Ready for some on your own? Good—have a crack at the five questions in this set, and then thoroughly review the explanations to see how you did.

1. A

Step 1: Get the Specs. We’ve got variables in Column A, which means choice D remains on the table. It’s also worth noticing the restrictions on the variables set out in the additional information: All four variables must be negative and decrease in size from s to p.

Step 2: Plan the Attack. FONZing seems to be a promising route, since making up numbers for the variables might help us to compare the expression in Column A to the number 1 in Column B. Just make sure to obey the restrictions in the additional information.

Step 3: Mine the Math. The rules of negative numbers come into play here: A negative × a negative = a positive, and a negative ÷ a negative equals a positive too. Those are really the only math concepts we’ll need, other than some simple arithmetic, to test out some numbers.

Step 4: Make the Comparison. Ready . . . set . . . FONZ! Since the variables must be negative, 0 and 1 are out, so let’s make s our simplest negative number, –1, and decrease from there: r = –2, q = –3, and p = –4. This gives . This is greater than 1 in Column B, so the answer could be A. If we pick other negative integers, such as –2, –3, –4, and –5, the result is the same: a positive number greater than 1. Just to be sure, though, we’d better try some fractions, because they sometimes cause funny things to happen. Let’s go with these:. This maintains the relationship in the additional information since is closer to 0 than , and hence bigger, and so on down the line. Filling in these for the variables yields the numerator , and the denominator . The entire fraction is then . Again, we get a positive number greater than 1 and can conclude that A is indeed correct.

2. B

Step 1: Get the Specs. Even though this may look intimidating, it’s nothing more than a made-up symbol problem. The expression in Column A and the whole number in Column B suggest that we can treat this as a disguised Problem Solving question testing whether we understand how made-up symbol problems work.

Step 2: Plan the Attack. There are no real shortcuts to made-up-symbol problems; we simply need to follow the rule to calculate the quantity in Column A and then compare what we get to the number in Column B.

Step 3: Mine the Math. Exponents and fractions seem to be the relevant math concepts, and is a special kind of fraction, called a reciprocal. We form the reciprocal of a number by flipping its numerator and denominator. If you’re up on these arithmetic concepts, you should have little problem following the rules to arrive at an answer.

Step 4: Make the Comparison. Let’s crunch through Column A and then compare what we get to 27. The first rule, a* = aa, means that whenever we see a number followed by an asterisk, we need to take that number to its own power. So 3* means we need to calculate 33 = 27. Anyone who stopped there would have been tempted to choose C, but of course we need to continue through the rest of the expression containing the # signs. The second rule, , means that whenever we see a number followed by a #, we need to take the reciprocal of that number. The expression 27# therefore means , which takes care of the first #. Performing the same for the second # brings us back to plain old 27, and doing it for the third and final time yields yet again as our final answer. That’s a lot of back and forth, but if you kept things straight, and took the reciprocal the right number of times (three), you’d find that 27 in Column B is bigger than in Column A. That means the second oval (what we call choice B) is the oval to click.

3. C

Step 1: Get the Specs. All of the information is found in the columns themselves, and the problem deals with prime numbers. There’s also this “inclusive/ exclusive” business, but that’s about all we’re up against. That means we’re basically looking at another Problem Solving question in disguise.

Step 2: Plan the Attack. They’re not asking us to come up with dozens of primes, so we may as well just do the work and count ‘em up.

Step 3: Mine the Math. Of course, to count them, we have to know what a prime number is, something we covered in the Math 101 chapter. A prime number is a number that has exactly two positive factors, 1 and itself. We also need to know that inclusive means we are to include the outer values of the set (if they apply), while exclusive means we must exclude the outer values of the set (even if they apply).

Step 4: Make the Comparison. Let’s get counting, beginning with Column A. As we just saw, the word inclusive means we have to include the 1 and the 7, if they apply. Seven is prime, so it counts, but 1 isn’t a prime number because it doesn’t have two distinct factors. So the numbers in the set defined in Column A are 2, 3, 5, and 7—four numbers total. The set defined in Column B contains these numbers: 13, 17, 19, and 23. The word exclusive means we have to exclude prime number 29 from our list, so there are four numbers in this set as well. Four primes in A, four primes in B, so the columns are equal, choice C.

