MathRevolution wrote:
[GMAT math practice question]
(Inequalities) Is \(\frac{m}{n} > \frac{(m+n)}{mn}\)?
1) \(m > n\)
2) \(m\) and \(n\) are integers greater than \(1\)
=>
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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https://www.mathrevolution.com/gmat/lesson for details.
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.
Since we have \(2\) variables (\(m\) and \(n\)) and \(0\) equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.
Conditions 1) & 2)
Since \(m\) and \(n\) are integers greater than \(1\),
The question \(\frac{m}{n} > \frac{(m+n)}{mn}\) is equivalent to \(m(m – 2) + (m – n) > 0\) for the following reason
\(\frac{m}{n} > \frac{(m+n)}{mn}\)
=> \(m^2 > m+n\), by multiplying both sides by \(mn\)
=> \(m^2 – m – n > 0\)
=> \(m^2 – 2m + m – n > 0\)
=> \(m(m – 2) + (m – n) > 0\)
We have \(m(m - 2) ≥ 0\), since \(m\) is an integer greater than or equal to \(2\) from condition 2).
We have \(m – n > 0\) from condition 1)
So we have \(m(m – 2) + (m – n) > 0\) and the answer is ‘yes’.
Since both conditions together yield a unique solution, they are sufficient.
Since this question is an inequality question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.
Condition 1)
If \(m = 4\) and \(n = 2\), then we have \(\frac{m}{n} = \frac{4}{2} = 2, \frac{(m+n)}{mn} = \frac{6}{8}\) and \(\frac{m}{n} > \frac{(m+n)}{mn},\) which means the answer is ‘yes’.
If \(m = 4\) and \(n = -2\), then we have \(\frac{m}{n} = \frac{4}{(-2)} = -2, \frac{(m+n)}{mn} = \frac{2}{(-8)} = \frac{-1}{4}\) and \(\frac{m}{n} < \frac{(m+n)}{mn},\) which means the answer is ‘no’.
Since condition 1) does not yield a unique solution, it is not sufficient
Condition 2)
If \(m = 4\) and \(n = 2\), then we have \(\frac{m}{n} = \frac{4}{2} = 2, \frac{(m+n)}{mn} = \frac{6}{8}\) and \(\frac{m}{n} > \frac{(m+n)}{mn}\), which means the answer is ‘yes’.
If \(m = 2\) and \(n = 4\), then we have \(\frac{m}{n} = \frac{2}{4} = \frac{1}{2}, \frac{(m+n)}{mn} = \frac{6}{8} = \frac{3}{4}\) and \(\frac{m}{n} < \frac{(m+n)}{mn}\), which means the answer is ‘no’.
Since condition 2) does not yield a unique solution, it is not sufficient
Therefore, C is the answer.
Answer: C
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.
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