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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.
Condition 1) tells that a = b or b =c or c = a. However, we have a≠b and b≠c from the original condition, so we have a = c.
Thus, condition 1) is sufficient.
Condition 2)
Assume (a^2+3a)/(a+1) = (b^2+3b)/(b+1) = (c^2+3c)/(c+1) = k
We have (a^2+3a) = k(a+1), (b^2+3b) = k(b+1) and (c^2+3c) = k(c+1)
When we subtract the first two equations, we have
(a^2+3a) - (b^2+3b) = k(a+1) - k(b+1)
=> (a^2-b^2+3a-3b) = ka+k-kb-k
=> (a^2-b^2) + 3(a-b) = ka-kb
=> (a^2-b^2) + 3(a-b) = k(a-b)
=> (a+b)(a-b) + 3(a-b) = k(a-b)
=> (a+b+3)(a-b) = k(a-b)
=> a+b+3 = k since a≠b
When we subtract the last two equations, we have
(b^2+3b) - (c^2+3c) = k(b+1) - k(c+1)
=> b^2-c^2+3b-3c = kb+k-kc-k
=> (b^2-c^2) + 3(b-c) = kb-kc
=> (b^2-c^2) + 3(b-c) = k(b-c)
=> (b+c)(b-c) + 3(b-c) = k(b-c)
=> (b+c+3)(b-c) = k(b-c)
=> b+c+3 = k since b ≠c
Since we have a+b+3 = k and b+c+3 = k, we have a = c.
Thus condition 1) is sufficient.
Therefore, D is the answer.
Answer: D
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 D is most likely to be the answer to this question, since each condition includes a ratio.
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.
