Explanation

In this question, we imagine that the profits were $100 in the first year. That would mean that in the second year, profits were , and that, in the third year, profits were . We want to know whether the profits were higher in the third year than the first year--namely, whether it's true that





Note that the first-year profits will cancel on both sides of this expression whatever they are, so choosing $100 caused no problems. So, we want to know whether On to the statements, looking at them separately first.

Statement (1) can be simplified a bit by subtracting 1 from both sides and getting the signs sorted out, giving us . We may not be quite sure whether that means We can consider cases. Since we are talking about percentages, x and y could range from a number very close to zero all the way up to 100. If we had , and , then and that would be a legal case. Then would be close to 2, would be close to 1, and the product of the two would be greater than 1. In such a case, the answer to the question posed would be "yes." Can we construct a case that is valid according to Statement (1) that yields a "no"? For ease of calculation, we can try x=50 and y=25. Then is and is . , which is still greater than 1 and still yields a "yes." Just trying two cases doesn't automatically make an analysis by cases complete, however, if we don't see a pattern that allows us to make a definitive conclusion. To try to make the effect of as large as possible, we'll choose a larger y. Say . Then and . And , so there we go--we managed to find a case in which the answer to the question, "Is ?" is "no." We don't have enough information to answer the question definitively, having obtained valid cases for yes and no, so Statement (1) is insufficient.

This part of the question is an example of how analysis by cases will sometimes generate a situation in which you are potentially stumped and not completely certain whether you are missing a relevant case. In this type of situation, you'll have found one or more cases that all yield a consistent result; you haven't found a case that yields a contrary result, and you're trying to decide whether that means there is no case that yields a contrary result, or you're just missing it. One technique is to be sure that you are trying as complete a range of numbers as possible: big, small, integer, fraction, positive, negative, even, odd. Another technique is to try to learn from the cases as you go; you might notice a pattern that allows you to finish the analysis. But sometimes it will be necessary in the interest of time to make a decision without being 100% certain.

Anyway, moving on to Statement (2): this statement bears some similarity to the question we are attempting to answer, . We can try multiplying out the left side of this expression to get









In other words, Statement (2) presents us with a fact that is exactly what we are looking for. Since Statement (2) tells us this inequality is true, the answer to the question posed will always be "yes." Statement (2) is therefore sufficient.

The correct answer is (B).