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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify the conditions, if necessary.

Condition 1)
Since x > y and the average of x and y is 9, we have x > 9 > y.
Thus, the median of x, 9, 9 and y is 9.
Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
Since ( x + y + 18 ) / 3 = 12 or x + y + 18 = 36, the average of x and y is 9.
Condition 2) is sufficient by the same reasoning as condition 1).

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).

This question is a CMT4(B) question: condition 1) is easy to work with and condition 2) is difficult to work with. For CMT4(B) questions, D is most likely to be the answer.