Recently, we took a look at a story problem dealing with ratios, and I finished up by giving you a second problem to test your skills. How did you do?
If you haven’t already, try the GMATPrep® problem below and then we’ll talk about it. Give yourself about 2 minutes. Go! Read more
How are you with story problems? Most math concepts can be presented in story form on the test and the GMAT test writers do like to get wordy with us. You’ve got a double task: you have to translate the words into math and then you still have to do the math! How can we get through these as efficiently as possible?
Try the GMATPrep® problem below and then we’ll talk about it. Give yourself about 2 minutes. Go!
Last time, I introduced you to a set of principles that tie together everything we need to do on the GMAT.
If you haven’t already read that article, go ahead and do so now.
Here’s our framework again:
Today, we’re going to try this out on a Data Sufficiency problem.
Try this DS problem from the GMATPrep® free exams.
I am super excited to announce a new edition of our GMAT Advanced Quant Strategy Guide! We worked hard on this book all of last year (yes, it takes a long time make a book!) and we hope that you find it to be a valuable addition to your GMAT preparation.
What is the Advanced Quant guide?
We created the Advanced Quant (AQ) guide a few years ago for people who want to get a top score (50 or 51) on the quant section of the GMAT.
Here’s the interesting thing: it doesn’t teach you a bunch of really hard math concepts. We teach all of those concepts in our five regular strategy guides (Algebra, Geometry, Word Problems, Number Properties, and Fractions, Decimals, & Percents). Instead, the AQ guide teaches you the next level of GMAT study: how to think your way through really hard quant problems.
What’s new in this edition?
A bunch of things! First, there are more than 50 brand-new, extremely hard problems. We actually removed some old ones that we thought were a bit too easy and replaced them with harder problems.
But that’s not all. Since the entire point of this book is how to solve better, we’ve updated some solutions to existing problems because we’ve discovered an even more efficient or effective way to solve.
We’ve also introduced a new organization method for working your way systematically through any quant problem. We’ve added or expanded lessons on test-taking strategies, such as testing cases on both problem solving and data sufficiency problems.
One student, who has already used the old version of AQ, asked whether we would provide a list indicating which questions are the new ones. I told him no. Not because I’m lazy or I don’t care, but because you don’t need such a list! If you’ve already tried the first edition and want to try this one, too, just start going through the book. If you hit a problem you remember, feel free to skip it. (Although maybe this is a chance to see if you really do remember what to do…and remember that we may offer an updated solution that you haven’t seen before.) If you hit a problem you don’t remember, then it doesn’t matter whether it’s old or new. It’s new to you right at this moment!
Who should use the AQ Strategy Guide?
First, you should have mastered most (if not all) of the material in our five main quant Strategy Guides. As I mentioned earlier, we do not actually teach you that math in this guide. We assume that you already know it.
As a general rule, we recommend that people avoid using this book until they’ve gotten to a score of at least 47 on a practice CAT. (Seriously. We say so right in the first chapter of the book!) I might let that slide a bit for certain students, but someone scoring below 45 likely does not have the underlying content knowledge needed to make the best use of the Advanced Quant lessons.
Note that, from an admissions standpoint, you may not necessarily need to score higher than 47. The scoring scale tops out at 51, so 47 is already quite high. Do a little research to see what you may need for the specific schools to which you plan to apply.
All right, that’s all I’ve got for you today. I’d love to hear what you think about the book. Which problem is your favorite? And which one do you think is the absolute hardest, most evil thing we could have given you? Let us know in the comments!
Check out our store to learn more about the new GMAT Advanced Quant Strategy Guide.
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.
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.
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.
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.
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.