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Last Updated on March 8, 2024
If you are new to the GRE, you may be surprised to learn that the GRE Quantitative Reasoning section consists of 4 different question types: multiple-choice questions, multiple-answer questions, fill-in-the-blank questions (also referred to as Numeric Entry), and Quantitative Comparison (QC) questions. You may have never encountered QC questions, but the good news is that QC problems should not be an issue for you with the right approach and solid Quant skills.
In this article, we will discuss what a QC question is, examine how to attack these questions, and walk through the solutions to some Quantitative Comparison examples. Additionally, you’ll get our top 3 tips for solving QC problems and the 5 keys to QC mastery.
Here are the topics we’ll cover:
- What is Quantitative Comparison in GRE Math?
- Tip #1: Don’t Be Afraid to Test Numbers
- Tip #2: Don’t Forget Your Quant Formulas and Rules
- Tip #3: Always Look to Simplify First and Solve Second
- Summary: 5 Keys to QC Mastery
- What’s Next?
First, let’s cover the basics of what Quantitative Comparison questions are and what they look like.
What is Quantitative Comparison in GRE Math?
Developing an understanding of Quantitative Comparison questions is essential, as roughly a third of the 27 Quant questions on the GRE fall into this category. In a typical GRE QC question, you are provided with some information, sometimes called the question stem, and two quantities: Quantity A and Quantity B. Your job is to decide the relationship between the two quantities, with the following standard answer choices, which you should memorize:
- Quantity A is greater than Quantity B.
- Quantity B is greater than Quantity A.
- Quantities A and B are equal.
- The relationship between the two quantities cannot be determined.
KEY FACT:
Roughly a third of the 27 Quant questions you see on the GRE fall will be Quantitative Comparison questions.
Let’s look at an example to introduce you to the essence of a QC question.
Example
x > 0 and x2 – x – 6 = 0
Solution:
The question stem tells us that x is greater than 0 and that x2 – x – 6 = 0. We need to compare Quantity A, which is x, to Quantity B, which is 2, and determine the relationship between them.
First, we want to solve the equation x2 – x – 6 = 0, so that we know the possible value(s) of x:
x2 – x – 6 = 0
(x – 3)(x + 2) = 0
x = 3 OR x = -2
We have two possible values of x, but since the question stem specifies that x > 0, we know that x must be 3, not -2.
Thus, Quantity A = 3 and Quantity B = 2. Therefore, Quantity A is greater than Quantity B.
Answer: A
Now that you have a basic understanding of how QC questions work, the following three tips will be extremely useful for you in solving Quantitative Comparison questions.
Tip #1: Don’t Be Afraid to Test Numbers
While many Quantitative Comparison questions can be answered by simplifying or solving algebraic expressions (as we did above) or using formulas, some cannot. In such cases, testing various possible values for variables may help determine which quantity is greater. When we plug in numbers, however, we do not do it blindly. We use the given information to guide us on which numbers to test, and then we can strategically consider testing the following types of numbers:
- Zero
- One
- Positive proper fraction
- Negative proper fraction
- Positive integer
- Negative integer
Memorize this list. Keep in mind that it is rare you will need to test all six types of numbers to arrive at a correct answer. In fact, the goal is to test the fewest numbers possible. Let’s practice this strategy with an example question:
Example 1
p is a positive number
Solution:
Suppose we were to look at these two quantities quickly. In that case, we might hastily (and incorrectly) conclude that Quantity B must be greater than Quantity A because a squared number is greater than the original number. However, as we will see, that reasoning is not always correct.
Since we know that p is positive, we test only the following number types from the list provided above:
- One
- Positive proper fraction
- Positive integer
If p = 1:
Quantity A: 1
Quantity B: (1)2 = 1
When p = 1, we see that the two quantities are equal, consistent with choice C.
If p = ½ (a positive proper fraction):
Quantity A: ½
Quantity B: (½)2 = ¼
When p is ½, Quantity A is greater than Quantity B, consistent with choice A.
Already, we see that there is not a consistent relationship between Quantity A and Quantity B. So, we should choose answer choice D: the relationship cannot be determined from the information given.
We can stop at this point, but let’s also try out one more number, a positive integer.
If p = 2:
Quantity A: 2
Quantity B = 22 = 4
When p = 2, quantity A is less than quantity B, consistent with choice B.
Answer: D
The main takeaway here is that when you’re in doubt, test numbers, using the list as your guide for which numbers to choose. Let’s try one more example.
Example 2
Solution:
The goal is to determine the relationship between the two quantities, choosing from the standard answer choices. Your knee-jerk reaction may be to say, “Well, if I multiply a number by 3, the answer will be bigger, so Quantity B is greater than Quantity A, and so I choose answer choice B.” However, you would be incorrect in this case, having fallen for a trap that has tripped up many GRE students. Here’s why:
If you let x = 1 or 2 or 3.71, or any positive number, you are correct that 3x is greater than x. But what if x = 0? Plug in 0 for x, and you find that Quantity A = 0 and Quantity B = 0, too! So, in this case, the two quantities are equal, and answer choice C appears to apply. Or, if x were a negative number like -2, -3.5, or -100, then x would be greater than 3x, in which case the answer would seem to be A.
