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.

Have you heard of the C-Trap? I’m not going to tell you what it is yet. Try this problem from GMATPrep® first and see whether you can avoid it

* “In a certain year, the difference between Mary’s and Jim’s annual salaries was twice the difference between Mary’s and Kate’s annual salaries. If Mary’s annual salary was the highest of the 3 people, what was the average (arithmetic mean) annual salary of the 3 people that year?

“(1) Jim’s annual salary was $30,000 that year.

“(2) Kate’s annual salary was $40,000 that year.”

I’m going to do something I normally never do at this point in an article: I’m going to tell you the correct answer. I’m not going to type the letter, though, so that your eye won’t inadvertently catch it while you’re still working on the problem. The correct answer is the second of the five data sufficiency answer choices.

How did you do? Did you pick that one? Or did you pick the trap answer, the third one?

Here’s where the C-Trap gets its name: on some questions, using the two statements together will be sufficient to answer the question. The trap is that using just one statement alone will also get you there—so you can’t pick answer (C), which says that neither statement alone works.

In the trickiest C-Traps, the two statements look almost the same (as they do in this problem), and the first one doesn’t work. You’re predisposed, then, to assume that the second statement, which seemingly supplies the “same” kind of information, also won’t work. Therefore, you don’t vet the second statement thoroughly enough before dismissing it—and you’ve just fallen into the trap.

How can you dig yourself out? First of all, just because two statements look similar, don’t assume that they either both work or both don’t. The test writers are really good at setting traps, so assume nothing.
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Recently, we talked about how to create Official Guide (OG) problem sets in order to practice for the test. I have one more component to add: track your work and analyze your results to help you prioritize your studies.

In the first half of this article, we talked about making problem sets from the roughly 1,500 problems that can be found in the three main OG books. These problems are generally regarded as the gold standard for GMAT study, but how do you keep track of your progress across so many different problems?

The best tool out there (okay, I’m biased) is our GMAT Navigator program, though you can also build your own tracking tool in Excel, if you prefer. I’ll talk about how to get the most out of Navigator, but I’ll also address what to include if you decide to build your own Excel tracker.

(Note: GMAT Navigator used to be called OG Archer. If you used OG Archer in the past, Navigator brings you all of that same functionality—it just has a new name.)

What is GMAT Navigator?

Navigator contains entries for every one of the problems in the OG13, Quant Supplement, and Verbal Supplement books. In fact, you can even look up problems from OG12. You can time yourself while you answer the question, input your answer, review written and video solutions, get statistics based on your performance, and more.

Everyone can access a free version of Navigator. Students in our courses or guided-self study programs have access to the full version of the program, which includes explanations for hundreds of the problems.

How Does Navigator Work?

First, have your OG books handy. The one thing the program does not contain is the full text of problems. (Copyright rules prevent this, unfortunately.)

When you sign on to Navigator, you’ll be presented with a quick tutorial showing you what’s included in the program and how to use it. Take about 10 minutes to browse through the instructions and get oriented.

When you reach the main page, your first task is to decide whether you want to be in Browse mode or Practice mode.

Practice mode is the default mode; you’ll spend most of your time in this mode. You’ll see an entry for the problem along with various tools (more on this below).

Browse mode will immediately show you the correct answer and the explanation. You might use this mode after finishing a set of questions, when you want to browse through the answers. Don’t reveal the answers and explanations before you’ve tried the problem yourself!

Here’s what you can do in Practice mode:
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Some Data Sufficiency questions present you with scenarios: stories that could play out in various complicated ways, depending on the statements. How do you get through these with a minimum of time and fuss?

Try the below problem. (Copyright: me! I was inspired by an OG problem; I’ll tell you which one at the end.)

* “During a week-long sale at a car dealership, the most number of cars sold on  any one day was 12. If at least 2 cars were sold each day, was the average daily number of cars sold during that week more than 6?

“(1) During that week, the second smallest number of cars sold on any one day was 4.

“(2) During that week, the median number of cars sold was 10.”

First, do you see why I described this as a “scenario” problem? All these different days… and some number of cars sold each day… and then they (I!) toss in average and median… and to top it all off, the problem asks for a range (more than 6). Sigh.

Okay, what do we do with this thing?

Because it’s Data Sufficiency, start by establishing the givens. Because it’s a scenario, Draw It Out.

Let’s see. The “highest” day was 12, but it doesn’t say which day of the week that was. So how can you draw this out?

