Math Workout for the GRE, 3rd Edition (2013)

Chapter 1. Introduction

ADVICE FOR THE FAINT OF HEART

Welcome to The Princeton Review’s Math Workout for the GRE, the one-stop shop for all of the mathematical knowledge and practice you’ll need to effectively tackle the Math section of the GRE.

You’ve bought this book, which means you may be one of many grad school candidates who are approaching the math, or quantitative, portion of the GRE with a little bit of trepidation. This might be for any of several reasons, including the following:

· You come in contact with the word “variable” only when it’s used to describe the weather.

· Your first thought about Pythagoras is that he might have been a character in The Lord of the Rings.

· You regard “standard deviation” as more of a psychological problem than a mathematical one.

If any of the above pertain to you, you’re definitely not alone.

But don’t worry, that’s what this book is all about. Its two main objectives are (1) to give you an overview of all of the math concepts you could see on the GRE, and (2) to give you simple strategies for handling even the most complex math you could encounter on test day.

WHAT KIND OF MATH DOES THE GRE ROUTINELY TEST?

The good news is that the GRE’s math sections don’t test anything that you learned after your sophomore year of high school, so the concepts aren’t extremely advanced.

The bad news is that the GRE’s math sections don’t test anything that you learned after your sophomore year of high school, so it may have been a long time since you studied them.

That’s largely why this book was written: to help you build up an impressive canon of math knowledge that will (1) help you score your best on the quantitative portion of the GRE, and (2) set you up to knock ’em dead at your next cocktail party.

The GRE supposedly was written so that graduate schools might get a better sense of an applicant’s ability to work in a post-graduate setting—a goal that is lofty and unrealistic at best. The test doesn’t even measure how intelligent you are; if you take a test-prep course and your score improves, does that mean you’re any smarter? Nope. Yet you can improve your score just by learning about what to expect on the GRE.

All the GRE really tests is how well you take the GRE.

Succeeding on the quantitative portion of the GRE—or any standardized math test, for that matter—is as much about relearning math concepts as it is about modifying the way you think. There are several very important skills to cultivate when you’re preparing to take the GRE, and each of them is attainable with the right guidance, a strong work ethic, and a healthy dose of optimism.

We’ll discuss the math basics you’ll need for the GRE, but if you need a quick reference, consult the glossary at the back of the book.

The Layout of the Test

Let’s talk about the different sections of the GRE. The GRE contains five scored sections:

· One 60-minute Analytical Writing section, which contains two essay questions

· Two 30-minute Verbal sections, which contain approximately 20 questions each

· Two 30-minute Math sections, which contain approximately 20 questions each

The first section will always be the Analytical Writing section, followed by the Math and Verbal sections, which can appear in any order. All of the Verbal questions are multiple choice. The Math questions are mostly multiple choice with some numeric entry questions, which require typing in an answer.

You will be able to see your Verbal and Math scores immediately upon completion of the test, but you will have to wait about two weeks before your Analytical Writing section is scored.

Scores are given on a scale from 130 to 170, in 1-point increments. The questions within each section are always worth the same amount of points. So the easy questions in a section are just as important to get right as the hard questions in a section.

Once you’ve completed one Math section, the GRE will use your score on that section to determine which questions to give you in the next Math section. The same applies for the two Verbal sections. (This doesn’t really affect how you will approach the test, so don’t worry about it too much.)

You will get a 1-minute break—enough time to close your eyes and catch a breath—between each section. You will also get a full 10-minute break after the first multiple choice section. Be sure to use it to visit the bathroom, take a drink of water, refresh your mind, and get ready for the rest of the exam.

Experimental Section

Here’s where ETS, the maker of the test, starts to get mean. In addition to the five scored sections listed above (one Analytical Writing, two Math, two Verbal), you may also have an unscored experimental section. This section is almost always a Math or Verbal section. It will look exactly like the other Math or Verbal sections, but it won’t count at all toward your score. ETS administers the experimental section to gather data on questions before they appear on real GREs.

Thus, after your Analytical Writing section you will probably see five—not four—multiple-choice sections: either three Verbal and two Math, or two Verbal and three Math, depending on whether you get a Verbal or Math experimental section. These sections can come in any order. You will have no way of knowing which section is experimental, so you need to do your best on all of them. Don’t waste time worrying about which sections count and which section does not.

