B) Let us denote the capacities of the barrels by v, x, y, z naming from the smallest one to the largest one. The median is 10 gallons. Therefore (x + y) / 2 = 10.
x + y = 20

The average is 8. Therefore (v + x + y + z) / 4 = 8.
v + (x + y) + z = 32
v + 20 + z = 32
v + z = 12

The capacity z must be no less than 10, since 10 is the median. Therefore the maximum possible capacity of the smallest barrel is 12 – 10 = 2 gallons (all the other barrels would have a capacity of 10 gallons in this case). The correct answer is (B).
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What about the set (1, 9, 11, 11)?


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 Post subject: Re: GMAT Statistics
PostPosted: Mon May 13, 2013 9:26 am 
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The set {1, 9, 11, 11} also satisfies the requirements of the original question statement:
Its median is (9 + 11)/2 = 20/2 = 10.
Its average is (1 + 9 + 11 + 11)/4 = 32/4 = 8.

BUT does the smallest barrel have the MAXIMUM POSSIBLE capacity for such requirements?

No, it does NOT. The capacity of the smallest barrel can be greater, e.g. 2, if the corresponding set is {2, 10, 10, 10}.

In GMAT questions, as also in other questions as well, you should always keep in mind what a question asks you. In other words, what your goal is.

P.S.
In case you have any concerns of whether the original explanation provides the MAXIMUM possible value or not, let's go over the key statement in it once again.

v + z = 12

This equality sets up the relation between the capacity of the smallest barrel and the capacity of the largest one. It's clear that the capacity of the smallest barrel is the greatest when the capacity of the largest barrel is the least.
The least possible value for the capacity of the largest barrel is 10 (because, otherwise it would NOT be the largest one). Therefore the maximum possible capacity of the smallest barrel is 2 gallons.

Could it be 2.5 or 3 gallons? NO, because the capacity of the largest one would be 9.5 or 9 gallons respectively. And so it would NOT be the largest one.


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 Post subject: Re: GMAT Statistics
PostPosted: Mon May 13, 2013 9:27 am 
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v:x:y:z = 1:10:10:11 then the median = 10, thus answer should be "A"?


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 Post subject: Re: GMAT Statistics
PostPosted: Mon May 13, 2013 9:27 am 
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questioner wrote:
v:x:y:z = 1:10:10:11 then the median = 10, thus answer should be "A"?

The set {1, 10, 10, 11} also satisfies the requirements of the original question statement:
Its median is (10 + 10)/2 = 20/2 = 10.
Its average is (1 + 10 + 10 + 11)/4 = 32/4 = 8.

BUT does the smallest barrel have the MAXIMUM POSSIBLE capacity for such requirements?

No, it does NOT. The capacity of the smallest barrel can be greater, e.g. 2, if the corresponding set is {2, 10, 10, 10}.

In GMAT questions, as also in any other questions as well, you should always keep in mind what a question asks you. In other words, what your goal is.


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 Post subject: Re: GMAT Statistics
PostPosted: Mon May 13, 2013 9:28 am 
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The answer is A. The largest can have a capacity of 11 gallons, and the smallest can have a capacity of 1 gallon. The other two can have a capacity of 10 gallons. This will give an average capacity of 8 gallons with the median capacity of all four barrels to be 10 gallons.


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 Post subject: Re: GMAT Statistics
PostPosted: Mon May 13, 2013 9:29 am 
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questioner wrote:
The largest can have a capacity of 11 gallons, and the smallest can have a capacity of 1 gallon. The other two can have a capacity of 10 gallons. This will give an average capacity of 8 gallons with the median capacity of all four barrels to be 10 gallons.
That's true.

BUT does the smallest barrel have the MAXIMUM POSSIBLE capacity for such requirements?

No, it does NOT. The capacity of the smallest barrel can be greater, e.g. 2, if the corresponding set is {2, 10, 10, 10}.


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