### GMAT Problem Solving Strategy: Test Cases

If you’re going to do a great job on the GMAT, then you’ve got to know how to Test Cases. This strategy will help you on countless quant problems.

This technique is especially useful for Data Sufficiency problems, but you can also use it on some Problem Solving problems, like the GMATPrep® problem below. Give yourself about 2 minutes. Go!

* “For which of the following functions *f* is *f*(*x*) = *f*(1 – *x*) for all *x*?

(A) | f(x) = 1 – x |

(B) | f(x) = 1 – x^{2} |

(C) | f(x) = x^{2} – (1 – x)^{2} |

(D) | f(x) = x^{2}(1 – x)^{2} |

(E) | f(x) = x / (1 – x)” |

Testing Cases is mostly what it sounds like: you will test various possible scenarios in order to narrow down the answer choices until you get to the one right answer. What’s the common characteristic that signals you can use this technique on problem solving?

The most common language will be something like “Which of the following must be true?” (or “could be true”).

The above problem doesn’t have that language, but it does have a variation: you need to find the answer choice for which the given equation is true “for all *x*,” which is the equivalent of asking for which answer choice the given equation is always, or must be, true.

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### When Your High School Algebra is Wrong: How the GMAT Breaks Systems of Equations Rules

*If you have two equations, you can solve for two variables.*

This rule is a cornerstone of algebra. It’s how we solve for values when we’re given a relationship between two unknowns:

*If I can buy 2 kumquats and 3 rutabagas for $16, and 3 kumquats and 1 rutabaga for $9, how much does 1 kumquat cost?*

We set up two equations:

2k + 4r = 16

3k + r = 9

Then we can use either substitution or elimination to solve. (Try it out yourself; answer* below).

On the GMAT, you’ll be using the “2 equations à 2 variables” rule to solve for a lot of word problems like the one above, especially in Problem Solving. Be careful, though! On the GMAT this rule doesn’t *always* apply, especially in Data Sufficiency. Here are some sneaky exceptions to the rule…

**2 Equations aren’t always 2 equations**

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### Tackling Max/Min Statistics on the GMAT (part 3)

Welcome to our third and final installment dedicated to those pesky maximize / minimize quant problems. If you haven’t yet reviewed the earlier installments, start with part 1 and work your way back up to this post.

I’d originally intended to do just a two-part series, but I found another GMATPrep® problem (from the free tests) covering this topic, so here you go:

“A set of 15 different integers has a median of 25 and a range of 25. What is the greatest possible integer that could be in this set?

“(A) 32

“(B) 37

“(C) 40

“(D) 43

“(E) 50”

Here’s the general process for answering quant questions—a process designed to make sure that you *understand* what’s going on and come up with the best *plan* before you dive in and *solve*:

Fifteen integers…that’s a little annoying because I don’t literally want to draw 15 blanks for 15 numbers. How can I shortcut this while still making sure that I’m not missing anything or causing myself to make a careless mistake?

Hmm. I could just work backwards: start from the answers and see what works. In this case, I’d want to start with answer (E), 50, since the problem asks for the greatest possible integer.

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### Tackling Max/Min Statistics on the GMAT (Part 2)

Last time, we discussed two GMATPrep® problems that simultaneously tested statistics and the concept of maximizing or minimizing a value. The GMAT could ask you to maximize or minimize just about anything, so the latter skill crosses many topics. Learn how to handle the nuances on these statistics problems and you’ll learn how to handle any max/min problem they might throw at you.

Feel comfortable with the two problems from the first part of this article? Then let’s kick it up a notch! The problem below was written by us (Manhattan Prep) and it’s complicated—possibly harder than anything you’ll see on the real GMAT. This problem, then, is for those who are looking for a really high quant score—or who subscribe to the philosophy that mastery includes trying stuff that’s harder than what you might see on the real test, so that you’re ready for anything.

Ready? Here you go:

“Both the average (arithmetic mean) and the median of a set of 7 numbers equal 20. If the smallest number in the set is 5 less than half the largest number, what is the largest possible number in the set?

“(A) 40

“(B) 38

“(C) 33

“(D) 32

“(E) 30”

Out of the letters A through E, which one is your favorite?

You may be thinking, “Huh? What a weird question. I don’t have a favorite.”

I don’t have one in the real world either, but I do for the GMAT, and you should, too. When you get stuck, you’re going to need to be able to let go, guess, and move on. If you haven’t been able to narrow down the answers at all, then you’ll have to make a random guess—in which case, you want to have your favorite letter ready to go.

