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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We’re going to kill two birds with one stone in this week’s article.

Inference questions pop up on both Critical Reasoning (CR) and Reading Comprehension (RC), so you definitely want to master these. Good news: the kind of thinking the test-writers want is the same for both question types. Learn how to do Inference questions on one type and you’ll know what you need to do for the other!

That’s actually only one bird. Here’s the second: both CR and RC can give you science-based text, and that science-y text can get pretty confusing. How can you avoid getting sucked into the technical detail, yet still be able to answer the question asked? Read on.

Try this GMATPrep® CR problem out (it’s from the free practice tests) and then we’ll talk about it. Give yourself about 2 minutes (though it’s okay to stretch to 2.5 minutes on a CR as long as you are making progress.)

“Increases in the level of high-density lipoprotein (HDL) in the human bloodstream lower bloodstream cholesterol levels by increasing the body’s capacity to rid itself of excess cholesterol. Levels of HDL in the bloodstream of some individuals are significantly increased by a program of regular exercise and weight reduction.

“Which of the following can be correctly inferred from the statements above?

“(A) Individuals who are underweight do not run any risk of developing high levels of cholesterol in the bloodstream.

“(B) Individuals who do not exercise regularly have a high risk of developing high levels of cholesterol in the bloodstream late in life.

“(C) Exercise and weight reduction are the most effective methods of lowering bloodstream cholesterol levels in humans.

“(D) A program of regular exercise and weight reduction lowers cholesterol levels in the bloodstream of some individuals.

“(E) Only regular exercise is necessary to decrease cholesterol levels in the bloodstream of individuals of average weight.”

Got an answer? (If not, pick one anyway. Pretend it’s the real test and just make a guess.) Before we dive into the solution, let’s talk a little bit about what Inference questions are asking us to do.

Inference questions are sometimes also called Draw a Conclusion questions. I don’t like that title, though, because it can be misleading. Think about a typical CR argument: they usually include a conclusion that is…well…not a solid conclusion. There are holes in the argument, and then they ask you to Strengthen it or Weaken it or something like that.
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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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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 + 0.1x = 1.1x. The largest district could be 1.1x but can’t be any larger than that.

Since you need to maximize the 10 remaining districts, set all 10 districts equal to 1.1x. As a result, there are (1.1x)(10) = 11x 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:

11x + x = 132,000

12x = 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.

Welcome to the third installment of our Core Sentence series. In part 1, we began learning how to strip an SC sentence (or any sentence!) down to the core sentence structure. In part 2, we took a look at a compound sentence structure.

Today, we’re going to look at yet another interesting sentence structure that is commonly used on the GMAT.

Try out this GMATPrep® problem from the free exams. (Note: as in the previous installments, I’m going to discuss aspects of our SC Process; if you haven’t learned it already, read about it before doing this problem.)

* “Many financial experts believe that policy makers at the Federal Reserve, now viewing the economy as balanced between moderate growth and low inflation, are almost certain to leave interest rates unchanged for the foreseeable future.

“(A) Reserve, now viewing the economy as balanced between moderate growth and low inflation, are

“(B) Reserve, now viewing the economy to be balanced between that of moderate growth and low inflation and are

“(C) Reserve who, now viewing the economy as balanced between moderate growth and low inflation, are

“(D) Reserve, who now view the economy to be balanced between that of moderate growth and low inflation, will be

“(E) Reserve, which now views the economy to be balanced between moderate growth and low inflation, is”

The First Glance didn’t tell me a lot on this one. In each case, there appears to be some kind of modifier going on, signaled either by the who / which language or by the comma, but I don’t have a good idea of what’s being tested. Time to read the sentence.

I don’t know about you, but the original sentence really doesn’t sound good to me. The difficulty, though, is that I don’t know exactly why. I just find myself thinking, “Ugh, I wouldn’t say it that way.”

Specifically, I don’t like the “now viewing” after the comma…but when I examined it a second time, I couldn’t find an actual error. That’s a good clue to me that I need to leave the answer in; they’re just trying to fool my ear (and almost succeeding!).

