Math Workout for the GRE ( PDFDrive ).pdf

 Editorial Rob Franek, Editor-in-Chief Casey Cornelius, VP Content Development Mary Beth Garrick, D...

 Editorial Rob Franek, Editor-in-Chief Casey Cornelius, VP Content Development Mary Beth Garrick, Director of Production Selena Coppock, Managing Editor Meave Shelton, Senior Editor Colleen Day, Editor Sarah Litt, Editor Aaron Riccio, Editor Orion McBean, Associate Editor Penguin Random House Publishing Team Tom Russell, VP, Publisher Alison Stoltzfus, Publishing Director Jake Eldred, Associate Managing Editor Ellen Reed, Production Manager Suzanne Lee, Designer The Princeton Review 555 W. 18th Street New York, NY 10011 E-mail: [email protected] Copyright © 2017 by TPR Education IP Holdings, LLC. All rights reserved. Published in the United States by Penguin Random House LLC, New York, and in Canada by Random House of Canada, a division of Penguin Random House Ltd., Toronto. Terms of Service: The Princeton Review Online Companion Tools (“Student Tools”) for retail books are available for only the two most recent editions of that book. Student Tools may be activated only twice per eligible book purchased for two consecutive 12- month periods, for a total of 24 months of access. Activation of Student Tools more than twice per book is in direct violation of these Terms of Service and may result in discontinuation of access to Student Tools Services. ISBN 9780451487865 Ebook ISBN 9781524710330 GRE is a registered trademark of Educational Testing Service, which is not affiliated with The Princeton Review. The Princeton Review is not affiliated with Princeton University. Editor: Meave Shelton Production Editors: Liz Rutzel, Kathy G. Carter, and Sara Kuperstein Production Artist: Deborah A. Silvestrini Cover art by Tetra Images / Alamy Stock Photo Cover design by Suzanne Lee v4.1 a Acknowledgments The Princeton Review would like to thank Doug French, author of the first edition, Kyle Fox and Kevin Kelly for their contributions to this edition, and John Fulmer, National Content Director for Graduate Programs, for his careful oversight. Special thanks to Adam Robinson, who conceived of and perfected the Joe Bloggs approach to standardized tests, and many of the other successful techniques used by The Princeton Review. Contents Cover Title Page Copyright Acknowledgments Register Your Book Online! 1 Introduction 2 Strategic Thinking for the GRE 3 Math Fundamentals 4 The Basics of Algebra 5 Turning Algebra into Arithmetic 6 Charts and Graphs 7 Math in the Real World 8 Geometry 9 The Rest of the Story 10 Sample Section 1 11 Sample Section 1: Answers and Explanations 12 Sample Section 2 13 Sample Section 2: Answers and Explanations 14 Glossary of Math Terms Register Your Book Online! 1 Go to PrincetonReview.com/​cracking 2 You’ll see a welcome page where you can register your book using the following ISBN: 9781524710330 3 After placing this free order, you’ll either be asked to log in or to answer a few simple questions in order to set up a new Princeton Review account. 4 Finally, click on the “Student Tools” tab located at the top of the screen. It may take an hour or two for your registration to go through, but after that, you’re good to go. If you have noticed potential content errors, please e-mail [email protected] with the full title of the book, its ISBN number (located above), and the page number of the error. Experiencing technical issues? Please e-mail [email protected] with the following information: your full name e-mail address used to register the book full book title and ISBN your computer OS (Mac or PC) and Internet browser (Firefox, Safari, Chrome, etc.) description of technical issue Once you’ve registered, you can… Download printable study plans for the content in this book, as well as a handy glossary of math terms Read important advice about the GRE and graduate school Access crucial information about the graduate school application process, including a timeline and checklist Check to see if there have been any corrections or updates to this edition Look For These Icons Throughout The Book Online Articles Proven Techniques Applied Strategies More Great Books 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 help you score your best on the quantitative portion. The GRE supposedly was written so that graduate schools might get a better sense of an applicant’s ability to work in a postgraduate 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. 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. Your Scores 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 scored Math or Verbal Section, the GRE will use your score on that section to determine the difficulty of the questions to give you in the next scored Math or Verbal section. This does not really affect how you will approach the test, so don’t worry about it too much. Experimental Section 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: Analytical Writing − 60 minutes Verbal − 30 minutes 10-minute break Math − 30 minutes Math − 30 minutes Verbal − 30 minutes Math − 30 minutes 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 only two of the three Math sections 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, on occasion 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, regardless of if you’ve seen an experimental section or not, you may also have an unscored Research section. At the beginning of this section, you will be told that the questions in the