A while back, we talked about the 4 GMAT math strategies that everyone needs to master. Today, I’ve got some additional practice for you with regard to one of those strategies: Testing Cases.

Try this GMATPrep® problem:

* ” If xy + z = x(y + z), which of the following must be true?

“(A) x = 0 and z = 0

“(B) x = 1 and y = 1

“(C) y = 1 and z = 0

“(D) x = 1 or y = 0

“(E) x = 1 or z = 0

How did it go?

This question is called a “theory” question: there are just variables, no real numbers, and the answer depends on some characteristic of a category of numbers, not a specific number or set of numbers. Problem solving theory questions also usually ask what must or could be true (or what must not be true). When we have these kinds of questions, we can use theory to solve—but that can get very confusing very quickly. Testing real numbers to “prove” the theory to yourself will make the work easier.

The question stem contains a given equation:

xy + z = x(y + z)

Whenever the problem gives you a complicated equation, make your life easier: try to simplify the equation before you do any more work.

xy + z = x(y + z)

xy + z = xy + xz

z =  xz

Very interesting! The y term subtracts completely out of the equation. What is the significance of that piece of info?

Nothing absolutely has to be true about the variable y. Glance at your answers. You can cross off (B), (C), and (D) right now!

Next, notice something. I stopped at z = xz. I didn’t divide both sides by z. Why?

In general, never divide by a variable unless you know that the variable does not equal zero. Dividing by zero is an “illegal” move in algebra—and it will cause you to lose a possible solution to the equation, increasing your chances of answering the problem incorrectly.

The best way to finish off this problem is to test possible cases. Notice a couple of things about the answers. First, they give you very specific possibilities to test; you don’t even have to come up with your own numbers to try. Second, answer (A) says that both pieces must be true (“and”) while answer (E) says “or.” Keep that in mind while working through the rest of the problem.

z =  xz

Let’s see. z = 0 would make this equation true, so that is one possibility. This shows up in both remaining answers.

If x = 0, then the right-hand side would become 0. In that case, z would also have to be 0 in order for the equation to be true. That matches answer (A).

If x = 1, then it doesn’t matter what z is; the equation will still be true. That matches answer (E).

Wait a second—what’s going on? Both answers can’t be correct.

Be careful about how you test cases. The question asks what MUST be true. Go back to the starting point that worked for both answers: z = 0.

It’s true that, for example, 0 = (3)(0).

Does z always have to equal 0? Can you come up with a case where z does not equal 0 but the equation is still true?

Try 2 = (1)(2). In this case, z = 2 and x = 1, and the equation is true. Here’s the key to the “and” vs. “or” language. If z = 0, then the equation is always 0 = 0, but if not, then x must be 1; in that case, the equation is z = z. In other words, either x = 1 OR z = 0.

The correct answer is (E).

The above reasoning also proves why answer (A) could be true but doesn’t always have to be true. If both variables are 0, then the equation works, but other combinations are also possible, such as z = 2 and x = 1.

Key Takeaways: Test Cases on Theory Problems

(1) If you didn’t simplify the original equation, and so didn’t know that y didn’t matter, then you still could’ve tested real numbers to narrow down the answers, but it would’ve taken longer. Whenever possible, simplify the given information to make your work easier.

(2) Must Be True problems are usually theory problems. Test some real numbers to help yourself understand the theory and knock out answers. Where possible, use the answer choices to help you decide what to test.

(3) Be careful about how you test those cases! On a must be true question, some or all of the wrong answers could be true some of the time; you’ll need to figure out how to test the cases in such a way that you figure out what must be true all the time, not just what could be true.

* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.

So you’ve been told over and over that guessing is an important part of the GMAT. But knowing you’re supposed to guess and knowing when you’re supposed to guess are two very different things. Here are a few guidelines for how to decide when to guess.

But first, know that there are two kinds of guesses: random guesses and educated guesses. Both have their place on the GMAT. Random guesses are best for the questions that are so tough, that you don’t even know where to get started. Educated guesses, on the other hand, are useful when you’ve made at least some progress, but aren’t going to get all the way to an answer in time.

Here are a few different scenarios that should end in a guess.

Scenario 1: I’ve read the question twice, and I have no idea what it’s asking.

This one is pretty straightforward. Don’t worry about whether the question is objectively easy or difficult. If it’s too hard for you, it’s not worth doing. In fact, it’s so not worth doing that it’s not even worth your time narrowing down answer choices to make an educated guess. In fact, if it’s that difficult, it may even be better for you to get it wrong!

To make the most of your random guesses, you should use the same answer choice every time. The difference is slight, but it does up your odds of getting some of these random guess right.

Scenario 2: I had a plan, but I hit a wall.

Often, when this happens, you haven’t yet spent 2 minutes on the problem. So why guess? Maybe now you have a better plan for how to get to the answer. I know this is hard to hear, but don’t do it! To stay on pace for the entire section, you have to stay disciplined and that means that you only have one chance to get each question right.

The good news is that no 1 question you get wrong will kill your score. But, 1 question can really hurt your score if you spend too long on it! Once you realize that your plan didn’t work, it’s time to make an educated guess. You’ve already spent more than a minute on this question (hopefully not more than 2!), and you probably have some sense of which answers are more likely to be right. Take another 15 seconds (no more!) and make your best educated guess.

Scenario 3: I got an answer, but it doesn’t match any of the answer choices.