Now, take a careful look at the answer choices. Since answer choices A, B, and C are each sometimes true, but not always, we cannot choose them. Thus, our only recourse is to select answer D: the relationship between the two quantities cannot be determined.
Answer: D
TTP PRO TIP:
Use strategic numbers to evaluate possible values of Quantity A and Quantity B when variables are present.
Let’s now see why it’s important to know all those math formulas.
Tip #2: Don’t Forget Your Quant Formulas and Rules
While this tip may seem obvious, it can’t go without saying: the better you know GRE Quant formulas and rules, the more effectively you will answer Quantitative Comparison questions.
For example, if you do not know exponent rules, how can you expect to answer a QC question on exponents? If you do not know geometry formulas, how can you answer a QC geometry question correctly?
TTP PRO TIP:
Know your geometry formulas and algebra rules!
To further highlight this idea, let’s consider the following QC question:
Example 3
The radius of circle A is 10, and the radius of circle B is 5.
Solution:
This problem is a great example to show that you may struggle with particular QC questions if you are not familiar with GRE Quant formulas. After all, if you do not know the formula for the area of a circle, how can you answer this question? Let’s now use that formula to evaluate quantities A and B.
We are given that the radius of circle A is 10, and the radius of circle B is 5. Let’s start by determining Quantity A, which is half the area of circle A.
Half the area of circle A = (½)(π)(radius)2
Half the area of circle A = (½)(π)(10)2 = 50π
Thus, Quantity A is 50π.
Next, let’s determine Quantity B, the area of circle B:
Area of circle B = (π)(radius)2
Area of circle B = (π)(5)2 = 25π
Thus, Quantity B is 25π.
We can now see that Quantity A is greater than Quantity B, since 50π is greater than 25π.
Answer: A
Let’s practice with one more example. This time, exponent rules will be a big help to us.
Example 4
n is a positive integer
Solution:
Although there are a few ways to manipulate the expressions in Quantity A and Quantity B algebraically, the quickest method is to combine the bases in each quantity using the following exponent rule:
(xa)(xb) = xa+b
Applying the above rule to the two quantities, we have the following:
Quantity A: (n4)(n2) = n6
Quantity B: (n5)(n1) = n6
We see that the two quantities are equal, so the answer is C. Once again, knowing your Quant rules pays off!
Answer: C
Let’s discuss one final QC tip: simplification.
Tip #3: Always Look to Simplify First and Solve Second
In certain QC questions, you will be provided with what looks like some very thorny information, which is meant to intimidate you. However, in these circumstances, you often can simplify the information presented and thus more easily evaluate the question. Let’s consider the following question to see this process in action.
Example 5
y > 0
z + y(149x + 298) = 263xy + 517y + z
Solution:
Looking at the question stem, you may initially be intimidated by the long string of math presented. However, let’s see whether we can simplify that equation to transform it into something more manageable.
First, we can start by subtracting z from both sides, giving us the following:
y(149x + 298) = 263xy + 517y
Next, we can distribute y into the parentheses, giving us the following:
149xy + 298y = 263xy + 517y
We can now see that Quantity A is equal to the left-hand side of the equation, and Quantity B is equal to the right-hand side of the equation.
Thus, Quantity A is equal to Quantity B, and the answer is C.
Answer: C
Let’s try one more.
Example 6
a + b + c = d
Solution:
Once again, we may not come to a correct answer if we do not simplify the information provided. Notice that Quantity A contains variables a and b, and Quantity B contains variables c and d. Thus, it may help to manipulate the given equation a + b + c = d to get a and b on one side and c and d on the other side.
If we subtract c from both sides of the equation, we have the following:
a + b + c = d
a + b = d – c
We can also simplify the quantities themselves. For example, if we distribute the negative sign in quantity B, we have the following:
-(c – d) = -c + d = d – c
Our two quantities are now as follows:
Quantity A: a + b
Quantity B: d – c
If we look at the manipulated given equation a + b = d – c, we see that the left side is identical to Quantity A, and the right side is identical to Quantity B.
Thus, the two quantities are equal. The correct answer is C.
Answer: C
Once again, had we not simplified the given information, it would have been difficult to efficiently arrive at the correct answer.
KEY FACT:
When the constraints or the quantities are complicated-looking, algebraic simplification can make the two quantities easier to compare to each other.
Summary: 5 Keys to QC Mastery
To sum up, here are the 5 key Quantitative Comparisons tips discussed in this article:
- Memorize the standard answer choices, which are the same for all QC questions.
- Memorize the list of key numbers to substitute if the same variable appears in both quantities.
- Know your math formulas. They are sometimes necessary, and they can always save you a bit of time.
- Keep an eye out for expressions and equations to simplify.
- Practice, practice, practice!
The more Quantitative Comparison questions you answer, the more comfortable you will be with the logic and approach you’ll need to get the correct answer. If you follow the expert tips and keys to mastery presented in this article, you will be well on your way to earning a great GRE Quant score on test day!
What’s Next?
Now that you’ve learned about Quantitative Comparison questions, why not check out 10 tips for learning for a great Quant score? Interested in improving your speed in the Quant sections? You can learn more about GRE Quant pacing in this article about timing strategies.