section are part of 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, such as entering your name into a drawing for a gift card, 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. A Quick Word About Answer Choices On the real GRE, answers will be designated by a circle, square, or numeric entry box. For the purposes of explaining concepts and answers to questions in this book, we are going to label all answer choices with a corresponding letter (A, B, C, D, E, etc.). So, for example, when we say the correct answer is (C), you know that the correct answer is the third option. It is useful to think about your answer choices in terms of these letters, as it will help keep you organized and allow you to eliminate answers efficiently. MATH OVERVIEW There are three main skills that we emphasize throughout this book: Don’t do the math in your head, 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. Don’t Do the Math in Your Head Many students are guilty of trying to solve GRE math questions by doing some quick calculations in their heads, or phantom drawing information relevant to the question on the test screen. This is what the test makers want you to do. They know if you do this, you will likely make careless and avoidable mistakes. Remember, your goal on the GRE is to get as many points as possible by answering questions correctly. There are no style points for getting the correct answer by doing all the calculations in your head. On test day, you will be provided with six pieces of scratch paper. To avoid careless mistakes and to maximize your score on the Math section, it is important that you use that scratch paper. When you see an equation, rewrite it on the paper. When there is a geometry figure, draw the figure on the paper. When you are doing calculations, chart the steps on your paper. Below is an example of a piece of scratch paper for two particular problems. This example is meant to provide an idea of how an organized piece of scratch paper may look. This student has written the question number and the answer choices on the left-hand side of the paper and left the remainder to show their work. If this example is a useful template for you, then we would suggest recreating it. But, if you have another method that you are more comfortable with, you should use that method. The method you use to track your problems is less important than making sure you always use the paper and avoid doing the math in your head. 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. 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 your time is limited, and on the GRE you can always come back to a question later. 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 get isn’t among the answer choices, move on. As we discussed before, you should take the easy test first. After you answer all the easy questions, you should work on the harder questions. Many times, you’ll find the question you thought was difficult, you actually just 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. Sometimes, a new perspective is all you need to make a previously confusing question, more clear. 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 that 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. Take a look at an example question below: 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 Approach this question one step at a time. The questions asks for the value of c + d. The first part of the question states that c is the greatest prime number less than 22. List the prime numbers less than 22, which are 2, 3, 5, 7, 11, 13, 17, and 19. The greatest number is 19, so c = 19. The question then states that d is the least prime number greater than 35. The next greatest number, 36, is not prime so d does not equal 36. However, the following number, 37, is prime. Therefore, the least prime number greater than 35 is 37, so d = 37. The question asks for the value of c + d, which is 19 + 37 = 56. The correct answer is (D). Quantitative Comparison We’ll refer to these questions as Quant Comp for short. These questions are variations of the basic multiple-choice question. You will sometimes be given a statement with information relevant to the problem. You will always be given two columns, labeled Quantity A and Quantity B, and each Quantity has a value underneath. You must select one of the following four answer choices: ○ 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. Choice (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 the information provided is not enough to determine if one Quantity is always greater than the other. Before we do a sample question, there’s one important thing you should know about (D): It can never be the answer for questions that contain only numeric values that could be calculated. 