If you have to think about what your favorite letter is, then you don’t have one yet. Pick it right now.

I’m serious. I’m not going to continue until you pick your favorite letter. Got it?

From now on, when you realize that you’re lost and you need to let go, pick your favorite letter *immediately* and move on. Don’t even think about it.

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### Break Your “Good” GMAT Study Habits! What Learning Science Can Teach Us About Effective GMAT Studying

*Distractions are bad. Routine, concentration, and hard work are good.* These all seem like common-sense rules for studying, right? Surprisingly (for many people, at least), learning science tells us that these “good habits” may actually be *hurting* your learning process!

When you were in college, your study process probably looked something like this: for a given class, you’d attend a lecture each week, do the readings (or at least most of them), and maybe turn in an assignment or problem set. Then, at the end of the semester, you’d spend a week furiously cramming all of that information to prepare for the test.

Since this is the way you’ve always studied, it’s probably how you’re approaching the GMAT, too. But I have bad news: this is not an effective approach for the GMAT!

Taking notes then cramming the night before the test is beneficial for tests that ask you to recite knowledge: “what were the major consequences of the Hawley-Smoot tariff” or “explain Heisenberg’s uncertainty principle.” You can hold a lot of facts -for a brief time – in your short-term memory when cramming. You memorize facts, you spit them out for the test… and then, if you’re like me, you find that you’ve forgotten half of what you memorized by the next semester.

**Why the GMAT is Different**

The GMAT doesn’t reward this style of studying because it’s not simply a test of facts or knowledge. The GMAT requires you to know a lot of rules, of course, but the main thing that it’s testing is your ability to apply those concepts to new problems, to adapt familiar patterns, and to use strategic decision-making. You’ll never see the same problem twice.

Shallow memorization is not nearly enough. You need *deep* conceptual understanding.

In *How We Learn****, science writer Benedict Carey outlines decades of research about how this kind of learning happens. Many of the findings go against what you probably thought were “good” study habits.

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### Tackling Max/Min Statistics on the GMAT (Part 1)

Blast from the past! I first discussed the problems in this series way back in 2009. I’m reviving the series now because too many people just aren’t comfortable handling the weird maximize / minimize problem variations that the GMAT sometimes tosses at us.

In this installment, we’re going to tackle two GMATPrep® questions. Next time, I’ll give you a super hard one from our own archives—just to see whether you learned the material as well as you thought you did.

Here’s your first GMATPrep problem. Go for it!

“*Three boxes of supplies have an average (arithmetic mean) weight of 7 kilograms and a median weight of 9 kilograms. What is the maximum possible weight, in kilograms, of the lightest box?

“(A) 1

“(B) 2

“(C) 3

“(D) 4

“(E) 5”

When you see the word *maximum *(or a synonym), sit up and take notice. This one word is going to be the determining factor in setting up this problem efficiently right from the beginning. (The word *minimum* or a synonym would also apply.)

When you’re asked to maximize (or minimize) one thing, you are going to have one or more decision points throughout the problem in which you are going to have to maximize or minimize some other variables. Good decisions at these points will ultimately lead to the desired maximum (or minimum) quantity.

This time, they want to maximize the lightest box. Step back from the problem a sec and picture three boxes sitting in front of you. You’re about to ship them off to a friend. Wrap your head around the dilemma: if you want to maximize the *lightest* box, what should you do to the other two boxes?

Note also that the problem provides some constraints. There are three boxes and the median weight is 9 kg. No variability there: the middle box must weigh 9 kg.

The three items also have an average weight of 7. The total weight, then, must be (7)(3) = 21 kg.

Subtract the middle box from the total to get the combined weight of the heaviest and lightest boxes: 21 – 9 = 12 kg.

The heaviest box has to be equal to or greater than 9 (because it is to the right of the median). Likewise, the lightest box has to be equal to or *smaller* than 9. In order to maximize the weight of the lightest box, what should you do to the heaviest box?

Minimize the weight of the heaviest box in order to maximize the weight of the lightest box. The smallest possible weight for the heaviest box is 9.

If the heaviest box is minimized to 9, and the heaviest and lightest must add up to 12, then the maximum weight for the lightest box is 3.

The correct answer is (C).

Make sense? If you’ve got it, try this harder GMATPrep problem. Set your timer for 2 minutes!

