Quantitative Comparison Strategies for the GRE

Quantitative Comparison problems are not like standard math problems you’ll find on the SAT or a common standardized test. Your job is not to find the correct answer amongst a group of answers (i.e. multiple choice) or to find the correct answer and write it in. You merely have to find out if one of the two expressions is larger, smaller, or equal to the other, or if such information is impossible to calculate. This may sound like a pain, but notice that you can get away with much less.
The art of quantitative comparison problems is getting away with the bare minimum to save time. Many quantitative comparisons are designed to look time consuming, which is a good indicator that there is a much faster way to solve the problem. Let’s look at some strategies that pinpoint those faster ways to solve quant comps: when to calculate, when not to calculate, and how to quickly compare variable expressions.

Understand the GRE Quantitative Comparison Answer Choices

GRE Quantitative Comparison questions always use the same four answer choices:

Answer	Meaning
A	Quantity A is greater.
B	Quantity B is greater.
C	The two quantities are equal.
D	The relationship cannot be determined from the information given.

Knowing these choices helps test takers spend more time comparing the quantities and less time rereading the answers.

Answer D does not mean that a calculation is difficult or that the test taker does not know how to solve it. It means that the information allows more than one relationship between the quantities. For example, Quantity A might be greater for one permitted value but smaller for another. D also applies when the quantities are equal in one valid case and unequal in another.

When both quantities have definite, well-defined numerical values, D can be eliminated. Even if those values are difficult to calculate, one must be greater or the two must be equal.

If, in your practice, you notice yourself doing endless calculations, you are doing unnecessary work. The GRE will not make you do endless calculations on paper, even if such a strategy appears to be the most obvious way to reach an answer.
Before we examine certain question types, let’s look at a couple simple examples to show how immediate calculation can be an inefficient strategy:
1. 3569                               OR                           3(10) + 5 (10²) + 6 (10¹) + 9 (10^0)
If I saw this problem without thinking, I might multiply out the second column (3 times 10³ is 3000, etc). Such an approach is self-defeating. There is a simple trick here. Notice that 3569 is the same thing as saying 3000 + 500 + 60 + 9, which is what the expression on the right is really saying. Calculation is not necessary, and I know that both expressions are equal.
2. 31 x 32 x 33 x 34 x 35                               OR                       32 x 33 x 34 x 35 x 36
Again, if you don’t quickly examine the two expressions intelligently, you might jump into calculation, which would take you quite a while (not to mention leave you vulnerable to errors). Since we are just comparing the two expressions, we can cross out the numbers that appear in both expressions, that is, 32, 33, 34, and 35. Thus, we are left with a simple comparison: 31 OR 36; now it’s quite clear that B is greater.

When presented with two baffling expressions, always think of ways to simplify before you calculate. Let’s check out this example:
1. 2,000,000 / 200, 000                        OR                             1,000 / 100
When you see many zeros in fractions like this, your first instinct should be to cross out matching zeros. If I have 2,000,000 in the numerator and 200,000 in the denominator, I should just eliminate five zeros from the top and 5 zeros from the bottom; now, my expression is simply 20/2 = 10. Same idea for column b: 1,000/100 = 10/1 = 10.

1. 4x +5                  OR                                3x +6
I could approach this problem a few ways. First, I could use the tried and true plug-in method, where I would test a few simple numbers (preferably something like -2, 0, 2, and .5–you want to use a positive, a negative, 0, and a fraction). Don’t forget, though, that you can manipulate both expressions to make the comparison simpler. As long as I add or subtract the same number or variable from these expressions, I’ll have the same relationship between the two expressions. Remember, when choosing numbers to add or subtract, our goal is to make the relationship simpler, so:
Subtracting 5 from both sides gives us:
4x OR 3x+1
Subtracting 3x from both sides gives us:
x OR 1
Now, look how simple it is? The comparison is any number (x) OR 1, which is clearly indeterminate. Our answer is D.

Let’s look at the problem we did above for the factoring method. We can actually use another method to figure out the answer. This just goes to show you that there is usually more than one way to solve a math problem. Though there is usually one “fastest” method, there are some rare cases when there are multiple quick methods to solving a problem.
1. 9^99 – 9^98             OR                     9 ^98
Why don’t we divide both sides by 9^98?
1. 9^99 – 9^98 / 9^98        OR           9 ^98 / 9^98
9^1 – 9^0                         OR                     9^0
9-1 OR 1 → 8 OR 1→A

Warning

Do not simplify by multiplication or division unless you know the quantity you are using is positive.

Let’s look at an example where you would not want to use this method:
3x                  OR               4x
If we tried to simplify by dividing by x, we’d think that we arrived at this:
3                   OR                  4
We might choose B as a result, but we’d be wrong. When in doubt, use your common sense. It’s easy to see that, when comparing 3x and 4x, the answer must be D because when x= 0, the values are equal, or when we use a negative number, 3x is larger.

