If x and y are positive integers, is

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

If x and y are positive integers, is \frac{x}{y}<\frac{x+2}{y+3}?
Statement #1: y > 20
Statement #2: x < 5

solution:

since x and y are positive, then we can cross multiply the questions:

is x(y+3) < y(x+2)?
is xy+3x < xy + 2y?
is 3x<2y?
is 3x-2y<0?

basically the question is asking if x<y?

from the statements, none is sufficient to answer the question alone. combining the two, we can conclusively conclude that x < y.
therefore answer is 'C'

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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.
Visit 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.

The question asks if \(3x < 2y\) as shown below.

\(\frac{x}{y} < \frac{x+2}{y+3}\)
\(⇔ x(y+3) < y(x+2)\) by multiplying \(y(y+3)\) since \(y(y+3)\) is positive
\(⇔ xy + 3x < xy + 2y\)
\(⇔ 3x < 2y\)

Since we have 2 variables (x and y) and 0 equations, C is most likely to be 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)
Condition 1) "x < 5" tells 3x < 15 and condition 2) "y > 20" tells 2y > 40.
Then we have 3x < 15 < 40 < 2y.
Thus, both conditions together 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)
Since we don't have any information about x, condition 1) is not sufficient.

Condition 2)
Since we don't have any information about x, condition 2) is not sufficient.

Therefore, C is the answer.

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.