GMAT Articles

Can You Divide by Variables?

- by Katharine Rudzitis

During math problems on the GMAT, test-takers might see problems like these:

x- 4x2 + 4x = 0

Does (3x)/(7x) equal 3/7?

For what values does [(x+3)(x-5)]/[(x+3)(x-6)] equal (x-5)/(x-6)?

It can be tempting to divide by x in these problems to get simpler equations, but this is not a correct strategy.

Why Can’t You Divide by X?

One essential rule in algebra is that dividing by 0 is undefined. It’s easy to see this in an example, like 10/0. What does this mean? The fraction stands for 10 ÷ 0, which tells us to cut 10 into 0 equal parts—that’s not possible. Since multiplication “undoes” division, look at the problem this way: what number times 0 gives back 10? Nothing!

The practice problems above have x terms instead of 0, so it might seem all right at first to divide by x and reduce these problems. However, we don’t know what number x represents, so we can’t divide by x in case it stands for 0.

How Can We Solve The Equations on the GMAT?

If we can’t divide by x, there must be another way to approach these problems. Let’s look at the first question again.

x- 4x2 + 4x = 0

We can’t divide both sides by x, but we can factor the left side.  

x(x- 4x + 4) = 

Now we know that x times (x- 4x + 4) equals 0, which means either x = 0 or (x- 4x + 4) = 0. If (x- 4x + 4) = 0, we’ll factor the quadratic equation to (x-2)(x-2) = 0. Now x = 0 or (x-2) = 0, which reduces to x=2. The only answer choices for x are 0 and 2.

Avoiding a Division Trap

The second problem shows an equation that looks easy to solve. Take a close look.

Does (3x)/(7x) equal 3/7?

If you plug in a few numbers, it looks like the equation holds.

(3*5)/(7*5) = 3/7 and (3*2)/(7*2) = 3/7

Without taking a moment to think, it seems like this equation is true. But what happens if x = 0? Then our fraction is undefined and we get 0 on top and 0 on bottom. Now it’s clear that the equation is true unless x = 0.

Simplifying Fractions by Cancelling

The third problem tests understanding of more complicated fractions.

For what values does [(x+3)(x-5)]/[(x+3)(x-6)] equal (x-5)/(x-6)?

As we saw above, we can’t cross out x terms without thinking first. We must make sure that the denominator is never 0, or else the fractions will be undefined. This means that (x + 3) and (x - 6) cannot equal 0, or in simpler form x is not equal to -3 or 6.

Let’s break this into cases. What happens when x is not equal to -3? Then (x + 3) is not equal to 0, and we can reduce the left side of the equation.

(x-5)/(x-6)

This is equal to the right side as long as the fractions are defined. Now let’s check what happens if x is not equal to 6. Then (x – 6) isn’t equal to 0. As long as the fractions are defined (neither denominator is 0), then the equations will be equal. Any value for x will work except for x = -3 or x = 6. If we’d just canceled out the terms, we might have missed one of these crucial values.

On the GMAT, always take a second to make sure that you’re not dividing by 0 or an x term that could be 0. Use factoring techniques to make these problems easier, or find values that x cannot be in order to find solutions. 

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