GRE Articles

GRE Quantitative Comparison Tip #1 - Algebra vs Plugging in

This article is our first post on GRE Quantitative Comparison (QC) methods. Let’s motivate this topic by looking at values in the two columns below:

Column A            Column B

16x + 4                  4x + 16

(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.

The two major strategies for QC problems that include variables are:

1)      Solving with algebra

2)      Solving by plugging in numbers

Both approaches have advantages and disadvantages, so we’ll try using both approaches to solve this problem.

Algebraic Strategy

Let’s start moving values around to solve for x, our variable.

Column A            Column B

16x + 4                  4x + 16

Our first step will be to subtract 4x from both sides.

Column A            Column B

12x + 4                  16

Now let’s subtract 4 from both sides.

Column A            Column B

12x                         12

The final step is to divide both sides by 12:

Column A            Column B

x                              1

We’ve reduced both columns as far as possible, so it’s time to look at the answer choices again. We can’t say for sure that x = 1, or that x is less than or greater than 1. The correct choice is (D) The relationship cannot be determined from the information given.

Plugging in Numbers Strategy

In order to use this method, we will choose values to plug in for x. Here’s the original problem.

Column A            Column B

16x + 4                  4x + 16

The key for this strategy is to choose values that represent a good selection of numbers, including 0, 1, positive/negative numbers, and large/small numbers. Smart choices for values to plug in could include 0, ±1, ± ½, and ± 100.

We’ll start with x = 0. Plugging in this value, we get

Column A            Column B

16(0) + 4               4(0) + 16

=4                           =16

Just based on this value, we have two possible answer choices left: (B) Quantity B is greater, or (D) The relationship cannot be determined from the information given. Let’s try another value to narrow down these choices.

When x = 1, the expressions become:

Column A            Column B

16(1) + 4               4(1) + 16

=20                         =20

Now the quantities are equal, which means that (B) Quantity B is greater cannot be the proper answer. The only possibility is (D) The relationship cannot be determined from the information given.

The important takeaway from this strategy is to plug in several different values instead of trying one number and then picking an answer. If we’d only plugged in x = 0, we could have chosen (B) and gotten the incorrect result.

Advantages and Drawbacks for Each Approach

We’ve showed that both methods can solve a problem, but each strategy has different pros and cons.

Plugging in different numbers has one major drawback: we cannot be sure of the right answer until we plug in multiple values.

The algebraic strategy is more challenging, since it requires test-takers to solve the equations. However, an algebraic approach will find the right answer in one go, without making test-takers try several different solutions.

In general, plugging in numbers may be a faster approach when students don’t feel confident in their algebra skills.

A Second Example

Let’s look at an example where plugging in numbers might not be the best approach.

Column A            Column B

x2 – 2x + 1                  -2

Some students might see an algebraic solution immediately, but others may not be so quick. If the answer doesn’t come to mind right away, the best plan is to begin plugging in values for x.

Plugging In Numbers Strategy

Let’s try x = 1:

Column A                   Column B

(1)2 – 2(1) + 1                   -2

=1 – 2 + 1                           -2

=0

It looks like the quantity in Column A is larger, but we have to check with another value. What if x = 0?

Column A                    Column B

(0)2 – 2(0) + 1                   -2

=1                                       -2

Again we’ve found that the value in Column A is larger. But is this ALWAYS true? We can’t be sure unless we keep trying different values, because the answer could also be (D) The relationship cannot be determined from the information given.

How Much Time Is Left?

It can be tough to decide between switching to another strategy or continuing to plug in numbers. The best way to choose is to think about how much time is left in the section. Do you feel good about your pacing, or are you falling behind? If you routinely run out of time with quantitative questions, it’s smarter to try another strategy. If you’re better with numbers but slow when it comes to Reading Comprehension, you probably have enough time to try a few more values.

Algebraic Strategy

Let’s move on to the algebraic strategy for solving this problem.

Column A                       Column B

x2 – 2x + 1                            -2

People who have practice in factoring quadratic equations should recognize the solution to this problem easily.  Let’s factor the quantity in Column A:

Column A                       Column B

x2 – 2x + 1                            -2

(x – 1)(x – 1)                        -2

=(x – 1)2

 

From basic principles of math, any value squared must be at least zero. There’s no way for (x – 1)2 to be negative. Now it’s clear that that Column A must be greater than Column B, so we can choose (A) Quantity A is greater.

Both strategies can be used to solve problems, but it’s important to know when one approach might be faster than another one. 

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