Explanation

The numbers here are positive, so the question is equivalent to . Or, maybe we'll get information that one of the fractions is smaller than one. Let's go to the data statements, separately first.

Statement (1) tells us that



This statement does not tell us enough information to be sufficient. We can see in analysis by cases. Case I: we can imagine that h is just slightly bigger than f, and g is much greater than e. That case would be allowed by Statement (1), because in that case we have a negative number over a positive number on the right side, yielding something that is negative. And in that case, the answer to the question is, "yes," because the denominators don't make any real difference, and g is much greater than e. Case II: h = 1, and f is a very small decimal, small enough to outweigh the fact that and make the left fraction larger than the right. That case yields the opposite answer, so we cannot answer definitively. Statement (1) is insufficient. Here we used some middle cases and some extreme cases - that's frequently useful in analysis by cases.

Statement (2) reeks of algebraic trickery. We can obtain



The thing on the right side is the same as what's inside the parentheses. So Statement (2) tells us about a fraction which, when squared, becomes smaller than it was originally. That means the fraction must be between 0 and 1. Therefore, the statement is equivalent to



This, in fact, establishes the very thing we are asked, as can be seen by multiplying both sides of this inequality by or both sides of the original inequality by . (Note that, in doing so, we have to consider whether we might need to flip the sign of the inequality. However, since all the variables are positive, both by and and positive, and therefore we know definitively that the direction of the inequality does not flip.) Therefore, this statement gives us precisely the data to answer the question definitively and is sufficient. The answer is (B).

Try a variation on this question: what if the direction of the inequality sign in Statement (2) were flipped? It would change the substance of what Statement (2) says, but not the sufficiency. In that case, the fraction would be greater than 1, the answer to the question would be definitively "no," and the answer would still be (B).

Again, in this question, the correct answer is (B).