The 15 QC questions appear before the 10 grid-ins in one of the two 30-minute math sections. The QCs are arranged by order of difficulty, from least to most difficult.

In addition to the directions provided above in general math instructions, Quantitative Comparisons have their own special instructions:

Note that though QC questions have only four possible answer choices while the grid has five answer spaces. Know the instructions!

QC questions ask you to look at two expressions or mathematical statements and choose which is of greater value. For example:

Answering (A) means you think that the expression in Column A is bigger. Answering (B) means you think that the expression in Column B is bigger. If the two expressions are equal, answer (C). If the relative size of the two expressions can’t be determined, then answer (D). The answer in the easy example above is (B), since 4 is bigger than 2.

In the example above, the values in Columns A and B were expressed by actual numbers. The columns might also contain:

Sometimes the values in the two columns will be represented in different ways. You may come upon a quantitative comparison that has an algebraic expression in Column A and a simple real number in Column B.

When the values in the two columns contain variables, algebraic expressions, values derived from a word problem, or geometric information, the problem will often also include additional information that defines the expressions in the columns or gives them some context. For example, the statement “All variables represent positive numbers” would limit and define what numbers a variable can represent.

In fact, let’s say you came upon the question:

The answer to this question would have to be (D), because if x = 0, then the two columns would be equal; if x = 1, then Column A would be greater; if x = –1, then Column B would be greater. But what if the question also included the following information?

From the information provided, you know that in this problem x can never equal 0 or any negative number. Instead, x must always be positive, which means that Column A will always be greater than Column B, making the answer to the question (A). Make sure to look at and understand the given information because it will always affect the outcome of the question.

If you answer a quantitative comparison with choice (A), (B), or (C), you are implicitly stating that the value of the expressions in each column can be determined in some relative way. After all, you can only claim that one column is greater than the other or that the two columns are equal if you know the relative values of two columns under every circumstance. In contrast, if you select answer choice (D) you are claiming that there is no possible way to know the value of at least one of the columns under every circumstance.

This difference among (A), (B), (C), and (D) has two important ramifications.

At first glance, you might think that of course Column A is bigger than Column B because Column A involves addition while B uses subtraction. But what if y represents a negative number? Then everything gets flipped and Column B will be bigger than Column A. Depending on which number you plug into the variables, you will receive difference answers, so the answer must be (D).

Note, however, that just showing that the value of one column can vary does not necessarily prove that the relative relationship between the two columns must vary. For example, let’s say you can prove that Column A can be either 3 or 7 (and nothing else), while Column B will always be 2. Even though Column A changes in value, its relative value will still always be greater than Column B, and so the answer will be (A).

Your job on QCs is to compare the relative size of the two columns, not the exact value of each column. While there will be many times when you do compute exact values to find the relative sizes, sometimes you won’t have to do any calculations at all. Always remember your QC priorities: do enough work so that you can successfully compare the two columns but no more. Of course, you must make sure you do enough work to compare the two expressions in every possible circumstance. Below are some techniques to help you in your efforts.

Working out a problem can take quite a bit of time and often isn’t necessary, as we just said. Take the following example:

To find the value of each of these expressions, you first would have to find a common denominator for each and then carry out the necessary steps. None of that is needed however, to quickly and accurately answer this question.

Since the only process in each expression is addition, you should automatically recognize that the column that contains bigger pieces will have the greater sum. Look at the pieces, and see if they can be easily compared. Both Column A and Column B have a ¹/₂, so at this point the columns are equal. The ⁷/₈ in Column B is greater than the ³ /₈ in Column A. It is possible to then quickly calculate that the ¹ /₄ in Column A is equal to ⁴/₁₆, and then see that the ⁹/ ₁₆ in Column B is larger. The fractions in Column B are all either larger or equal to their counterparts in Column A, so the value of Column B as a whole must also be larger. There’s the answer, and without having to do any math more complicated than converting ¹ /₄ to ⁴/₁₆.

