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Consecutive XYZ

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Consecutive XYZ

by sumanr84 » Fri Feb 12, 2010 8:08 am
If x, y, and z are integers, and x < y < z, is z - y = y - x?

(1) The mean of the set {x, y, z, 4} is greater than the mean of the set {x, y, z}.
(2) The median of the set {x, y, z, 4} is less than the median of the set {x, y, z}.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

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by Osirus@VeritasPrep » Fri Feb 12, 2010 8:14 am
If a set of integers are consecutive integers, then the mean and median of the set will be equal.

Statement 1: insufficient, we are only given information about the mean of the set. Fromt his information we can see that 4 is greater than the mean of set (x, y, z). We know this because if a number greater than the mean is inserted into a set the mean will increase.

Statement 2: Insufficient, we are only given information about the median of the set. From this information we can see that 4 is less than the median of the set (x, y, z).

Combining 1 and 2: sufficient: From the two statements we learn that the median and mean are not equal. Therefore the integers are not consecutive. Choose C
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by sumanr84 » Fri Feb 12, 2010 11:49 pm
Osirus has done perfectly. Here is how MGMAT soln goes for those who want detailed expln,

If z - y = y - x, then the three numbers x, y, and z are evenly spaced on the number line (i.e., they are in arithmetic progression). The key to this observation is the fact that, in any set whose elements are in arithmetic progression (no matter how many or how few numbers are in the set), the median and mean are equal.

(1) INSUFFICIENT: Adding a number higher than a set's mean increases the mean, and adding a number lower than the set's mean will decrease the mean. (Adding a number equal to the mean will not alter the overall mean.) Therefore, this statement means that the number 4 is greater than the mean of the original set {x, y, z}. However, this fact gives no insight into whether x, y, z are in arithmetic progression.

Sample cases: {1, 2, 3} gives a YES answer, and {1, 2, 4} gives a NO answer.

(2) INSUFFICIENT: Adding a number lower than a set's median either decreases or does not change the median; adding a number higher than the set's median either increases or does not change the median. (Adding a number equal to the median will not change the median.) Therefore, this statement means that the number 4 is less than the median of the original set {x, y, z}. However, this fact gives no insight into whether x, y, z are in arithmetic progression.

Sample cases: {5, 6, 7} gives a YES answer, and {5, 6, 8} gives a NO answer.

(1) AND (2) SUFFICIENT: Both statements together show that (mean) < 4 < (median). This means that the mean and median are not equal, and so the set is not in arithmetic progression, which means that z - y can NOT equal y - x.

The correct answer is C.
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