This is another painful one, but it’s an almost identical situation to Scenario 2. It means you either made a calculation error somewhere along the way, or you set the problem up incorrectly to begin with. In an untimed setting, both of these problems would have the same solution: go back over your work and find the mistake. On the GMAT, however, that process is too time-consuming. Plus, even once you find your mistake, you still have to redo all the work!

Once again, though it might hurt, it’s still in your best interest to let the question go. If you can narrow down the answer choices, great (though don’t spend longer than 15 or 20 seconds doing so). If not, don’t worry about it. Just make a random guess and vow to be more careful on the next one (and all the rest after that!).

Scenario 4: I checked my pacing chart and I’m more than 2 minutes behind.

Pacing problems are best dealt with early. If you’re more than 2 minutes behind, don’t wait until another 5 questions have passed and you realize you’re 5 minutes behind. At this point, you want to find a question in the next 5 that you can guess randomly on. The quicker you can identify a good candidate to skip, the more time you can make up.

This is another scenario where random guessing is best. Educated guessing takes time, and we’re trying to save as much time as possible. Look for questions that take a long time to read, or that deal with topics you’re not as strong in, but most importantly, just make the decision and pick up the time.

Wrap Up

Remember, this test is not like high school exams; it’s not designed to have every question answered. This test is about consistency on questions you know how to do. Knowing when to get out of a question is one of the most fundamental parts of a good score. The better you are at limiting time spent on really difficult questions, the more time you have to answer questions you know how to do.

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Right now, you might be thinking, “Wait, what? I don’t actively want to get stuff wrong!”

In fact, yes, you do. Let me take you on what might seem like a tangent for a moment.

Would you agree that one of the marks of a strong business person is the ability to tell the difference between good opportunities and bad ones? And the ability to capitalize on those good opportunities while letting the bad ones go?

Yes, of course—that’s a basic definition of business. What does that have to do with the GMAT?

The GMAT is a test of your business skills. They don’t really care how great you are with geometry or whether you know every obscure grammar rule in the book. They care whether you can distinguish between good and bad opportunities and whether you can drop the bad ones without a backward glance.

If you want to maximize your score on the GMAT, then you will have a short-list of topics that you want to get wrong fast on the test. My top three in math are combinatorics, 3-D geometry, and anything with roman numerals.

How do you decide what your categories should be? Let’s talk.

But I don’t really want to get stuff wrong… that’s just a metaphor, right?

No, it’s not a metaphor. I really want you to plan how and what you’re going to get wrong! If you haven’t already, read my post about what the GMAT really tests. (You can go ahead and read it right now; I’ll wait.)

In a nutshell, the GMAT is set up to force us to get some of the questions wrong. No matter what you can do, they’ll just give you something harder.

Ultimately, they want to see whether you have the makings of a good business person. One way to test that is to force you into a situation where your choice is between spending extra time and mental energy on something that’s too hard—likely causing yourself to run out of time and energy before the test is over—and cutting yourself off when appropriate.

How do I cut myself off?

First of all, put yourself in this mindset:

You’re at the office, working on a group project.

A colleague of yours is the project manager.

The manager annoys you because he (or she) keeps assigning too many tasks, some of which are not all that important.

Sometimes, you’re rolling your eyes when your colleague tosses a certain piece of work at you; you’re thinking, “Seriously, the client meeting is in 3 days. This is NOT the best use of our remaining time.”

Got that? Okay, now during the test, put yourself in that mindset. The test itself is your annoying colleague. When he drops a roman numeral question in your lap, or a 4-line sentence correction with every last word underlined, you’re already rolling your eyes and thinking, “Are you serious? Come on.”

Here’s the key step: let yourself get just a little annoyed—but with the test, not yourself. You’re not feeling badly that you don’t like the problem; you don’t feel as though you’re falling short. No way! Instead, your colleague is trying to get you to do something that is clearly a waste of time. Roll your eyes. To appease your colleague, figure out whether there’s enough here for you to make an educated guess. Then pick something and move on to more important tasks.

How do I know when to cut myself off?

Quick: name your top three annoyances in quant. Now do the same in verbal. Here’s another one of mine: an RC detail EXCEPT question on a really technical topic with very long answer choices. (In other words, I have to find the four wrong answers in order to find the one right answer… and the topic area is very long and annoying.)

That’s your starting point: you already know you dread these areas. Back this up with data: make sure that these really are the worst ones for you. “Worst” is defined as “I rarely get these right and even when I do, I still use too much time and brain energy.”

Next, check to see how commonly tested the particular topic or question type is. You can’t afford to blow off algebra—that’s too broad a topic. You can, though, blow off sequences.

For some topics, you do want to try to be able to answer lower-level questions. For instance, if one of my students just hates polygons (triangles, squares, rectangles), he has my blessing to blow off harder questions—the ones that combine shapes, for example, or that move into the 3-D arena. He does need to learn the more basic formulas, though, so that he isn’t missing too many lower-level questions.

Your particular mix of pet peeves will almost certainly change over time. Initially, I had some other things at the top of my list, such as weighted averages. Then, I discovered a much better way to do those problems, so 3-D geometry took its place.

Some topics, though, will always be weaknesses. I’ve never liked combinatorics and doubt I ever will. That’s perfectly fine, particularly when the topic is not that commonly tested anyway!

Sound off in the comments below: what areas do you hate the most? Your new strategy is to get those wrong fast and redirect that time and mental energy elsewhere!