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. Point E lies in square ABCD and the area of square ABCD is 100. Quantity A Quantity B The length of line DE 15 ○ 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 The question asks about a square but does not provide the figure, so draw the square and label the points and lengths as specified in the question. The question gives the area of the square, and the area of a square is defined as s2, where s is the side of the square. Therefore, each side of the square is 10. Quantity A asks for the length of line DE. The question specifies that point E lies in the square but does not state where, so find the greatest possible length of line DE. The longest line in a square is the diagonal between two points, so draw a line from point D diagonally to the point opposite it, and assume that point E is just slightly inside that point. Therefore, the greatest length possible for DE is a length that is a little bit less than the diagonal of the square. Find the diagonal of the square. Because each corner in a square is 90°, and a diagonal line cuts the angles in half, the end result is a 45 : 45 : 90 right triangle. The side lengths of a 45 : 45 : 90 triangle are x : x: x. The value of x is 10, so the length of the diagonal is 10. The value of is approximately 1.4, so 10 × 1.4 = 14. Therefore, Quantity A is, at most, a little less than 14. The value of Quantity B is 15. Quantity B is always greater than Quantity A, so the correct answer is (B). All That Apply These are multiple-choice questions that have from three to eight answer choices, and you will have to select all the answers that correctly answer the question. The answer choices for these questions are always represented by 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. Keep track of your work and check each answer choice individually. Make sure you have selected all the appropriate answer choices before moving on to the next question. 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 This is an All that Apply question, so x could be equal to more than one number. The problem states that 1213 is an integer when divided by x, so x divides into 1213 evenly. The best way to tackle questions with great exponential values is to break the base numbers down into their prime factors. 12 is equal to 2 × 2 × 3, so its prime factors are 2, 2, and 3. So, the prime factorization of 1213 is (22 × 3)13. Apply the Power- Multiply rule of exponents to find that the prime factorization of 1213 is 226 × 313. (Don’t worry if you don’t know what the Power-Multiply rule is—we’ll go over that later. For now, just take our word for it.) Because the problem states that 1213 divided by x is an integer, the prime factorization of 1213 must cancel the prime factorization of x entirely when divided by each other. Evaluate the answer choices, looking for values of x that divide evenly into the prime factorization of 1213. Try (A). The prime factorization of 8 is 23, which cancels out completely when dividing 226 × 313, so keep (A). Try (B). If x = 10, the prime factorization of x is 2 × 5. This is not divisible by 1213 because the prime factors of 1213 do not include 5. Eliminate (B). Now look at (C). The prime factorization of 36 is 22 × 32, which divides evenly into 226 × 313. Keep (C). Evaluate (D). The prime factorization of 64 is 26, which divides evenly into 226 × 313. Keep (D). The prime factorization of 112 is 24 × 7, which does not divide evenly into 226 × 313, so eliminate (E). Finally, for (F), the prime factorization of 432 is 24 × 33 which divides evenly into 226 × 313, so keep (F). The correct answer is (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.  If Jamal is charged $236.30 to rent a car and that charge consists of a flat fee of $95 and a charge of $0.075 for every tenth of a mile driven, how many miles did Jamal drive the rental car? Here’s How to Crack It 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. 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 to find that the answer is 188.4 miles. If the correct answer should be given as a fraction, the space to fill in the answer will be represented by 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: Also, note that when entering numbers into the fraction in this fashion, the fraction does not need to be reduced to the simplest form. So, where applicable, if you have the correct answer you can save yourself some time by not reducing the fraction to its simplest form, unless otherwise noted by the directions of the question. 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 work only 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.  Check out our downloadable study plans for the content in this book! Register your book online for access. 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. 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, which is published by ETS, 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. 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. 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. Go to the GRE section at PrincetonReview.com where you will find a free practice GRE exam, 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! Chapter 2 Strategic Thinking for the GRE WHY ARE YOU HERE? Some people describe themselves as “bad at math.” These people believe that math is beyond their abilities and have a high level of anxiety about the Math section of the GRE. Maybe you picked up this book because you’re one of these people and the very idea of engaging in a test of your math skills makes you nervous. Other people are comfortable with math. They feel at home inside the numbers and are confident in their ability to execute on any kind of math problem. Either way, both kinds of people often have the same misconceptions about the GRE. And both people tend to approach a math question on the GRE the same way. What’s worse is that ETS knows both things, and actively uses them to their advantage on test day. What Do Most People Think about the GRE? Most people have a few perceptions about the GRE: The GRE is constructed to be fair. Knowing