“*A certain city with a population of 132,000 is to be divided into 11 voting districts, and no district is to have a population that is more than 10 percent greater than the population of any other district. What is the minimum possible population that the least populated district could have?

“(A) 10,700

“(B) 10,800

“(C) 10,900

“(D) 11,000

“(E) 11,100”

Hmm. There are 11 voting districts, each with some number of people. We’re asked to find the *minimum* possible population in the *least* populated district—that is, the smallest population that any one district could possibly have.

Let’s say that District 1 has the minimum population. Because all 11 districts have to add up to 132,000 people, you’d need to *maximize* the population in Districts 2 through 10. How? Now, you need more information from the problem:

“no district is to have a population that is *more than 10 percent greater* than the population of any other district”

So, if the smallest district has 100 people, then the largest district could have up to 10% more, or 110 people, but it can’t have any more than that. If the smallest district has 500 people, then the largest district could have up to 550 people but that’s it.

How can you use that to figure out how to split up the 132,000 people?

In the given problem, the number of people in the smallest district is unknown, so let’s call that *x*. If the smallest district is *x*, then calculate 10% and add that figure to *x*: *x* + 0.1*x* = 1.1*x*. The largest district could be 1.1*x* but can’t be any larger than that.

Since you need to maximize the 10 remaining districts, set all 10 districts equal to 1.1*x*. As a result, there are (1.1*x*)(10) = 11*x* people in the 10 maximized districts (Districts 2 through 10), as well as the original *x *people in the minimized district (District 1).

The problem indicated that all 11 districts add up to 132,000, so write that out mathematically:

11*x* + *x* = 132,000

12*x* = 132,000

*x* = 11,000

The correct answer is (D).

Practice this process with any max/min problems you’ve seen recently and join me next time, when we’ll tackle a super hard problem.

### Key Takeaways for Max/Min Problems:

(1) Figure out what variables are “in play”: what can you manipulate in the problem? Some of those variables will need to be maximized and some minimized in order to get to the desired answer. Figure out which is which at each step along the way.

(2) Did you make a mistake—maximize when you should have minimized or vice versa? Go through the logic again, step by step, to figure out where you were led astray and why you should have done the opposite of what you did. (This is a good process in general whenever you make a mistake: figure out why you made the mistake you made, as well as how to do the work correctly next time.)

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

### What is the GMAT? An Introduction to the GMAT Exam

The Graduate Management Admission Test, better known as the GMAT®, is a standardized test used in the admissions process for business school and other specialized Master’s programs. The exam measures certain skills that the business schools care about, most notably Executive Reasoning skills. It does not test any specific business knowledge.

### When is the GMAT given?

You can take the GMAT year-round, nearly any day of the week (though they limit you to 5 sittings in a 12-month period and require a wait of 31 days between tests). The exam is given on a computer and is known as a “CAT.”

### What is a CAT?

A CAT is a computer-adaptive test: the test actually adapts itself to you while you’re taking it! Two of the four sections on the GMAT, the Quantitative and Verbal sections, are adaptive. Each of these two sections begins with a random, approximately medium-level question. The computer chooses each subsequent question based upon your collective performance to that point in the section.

The practical implications are important. First, every test taker will take a different exam with a different mix of questions, but the test feels hard for everyone, since the test will just keep getting harder until it finds a particular person’s limit. Second, the scoring is pretty peculiar; it’s important to understand how the scoring works.

Want to try your hand at a practice test? Take our free, full-length practice exam here.

### How is the GMAT Scored?

Tests you took in school were generally based on the percentage of questions answered correctly: the more you got right, the higher the score you received. As a result, you have been trained to take your time and try to get everything right when you take a test. This general strategy *does not work* on computer-adaptive sections of the GMAT because, strangely enough, the quant and verbal scores are not based on the percentage of questions answered correctly. On the GMAT*, most people answer similar percentages of questions correctly, typically in the 50% to 70% range (even at higher scoring levels!).*

How is that possible? The first thing to know: the GMAT is not a school test. The quant section is not *really* a math test, and the verbal section is not *really* a grammar test. Of course, you do need to know how to handle those topics. The test writers are really interested, however, in knowing how good you are at making decisions and managing scarce resources. (That’s the second time we’ve linked to that same article. Go read it!)

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### GMAT Data Sufficiency Strategy: Test Cases

If you’re going to do a great job on Data Sufficiency, then you’ve got to know how to Test Cases. This strategy will help you on countless DS problems.

