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 – x2|
|(C)||f(x) = x2 – (1 – x)2|
|(D)||f(x) = x2(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.
Lately, we’ve been talking about how to decide which test to take, as well as what to do if you decide to stick with the GMAT. What if you decide to switch from the GMAT to the GRE? That’s what we’ll tackle today! (Next time, we’ll talk about what to do if you want to switch from the GRE to the GMAT.)
How do I study?
The overall way that you want to study doesn’t actually change that much; rather, you’ll just need to change what you are studying, as discussed later in this article.
First, you’ll need to determine whether the way that you’ve already been studying is actually the optimal way. If not, then you’ll need to make some changes, regardless of whether you stick with the GMAT or switch to the GRE.
The GMAT and the GRE are both executive reasoning tests; that is, the test makers want to know how you think and make decisions. You of course need to know content (certain facts, rules, formulas) in order to do well on either test, but that level of study is not enough; you also need to lift yourself to a second level of understanding that allows you to think your way through these sometimes bizarrely-worded problems as effectively and efficiently as possible.
Follow the two links I put in the last paragraph. Take some time to just think about the concepts presented there. Has this been your approach to studying so far? If so, great. Keep thinking and working in that way.
If not, however, recognize that you’re going to need to start studying with this new mindset, regardless of whether you take the GMAT or the GRE.
What are my strengths and weaknesses?
Any time you’re developing or revising a study plan, you’ll want to put together a solid analysis of your strengths and weaknesses.
If you have been studying for the GMAT for a while, then you should have some practice CAT data. (If not, or if it has been more than 6 weeks since you last took a CAT, then you’ll need to take one to get the data. Make sure to take the test under official conditions, including the essay and IR sections, length of breaks, and so on.)
So you’ve taken the test and you aren’t entirely happy with your score. How do you decide whether to re-take the test?
It might be the case that your score is close to what you wanted, but not quite all the way there. Alternatively, you may be trying to decide whether to stick with the GMAT or switch to the GRE (and, if so, I recommend you follow that link I just inserted).
If you already know that you do want to stick with the GMAT, read on.
Should I re-take?
There are two main reasons someone might want to go for a higher score. The most common is that you think a better score will improve your chances of getting into business school or of obtaining certain internships once in school. Some people also feel that achieving a certain score is a personal goal and they want to meet that challenge.
If you’re trying to gauge whether a better score will make a big difference, start researching. What’s the average or median score for last year’s incoming class at your preferred schools? (Look at whatever data the school publishes—different schools might publish data in different forms.) Are you in range? Are you strong? If you are already above the average or median at that school, then adding 30 points might not make as big a difference as, say, earning a promotion at work.
Check GPA statistics as well. You have a little leeway for your GMAT score to be lower if your GPA is higher than the average for admitted students; if your GPA is lower, however, then it would be better to have an above-average GMAT. (Also, all of this just means that you have a chance, not that you’ll definitely get in. These are only two of many parts to your application!)
Most business schools now accept both the GMAT and the GRE, so which one should you take? I’ve written on the topic before but it’s been nearly a year and I’ve got some updates.
The conventional wisdom has been that the math is easier on the GRE. Though many schools do accept the GRE, rumors abound that students who take this test are at a bit of a disadvantage because they are expected to do better on the (easier) quant section. Anecdotally, we have heard a few admissions officers admit that they do think about this (strictly off the record, of course). Most admissions officers, though, have said this doesn’t matter to them at all, including several officers at the top 10 schools.
So we’ve come up with a series of decisions to help you make this choice. The first three questions are “deal-breakers”—that is, a certain answer will point you definitively to a specific test (the GMAT, as it happens). The fourth question is…murkier. We’ll address that in a little bit.
#1: Do all of “your” schools accept the GRE?
This one is obvious. All business schools (that ask for a standardized test score) accept the GMAT. Most—but not all—accept the GRE. If you want to apply to any schools that require the GMAT, such as London Business School MBA (at the time of this publication), then you’ll be taking the GMAT.
#2: Do any of “your” schools prefer the GMAT?
Most schools that accept both tests don’t express a preference between the two. Some schools, though, do say that the prefer the test. They publish this preference right on their web site, so go look up all of your schools and see what they say about the GMAT / GRE requirement for admissions.
As of the date of this article, Columbia, Haas (Berkeley) and Anderson (UCLA) all state that they prefer the GMAT, even though they do accept the GRE. If you want to apply to one of these schools, I recommend that you take the GMAT. (Note: these aren’t the only three schools that prefer the GMAT; I just picked out the three most well-known ones that do. You still need to research your schools!)
#3: Do you want to go into banking or management consulting after b-school?
The major banks and consulting firms ask for GMAT scores when you apply. (Some of them even ask for undergraduate GPA and SAT scores. I think that data is irrelevant after someone has a b-school GPA and GMAT scores but I’m not the one making the hiring decisions!)
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?
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.
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.
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?
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.
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?
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?
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.
So you’ve just taken a practice test. Chances are, you didn’t get a perfect 800. (If you did, stop studying and come work for us!). You probably didn’t even get a score that you like yet. That doesn’t mean that you’ve failed, though. In fact, you’ve barely started, because…
REVIEWING IS THE MOST IMPORTANT PART OF THE PROCESS!!
Most people take tests, then look at the score, then click on the explanations to the ones they got wrong. Their review process takes about 15 minutes, and just involves “oh, I did that wrong. Oh, that’s the right answer.” This kind of review process teaches you next to nothing about how to do better on the next one.
So here’s what you need to do. The test gave you a score from 200-800 on accuracy, but you need to give yourself your own score on your review process. Here’s how it works…
For every single question – not just the ones you got wrong! – you should be going back and re-solving. Take yourself through this checklist for quant problems:
1) Did I fully understand the concept and the rules behind it? +1
Give yourself a point if you could tell that a question was asking about DIVISIBILITY, or understood the RATE x TIME = DISTANCE relationship.
2) Did I understand what the question was asking for? +1
Did you rephrase DS questions to pinpoint what they were really asking for? Did you notice that it asked for “Amy’s age in 5 years,” and wrote down A + 5 instead of just A? Did you understand what it means when they ask for “x in terms of y and z”?
3) Did you solve it correctly? up to +5
Give yourself up to 5 points if you solved correctly the first time and got the right answer. Subtract a point or two if you took longer than you should have, or made a mistake before ultimately correcting it. Only give yourself +1 for a random lucky guess and +2 for an educated guess.
4) … or if you didn’t solve correctly, did you make a good decision to skip? +2
You’re not going to be able to solve every question on the GMAT, because you’re always going to run into questions that are above your ability level – that’s how the test is designed! So you should pat yourself on the back whenever you recognize that a question is too hard to solve, and you make the decision not to attempt it. Lock in an educated guess and save that extra time for a problem that is doable for you.