Which one of the expressions x^2y and x^2/y is greater? (1) y > x (2)

Official Explanation

We can reword the question as either “Is \(x^2y > \frac{x^2}{y}\)?” or “Is \(x^2y < \frac{x^2}{y}\)?” A valid assumption is \(x^2y ≠ \frac{x^2}{y}\) since the original question asks for the greater one of the two. Let's work on “Is \(x^2y > \frac{x^2}{y}\)?” Dividing this inequality by the positive value \(x^2\) changes the question to “Is \(y > \frac{1}{y}\)?” Since the inequality in the question no longer has \(x^2\), any further information about \(x\) is irrelevant. So, Statement (1) is not required.

From Statement (2) alone, we have that \(y < –1\). Hence, \(y\) is negative. Dividing both sides of this inequality by \(–y\), a positive number, yields \(–1 < \frac{1}{y}\). Combining the inequalities \(y < –1\) and \(–1 < \frac{1}{y}\) yields \(y < –1 < \frac{1}{y}\) from which we have \(y < \frac{1}{y}\). Hence, Statement (2) alone answers the question. Hence, the answer is (B). The answer to the question is “No. \(y\) is not greater than \(\frac{1}{y}\).” So, the answer to the original question is “No. \(x^2y\) is not greater than \(\frac{x^2}{y}\),” or “\(\frac{x^2}{y}\) is the greater one.”

Answer: B