Because I’m not certain what to examine and because I know that there may be something going on with modifiers, I’m going to strip the original sentence down to the core:

Here’s the core:

Many experts believe that policy makers are almost certain to leave interest rates unchanged.

This sentence uses what we call a “Subject-Verb-THAT” structure. When you see the word that immediately after a verb, expect another subject and verb (and possibly object) to come after. The full core will be Subject-Verb-THAT-Subject-Verb(-Object).

Back to the problem: notice where the underline falls. The Subject-Verb-THAT-Subject part is not underlined, but the second verb is, and it’s the last underlined word. Check the core sentence with the different options in the answers:

Many experts believe that policy makers __________ almost certain to leave rates unchanged.

(A) Many experts believe that policy makers are almost certain to leave rates unchanged.

(B) Many experts believe that policy makers and are almost certain to leave rates unchanged.

(C) Many experts believe that policy makers.

(D) Many experts believe that policy makers will be almost certain to leave rates unchanged.

(E) Many experts believe that policy makers is almost certain to leave rates unchanged.

Excellent! First, answer (E) is wrong because it uses a singular verb to match with the plural policy makers.

Next, notice that answer (B) tosses the conjunction and into the mix. A sentence can have two verbs, in which case you could connect them with an and, but this answer just tosses in a random and between the subject and the verb. Answer (B) is also incorrect.

Answer (C) is tricky! At first, it might look like the core is the same as answer (A)’s core. It’s not. Notice the lack of a comma before the word who. Take a look at this example:

The cat thought that the dog who lived next door was really annoying.

What’s the core sentence here? This still has a subject-verb-THAT-subject-verb(-object) set-up. It also has a modifier that contains its own verb—but this verb is not part of the core sentence:

The cat thought that the dog [who lived next door] was really annoying.

Answer (C) has this same structure:

Many experts believe that policy makers [who are almost certain to leave rates unchanged]…

Where’s the main verb that goes with policy makers? It isn’t there at all. Answer (C) is a sentence fragment.

We’re down to answers (A) and (D). Both cores are solid, so we’ll have to dig a little deeper. So far, we’ve been ignoring the modifier in the middle of the sentence. Let’s take a look; compare the two answers directly:

“(A) Reserve, now viewing the economy as balanced between moderate growth and low inflation, are

“(D) Reserve, who now view the economy to be balanced between that of moderate growth and low inflation, will be”

Probably the most obvious difference is are vs. will be. I don’t like this one though because I think either tense can logically finish the sentence. I’m going to look for something else.

There are two other big differences. First, there’s an idiom. Is it view as or view to be? If you’re not sure, there’s also a comparison issue. Is the economy balanced between growth and inflation? Or between that of growth and inflation?

The that of structure should be referring to another noun somewhere else: She likes her brother’s house more than she likes that of her sister. In this case, that of refers to house.

What does that of refer to in answer (D)?

I’m not really sure. The economy? The Federal Reserve? These don’t make sense. The two things that are balanced are, in fact, the growth and the inflation; that of is unnecessary. Answer (D) is incorrect.

The correct answer is (A).

The correct idiom is view as, so answers (B), (D), and (E) are all incorrect based on the idiom.

Key Takeaways: Strip the sentence to the Core

(1) When you see the word that immediately following a verb, then you have a Subject-Verb-THAT-Subject-Verb(-Object) structure. Check the core sentence to make sure that all of the necessary pieces are present. Also make sure, as always, that the subjects and verbs match.

(2) If you still have two or more answers left after dealing with the core sentence, then check any modifiers. The two main modifier issues are bad placement (which makes them seem to be pointing to the wrong thing) or meaning issues. In this case, the modifier tossed in a couple of extraneous words that messed up the meaning of the between X and Y idiom.

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

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.

This is the original version of a piece that has since been updated. See Stacey’s latest tips on maximizing the last two weeks before your GMAT.

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 Read more

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.

This is the original version of a piece that has since been updated. See Stacey’s latest tips on maximizing the last two weeks before your GMAT. 

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?

