First off, here’s a tricky algebraic QC:

Column A | Column B |

A. The quantity in Column A is greater

B. The quantity in Column B is greater

C. The two quantities are equal

D. The relationship cannot be determined from the information given

The average (arithmetic mean) of 3, 3, 5, 6 and

Column A | Column B |

A. The quantity in Column A is greater

B. The quantity in Column B is greater

C. The two quantities are equal

D. The relationship cannot be determined from the information given

Suppose you have an equation,

Now, suppose you have an inequality,

OK, probably all of that was review, but we need that as our ground rules.

A powerful method for solving Quantitative Comparison

For some folks, especially folks with good mathematical intuition, what I am going to explain here will be patently obvious — probably you have already thought of it. But, for folks for whom math is a constant struggle and not particularly intuitive, this new method might bring a landslide of insights.

You see, one way to think about an ordinary variable like

As you know, the GRE Quantitative Comparisons set a very different task. We are not trying to solve for a number in the ordinary sense. Instead, we are trying to figure out the relationship between columns A & B —- either A

I will suggest introducing a “mystery symbol”, which I will denote by the arbitrary combination ???. This is a holder, not for an unknown number, but for an unknown relation. We could begin any GRE QC by sticking this mystery symbol between the two columns.

What’s the point of that? Now, we just have a funky symbol between the two columns! How does this help us? Well, think about it. Whether ??? stands for < or > or =, there are certain mathematical transactions that are 100% legal. Here is a summary of what we can do to both sides of ???:

a. We can add or subtract any number to both sides of ???.

b. We can multiply or divide both sides of ??? by any positive number.

This opens up a panoply of ways of handling the two columns.

**Q1**. I will demonstrate a solution to the QC at the top of the page using this “mystery symbol” method. Begin with the mystery symbol between the content of the columns.

First, because I can subtract any number from both sides, I will subtract

Now, I will subtract

I am allowed to divide by any positive number, so I will divide by

Well, we were told that

**Answer = (B)**

**Q2**. If the average of 5 numbers is 2, then the sum of those numbers must be 5*2 = 10.

We can solve for x:

3+3+5+6+x = 10

17 + x = 10

x = -7

X is greater than -8.

**Answer = (A)**

Here, I made up the arbitrary symbol ??? to demonstrate the idea of handling an unknown relationship, but of course, you don’t actually need this symbol once you understand the idea. You can leave the space between the columns blank and simply follow guidelines (a) & (b) above, doing simultaneous operations on both columns until they are simplified to a form that allows you to decide directly.

The problem at the top of this article was a relative easy problem — folks with good number sense might have been able to solve it by inspection. Here’s a somewhat more challenging problem in which the same principle can be used with powerful effect.