Here is how a typical GRE might look:

Remember, the Analytical Writing section will always be first, and it will never be experimental. In the above example, the two Verbal sections will be scored, but out of the three Math sections only two will be scored. One of the three is an experimental section, but we don’t know which one. Of course, on your GRE you might see three Verbal sections instead, meaning one of your Verbal sections is experimental, and they may come in any order. Be flexible, and you’ll be ready for the test no matter the order of the sections. In fact, the test makers may not even include an experimental section! If so, count your lucky stars that you didn’t have to waste your time on a meaningless section.

Research Section

At the end of the test, you may also have an unscored Research section. At the beginning of this section, you will be told that it is an unscored Research section, used only to help develop and test questions for the GRE. If you want to skip it, you have the option of skipping it. They normally offer some sort of financial incentive to induce people to take it, but by that point in the test you will probably be exhausted. Take it if you like, but also feel free to just go ahead and decline, get your scores, and go home.

MATH OVERVIEW

There are three main skills that we will emphasize throughout this book: keep your hand moving, take the easy test first, and be prepared to walk away. These are not necessarily what you would naturally do while taking a test, so you’ll have to force yourself to apply these skills as you work through the problems in this book and as you take practice tests. If you do, you’ll find that once you get to the real test your body and brain already know how to tackle each question, and you’ll be able to breathe a bit easier.

Most people assume that the Math section is about thinking. It’s not. It’s about doing. When you get stuck, resist the urge to sit and stare. Instead, get your hand moving.

Keep Your Hand Moving

You’ll get about six pieces of scratch paper to use for the test. Use them. Use all of them, and then get more. Don’t just use scratch paper for multiplying or doing long division. Use your scratch paper for every single part of the problem, from beginning to end.

You will not solve the problems in your head, and you will not solve them on the screen. You will solve them on the scratch paper. As soon as you click Next, and see a question, start writing.

We’ll go into exactly what to write for each question later in the book. For now, you should get into the habit of writing the following:

· Question Number: You may need to leave a question and come back to it. If so, you may have some calculations that you can reuse. If you’ve got the question number written down, you can easily see what work goes with which question.

· Answer Choices: For most questions, this will mean simply writing A B C D E vertically on the left side of your scratch paper. As you work each question, you may realize that certain answer choices are definitely wrong. In that case, cross them off on your scratch paper.

· Problem Set Up: On the left side of your paper, next to the answer choices, write down information as you read the problem. If it’s a geometry question, redraw the figure. If the question says that Bob has 142 oranges in his grove, and Sue has 219 oranges in her grove, then immediately write down “Bob = 142 oranges” and “Sue = 219 oranges.” Don’t keep anything in your head.

· Calculations: As you work through the problem, you will probably need to the use the onscreen calculator. As you do so, write down every single calculation on your piece of paper, off to the right side. Don’t enter in 216 × 3 ÷ 4 all at once. First, do 216 × 3 on the calculator, and write down the result, 648. Then, do 648 ÷ 4, and write down the result, 162. It’s easy to make small mistakes with the calculator if you’re not careful.

· A Horizontal Line: After each problem, draw a line to separate it from the next problem. This will keep your work organized, and prevent you from accidentally using numbers or information for one problem while solving another.

Your scratch paper could look something like this:

Take the Easy Test First

All questions within a given section are worth the same amount. Many people rush through the easy questions so they can spend more time on the hard questions. However, if easy questions are worth just as much as hard questions, why not focus just as much on them?

There are a certain number of questions on the GRE that you can easily answer correctly. As soon as you read through them, you know what they’re asking and how to get to the answer. Your job is to answer all of those questions first. Don’t rush through them, because you can’t afford to get these questions wrong. These are practically free points, as long as you’re paying attention. Use your scratch paper and read carefully.

Save the hard questions for later. You can always return to them, even if it’s just for a last-second guess. The goal with your first pass through any section is to get as many points as you can, without any mistakes. Once you’ve done that, you can use the time remaining to return to the other, harder questions. You’ll find that after a second look, some of the hard questions are easier than you initially thought. Go ahead and do those questions now. Some of the questions you thought were going to be hard are, in fact, hard questions. Leave those. You’ll come back with any time remaining and either work through them or eliminate answers and guess.

Easy questions are worth the same as hard questions. Work easy questions carefully, so you don’t get any wrong.

Be Prepared to Walk Away

At the top of the screen are buttons labeled Mark, Review, and Next. Any question you’re not sure about, click Mark, then click Next and move on. If you click on Review, you’ll see a screen like this:

Here you can see every question you haven’t answered, and every question you marked to come back to later. If you need to return to any question, you can click on that question on the review screen and you’ll be brought right to it.

