Solution:
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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Now we will solve this DS question using the Variable Approach.
Let’s apply the 3 steps suggested previously.
Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
Modify the original condition and question: \(\frac{3mr\ -\ nt}{4nt\ -\ 7mr}\) .
=> Dividing the numerator and denominator by 'nt' gives us:
=> \(\frac{\frac{3mr\ -\ nt}{nt}}{\frac{4nt\ -\ 7mr}{nt}}\)
OR
=> \(\frac{\frac{3mr\ }{nt}-1}{4-\frac{7mr}{nt}}\)
OR
=> \(\frac{3\cdot\frac{m\ }{r}\cdot\frac{n\ }{t}-1}{4-7\cdot\frac{m\ }{r}\cdot\frac{n\ }{t}}\)
So, we are looking for the value of \(\frac{m}{n}\) and \(\frac{r}{t}\) .
Thus, let’s look at the condition (1). It tells us that \(\frac{m}{n}\) = \(\frac{4}{3}\) and \(\frac{r}{t}\) = \(\frac{9}{14}\), which is exactly what we are looking for.
=> Substituting the values, we get:
=> \(\frac{3\cdot\frac{4}{3}\cdot\frac{9}{14}-1}{4-7\cdot\frac{4}{3}\cdot\frac{9}{14}} = \frac{4\cdot\frac{9}{14}-1}{4-\frac{4}{3}\cdot\frac{9}{2}} = \frac{\frac{36}{14}-1}{4-\frac{36}{6}} =
\frac{\frac{36}{14}-\frac{14}{14}}{4-6} = \frac{\frac{22}{14}}{-2} = \frac{\frac{11}{7}}{-2}\ = \ -\frac{11}{14}\)
Since the answer is unique, and the condition is sufficient, according to CMT 2, which states that the number of answers must be one.
NOTE: We ideally don't have to solve for the value once we know that required ratios are given in the condition.
Condition (2) tells us that m, n, r, and t are natural numbers.
However, we cannot determine the unique values of m, n, r, and t to get the value of \(\frac{m}{n}\) and \(\frac{r}{t}\).
So, the condition is not sufficient, according to CMT 2, which states that the number of answers must be one.
Also, remember that if one condition has a statement with a ratio and another condition has a statement with numbers, then the condition with a ratio is more likely to be the answer. A is the correct answer because the condition with a ratio is the answer.
Condition (1) ALONE is sufficient.
Therefore, A is the correct answer.
Answer: A
