Bunuel wrote:

If x and y are positive integers, does x/y = 7/9?

(1) x = 103

(2) 11/13 < x/y < 8/9

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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider 1) & 2) first, since we can save time by first checking whether conditions 1) and 2) are sufficient, when taken together.

Conditions 1) & 2)

We don't have an integer solution of the equation \(\frac{x}{y} = \frac{7}{9} ⇔ 9x = 7y ⇔ 9 \cdot 103 = 7y\) if \(x = 103\).

Both conditions together are sufficient.

Since this question is a statistics 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)

We can apply the same logic for both conditions together to the condition 1).

We don't have a solution and the answer is "no", The condition 1) is sufficient by CMT (Common Mistake Type) 1, since the answer "no" also means the condition is sufficient.

Condition 2)

\(\frac{11}{13} < \frac{x}{y} < \frac{8}{9}\)

\(\frac{11 \cdot 9}{13 \cdot 9} < \frac{x}{y} < \frac{8 \cdot 13}{9 \cdot 13}\)

\(\frac{99}{117} < \frac{x}{y} < \frac{104}{117}\)

We have lots of solutions such as \(x = 100, y = 117\) or \(x = 101, y = 117\).

Since we don't have a unique solution, the condition 2) is not sufficient.

Therefore, the answer is A.

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.

from statement 2 it is clear that x/y will never be equal to 7/9. Hence the statement is sufficient to give a clear NO to our question stem.