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