Why is all this so important? Because you can always move on. If you read a question and you don’t immediately know what to do, move on. If a question seems particularly difficult, move on. If you start working through a question and realize you aren’t getting any closer to the answer, move on. If you work through a question and the answer you got isn’t among the answer choices, move on.

Your first pass is all of the easy questions. Your second pass is the harder questions, and these are the questions you’re most likely to misread. Once you’ve read a question one way, it’s hard to get your brain to read that question any other way. So if you’re not sure what the question is asking, if you realize you’re doing a lot more math than you normally do for GRE questions, or if you get an answer that isn’t one of the answer choices, then move on. Do a couple other questions, give your brain a chance to shift gears, and then come back to it. Don’t look back at your old scratch paper yet. Reread the question and take notes on your scratch paper as if it’s the first time you’ve seen this question.

Any time you stall while working on a question, move on. Do not let yourself get stuck. Sitting and staring blankly at the computer screen does not help. Move to another question, and keep your hand moving.

QUESTION TYPES

There are four types of math questions on the GRE. Once you know how these questions work, you’ll save yourself the time of rereading the instructions each time they appear. We’re going to show you a sample problem for each question type. Don’t worry if you don’t know how to solve these yet; these are here mostly for you to see the format for each question type.

Multiple Choice

You’ve seen these questions before. You’ve probably answered them for most of your life. Multiple-choice questions are questions which have five answer choices. You have to select one answer choice and then click Next.

The answer bubbles for these questions will always be round. Whenever a question has circular bubbles, you must select one and only one answer and then click Next to continue.

Get used to thinking of each answer choice as (A), (B), (C), (D), or (E). As soon as you see a multiple-choice question, write down A B C D E vertically on the left side of your scratch paper.

Question 1 of 4

If c is the greatest prime number less than 22, and d is the least prime number greater than 35, then c + d =

33

41

50

56

58

Here’s How to Crack It

Since this is a multiple-choice question, write down the question number and A B C D E on your scratch paper. The question says that c is the greatest prime number smaller than 22. Check each number less than 22, starting with 21. Is 21 prime? Nope. Neither is 20. 19 is prime, so 19 is the greatest prime number that is still less than 22. Write down c = 19. Now try to find d. It must be greater than 35, and prime, so it can’t be 36. The next number, 37, is prime, because the only numbers we can divide 37 by are 37 and 1, so write down d = 37. The question asks for c + d, which is 19 + 37 = 56, answer (D).

Quantitative Comparison

We’ll normally call these questions Quant Comp for short. These questions are variations of the basic multiple-choice question. You will be given Quantity A in one column and Quantity B in another column, and you must select one of four answers:

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

These answer choices are the same for every single Quantitative Comparison question. (A) means that Quantity A is always greater, (B) means that Quantity B is always greater, (C) means that the two quantities are always equal, and (D) means that we’re not sure: Sometimes Quantity A is bigger, sometimes Quantity B is bigger, or sometimes they’re the same.

Since these questions have round answer bubbles, you’ll select only one answer: (A), (B), (C), or (D). As soon as you see one of these questions, write down A B C D vertically on the left side of your scratch paper.

Before we do a sample question, there’s one important thing you should know about answer choice (D): It can never be the answer for straight calculation questions. For instance, if Quantity A is 10¹² and Quantity B is 5²⁴, then a calculator could solve that question, right? Whatever the answer is, it will always be one particular number for Quantity A, and one particular number for Quantity B. Sure, those numbers are super hard to find without the calculator, and we would have to use some clever tricks to actually find the answer, but the answer can’t be (D). If there are no variables, and the problem is simply about doing calculations, the answer can’t be (D) for a Quant Comp question, because the relationship can be determined, even if it may be a pain to determine it.

Question 2 of 4

Point E lies in square ABCD and the area of square ABCD is 100.