Try this GMATPrep® problem from the free exams. Give yourself about 2 minutes. Go!

* “On the number line, if the number *k* is to the left of the number *t*, is the product *kt* to the right of *t*?

“(1) *t* < 0

“(2) *k* < 1”

If visualizing things helps you wrap your brain around the math (it certainly helps me), sketch out a number line:

*k* is somewhere to the left of *t*, but the two actual values could be anything. Both could be positive or both negative, or *k* could be negative and *t* positive. One of the two could even be zero.

The question asks whether *kt* is to the right of *t*. That is, is the product *kt* greater than *t* by itself?

There are a million possibilities for the values of *k* and* t*, so this question is what we call a theory question: are there certain characteristics of various numbers that would produce a consistent answer? Common characteristics tested on theory problems include positive, negative, zero, simple fractions, odds, evens, primes—basically, number properties.

“(1) *t* < 0

This problem appears to be testing positive and negative, since the statement specifies that one of the values must be negative. Test some real numbers, always making sure that *t* is negative.

Case #1:

t |
k |
Valid case? |
Is kt > t? |

-1 | -2 | Valid: t < 0 and k < t |
2 > -1? Yes. |

Testing Cases involves three consistent steps:

First, choose numbers to test in the problem

Second, make sure that you have selected a valid case. All of the givens must be true using your selected numbers.

Third, answer the question.

In this case, the answer is Yes. Now, your next strategy comes into play: try to prove the statement *insufficient*.

How? Ask yourself what numbers you could try that would give you the opposite answer. The first time, you got a Yes. Can you get a No?

Case #2:

t |
k |
Valid case? |
Is kt > t? |

-1 | 2 | Invalid! k is not less than t! |

Careful: this is where you might make a mistake. In trying to find the opposite case, you might try a mix of numbers that is invalid. Always make sure that you have a valid case before you actually try to answer the question. Discard case 2.

Case #3:

t |
k |
Valid case? |
Is kt > t? |

-1 | -5 | Valid: t < 0 and k < t |
5 > -1? Yes. |

Hmm. We got another Yes answer. What does this mean? If you can’t come up with the opposite answer, see if you can understand why. According to this statement, *t* is always negative. Since *k* must be smaller than *t*, *k* will also always be negative.

The product *kt*, then, will be the product of two negative numbers, which is always positive. As a result, *kt* must always be larger than *t*, since *kt* is positive and *t *is negative.

Okay, statement (1) is sufficient. Cross off answers BCE and check out statement (2):

“(2) *k* < 1”

You know the drill. Test cases again!

Case #1:

k |
t |
Valid case? |
Is kt > t? |

0 | 1 | Valid: k < 1 and k < t |
0 > 1? No. |

You’ve got a No answer. Try to find a Yes.

Case #2:

k |
t |
Valid case? |
Is kt > t? |

-1 | 1 | Valid: k < 1 and k < t |
-1 > 1? No. |

Hmm. I got another No. What needs to happen to make *kt* > *t*? Remember what happened when you were testing statement (1): try making them both negative!

In fact, when you’re testing statement (2), see whether any of the cases you already tested for statement (1) are still valid for statement (2). If so, you can save yourself some work. Ideally, the below would be your path for statement (2), not what I first showed above:

“(2) *k* < 1”

Case #1:

k |
t |
Valid case? |
Is kt > t? |

-2 | -1 | Valid: k < 1 and k < t |
Same case, still Yes. |

All you have to do is make sure that the case is valid. If so, you’ve already done the math, so you know that the answer is the same (in this case, Yes).

Now, try to find your opposite answer: can you get a No?

Case #2: Try something I couldn’t try before. *k* could be positive or even 0…

k |
t |
Valid case? |
Is kt > t? |

0 | 1 | Valid: k < 1 and k < t |
0 > 1? No. |

A Yes and a No add up to an insufficient answer. Eliminate answer (D).

The correct answer is (A).

Guess what? The technique can also work on some Problem Solving problems. Try it out on the following GMATPrep problem, then join me next week to discuss the answer:

* “For which of the following functions *f* is *f*(*x*) = *f*(1 – *x*) for all *x*?