the content is enough to get most questions right. ETS wants you to get the questions right if you have the knowledge needed. Unfortunately, none of these is true. The Truth about the GRE The GRE is constructed to make you miss questions you should get right. You can completely understand the content behind the question and still get it wrong. In fact, almost all GRE test takers miss some questions this way. ETS wants to trick you into getting the question wrong, even if you understand the concepts. The Anatomy of an Average Test Taker Those misconceptions are ones of the average test taker. Imagine this scenario. It’s test day. You’ve done all the important things you need to do to succeed—you’ve studied, gotten a good night’s sleep, eaten a good breakfast—and you are ready to go. Shortly into the test, a math question appears. You read the question, look at the answer choices, think you know the answer, do some quick calculations in your head, select your answer, and move on to the next question. If this sounds like you, then you have done exactly what ETS wants you to do. Don’t Be Average One of the single best lessons you can learn to succeed on the GRE is to not do what ETS wants you to do. ETS writes questions and answer choices designed to trip up the average test taker. The more you know about how ETS constructs questions and answer choices, and about how the average student responds to those questions and answer choices, the greater your chances of having a successful test day. So, what are the types of things that you can do to not be an average test taker? THINK LIKE A TEST WRITER Putting yourself inside the mind of an ETS question writer is a great first step to being able to succeed on the GRE. Let’s imagine a test writer sits down to create a new math question for the GRE and they come up with the following question and correct answer. A store has a clearance on six-packs of gum for $2.70. If this price is 10% less per pack than purchasing individual packs at the original price, then what is the original price of a single pack of gum? Correct Answer: $0.50 The question writer has produced a question with a difficulty level of easy to medium. But, now the test writer must create 4 incorrect answer choices. The test writer could pick values at random, but that is not what test writers do. Instead, test writers try to predict how a student might make a mistake on the problem and use those mistakes to make their correct answer choices. For instance, a student may have quickly read this problem and not realized that the store is having a clearance on six-packs of gum for $2.70 and assumed that the clearance price was for a single pack of gum. This student will immediately realize that if a clearance price of $2.70 is a discount of 10%, then the original price must be $3.00. So, the test writer will make $3.00 an answer choice. What if a student misses that the question is asking for the original price of a single pack of gum? Well, that student will divide the clearance price of $2.70 by the six packs of gum, and get an answer of $0.45. So, the test writer will make $0.45 an answer choice. And for the student who is rushing and therefore sees only the phrase six-packs, $2.70, 10% less per pack, and price of a single pack? Well that student will divide $2.70 by 6 and then subtract 10%, which yields $0.405. The test writer will round up and write $0.41 as an answer choice. And what about the student who doesn’t quite fully understand how to calculate a savings of 10%? Well, that student may multiply the clearance price of $2.70 by 10% to find $0.27. They may then add $0.27 to $2.70 to yield $2.97. The completed question, with all five answer choices, looks like this: A store has a clearance on six-packs of gum for $2.70. If this price is 10% less per pack than purchasing individual packs at the original price, then what is the original price of a single pack of gum?  ○ $0.41 ○ $0.45 ○ $0.50 ○ $2.97 ○ $3.00 The test writer has now created a GRE question that includes trap answers for common mistakes that a student might make. The good news for you is that now that you know a little bit about how a test writer constructs wrong answer choices, you can use this knowledge to help you eliminate incorrect answer choices. Process of Elimination (POE) When test writers create the question, they already know the correct answer. They must create the incorrect answer choices for multiple-choice questions. Because, for multiple-choice questions, there is only one correct answer choice, you know that most of the answer choices are going to be incorrect. So, sometimes, it is easier to work to find and eliminate bad answer choices than it is to find the correct answer choice. This is a strategy that not-average test takers will often employ. We’ll use this icon throughout the book to highlight proven techniques like POE. Consider the following question. In a building with 100 occupants, the number of marketing firms that are occupants is the number of law firms that are occupants. If the building is occupied by only marketing firms and law firms, how many law firms are occupants in the building? ○ 75 ○ 66 ○ 60 ○ 40 ○ 33 Stop! Before you do any calculations, see if you can use what you know now about how test writers create wrong answers to do some Process of Elimination for incorrect answer choices. You’ll see this icon near examples of the strategies  in action. The problem states that