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Here’s How to Crack It

Since this is a Quant Comp question, write down A B C D vertically on the left side of your paper. Draw square ABCD to the right of the answer choices. The area of ABCD is 100, which means each side must be 10. Label the sides on your figure. The problem states that point E is somewhere within the square, but it doesn’t say exactly where. It could, for instance, be really close to point D, only 1 unit away. In that case, DE would definitely be smaller than 15, so cross off answers (A) and (C) on your scratch paper: Quantity A isn’t always greater, and the two quantities aren’t always equal. Now we have to find out if we can make Quantity A greater than Quantity B. What if we put point E all the way on the other side of the square? Go ahead and redraw the square, but now put E next to point B. How far away are E and D now? Well, since it’s a square, the distance from E to D makes a 45-45-90 triangle, which means that the distance is less than 10 ≈ 10 × 1.4 = 14. In that case, DE is still less than 15, so the answer is (B). No matter where we put point E, it is always less than 15 units away from point D. Your scratch paper should look something like this:

All That Apply

These are multiple-choice questions which will have any number of answer choices (generally three or five, but sometimes more), and you will have to select all the answers that apply to the question. The answer choices for these questions are always squares. The question will typically state to select all values or statements that apply.

Note that there’s no partial credit for these questions. You must choose every single answer that works, or you get no credit for that question. There will always be at least one answer for these questions, but there may be only one answer that works. Or two. Or three. Or all eight answer choices could work.

When you see these questions, write down letters (A, B, etc.) for each answer choice. As you work through the problem, put a check mark next to each answer choice that works, and cross off any answer choice that doesn’t work. Before you click Next, double check that you have selected all the answers you have written on your scratch paper, and none of the answers that you crossed off.

Question 3 of 4

If is an integer, which of the following could be the value of x ?

Indicate all such values.

8

10

36

64

112

432

Here’s How to Crack It

Since this is an All That Apply question with six answer choices, write down A B C D E F vertically on the left side of your paper. Notice that they’re asking about 12¹³? That’s way too big to enter into our calculator, so we’ll have to do something other than just brute force calculation. For the fraction to be an integer, x will have to be a factor of 12¹³. For instance, if x were 12, then the fraction would be an integer because the 12 in the denominator would cancel out with one of the 12s in the numerator. x could also be 2, because 2 is one of the prime factors of 12.

Let’s break 12 down into prime factors. Once we know the prime factors of 12¹³, then we’ll know that any answer choice that has some of those same prime factors is definitely a factor of 12¹³. The prime factors of 12 are 2 × 2 × 3 = 2² × 3. Therefore the prime factors of 12¹³ are (2² × 3)¹³ = 2²⁶ × 3¹³. So any number with up to 26 2s as prime factors will fit into 12¹³, as will any number with up to 13 3s as prime factors.

(A) is 8. The prime factors of 8 are 2 × 2 × 2. Since 12¹³ definitely contains 2³, put a check next to (A). (B) is 10, which is the same as 2 × 5. Although that 2 will fit into 12¹³, 5 will not. Since 12¹³ is only made up of 2s and 3s, that 5 has nothing to cancel. Cross off answer (B), because x cannot be 10; if it were, the fraction would not be an integer. (C) is 36, which has prime factors of 2² × 3³, so put a check mark next to it. (D) is 64, which is 8 × 8, which is 2³ × 2³, which is 2⁶. Put a check mark next to it. (E) is 112. Since 112 is even, start by dividing by 2: 112 = 2 × 56 = 2 × 8 × 7 = 2 × 2³ × 7 = 2⁴ × 7. Although the 2⁴ will divide evenly into 12¹³, that 7 will not, because 7 is not a prime factor of 12¹³. Cross off (E). The only one left is (F), which is 432. Break down 432 to its prime factors and you’ll get 2⁴ × 3³, which definitely fits into 12¹³. The answers are therefore (A), (C), (D), and (F).

Numeric Entry

These questions don’t give you any answer choices at all. Instead, you’ll be given a question and an empty box to type a number in. Your answer could be an integer, a decimal, positive, or negative. Never round your answer unless the question asks you to, or it’s a question that cannot have decimal answers (the number of children on a school bus, for instance).

The GRE will give you the correct units—they’ll be right there next to the box. So before you submit your answer, be sure that it uses the proper units. Be extra careful if problems involve dollars and cents, ounces and pounds, feet and inches, percents, and other common increments that have sub-increments.

Question 4 of 4

If Jamal is charged $236.30 to rent a car and that charge consists of a flat fee of $95 and charge of $0.075 for every tenth of a mile driven, how many miles did Jamal drive the rental car?

miles

Click on the answer box, then type in a number. Backspace to erase.