In fact, the last problem could have been made even simpler. Instead of immediately comparing the two expressions, it would have been better to simplify the two columns by subtracting ¹/₂ from each side. Then you would only have to deal with two fractions on each side.

The simpler the expression, the more likely you are to see how to compare them quickly.

Make sure that when you employ this technique you do so evenly on each column. If you inadvertently perform an operation on only one column, you will not preserve the relative values of the two columns and you will get the question wrong.

As the cliché goes, you can’t compare apples and oranges. But there are times in QCs comparisons when it seems as if you’re being asked to do just that. In such situations, don’t panic: these questions are designed so that by applying some knowledge of math you will be able to make the columns comparable. For example:

At first glance, these two columns look very different, and as if they can’t be compared. But if you simply multiply out the first column, it will resemble the second:

Whenever you get into a situation in which you don’t know how to compare the two columns, see if there might be a way for you to make the two columns similar.

Sometimes the best way to compare two columns that have variables is to choose substitute numbers for those variables. Choosing the proper substitute numbers takes some skill and practice. First, you should try to use numbers that are easy to calculate. Second, you must make sure that the numbers you choose to substitute represent all possible circumstances. Remember, when you are answering QCs you are not trying to figure out a single answer. You must also figure out if you can arrive at a definitive answer. Take the following example:

Let’s plug in two easy-to-use numbers that fit the definition in the given information; how about 3 for p (which makes p² = 9) and 2 for q (making q² = 4)? Plugging the numbers into Column A, we get 9 – 4 = 5. Column A is greater than Column B.

Before we congratulate ourselves, it’s necessary to make sure that Column A is always larger than Column B, not just in this single instance with these particular numbers. Let’s try negative numbers (remember, p has to be larger than q, meaning p has to be less negative than q). A good move is to take your original numbers and make them negative. In this case, we also have to remember to keep p greater than q, so we must also switch the numerical values of the two variables: p = –2 and q = –3. If we plug these new numbers into Column A we get (–2)² – (–3)² = 4 – 9 = (–5). In this instance, Column A is smaller than Column B. In other words, Column A can be larger than Column B in some instances and smaller in others, meaning that there is no definite answer and the answer must be (D).

Whenever you plug in numbers for QCs, you must be aware that it is possible that other numbers might yield a different result. So, any time you plug in numbers, you must make certain to try all of the different possible numbers. Therefore, each time you plug in numbers make sure to:

If any of these options yields a comparative result different from the others, your answer should be (D).

Because QCs have four rather than five multiple-choice possibilities, ETS adjusted the guessing penalty so that you lose ¹/₃ of a point rather than ¹/ ₄ of a point for a wrong answer. This means that you still have to eliminate at least one possibility on a QC in order to make guessing worthwhile.

Sometimes eliminating an answer choice is easier than it looks. Remember that for the answer to be (D), you must not be able to find the exact value of either side. Therefore, if you know that each side has an exact value, even if you don’t know how to determine what those values are, you do know that (D) cannot be the answer. At that point, the guessing odds are in your favor. Take the following example:

This is a pretty easy problem, but let’s say that you didn’t have enough time to cross-multiply and figure out that column (B) is greater. Even so, from looking at these two columns you should instantaneously know that their respective values are set in stone, meaning (D) cannot be the answer.

The 15 QCs are found in the same 30-minute section as the 10 GIs. As you already know, both of these question-type groups are organized by difficulty. Depending on your target score, you can manage your time by skipping over questions that you just can’t figure out. Remember, you should make sure to take a look at every problem in a section to see if you can answer it or put yourself in position to guess, but don’t struggle to answer questions that are too difficult. You should use the location of a question as a hint of its probable difficulty, but you shouldn’t just skip a question without ever looking at it. If you get to the end of the QCs and haven’t answered a few questions, don’t worry. Just move on to the GIs and answer what you can. If time allows, when you’ve answered everything you definitely can, then come back to the questions you skipped and give them another try.