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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.
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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 conditions if necessary.
The question 1/(2x-1) + 1/(2y-1) is equivalent to (2x+2y-2)/(4xy-2x-2y+1) for the following reason.
1/(2x-1) + 1/(2y-1)
= (2y-1)/[(2x-1)(2y-1)] + (2x-1)/[(2x-1)(2y-1)]
= (2x+2y-2)/(4xy-2x-2y+1)
Condition 2)
Since x = 4y = 1, we have 4xy = 1 and (2x+2y-2)/(4xy-2x-2y+1) = (2x+2y-2)/(1-2x-2y+1) = (2x+2y-2)/(-2x-2y+2) = {2(x+y-1)}/{(-2)(x+y-1)} = -1.
Since condition 2) yields a unique solution, it is sufficient.
Condition 1)
1/(2x+1) + 1/(2y+1) = 1
=> (2y+1)/[(2x+1)(2y+1)] + (2x+1)/[(2x+1)(2y+1)] = 1
=> (2x+1+2y+1)/[(2x+1)(2y+1)] = 1
=> (2x+2y+2)/(4xy+2x+2y+1) = 1
=> 2x+2y+2 = 4xy+2x+2y+1
=> 4xy = 1.
Then we have (2x+2y-2)/(4xy-2x-2y+1) = (2x+2y-2)/(1-2x-2y+1) = (2x+2y-2)/(-2x-2y+2) = {2(x+y-1)}/{(-2)(x+y-1)} = -1.
Since condition 1) yields a unique solution, it is sufficient.
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 2) is easy to work with, and condition 1) is difficult to work with. For CMT4(B) questions, D is most likely to be the answer.
