M18-30

Official Solution:


Is \(xy > \frac{x}{y}\)?

We can rephrase the question as follows:

Is \(xy - \frac{x}{y} > 0\)?

Is \(x(y-\frac{1}{y}) > 0\)?

(1) \(0 < y < 1\).

Since \(y\) is a fraction between 0 and 1, \(y < \frac{1}{y}\). Hence, \(y-\frac{1}{y} < 0\). Now, if \(x > 0\) then \(x(y-\frac{1}{y}) < 0\) and we'd have a NO answer to the question. However, if \(x < 0\), then \(x(y-\frac{1}{y}) > 0\) and we'd have a YES answer to the question. Not sufficient.

(2) \(xy > 1\).

If \(x=y=2\) then the answer is YES. However, if \(x=4\) and \(y=\frac{1}{2}\) then the answer is NO. Not sufficient.

(1)+(2) Since from (1) \(y > 0\) and from (2) \(xy > 1\), then \(x > 0\) too. Thus, we have that \(x > 0\) and \((y-\frac{1}{y}) < 0\) (from 1), which means that their product \(x(y-\frac{1}{y}) < 0\) and we have a NO answer to the question. Sufficient.


Answer: C