# GRE Quantitative Comparison Tip #3 – Logic over Algebra

In previous articles (Algebra vs Plugging in and Finding Equalitywe showed several strategies for approaching GRE Quantitative Comparison problems. As a refresher, the two methods are solving using algebra, or solving by plugging in values for the variable. While the algebraic method can be more reliable, plugging in numbers can help eliminate answer choices right away.

A Third Approach - Logic

It’s time to learn a new strategy: the logical approach. Instead of jumping in and using algebra or plugging in some values, the logical approach means stopping for a moment to think about the problem and what types of answers we could get. Let’s look at this problem to see why we’d use the logical approach.

x > 4

Column A            Column B

11x + 17                47 + x

2                           2

(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.

First, let’s see why using algebra might not be the best strategy.

Algebraic Approach

We see that both sides have a 2 in the denominator, so let’s multiply by 2 to remove it.

x > 4

Column A            Column B

11x + 17                47 + x

Now we’ll subtract x from both sides.

x > 4

Column A            Column B

10x + 17                47

Next we subtract 17 from both columns sides to isolate the x term.

x > 4

Column A            Column B

10x                         30

Finally, we divide by 10:

x > 4

Column A            Column B

x                              3

After we’ve reduced both sides, we need to compare x with 4. We know x > 4, so Column A must be greater than Column B and the right answer is A.

Using the Logical Approach

It took us four steps to solve the problem using our algebraic strategy, but there’s a faster way to figure out what the right answer should be.

Take another look at the problem:

x > 4

Column A            Column B

(11x + 17)/2        (47 + x)/2

The problem tells us that x must be greater than or equal to 4, so we can use that information to rule out answer choices right away.  The lowest value x could be is 4, so let’s see how the problem looks when we substitute 4 for x:

Column A            Column B

(11(4) + 17)/2     (47 + (4))/2

=(44 + 17)/2        (47 + (4))/2

=    (61)/2                (51)/2

Without factoring, eliminating denominators, or any of the steps we took using the algebraic approach, it’s now clear that Column A has a larger value than Column B. Since x has to be at least 4, this relationship must stay the same for all values of x. The right answer is A.

Logic and Positive/Negative Numbers

Here's a slightly different problem.

(18x)/(10x-4y) = 3

Column A            Column B

x                              y

Before even trying an algebraic approach, let’s take a moment to think about this problem. From the givens, we see that

(18x)/(10x-4y)  =    3

Ignoring what’s in the denominator, let’s think about factors of 18: 3 x 6 = 18, or 2 x 9, or 1 x 18. Here we see that 18x divided by something is 3, but that means something is equal to 6x.

10x - 4y = 6x  (now solve)

-4y = -4x

y = x

Let’s make sure this works by replacing y with x in the original problem:

(18x)/(10x-4y)  =   (18x)/(6x)  = 3

This equation is true, so we can continue replacing y with x:

Column A            Column B

x                              x

No matter what we choose for x, these two columns must be equal. The correct answer is C.

We only had to solve one equation with this method, instead of going through more steps with the algebraic strategy. Plugging in numbers could have taken even more time, because we’d have to choose values for x and y.  Even though the algebraic approach may seem more trustworthy, something you can save time by using some logic.