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Consistently and overwhelmingly, the evidence showed that experts are always made, not born. (“The Making of an Expert” by K. Anders Ericsson, Michael J. Prietula, and Edward T. Cokely, Harvard Business Review, July-August 2007)

Standardized test-taking is a skill–like winning a chess game, swinging a golf club, or playing a Bach concerto. And to master a skill, you need high-quality practice. Of course, the more content you know the better, but no matter how much you study for the GMAT, you won’t improve without practice. (I tried reading a book about snowboarding before my first time on the slopes, with predictably laughable results.) According to the scientific research, the most efficient and most effective kind of practice-the way Tiger Woods become the golfer he is today–is called “Deliberate Practice.”

If you spend time reading motivational blogs such as LifeHacker you’ll see many articles about “Deliberate Practice.” You may have even heard of whole books–Talent is Overrated by Geoffrey Colvin or Outliers by Malcolm Gladwell–about exceptional individuals such as Bobby Fischer and Tiger Woods. All those blogs, as well as Colvin and Gladwell, base their ideas on the research of K. Anders Ericsson, a Professor of Psychology at Florida State University and probably the world’s number-one expert on expertise. His good-news thesis can be summed up as follows:

New research shows that outstanding performance is the product of years of deliberate practice and coaching, not of any innate talent or skill. (Ericsson et al., “The Making of an Expert”)

First of all, relax. You may have heard about Ericsson’s 10,000 hour rule. Apparently, it takes about 10 years and 10,000 hours of “deliberate practice” to achieve true mastery. Yes, Tiger Woods, Bobby Fischer, Mozart, and other one-in-a-million people needed 10,000 hours to get to where they are. Luckily, the GMAT is much less difficult to master than golf, chess, or composition. Also, you’re not looking to be one in a million–at best 1 in 100 (a score of 760-800)–so you don’t need 10,000 hours. Maybe a few hundred hours, depending on how much you want to improve.

But what is “Deliberate Practice?”  And how do you apply it to the GMAT? At the end of this article, I’ve given you a few links, but to save you time, I’ve pulled my favorite Ericsson quotes and applied them to the GMAT:

1) Get motivated.

The most cited condition concerns the subjects’ motivation to attend to the task and exert effort to improve their performance. (“The Role of Deliberate Practice in the Acquisition of Expert Performance” by K. Anders Ericsson, Ralf Th. Krampe, and Clemens Tesch-Romer. Psychological Review. 1993, Vol. 100. No. 3)

Moving outside your traditional comfort zone of achievement requires substantial motivation and sacrifice, but it’s a necessary discipline. (Ericsson et al., “The Making of an Expert”)

If you’re reading this, you want a higher GMAT score. You’re already motivated. If you need more motivation, research schools. Take a diagnostic test and see how far you are from your dream school’s median. After that, the best way to get motivated is to sign up for the real GMAT a few months from now. (How many people don’t lose weight until they schedule the wedding or high school reunion?) Read more

Imagine two students sit down to study GMAT questions together. The first takes out 100 addition questions and gets all of them right. The other takes out 100 of the most-difficult, 800-level GMAT questions one can find, and gets all of them wrong. Who benefits more from this type of studying? It’s an absurd thought experiment since it’s fairly obvious that neither of these students is benefitting much from their study method. But over my years of teaching the GMAT, I’ve seen far too many students who fit too closely into one of these two camps. Students who are great at quant but not at verbal, yet spend all of their time doing quant questions because they are “more fun”. Other students are determined to score 750 and spend all of their time and effort doing as many 700-800 level questions as they can find, not seeing an improvement, and thinking that the solution is to see more 700-800 level questions. This isn’t some profound discovery, but too many students miss this critical point:

You get better at the GMAT by identifying a weakness, learning a better/faster method to attack that weakness, and practicing that method until it becomes habit. Repeat.