the number of marketing firms is the number of law firms, so the number of marketing firms is less than the number of law firms. There are 100 total occupants, and because there are more law firms than marketing firms, it is impossible for there to be 50 or fewer law firms in the building, so eliminate (D) and (E) because they are less than 50. Now, look at the remaining choices. Do any of them seem like potential trap answer choices? Choice (B) is close to of 100, and the question does use both of those numbers. However, the question states that the number of marketing firms that are occupants is the number of law firms that are occupants, not that the number of law firms is the total number of occupants. This is a trap answer for a student who is rushing. Eliminate (B). Without doing a single piece of math on this question, you have now successfully narrowed the potential answer choices down to two. Even if you did not know how to solve this problem, you have a fifty-fifty chance of getting the question correct, just because you thought like a test writer. Well done. Turn Algebra into Arithmetic Later in this book, we will discuss Plugging In, a strategy for turning algebra into arithmetic. The basics of the strategy are that when a problem contains a variable and could be solved using algebra, you should plug in actual numbers for the variables and then solve. A lot of times, this will turn an algebra question into an arithmetic question. We are generally much better at arithmetic than we are at algebra, which works in your favor. While we’ll discuss this strategy in much more detail later in this book, we felt the need to provide a preamble to it here. Why? Because Plugging In, and all its variations, is one of the most powerful tools in your belt to turn questions that may otherwise be difficult or time-consuming into questions that you can answer with relative ease. You should take extra effort to be very comfortable with this strategy, as it is one of the best ways to ensure that you are not being an average test taker and not doing what ETS wants you to do. In short, Plugging In is an essential tool to the test taker seeking to not be average. Be Aware of All Types of Numbers Part of successfully navigating Plugging In and not being an average student is to show awareness of all kinds of different numbers. What do we mean by that? Consider the following. If a GRE question gave the variable x and there were no restrictions on what x could be, what number would you plug in for x ? Many test takers will plug in 2, 3, 5, 10, or some other common number. But x could also be −2, −3, 0, , 10 , or any other number you could imagine. Test writers will often rely on test takers to not consider all possible types of numbers when creating a question that contains variables. The more aware you are of the types of numbers available to you on any given question, the better chance you have of avoiding choosing one of ETS’s trap answers. This is not to say that you should ignore the common numbers. You shouldn’t. You should always work with the common numbers first to eliminate as many answer choices as possible. But, after you’ve plugged in a common number, if you still have a couple of answer choices remaining, plugging in a less common number is a good strategy to try to eliminate more answer choices. Bite-Sized Pieces Many of the math questions on the GRE read like verbal questions. These word problems are often long and contain a lot of information to process. The average test taker approaches this type of problem by trying to do all the steps at once, finding a shortcut, or trying to keep track of everything in their head. These are bad strategies. Avoid them. Instead, approach these problems by breaking them into bite-sized pieces. By breaking the question into smaller parts that you can handle individually, you will stay more organized and run less risk of making a careless mistake. Look at the following example. Point B is 18 miles east of point A, and point C is 6 miles west of point B. If point D is halfway between points B and C, and point E is halfway between points D and B, how far, in miles, is point E from point D ? At first glance, there is a lot of information in this question. Don’t try to answer the question all at once. Instead, break the question down into bite-sized pieces. The question begins by stating that point B is 18 miles east of point A, so draw a line with points A and B on the ends and the length labeled 18. Point C is 6 miles west of point B, so draw another line from point B to point C and label the length 6. Point D is halfway between points B and C, so put a point between points B and C and label it D. Because point D is halfway between B and C and the length of BC is 6, the lengths of BD and DC are 3. Point E is halfway between D and B, so put a point between D and B and label it E. The distance between D and B is 3, and E is halfway between them, so the distance from point D to point E is 1.5 miles. By breaking this question down into bite-sized pieces, it was easy to keep organized and to work your way to the correct answer. Ballparking Occasionally, the GRE will present to you a problem that contains strange numbers, such as long decimals or numbers that don’t appear to divide evenly. Many times, those questions will ask for a nonspecific value. When confronted with a question like those, the writers at ETS are hoping that you spend a considerable amount