Quarter Wit, Quarter Wisdom: When Can You Divide by a Variable?

Quarter Wit, Quarter WisdomWe have often come across test takers confused about division by a variable. When is it allowed, when is it not allowed? Why is it allowed in some cases and not in others? What are the constraints we need to look out for?

For example:

Is division by x allowed here: x^2 = 10x?
Is division by x allowed here: y = 4x?
Is division by x allowed here: x^2 < 4x?

Let’s take a detailed look at all these questions today.

The basic guidelines:

  1. Division by 0 is not allowed, hence you cannot divide by a variable until and unless we know that it cannot be 0.
  2. In the case of an inequality, when you divide by a negative number, the sign of the inequality flips. So we cannot divide by a variable until and unless we know that it cannot be 0 AND whether it is positive or negative.

Let’s look at the three questions given above and try to solve them using these guidelines:

Is division by x allowed here: x^2 = 10x?

The first thing to find out here is whether or not x can equal 0.

Case 1: If no other information has been given, then x can be 0 and we cannot divide by it. This is how we proceed in that case:

x^2 – 10x = 0
x(x – 10) = 0
x = 0 or 10

Case 2: If the question stem tells us that x is not 0, then we can divide by x.

x^2/x = 10x/x
x = 10

Obviously, we don’t get the second solution (x = 0) in this case, as we already know that x cannot be 0. Now let’s look at the second problem:

Is division by x allowed here: y = 4x?

Again, this is an equation and we need to know whether or not x can equal 0.

Case 1: If x can be 0, you cannot divide by it. In this case, x = 0 and y = 0 is one of the infinite possible solutions.

Case 2: If the question stem states that x cannot be 0, then we can do the following:

y/x = 4

Now let’s look at the final question:

Is division by x allowed here: x^2 > -4x?

Here, we have an inequality. Before deciding whether we can divide by x or not, we need to know not only whether x can be equal to 0, but also whether x is positive or negative.

Case 1: If we know nothing about the possible values that x can take, then this is how we proceed:

x^2 + 4x > 0
x(x + 4) > 0

Now we can use the method discussed in the first problem to arrive at the range of x.

x > 0 or x < -4

Case 2: If we know that x is positive, then we can proceed like this:

x^2/x > -4x/x
x > -4

Since we are given that x is positive, we know that that x > 0 (looking at the two options above).

Case 3: If we know that x is negative, then this is how we will proceed:

x^2/x < -4x/x (we flip the sign of the inequality because we divide by x, which is negative)
x < -4

The results obtained are logical, right? When x can be anywhere on the number line, we get the range as x > 0 or x < -4.

If x has to be positive, the range is x > 0.
If x has to be negative, the range is x < -4.

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Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

GMAT Tip of the Week: Zero Out

GMAT prep

Raise your hand if you’ve cringed this week as someone wished you a happy “Oh-Ten”, which, though technically correct (the year 2010 has an “oh” in front of the “ten”) is practically wrong (they didn’t as you how “oh-oh-nine” ended up for you). That zero that we’ve been used to including in front of the year for the last decade is officially out. Spread the word!

There is one other way in which the zero can actually help you out of a pretty common awful situation. On most automated answering services for corporate phone numbers (“press one for English, two for Spanish, etc.), hitting zero repeatedly will put you through to an operator much more quickly, and you can effectively “zero out” of the frustrating phone mail system.

On the GMAT, you’ll also have opportunities to “zero out” of some tight spots. Consider the question:

If 3^x * 4^y = 531,441, and x and y are both integers, what is the value of y?

Seemingly, this question would require an awful amount of trial-and-error as you attempt to find a combination of x and y to satisfy the equation. A more astute look, however, proposes a fairly significant problem: Because the equation produces an odd number, 3^x cannot be multiplied by an even number, so any multiple of 4 in the 4^y term would ruin the equation.

Therein lies your clue – the 4^y term must somehow be made odd in order for the equation to hold. Taking 4 to any positive exponent would produce an even number – by definition, an even number multiplied by itself is going to produce an even product – but exponential properties provide us with an opportunity to “zero out”:

Any number to the power of 0 (x^0) will equal 1. (note: 0, itself, is the exception to this rule…and to many rules. We’ll explain this one below.)

You may remember this rule from high school, and you may remember it being a little awkward if you simply were told to memorize it. Here’s why it’s true:

x^1 * x^(-1) = x^0 (we know this because of the rules for multiplying exponential terms that have the same base)
x^1 * x^(-1) = x/x (we know this because a negative exponent means that you move that term to the denominator, another exponent rule)

Therefore:

x^0 = x/x, and we know that x/x is going to equal 1

This won’t work for 0, because you can’t divide by 0, and this is why 0 is the exception to this rule.

Because taking 4^0 is our only hope to make that term odd, y must be 0 so that the equation can hold, and 0 is therefore the answer to that question.

To summarize, setting an exponent equal to zero is a remove an “undesirable” exponent term by setting it to 1. Be aware of this unique property and you’ll be able to “zero out” of a seemingly-paradoxical or convoluted exponent problem.

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GMAT Tip of the Week

From Zero to Hero

(This is one of a series of GMAT tips that we offer on our blog.)

The number 0 is a tricky one on the GMAT, as its unique properties are often either the key to unlocking a difficult solution, or the trap in to which a seemingly logical solution can lead you. Learning the properties of zero (keep in mind that it is an even number) is an important skill, particularly on data sufficiency problems. Even more importantly, never forget to consider zero as a potential value for a variable, as it often produces surprising results. Consider the case of zero as an exponent:

x^0 is, by definition, equal to 1. This concept may have bothered you in high school, as it seems almost arbitrary that an exponent without value would automatically lead to a value of 1. But noting the properties of exponents can help you to prove and more easily embrace and remember this useful device: take, for example, the expression x^2 * x^-2. You could rearrange this two ways:

a) (x^2) / (x^2) –> The negative exponent simply moves that term to the denominator

b) x^(2-2), or x^0 –> When multiplying terms with the same base, taken to different exponents, you simply add the exponents

Because we can prove that (x^2) / (x^2) must be equal to 1, and that the two expressions above are equal to each other, we can prove that x^0 = 1.

Now here comes the payoff – because x^0 is equal to 1, it’s the ultimate in cop-out solutions to difficult problems. Say that a question asks:

For what value of x will 5^x be a factor of 2^10?

2^10 is not divisible by 5 (its only prime factor is 2), but the question might seem to require you to multiply that value out, as well as some potential values of 5^x, in futility to prove that point. However, if 0 is an option, it will set the term equal to 1, which is a factor of any integer, and your work is already done.

Note that the use of 0 as an exponent is always quick and often the only solution to an exponent/divisibility problem. Keeping this device in mind will save you time on the GMAT, as well as enable you to solve some fairly difficult problems.