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GRE Quantitative Comparison Tip #2 – Finding Equality

In our earlier post, we worked through a GRE practice problem using both an algebraic method and by plugging in values for x.  This post will focus on an additional strategy to use while plugging in numbers.

To get started, let’s look at this problem:

Column A            Column B

3x2 + 8x + 4         4x2 -12x + 4

(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.

While there may be a way to factor these equations, it could also save time to choose values for x. The key concept to keep in mind is finding equality: is there a nice number we can plug in to make both columns equal?

By “nice,” we mean a small integer like 0, ±1, or something similar. There’s no point in wasting precious minutes by thinking about the perfect number to choose.

Finding Equality by Plugging in Values

In our two equations, there’s a constant term of 4:

Column A            Column B

3x2 + 8x +4         4x2 -12x + 4

If we set x = 0, we should be able to make both sides equal.

Column A            Column B

3(0)2 + 8(0) + 4   4(0)2 -12(0) + 4

=4                           =4

That only took a few seconds, and we barely did any math. Here’s the payoff: while we haven’t yet solved the problem, we instantly know that the answer is either (C) The two quantities are equal, or (D) The relationship cannot be determined from the information given.

To get our final answer, we should plug in more values. If we can find a value that makes the columns unequal, we’re left with Choice D. If the columns are equal again, Choice C looks more likely.

Let’s try another simple number: x = 1:

Column A            Column B

3(1)2 + 8(1) + 4   4(1)2 -12(1) + 4

=3 + 8 + 4             =4 – 12 + 4

=15                        = -4

These two columns aren’t equal, which means our choice has to be D.

As you saw, it was far quicker for us to make these columns equal than to blindly plug in numbers. Once you can make the quantities equal, you immediately rule out answers A and B, which leaves you with only two other options. This can save valuable time during quantitative problems.

Take a Shortcut by Finding Equality

Here’s another problem:

Column A                 Column B

(x+3)(x-5)(x-2)       (x-9)(x+3)(x+1)

If we tried to use the algebraic method for this problem, we would need to multiply out both columns and then simplify before we even try to solve the problem. Finding equality will help us eliminate several answer choices at once.

Get Zero on Both Sides

 We can see that both columns include x+3:

Column A                  Column B

(x+3)(x-5)(x-2)       (x-9)(x+3)(x+1)

There’s a fast way to make these equations equal. Let’s set x = -3. Now look at what happens to those values:

Column A                 Column B

(-3+3)(-3-5)(-3-2)   (-3-9)(-3+3)(-3+1)

We don’t even need to look at the other terms. We know that (-3+3) = 0, and if you multiply something by 0 you still get 0.

Column A                 Column B

(0)(-3-5)(-3-2)     (-3-9)(0)(-3+1)

The answer has to be C or D. Let’s try another value to narrow these choices down further. Here’s the problem again.

Column A                 Column B

(x+3)(x-5)(x-2)       (x-9)(x+3)(x+1)

Get Zero on One Side

We can select a value to make one column reduce to 0, and then we’ll only have to multiply numbers in the other column. Let’s set x = 2:

Column A              Column B

(2+3)(2-5)(2-2)    (2-9)(2+3)(2+1)

Now it’s easy to see that Column A will reduce to 0, but Column B will not turn out to be 0. It doesn’t even matter what we get for Column B, because we’ve eliminated answer C—all that’s left is D. Let’s solve for Column B just to check our work.

Column A                 Column B

(2+3)(2-5)(2-2)       (2-9)(2+3)(2+1)

=0                               =(-7)(5)(3)

                                    = -105

These columns clearly aren’t equal, so the correct answer is D.

When you plug in values to solve a problem, make sure to look for smart choices that yield equality and eliminate two answer choices at once. You can see more practice problems in the free videos in our Quantitative Comparison module. 

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