of time working with difficult numbers. At these times, a good strategy to remember is to use Ballparking. With this technique, a number is designed to make difficult numbers easier to work with by rounding or approximating them to a more favorable number. After ballparking the numbers in a question, the correct answer will be the one that is closest to the value that you determined from your ballparked numbers. At various points throughout this book, we present examples of how to use the technique of ballparking. PRACTICE You should take the time to sharpen your GRE math skills if you want to succeed. If you don’t spend much time working with numbers, or if all your numerical calculations are done by calculator or spreadsheet, it is worth your time to go out of your way to become comfortable with numbers again. Here are a couple ways that you can reintroduce numbers into your daily life that are relevant to the GRE. Figure out a 15% tip by calculating 10% and adding half again (because 10% plus 5% equals 15%). Take a few measurements and find out exactly how much storage space that old armoire has. Calculate the exact miles per gallon your car got on that last trip to your sister’s house. Figure out what fraction of your monthly budget is taken up by housing expenses, food, utilities, or loan payments. It also should go without saying that probably the best action you can take is to use the strategies in this book and do the practice problems correctly. While it is true that speed is important when taking the GRE, so is accuracy. You will get faster at employing the techniques we outline here the more familiar you become with them. But you will become faster and more accurate only if you employ the techniques with precision. So, do not feel rushed to work through a practice set in this book. Work consistently and work accurately and you will eventually begin to work at a faster pace. THE CALCULATOR As we mentioned before, on the GRE you’ll be given an on-screen calculator. The calculator program on the GRE is a rudimentary one that gives you the five basic operations: addition, subtraction, multiplication, division, and square root, plus a decimal function and a positive/negative feature. It follows the order of operations, or PEMDAS (more on this topic in Chapter 3). The calculator also has the ability to transfer the answer you’ve calculated directly into the answer box for certain questions. The on-screen calculator can be a huge advantage—if it’s used correctly! As you might have realized by this point, ETS is not exactly looking out for your best interests. Giving you a calculator might seem like an altruistic act, but rest assured that ETS knows that there are certain ways in which calculator use can be exploited. Keep in mind the following: 1. Calculators Can’t Think. Calculators are good for one thing and one thing only: calculation. You still have to figure out how to set up the problem correctly. If you’re not sure what to calculate, then a calculator isn’t helpful. For example, if you do a percent calculation on your calculator and then hit “Transfer Display,” you must remember to move the decimal point accordingly, depending on whether the question asks for a percent or a decimal. 2. The Calculator Can Be a Liability. ETS will give you questions that you can solve with a calculator, but the calculator can actually be a liability. You will be tempted to use it. For example, students who are uncomfortable adding, subtracting, multiplying, or dividing fractions may be tempted to convert all fractions to decimals using the calculator. Don’t do it. You are better off mastering fractions than avoiding them. Working with exponents and square roots is another way in which the calculator will be tempting but may yield really big and awkward numbers or long decimals. You are much better off learning the rules of manipulating exponents and square roots. Most of these problems will be faster and cleaner to solve with rules than with a calculator. The questions may also use numbers that are too big for the calculator. Time spent trying to get an answer out of a calculator for problems involving really big numbers will be time wasted. Find another way around. 3. A Calculator Won’t Make You Faster. Having a calculator should make you more accurate, but not necessarily faster. You still need to take time to read each problem carefully and set it up. Don’t expect to blast through problems just because you have a calculator. 4. The Calculator Is No Excuse for Not Using Scratch Paper. Scratch paper is where good technique happens. Working problems by hand on scratch paper will help to avoid careless errors or skipped steps. Just because you can do multiple functions in a row on your calculator does not mean that you should be solving problems on your calculator. Use the calculator to do simple calculations that would otherwise take you time to solve. Make sure you are still writing steps out on your scratch paper, labeling results, and using set-ups (we’ll go into this in more depth later). Accuracy is more important than speed! Of course, you should not fear the calculator; by all means, use it and be grateful for it. Having a calculator should help you eliminate all those careless math mistakes. Chapter 3 Math Fundamentals DEALING WITH NUMBERS Whenever you decide to learn a new language, what do they start with on the very first day? Vocabulary. Well, math has as much of its own lexicon as