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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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Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to find the value of x + y.
Follow the second and the third step: From the original condition, we have 2 variables (x and y). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.
Recall 3 Principles and choose C as the most likely answer. Let’s look at both conditions (1) and (2) together.
Conditions (1) and (2) tell us that x and y are positive integers and 1/x+1/y=1/5, from which we get 1/x+1/y=1/5, y/xy + x/xy = 1/5, y + x/xy=1/5, 5y + 5x = xy, xy – 5x – 5y = 0, or xy – 5x – 5y + 25 = 25. We can factor xy – 5x – 5y + 25 = 25 as follows: (xy – 5x) + (-5y + 25) = 25, x(y – 5) + (-5)(y – 5) = 25, which is equal to (x - 5)(y - 5) = 25. We have 3 possible cases: x – 5 = 25, and y – 5 = 1 / x – 5 = 5, and y – 5 = 5 / x – 5 = 1, and y – 5 = 25, since x and y are positive integers from condition (1).
Thus, the possible pairs of x and y are x = 6, and y = 30 / x = 10, and y = 10 / and x = 30, and y = 6.
If x = 6 and y = 30, then x + y = 36.
If x = 10 and y = 10, then x + y = 20.
The answer is not unique, and both conditions (1) and (2) combined are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Both conditions (1) and (2) together are not sufficient.
Therefore, E is the correct answer.
Answer: E
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in Common Mistake Types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