4. C

Step 1: Get the Specs. No computer—let alone person—in its right mind would be happy crunching these numbers, which means there must be a logical way into it. There’s no additional information, and the columns contain pure numbers, no variables, so we can chop D from the get-go. But we’d better find a shortcut, since there’s no way we can possibly calculate these expressions. The shortcut to employ is suggested by the look of the quantities in the two columns. They look kind of similar, don’t they, what with all those 500s and a couple of 99th powers floating around? That can only mean one thing . . .

Step 2: Plan the Attack. Mirroring! These expressions look so similar that there simply has to be a way to make them resemble one another even further, making a comparison possible. But that’s gonna require some math chops, which is another way to say it’s time for Step 3:

Step 3: Mine the Math. Clearly exponents is the name of the game, and here’s the main thing you need to know about them to get over the hump: x ^y = (x)(x^{y – 1}). For example, since x¹⁰ is the same as x multiplied by itself 10 times, it’s also the same as (x) × (x multiplied by itself 9 times). Either way, you get x times itself 10 times. Why does this help us, you ask? Here’s why:

Step 4: Make the Comparison. 500¹⁰⁰ = (500)(500⁹⁹). Substituting that for 500100 in Column A gives us:

Factoring out the common 500⁹⁹ term results in:

which equals:

That’s the same as Column B, even though the terms are reversed. (The order of terms doesn’t matter for multiplication: A × B = B × A.) The quantities are equal, so C is correct.

5. D

Step 1: Get the Specs. Next up is a standard deviation question. As we discussed in our Math 101 chapter, you’ll never be asked to actually calculate standard deviation since the formula is too complex to be tested on the GRE. We have to assume therefore that the question is merely testing our understanding of the mechanics—that is, the principle behind it. We’ll need to mine our reservoir of math concepts for that principle, but first let’s get our plan of attack in place.

Step 2: Plan the Attack. We’ll use the numbers given but not expect to calculate the actual standard deviations for the two columns. Instead, we’ll rely on math logic to determine the relationship between them. First we have to know what standard deviation is, and that’s the purpose of Step 3.

Step 3: Mine the Math. Standard deviation is a measure of the spread of a group of numbers. In other words, a group whose numbers are all close to the same value has a smaller standard deviation than a group whose numbers differ widely from one another.

Step 4: Make the Comparison. Based on the definition in Step 3, our task is to figure out which set of numbers is more spread out. Alfonso’s performance is better defined: We know that his test scores are 90, 90, 90, 100, and 100. That gives us a good sense of the spread of his scores, but we’ll need to compare that to the spread of Nancy’s scores to get our answer. Her scores are less defined. While we know she definitely received 85, 85, and 85 on her first three tests, we don’t know exactly what she got on the last two; all we know is that they average out to 80.

Now, anyone who assumed that means her last two scores were 80 and 80 would be likely to choose B as the answer, since the spread of 85, 85, 85, 80, and 80 is less than the spread of 90, 90, 90, 100, and 100. That’s because 80 is closer to 85 than 100 is to 90. The problem is, we don’t know that Nancy’s final two scores are 80 and 80. They could be, in which case her standard deviation would be smaller than Alfonso’s, and B would be correct. But we have to use some cleverness and ask ourselves how else her final two scores could possibly average out to 80. The other extreme, given that the tests are out of 100, is that she got 100 on one test and 60 on the other. Those numbers also average out to 80, keeping with the requirement in the additional information, but change her test score distribution to 85, 85, 85, 100, and 60. Now Nancy’s last two scores differ from her previous scores by 15 and 25 points, respectively. This is greater than the 10-point spread in Alfonso’s scores, so in this scenario Nancy has the greater standard deviation. The answer must be D, since we get two different results from two different but equally plausible scenarios.

So there you have it: the wonderful world of Quantitative Comparisons, in a bit more than a nutshell. Sure, we covered a lot of ground in this chapter, but the fundamentals and step method will sink in as you get more practice with this question type. A little extra work on QCs is warranted since QCs make up the majority of GRE math questions: approximately 14 compared to 10 Problem Solving and only 4 Data Interpretation questions. It is to this final math question type, Data Interpretation, that we turn our attention now.