Neither statement provides information about a specific day of the week, either. Rather, they provide information about the least number of sales and the median number of sales.

The use of median is interesting. How do you normally organize numbers when you’re dealing with median?

Bingo! Try organizing the number of sales from smallest to largest. Draw out 7 slots (one for each day) and add the information given in the question stem:

Now, what about that question? It asks not for the average, but whether the average number of daily sales for the week is more than 6. Does that give you any ideas for an approach to take?

Because it’s a yes/no question, you want to try to “prove” both yes and no for each statement. If you can show that a statement will give you both a yes and a no, then you know that statement is not sufficient. Try this out with statement 1

(1) During that week, the least number of cars sold on any one day was 4.

Draw out a version of the scenario that includes statement (1):

Can you find a way to make the average less than 6? Keep the first day at 2 and make the other days as small as possible:

The sum of the numbers is 34. The average is 34 / 7 = a little smaller than 5.

Can you also make the average greater than 6? Try making all the numbers as big as possible:

(Note: if you’re not sure whether the smallest day could be 4—the wording is a little weird—err on the cautious side and make it 3.)

You may be able to eyeball that and tell it will be greater than 6. If not, calculate: the sum is 67, so the average is just under 10.

Statement (1) is not sufficient because the average might be greater than or less than 6. Cross off answers (A) and (D).

Now, move to statement (2):

(2) During that week, the median number of cars sold was 10.

Again, draw out the scenario (using only the second statement this time!).

Can you make the average less than 6? Test the smallest numbers you can. The three lowest days could each be 2. Then, the next three days could each be 10.

The sum is 6 + 30 + 12 = 48. The average is 48 / 7 = just under 7, but bigger than 6. The numbers cannot be made any smaller—you have to have a minimum of 2 a day. Once you hit the median of 10 in the middle slot, you have to have something greater than or equal to the median for the remaining slots to the right.

The smallest possible average is still bigger than 6, so this statement is sufficient to answer the question. The correct answer is (B).

Oh, and the OG question is DS #121 from OG13. If you think you’ve got the concept, test yourself on the OG problem.

Key Takeaway: Draw Out Scenarios

(1) Sometimes, these scenarios are so elaborate that people are paralyzed. Pretend your boss just asked you to figure this out. What would you do? You’d just start drawing out possibilities till you figured it out.

(2) On Yes/No DS questions, try to get a Yes answer and a No answer. As soon as you do that, you can label the statement Not Sufficient and move on.

(3) After a while, you might have to go back to your boss and say, “Sorry, I can’t figure this out.” (Translation: you might have to give up and guess.) There isn’t a fantastic way to guess on this one, though I probably wouldn’t guess (E). The statements don’t look obviously helpful at first glance… which means probably at least one of them is!

Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We’re not kidding! Check out our upcoming courses here.

I’m reviving an old article I first wrote five years ago (time flies!) because the topic is so important. I hope that no one ever again experiences a significant GMAT score drop on the real test (or even a practice one!), but the reality is that this does happen. The big question: now what? Read more

You’ve heard a million times that you’re supposed to create Official Guide (OG) problem sets in order to practice for the test. But how do you actually do so in a way that will help you get the most out of your study?

Fear not! This article is coming to your rescue.

Initially, when you’re studying a new topic or problem type, you won’t do sets of problems; instead, you’ll just try one problem at a time. As you gain experience, though, you’re going to want to do 3 problems in a row, or 5, or 10.

Why?

Because the real test will never give you just one problem!

The GMAT will give you many questions in a row and they’ll be all jumbled up—an SC, then a couple of CRs, then back to another SC (that tests different grammar rules than the first one), and so on.

You want to practice two things:

(1) Jumping around among question types and topics

(2) Managing your timing and mental energy among a group of questions

When do I start doing problem sets?

You’re going to use problem sets to test your skills, so you’ve got to develop some of those skills first. If you’re using our Strategy Guides to study, then at the end of one chapter, you’ll do only two or three OG problems to make sure that you understood the material in the chapter.

Later, though, when you finish the Guide, do a set of problems that mix topics (and question types) from that entire book. Make sure you can distinguish between the similar-but-not-quite-the-same topics in that book, and also practice your skills on both problem solving and data sufficiency. As you finish subsequent Guides, your sets can include problems from everything you’ve done so far. Keep mixing it up!

How do I make the sets?

You’ll need to balance three things when you create a problem set:

(1) Number of problems. Initially, start out with about 3 to 5 problems. As you gain experience and add topics, you’ll increase the size of the sets—we’ll talk more about this a little later.

