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Manhattan GMAT -700+ Challenge Problem - August 7, 2006

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Manhattan GMAT -700+ Challenge Problem - August 7, 2006

by Kevin » Mon Aug 07, 2006 8:15 am
Most Manhattan GMAT students are trying to break the 700 barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you'll WANT to see, when you are working at that level. Try to solve this 700+ level problem (I'll post the solution next Monday).

Question: The Middle Member
Set A, Set B, and Set C each contain only positive integers. If Set A is composed entirely of all the members of Set B plus all the members of Set C, is the median of Set B greater than the median of Set A?

(1) The mean of Set A is greater than the median of Set B.
(2) The median of Set A is greater than the median of Set C.

(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.
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by 800guy » Tue Aug 08, 2006 7:20 pm
anyone here good at set problems? i really suck at them...
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by Kevin » Mon Aug 14, 2006 11:26 am
Answer
From the question stem, we know that Set A is composed entirely of all the members of Set B plus all the members of Set C.

The question asks us to compare the median of Set A (the combined set) and the median of Set B (one of the smaller sets).

Statement (1) tells us that the mean of Set A is greater than the median of Set B. This gives us no useful information to compare the medians of the two sets. To see this, consider the following:

Set B: { 1, 1, 2 }
Set C: { 4, 7 }
Set A: { 1, 1, 2, 4, 7 }

In the example above, the mean of Set A (3) is greater than the median of Set B (1) and the median of Set A (2) is GREATER than the median of Set B (1).

However, consider the following example:

Set B: { 4, 5, 6 }
Set C: { 1, 2, 3, 21 }
Set A: { 1, 2, 3, 4, 5, 6, 21 }

Here the mean of Set A (6) is greater than the median of Set B (5) and the median of Set A (4) is LESS than the median of Set B (5).

This demonstrates that Statement (1) alone does is not sufficient toanswer the question.

Let's consider Statement (2) alone: The median of Set A is greater than the median of Set C.

By definition, the median of the combined set (A) must be any value at or between the medians of the two smaller sets (B and C).

Test this out and you'll see that it is always true. Thus, before considering Statement (2), we have three possibilities

Possibility 1: The median of Set A is greater than the median of Set B but less than the median of Set C.

Image

Possibility 2: The median of Set A is greater than the median of Set C but less than the median of Set B.

Image

Possibility 3: The median of Set A is equal to the median of Set B or the median of Set C.

Statement (2) tells us that the median of Set A is greater than the median of Set C. This eliminates Possibility 1, but we are still left with Possibility 2 and Possibility 3. The median of Set B may be greater than OR equal to the median of Set A.

Thus, using Statement (2) we cannot determine whether the median of Set B is greater than the median of Set A.

Combining Statements (1) and (2) still does not yield an answer to the question, since Statement (1) gives no relevant information that compares the two medians and Statement (2) leaves open more than one possibility.

Therefore, the correct answer is Choice (E): Statements (1) and (2) TOGETHER are NOT sufficient.
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Manhattan GMAT
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by chiru1849 » Sun Feb 20, 2011 7:43 pm
Isn't the answer 'B'. Given statement 2 we know that the median of Set A is greater than the Median of Set C. Since Set A is a combination of numbers in Set 'B' and Set 'C' for Set'A' to have a median greater than Set'C' its median must also be lesser than Set 'B'.
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by chiru1849 » Sun Feb 20, 2011 7:44 pm
Isn't the answer 'B'. Given statement 2 we know that the median of Set A is greater than the Median of Set C. Since Set A is a combination of numbers in Set 'B' and Set 'C' for Set'A' to have a median greater than Set'C' its median must also be lesser than Set 'B'.
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