I. Roman numeral Quant problems aren’t a whole lot of fun.
II. A lot of my students choose to skip them entirely, which is much smarter than wasting five minutes wondering what to do!
III. However, it’s possible to turn this rare and tricky problem type into an opportunity.
Read on, and learn why many GMAT high-scorers love Roman numeral problems. Read more
There are really only a dozen different Critical Reasoning problems in the Official Guide to the GMAT. The test writers recycle the same basic argument structures over and over, and they use the same right answers over and over, too. Even though the topics change — an argument might be about school funding the first time you see it, and industrial efficiency the next — you can sometimes recognize the underlying structure, outsmart the test, and earn some well-deserved points on the Verbal section. Read more
Most second-round deadlines are in early January, so around now, a lot of people are asking me how to eke out the last 30 to 80 points they need to reach their goal.
Let’s talk about what to do to try to lift your score that last bit in the final 2 months of your study.
Is this article for me?
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.
This is going to be a short post. It will also possibly have the biggest impact on your study of anything you do all day (or all month!).
When people ramp up to study for the GMAT, they typically find the time to study by cutting down on other activities—no more Thursday night happy hour with the gang or Sunday brunch with the family until the test is over.
There are two activities, though, that you should never cut—and, unfortunately, I talk to students every day who do cut these two activities. I hear this so much that I abandoned what I was going to cover today and wrote this instead. We’re not going to cover any problems or discuss specific test strategies in this article. We’re going to discuss something infinitely more important!
#1: You must get a full night’s sleep
Period. Never cut your sleep in order to study for this test. NEVER.
Your brain does not work as well when trying to function on less sleep than it needs. You know this already. Think back to those times that you pulled an all-nighter to study for a final or get a client presentation out the door. You may have felt as though you were flying high in the moment, adrenaline coursing through your veins. Afterwards, though, your brain felt fuzzy and slow. Worse, you don’t really have great memories of exactly what you did—maybe you did okay on the test that morning, but afterwards, it was as though you’d never studied the material at all.
There are two broad (and very negative) symptoms of this mental fatigue that you need to avoid when studying for the GMAT (and doing other mentally-taxing things in life). First, when you are mentally fatigued, you can’t function as well as normal in the moment. You’re going to make more careless mistakes and you’re just going to think more slowly and painfully than usual.
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
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.