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Re: If xФy = (2x - y)/(2y - x), where x ≠ 2y, then is a≠b > b≠a ?
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31 Dec 2015, 21:05
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.
If xФy = (2x - y)/(2y - x), where x ≠ 2y, then is aФb > bФa ?
(1) a < b
(2) 2a < b
If we modify the original condition and the question, we can see that the question is asking if (2a-b)/(2b-a)>(2b-a)/(2a-b). If we multiply both sides by (2b-a)^2(2a-b)^2, the question can be modified to(2a-b)^3(2b-a)>(2b-a)^3(2a-b)? Then, the question changes to whether (2a-b)^3(2b-a)-(2b-a)^3(2a-b)>0 is true. Then, we get (2a-b)(2b-a)[(2b-a)^2-(2a-b)^2]>0? Essentially, if we further modify the question, we can see that the question is asking 3(2a-b)(2b-a)(3b^2+3a^2)>0? There are 2 variables (a and b), and in order to match the number of variables and the number of equations, we need 2 equations. Since the condition 1) and 2) each has 1 equation, there is high chance C is the answer. However, using both the condition 1) and 2), we can see that there is no way we can find out the relationship between 2b and a. Therefore, the correct answer is E.
For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.