“(A) *f*(*x*) = 1 – *x*

“(B) *f*(*x*) = 1 – *x*^{2}

“(C) *f*(*x*) = *x*^{2} – (1 – *x*)^{2}

“(D) *f*(*x*) = *x*^{2}(1 – *x*)^{2}

“(E)

### Key Takeaways: Test Cases on Data Sufficiency

(1) When DS asks you a “theory” question, test cases. Theory questions allow multiple possible scenarios, or cases. Your goal is to see whether the given information provides a consistent answer.

(2) Specifically, try to disprove the statement: if you can find one Yes and one No answer, then you’re done with that statement. You know it’s insufficient. If you keep trying different kinds of numbers but getting the same answer, see whether you can think through the theory to prove to yourself that the statement really does always work. (If you can’t, but the numbers you try keep giving you one consistent answer, just go ahead and assume that the statement is sufficient. If you’ve made a mistake, you can learn from it later.)

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

*Studying for the GMAT? Take our free GMAT practice exam or sign up for a free GMAT trial class running all the time near you, or online. And, be sure to find us on Facebook and Google+,LinkedIn, and follow us on Twitter!*

### The Last 14 Days before your GMAT, Part 2: Review

As we discussed in the first half of this series, Building Your Game Plan, during the last 7 to 14 days before you take the real test, your entire study focus changes. In this article, we’re going to discuss the second half of this process: how to review. (If you haven’t already read the first half, do so before you continue with this part.)

### What to Review

Part of the game planning process is determining your strengths and weaknesses. Map these against the frequency with which various topics or question types tend to be tested on the real exam. You want to spend the bulk of your time reviewing the material that is most likely to appear on the test.

If an infrequently-tested area is also a weakness (I’m looking at you, combinatorics), drop it entirely. If you get an easier one on the test, try it for up to 2 minutes. If you get a hard one, call that one of your freebies: guess quickly and use that time elsewhere.

If you’re not sure how frequently a particular type of content or question appears on the exam, ask on the forums. I’m not going to provide a list in this article because these frequencies can change over time; I don’t want people reading this in future to be misled when things do change. The General GMAT Strategy folder in our own forums has a bookmarked thread—it always sits at the top—that discusses this very issue. If I’ve commented on the topic or question type mix within (at least) the past 6 months, then you’re up to date.

### How to Review

How you review is going to vary somewhat depending upon whether you’re reviewing a strength or a weakness. You do NOT want to do the same kind of review for everything, but you DO want to review both strengths and weaknesses. I’m going to make a distinction between the following categories:

*Easier-for-you:* you find the question fairly straightforward and you expect to answer it correctly without needing extra time, though you may sometimes make a careless mistake.

*Harder-for-you:* this question is more of a struggle, though you still will answer some of these correctly.

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### The Last 14 Days before your GMAT, Part 1: Building Your Game Plan

What’s the optimal way to spend your last 14 days before the real test? Several students have asked me this question recently, so that’s what we’re going to discuss today! There are two levels to this discussion: building a Game Plan and how to Review. We’ll discuss the former topic in the first half of this article and the latter in the second half.

### What is a Game Plan?

For the past several months, you’ve been focused on lifting your score. During the last two weeks before your test, your focus needs to shift: your skills are what they are and your score is what it is. These things are not going to change an enormous amount in the last two weeks.

What could happen, though, is that your score actually *drops* on test day because you spent the last couple of weeks trying to build up a few weaknesses and you forgot a bunch of strategies that you last reviewed 5 weeks ago. You can tweak some weaknesses in the final two weeks, but now you need to focus on the big picture.

(If your immediate reaction to this is, “But my score is nowhere near where I want it to be!!” then be smart and postpone your test. You’re not going to have a huge score increase in just 2 weeks.)

Your Game Plan will help you to make certain decisions quickly during the test. When is it a good idea to spend an extra 20 or 30 seconds on a problem? When should you decide to make an educated guess? When should you cut yourself off completely, guess immediately, and move on? What should you do if you find yourself ahead or behind on your timing? (We’ll discuss the answers to these questions later in the article.)

Your Game Plan will also help you to prioritize your review based upon your strengths and weaknesses. You’re going to review your major strategies in all areas, the major content you need to know (don’t try to cram everything into your brain; review the stuff that shows up the most!), your pacing, your educated guessing strategies, and so on. As you do that, the data you gather will help you to tweak your game plan further.

### Building Your Game Plan

Your Game Plan is a dynamic thing. You perfect it a little bit more every few days as you gather more data and continue to review.

*What Does My Gut Say?*

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