Hello Chethan,
This is a good question for you to understand the dos and don’ts of equations and inequalities, simultaneously.
In any equation, you can add, subtract, multiply and divide both sides by the same value without worrying about the equation changing.
In an inequality, you may add and subtract both sides by the same number without changing the inequality. However, when you are multiplying or dividing both sides by a value, you have to be mindful of the signs. This is why we always advise students not to cross multiply or cancel out variables if they do not know the signs. Because,
Cross multiplying is nothing but multiplying both sides by the denominator.
Cancelling is nothing but dividing both sides by the numerator.
So, if you do not know whether the denominator is positive or negative, you cannot just cross-multiply; similar rule holds true for cancelling out terms as well. Remember, this applies for an inequality.
In an equation, you are free to do all operations that you want to, as long as it is within the mathematical scheme of things.
If \(\frac{x}{2}\) = \(\frac{3}{y}\), we can cross multiply and rewrite the equation as xy = 6 (assuming that y is not zero).
From this, we understand that the product of x & y is 6. If the product of a pair of numbers is a constant, increasing one will result in a reduction of the other AND vice-versa.
From statement I alone, y≥3. This means, the minimum value of y is 3. When y = 3, x = 2, so x<y. y can only increase and hence x will only reduce. We can conclude that x will always be less than y.
Statement I alone is sufficient. Possible answer options at this stage are A or D. Answer options B, C and E can be eliminated.
From statement II alone, y≤4. If y = 3, x =2 and x<y. If y = 1, x = 6 and x>y.
This happens because y will be greater than x till a certain value. Beyond this tipping point, x will be greater than x. This is why statement II alone is insufficient.
Answer option D can be eliminated. The correct answer option is A.
Note that GMAT may not exactly frame the question like how it is given. If there’s a variable or an expression in the denominator, GMAT will always make sure that it will specify that the term in the denominator cannot be ZERO.
Hope that helps!