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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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Condition 2)
The minimum possible value of x is 3, and the minimum possible value of y is 5 since y is a prime number.
Then we have z = 2*3*5 = 30, which is the possible maximum value of z.
x = 3, y = 5 and z = 30 are the unique tuple of solutions.
Since condition 2) yields a unique solution, it is sufficient.
Condition 1)
When we take reciprocals, we have 1/30 < 1/z < 1/y < 1/x ≤ 1/3
We have 1/2 < 1/x + 1/y < 1/x + 1/x = 2/x from 1/x + 1/y = 1/2 + 1/z.
Then we have x = 3 since x < 4 and x ≥ 3.
Since 1/x + 1/y = 1/2 + 1/z > 1/2, we have 1/3 + 1/y > 1/2 or 1/y > 1/6.
Therefore, we have y < 6.
Since we have 3 = x < y < 6 and y is a prime number, we have y = 5.
Then, we have 1/z = (1/x + 1/y) - 1/2 = (1/3 + 1/5) - 1/2 = 8/15 - 30/60 = 1/15 or z = 15. Since condition 1) yields a unique solution, it is sufficient.
Since each condition yields a unique solution, the answer is D.
Answer: D
In cases where 3 or more additional equations are required, such as for original conditions with "3 variables", or "4 variables and 1 equation", or "5 variables and 2 equations", conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
Note: Since this question is a CMT 4(B) question because condition 2) is easy to understand and condition 1) is hard. When one condition is easy to understand, and the other is hard, D is most likely the answer.