Factoring is another popular way to simplify both expressions in order to make a comparison easier. Factoring doesn’t just mean pulling an x out of an expression. You can, and should, factor with constants (real, known numbers) in order to simplify. Let’s check out a few examples:
Example 1. 9^99 – 9^98          OR                   9 ^98
Hmm. There’s no way I’m calculating this one. Let’s try to factor out a 9^98. It helps to remember your basic multiplication and exponent rules, e.g. (x^5)(x)= x^6
9^98 (9 – 1)                  OR                            9^98
= 9^98 (8)                     OR                           9^98
What’s bigger: a huge number times 8 or that huge number by itself? Clearly, A is bigger.
Example 2. 5x + 5y / x + y         OR                   5
Here’s a chance for some good ol’ fashioned factoring.
5( x +y ) / x +y                           OR                     5
(X+Y) on top and bottom cancel, giving you: 5 OR 5
Easy choice. They’re equal.

Testing numbers can help reveal whether the relationship between two quantities changes. Start with a simple value that satisfies every condition in the problem. Then look for another permitted value that could produce a different result.

The mnemonic ZONEF can help test takers choose useful cases:

• Z: Zero
• O: One
• N: Negative numbers
• E: Extremes, such as large values or values near a stated boundary
• F: Fractions, especially values between 0 and 1

These are options, not a required checklist. If a variable must be positive, zero and negative values are excluded. If it must be an integer, noninteger fractions are excluded. When variables are connected by an equation, every test case must satisfy that equation as well.

Two valid cases with different relationships prove that the answer is D. Several cases with the same relationship do not, by themselves, prove A, B, or C. Those answers require reasoning that establishes the relationship for every permitted case.

The following examples show how zero, negative numbers, and fractions can affect a comparison.

Example 1: If x>0, y> 0, z= 0…
3z (2x +5y)                 OR                  3x (2z+5y)
When a zero is on the outside, as in our A value, the whole value is zero. Since all the other values are positive, we can be confident that our second value is larger.
Example 2: If x<0, y >0, z=0
3z (2x +5y)             OR                       3x (2z+5y)
We still have zero for our first value, but we’re not in the clear quite yet. When negative numbers are involved, always test them. Testing our second value gives us 3(-1) (2*0 + 5*2). What we have is a negative multiplied by a positive, so we know the answer is negative. Our first value is larger.
Example 3: If 3x= 4y
x                               OR                         y
We know that if x and y are positive, then x is greater than y, e.g. if x is 4 then y is 3. But what if they are negative? If x is -4, then y is -3. So in that case, y is greater. And, what if x is zero? Well in that case, y is zero also, so both values are equal.
It turns out that the answer must be D.

A small table can keep the cases organized. Each pair below satisfies the original condition, 3x = 4y.

x	y	Relationship
4	3	Quantity A is greater.
−4	−3	Quantity B is greater.
0	0	The quantities are equal.

The first two rows are enough to establish answer D. The third row shows another permitted outcome, but further testing is unnecessary once the relationship changes.

Example 4: If x > 0 and x does NOT equal 1…
x ²                        OR           x
So x must be positive and cannot be one. So in that case, it must be A right? Common Not so fast. Remember, our special numbers to test are negatives, positives, zeroes, and of course, fractions. Fractions have some very special properties. If we multiply a fraction by itself, it happens to become smaller, not bigger (.5 * .5 = .25).
So in that case, our answer is D.
There you have it. Try using different strategies to figure out which work best for you; remember, speed and accuracy are crucial!

Avoid Assumptions About Geometric Figures

A geometric figure that looks like a familiar shape may not have all of that shape’s properties. Test takers should rely on the stated information and markings rather than assume that an angle is a right angle or that two lengths are equal.

For example, suppose a triangle has two sides of lengths 3 and 4. Quantity A is the third side’s length, and Quantity B is 5. Without a stated or marked right angle between the sides of lengths 3 and 4, the triangle cannot automatically be treated as a 3-4-5 right triangle.

The third side could be 4, 5, or 6; each value satisfies the triangle inequality. Quantity A can therefore be smaller than, equal to, or greater than Quantity B. The answer is D.

Before selecting an answer, test takers should check whether the comparison follows from the given information or from an assumption about the drawing.

Eliminate Answer Choices Before Guessing

Even an unfinished comparison can provide enough information to eliminate answers. A single valid test case narrows the possibilities:

Result from one valid case	Answers that remain possible
Quantity A is greater.	A or D
Quantity B is greater.	B or D
The quantities are equal.	C or D

If another valid case produces a different relationship, choose D. If time is running short before the relationship can be established, guess from the remaining choices rather than reconsidering answers already eliminated.