Note that this doesn’t mean that you have to do 50 rate questions and by question 50, you’ll be a master at determining the train schedule between two different towns. Nor do you need to do every question in every GMAT-related book you can get your hands on. If you’ve been to a Manhattan class, you’ve seen first-hand that our instructors’ goal is not to do as many questions as we can cram into a class. There are some topics in class where we only look at 4-5 questions, but we spend an hour breaking down the methods, key words, traps, and wrong answer choices that will be similar to the methods, key words, traps, and wrong answer choices that students will one day see on the real test. The goal is never to see why Answer Choice E is a trap answer. It’s to see why Answer Choice E fits into a certain category of trap answers and learn how to avoid that category of trap answers come test day.

So how does this relate to your own studying? Let’s talk about what a productive 1-hour study session might look like by examining what many of my own study sessions looked like while I was studying for my GMAT.

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I know what Statement 2 is telling me; it’s saying ˜Become a carpenter!’

Why is this question here? Why am I here?  When’s the civil service exam?  Garbage men still have a union. . .

Have you lived that movie?  Paranoia is only human and the old saying is true: Just because you’re paranoid doesn’t mean they’re not out to get you.  Paranoia is a primal reaction, developed to help protect humans from animals with sharp, pointy teeth.  Unfortunately, it is not helpful when one is facing questions with sharp, pointy teeth.  Even though the GMAT is out to get you.  Failing to control your paranoia is a hidden reason for underperforming on the actual exam.

On this blog, I and others have discussed many factors crucial for success: foundation skills, strategies, timing, precision, and so forth.  And it’s like I say about L.A.—everything you ever read [here] about it is true.  However, after honing these skills, after achieving mastery, too many test takers succumb to their paranoia and thus revert when taking the actual exam, especially for the first time.  Even 99^th percentile skills will crumble if undermined by irrational panic and the results will not be gratifying.  (Have you ever watched the Chicago Cubs play a post season series?)   To succeed, folks must understand the difference between dispassionate, objective analysis—I’ve never gotten a combinatrics question right in life, why do I think I’ll have a divine inspiration today?—and irrelevant fear—They’re going to tattoo a scarlet L on my forehead.  Just as folks plan question and timing strategies, they must develop tools to banish their internally generated negative visualizations.

How do you tell the difference?  Objective analysis responds to the stimuli on the monitor.  Paranoia is a response to internal doubts.  (Notice how this is parallel to the nature of the exam—search for the answer on the screen, not in the opinions in your head.)  Sometimes, after you’ve read a question twice (everyone has a sinking feeling the first time), you hear yourself singing, I’ve got the ˜I don’t know where I’m going but I’m going nowhere in a hurry’ blues.  That’s the truth, not paranoia.  Bail out.  As one of my acting coaches used to say, Only schizophrenics don’t react to the reality around them.  Conversely, paranoia is when your thoughts of impending disaster revolve around your supposed shortcomings rather than the material on the screen.  As I’ve said before, if while taking the exam you find yourself thinking about how big a dumb ass you are, check the question—if it doesn’t read, Which of the following best describes how big a dumb ass you are?, you’re thinking about the wrong thing.  That is paranoia.  No kidding—you knew that.

Well then, why do people recognize the difference between analysis and paranoia but still succumb to the latter?  Because they try to do the impossible.  They try not to have thoughts of failure.  That’s impossible—you can’t override human nature.  I have feelings of paranoia, even though I’ve always scored in the 99^th percentile.  I still have them”even though I don’t really care about my score anymore.  Instead, you have to recognize irrationality in yourself and laugh it off.  I say to myself, Save some of that craziness for menopause.  Then I giggle, read the question again, and really listen to the words.  And if I still don’t get it, I say, Screw them if they can’t take a joke.  And bail out.

Maybe some of you can’t make jokes to yourself during the exam because you’re worried about your entire future.  That’s part of the problem—if a chunk (or all) of your mind is thinking about things other than the words on the monitor, it will lower your score.  It’s the difference between worrying about being the hero or the goat and just seeing the ball and hitting the ball.  Feelings of failure while taking the exam are like stage fright.  That’s what stage fright is—standing up there thinking you look like an idiot.  You say, No, it’s much different—they give me a piece of paper that says I’m an idiot.  No.  Really.  It’s the same.  So, I’ve got another suggestion for you, if you didn’t like the first one.

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