This page is part of my unofficial solutions manual to the GRE Paper Practice Book (2e), a free resource available on the ETS website. They publish the questions; I explain the answers. If you haven’t worked through the Practice Book, give Section 6 a shot before reading this!

6.19: State Populations, Pt. 3

Yet another percent-difference question! If you’re at all shaky on this concept, consider this your warning: the GRE really likes this type of problem. As I noted in problem 6.13, percent-difference scenarios can be modeled using the following equation:

This is true even if we’re comparing two different quantities, rather than tracking a change over time. Here are two examples to make sure you’re rock solid on this point. (If you’re comfortable working with percent change, feel free to skip down to the end of the post for the answer to 6.19).

Percent Change and Percent Difference

Suppose that an electronics retailer — let’s call them Best Bargain — normally sells the SwitchBox 4 game console for $300. But this week they’re running a special sale and offering the SwitchBox for $240. What percent discount does this represent? Well, the base here is our original retail price, and the change is the difference in dollars between the retail price and the sale price. So the console is

cheaper during the sale than it is normally.

Now suppose that it’s actually two stores offering the same item at different prices: Best Bargain offers it for $300, and their competitor Circuit Center sells it for $240. If we want to know the percent difference between these prices, we can still use the percent-change equation. The only catch is that we must be careful to choose the correct base. Ordinarily, in questions involving discounts or markups, the base is the original retail price. Here, however, there is no “original” price. Instead, two different questions could be asked:

• The console is what percent less expensive at Circuit Center than at Best Bargain?
• The console is what percent more expensive at Best Bargain than at Circuit Center?

In each case, the wording of the question tells us which quantity to use. “Less expensive” means that the higher price will be used as our point of comparison, so the higher number goes in the denominator. “More expensive” tells us that the lower price is our basis of comparison.

So for the first question (“The console is what percent less expensive at CC than at BB?”), our equation looks like this:

At a price of $240, the console is 20 percent less expensive than at $300.

For the second question (“The console is what percent more expensive at BB than at CC?”), we use the following equation:

At a price of $300, the console is 25 percent more expensive than at $240.

Back to Problem 6.19

Armed with that knowledge, we can see how to approach 6.19. The problem asks us to compare two groups of categories (“supercategories,” if you will) in terms of how many states they contain. The two supercategories contain

and

states respectively.

Here’s where the percent-change math comes in. The phrase “greater than” tells us that the base (i.e., the denominator) will be the smaller of the two numbers. In this case, that’s the number of states in the smaller supercategory (11). So the percent difference is

which corresponds to answer choice (A).

Math Review Reference

For more on this topic, see the following sections of the GRE Math Review:

• 1.7: Percent (pp. 9-12)
• 4.1: Graphical Methods for Describing Data (pp. 62-64)