Here’s How to Crack It

Start at the beginning: If Jamal paid a total of $236.30 and $95 of that was the flat fee, then the cost of the mileage must have been $236.30 – $95.00, or $141.30. Now, here’s the crucial conversion: If the rental company charged $0.075 per tenth of a mile, then each mile cost $0.075 × 10, or $0.75. Divide $141.30 by $0.75, and your work is done. The answer is 188.4 miles.

If the answer will be in the form of a fraction, you will see two boxes, one on top of the other, like this:

Each of these boxes can hold a maximum of five characters, so your fraction can get pretty complex. For example, if you solve a problem and the answer is , you should enter the numerator and denominator separately, like this:

HOW TO USE THIS BOOK

In Math Workout for the GRE, we focus solely on the math portions of the GRE. This book includes more than two hundred sample questions (including two sample GRE quantitative sections, complete with answers and explanations) on which to practice all of the new techniques you learn. This book is called a “workout” because if you resolve to “feel the burn” of diligent mental exertion, you won’t just memorize a bunch of new techniques. Instead, you’ll absorb them into your subconscious so completely that you will use them automatically.

Trust the Techniques

As we’ll discuss in Chapter 2, you’re about to do a lot of work toward changing the way you think about taking this test. To do that, you should be prepared to let go of a number of presumptions and give yourself over to the techniques, which we’ve designed to conserve your thinking power and greatly reduce the chance that you’ll make careless errors come test day. Some of the techniques might seem a little strange or counterintuitive at first, but trust us: Part of the secret to a better score on a standardized test is to think in a non-standardized way.

When we encounter stress, we are hard-wired to fall back on our instincts to protect ourselves. If you start to feel anxious as you take the GRE, you might be tempted to abandon the new techniques in favor of whatever methods you used in order to get through high school and college—methods that won’t be as useful.

So when you work with practice questions, be sure to practice using our techniques over and over again. Once you see them working, you’ll build enough faith in them to let them replace your old habits. Soon you’ll summon them without thinking.

Set Up a Schedule—and Stick to It

When you’ve registered to take the GRE, it’s important to keep preparing for the test almost every day. Cramming for eight hours on a Sunday and then leaving the book alone for a week won’t be very useful because, like anything else, your new skills will atrophy with disuse. It will be far more effective if you set aside one hour per day to study.

When you set up your work regimen, keep these things in mind.

· As you work, look for patterns in the types of questions that you frequently answer correctly and patterns in the types you keep getting wrong. This will help you pinpoint your strengths and weaknesses and guide you to the areas in which you need more practice.

· Again, be sure to use the new techniques. If you read up about all these cool new methods for subverting the GRE and then just go back to your same old ways when it’s time to try practice problems, you won’t learn anything. All you’ll do is further the same old bad habits.

· Practice under conditions that are as close to the real-life test situation as possible. This means that you should only work when you feel mentally fresh enough to absorb the benefits of what you’re doing. If you come home late, don’t stay up until the wee hours reading and fighting off yawns. If you can’t absorb anything from the process, you’re just doing homework for the sake of getting it done.

Other Resources

Keep in mind that there are many other tools available to you so that you can practice all the new techniques you’re about to learn.

POWERPREP II Software

The GRE website (www.ets.org/gre) has a link to the POWERPREP II software. This free program contains two GRE tests which you can take on your own computer. It’s a great way to get used to the computerized format of the test and try various questions and essay prompts. However, the number of questions it has is limited, so you should probably save at least one of the tests until you’ve worked through most of this book.

If you want to brush up on
your basic math skills, you
can also get Math Smart,
which takes the time to
explain, in step-by-step
detail, mathematical
concepts from the most
basic to the most complex

Books

The most important book (besides this one, of course!) to check out is The Official Guide to the GRE revised General Test. This book is published by ETS, and contains questions for every single question type, Math and Verbal, and practice essay prompts. It also contains a CD with a copy of POWERPREP II software.

We at The Princeton Review have other helpful GRE titles to offer you, too, including the Verbal Workout for the GRE, (the sister to this book) and the larger and more comprehensive Cracking the GRE. If you’re really pressed for time, the short Crash Course for the GRE will give you a quick overview of what you need to know for the test.

On the Web

Books are great learning resources, but they can’t replicate the process of working with a computer interface. That’s why The Princeton Review has developed several online test-prep resources. At PrincetonReview.com, go to the GRE section of our website and you will find a free, practice GRE exam there, along with lots of helpful articles and information.

To find out more, surf over to PrincetonReview.com or call 1-800-2REVIEW.

Above all: Keep practicing and stay focused. Good luck!