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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 x/(x+y) + y/(x-y) is equivalent to (x^2+y^2)/(x^2-y^2) for the following reason
x/(x+y) + y/(x-y)
=> x(x-y)/(x+y)(x-y) + y(x+y)/(x+y)(x-y)
=> (^x2-xy+xy+y^2)/(x^2-y^2)
=> (x^2+y^2)/(x^2-y^2)
=> (x^2/y^2+1)/(x^2/y^2-1) by dividing the top and bottom by y^2
=> [(x/y)^2+1)/[(x/y)^2-1]
When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that A is most likely to be the answer to this question.
Condition 1)
The condition (x+y):y = 3:1 is equivalent to x = 2y since x + y = 3y from (x+y):y = 3:1.
Then (x^2+y^2)/(x^2-y^2) = ((2y)^2+y^2)/((2y)^2-y^2) = (4y^2+y^2)/(4y^2-y^2) = 5y^2/3y^2 = 5/3.
Since condition 1) yields a unique solution, it is sufficient.
Condition 2)
If x = 5 and y = 3, then we have x/(x+y) + y/(x-y) = 5/8 + 3/2 = 5/8 + 12/8 = 17/8.
If x = 6 and y = 2, then we have x/(x+y) + y/(x-y) = 6/8 + 2/4 = 3/4 + 2/4 = 5/4.
Since condition 2) does not yield a unique solution, it is not sufficient.
Therefore, A is the answer.
Answer: A