(2) Type of problem and content.
(a) For quant, always do a mix of Problem Solving (PS) and Data Sufficiency (DS). For verbal, mix at least two of the three types; you can include all three types in larger sets.
(b) Do not do a set of 3 or more questions all from the same chapter or content area—for example, don’t do 3 exponents questions in a row. You know exactly what you’re about to get and the real test will never be this nice to you.

(3) Difficulty level.
(a) Include a mix of easier, medium, and harder questions in your set. For all types except Reading Comprehension, the OG places problems in roughly increasing order of difficulty. On average, aroblem 3 is easier than a problem 50, which is easier than a problem 102. (This does not mean that problem 5 is necessarily harder than problem 3. In general, higher question numbers represent harder questions, but the increase is not linear from problem to problem.)
(b) Note: your personal strengths and weaknesses will affect how you perceive the problems—you might think a lower-numbered problem is hard or a higher-numbered problem is easy. They are… for you! Expect that kind of outcome sometimes.

Timing!

Next, calculate how much time to give yourself to do the problem set.
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Stop! Before you dive in and start calculating on a math problem, reflect for a moment. How can you set up the work to minimize the number of annoying calculations?

Try the below Percent problem from the free question set that comes with your GMATPrep® software. The problem itself isn’t super hard but the calculations can become time-consuming. If you find the problem easy, don’t dismiss it. Instead, ask yourself: how can you get to the answer with an absolute minimum of annoying calculations?

* ” The table above shows the results of a recent school board election in which the candidate with the higher total number of votes from the five districts was declared the winner. Which district had the greatest number of votes for the winner?

“(A) 1

“(B) 2

“(C) 3

“(D) 4

“(E) 5”

Ugh. We have to figure out what they’re talking about in the first place!

The first sentence of the problem describes the table. It shows 5 different districts with a number of votes, a percentage of votes for one candidate and a percentage of votes for a different candidate.

Hmm. So there were two candidates, P and Q, and the one who won the election received the most votes overall. The problem doesn’t say who that was. I could calculate that from the given data, but I’m not going to do so now! I’m only going to do that if I have to.

Let’s see. The problem then asks which district had the greatest number of votes for the winner. Ugh. I am going to have to figure out whether P or Q won. Let your annoyance guide you: is there a way to tell who won without actually calculating all the votes?

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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!

Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We’re not kidding! Check out our upcoming courses here.

Last week, we talked about the first two elements of getting the most out of your CATs.

#1: How NOT to use your practice CATs

#2: How to analyze your strengths and weaknesses with respect to timing

This week, we’re going to dive even further into strengths and weaknesses using the Assessment Reports.

#3: Run the reports.

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A couple of months ago, we talked about what to do when a geometry problem pops up on the screen. Do you remember the basic steps? Try to implement them on the below GMATPrep® problem from the free tests.

* ”In the xy-plane, what is the y-intercept of line L?

“(1) The slope of line L is 3 times its y-intercept
“(2) The x-intercept of line L is – 1/3”

My title (3 Steps to Better Geometry) is doing double-duty. First, here’s the general 3-step process for any quant problem, geometry included:

All geometry problems also have three standard strategies that fit into that process.

First, pick up your pen and start drawing! If they give you a diagram, redraw it on your scrap paper. If they don’t (as in the above problem), draw yourself a diagram anyway. This is part of your Glance-Read-Jot step.

Second, identify the “wanted” element and mark this element on your diagram. You’ll do this as part of the Glance-Read-Jot step, but do it last so that it leads you into the Reflect-Organize stage. Where am I trying to go? How can I get there?

Third, start Working! Infer from the given information. Geometry on the GMAT can be a bit like the proofs that we learned to do in high school. You’re given a couple of pieces of info to start and you have to figure out the 4 or 5 steps that will get you over to the answer, or what you’re trying to “prove.”

Let’s dive into this problem. They’re talking about a coordinate plane, so you know the first step: draw a coordinate plane on your scrap paper. The question indicates that there’s a line L, but you don’t know anything else about it, so you can’t actually draw it. You do know, though, that they want to know the y-intercept. What does that mean?

They want to know where line L crosses the y-axis. What are the possibilities?

Infinite, really. The line could slant up or down or it could be horizontal. In any of those cases, it could cross anywhere. In fact, the line could even be vertical, in which case it would either be right on the y-axis or it wouldn’t cross the y-axis at all. Hmm.
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