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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.
So, we will first simply the original condition as follows:
a(1/b + 1/c) +b(1/c + 1/a) + c(1/a + 1/b)
= a/b + a/c + b/c + b/a + c/a + c/b
= (b+c)/a + (a+c)/b + (a+b)/c
= (a+b+c-a)/a + (a+b+c-b)/b + (a+b+c-c)/c
= (a+b+c)/a - a/a + (a+b+c)/b - b/b + (a+b+c)/c - c/c
= (a+b+c)[(1/a)+(1/b)+(1/c)] - 3
If a+b+c= 0, then we have (a+b+c)[(1/a)+(1/b)+(1/c)] - 3 = -3.
Condition 1)
|a+b+c| ≤ 0 means a+b+c=0 since |a+b+c| ≥ 0
So, we have a(1/b + 1/c) +b(1/c + 1/a) + c(1/a + 1/b) = -3.
Since condition 1) yields a unique solution, it is sufficient.
Condition 2)
Since we have a+b+c=0, we have a(1/b + 1/c) +b(1/c + 1/a) + c(1/a + 1/b) = -3.
Since condition 2) yields a unique solution, it is also 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).