any country’s mother tongue, so now is as good a time as any to familiarize yourself with the terminology. These vocabulary words are rather simple to learn—or relearn—but they’re also very important. Any of the terms you’ll read about in this chapter could show up in a GRE math question, so you should know what the test is talking about. (For a more lengthy list, you can consult the glossary in Chapter 14.) We’ll start our review with the backbone of all Arabic numerals: the digit. Digits You might think there are an infinite number of digits in the world, but in fact there are only ten: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This is the mathematical “alphabet” that serves as the building block from which all numbers are constructed. Modern math uses digits in a decile system, meaning that every digit in a number represents a multiple of ten. For example, 1,423.795 = (1 × 1,000) + (4 × 100) + (2 × 10) + (3 × 1) + (7 × 0.1) + (9 × 0.01) + (5 × 0.001). You can refer to each place as follows: 1 occupies the thousands place. 4 occupies the hundreds place. 2 occupies the tens place. 3 occupies the ones, or units, place. 7 occupies the tenths place, so it’s equivalent to seven tenths, or. 9 occupies the hundredths place, so it’s equivalent to nine hundredths, or. 5 occupies the thousandths place, so it’s equivalent to five thousandths, or. When all the digits are situated to the left of the decimal place, you’ve got yourself an integer. Integers When we first learn about addition and subtraction, we start with integers, which are the numbers you see on a number line. Integers and digits are not the same thing; for example, 39 is an integer that contains two digits, 3 and 9. Also, integers are not the same as whole numbers, because whole numbers are non-negative, which include zero. Conversely, integers include negative numbers. Any integer is considered greater than all of the integers to its left on the number line. So just as 5 is greater than 3 (which can be written as 5 > 3), 0 is greater than −4, and −4 is greater than −10. (For more about greater than, less than, and solving for inequalities, see Chapter 4.) Consecutive Integers and Sequences Integers can be listed consecutively (such as 3, 4, 5, 6…) or in patterned sequences such as odds (1, 3, 5, 7…), evens (2, 4, 6, 8…), and multiples of 6 (6, 12, 18, 24…). The numbers in these progressions always get larger, except when explicitly noted otherwise. Note also that, because zero is an integer, a list of consecutive integers that progresses from negative to positive numbers must include it (−2, −1, 0, 1…). Zero Zero is a special little number that deserves your attention. It isn’t positive or negative, but it is even. (So a list of consecutive even integers might look like −4, −2, 0, 2, 4….) Zero might also seem insignificant because it’s what’s called the additive identity, which basically means that adding zero to any other number doesn’t change anything. (This will be an important consideration when you start plugging numbers into problems in Chapter 5.) Positives and Negatives On either side of zero, you’ll find positive and negative numbers. For the GRE, the best thing to know about positives and negatives is what happens when you multiply them together. A positive times a positive yields a positive (3 × 5 = 15). A positive times a negative yields a negative (3 × −5 = −15). A negative times a negative yields a positive (−3 × −5 = 15). Even and Odd As you might have guessed from our talk of integers above, even numbers (which include zero) are multiples of 2, and odd numbers are not multiples of 2. If you were to experiment with the properties of these numbers, you would find that any number times an even number yields an even number the product of two or more odd numbers is always odd  the sum of two or more even numbers is always even the sum of two odd numbers is always even the sum of an even number and an odd number is always odd Obviously, there’s no need to memorize stuff like this. If you’re ever in a bind, try working with real numbers. After all, if you want to know what you get when you multiply two odd numbers, you can just pick two odd numbers—like 3 and 7, for example—and multiply them. You’ll see that the product is 21, which is also odd. Digits Quick Quiz Question 1 of 3 If x, y, and z are consecutive even integers and x < 0 and z > 0, then xyz = Question 2 of 3 The hundreds digit and ones digit of a three-digit number are interchanged so that the new number is 396 less than the old number. Which of the following could be the number? ○ 293 ○ 327 ○ 548 ○ 713 ○ 801 Question 3 of 3 a, b, and c are consecutive digits, and a > b > c Quantity A Quantity B abc a+b+c ○ Quantity A is greater. ○ Quantity B is greater. ○ The two quantities are equal. ○ The relationship cannot be determined from the information given. Explanations for Digits Quick Quiz  1. If x, y, and z are consecutive even integers and x < 0 and z > 0, then x must be −2, y must be 0, and z must be 2. Therefore, their product is 0, and you would enter this number into the box. 2. Take the answer choices and switch the hundreds digit and ones digit. When the result is 396 less than the old number, you have a winner. Choices (A), (B), and (C) are out, because their ones digits are greater than their hundreds digits; therefore, the result will be greater (for example, 293 becomes 392). If you rearrange 713, the result is 317, which is 396 less than 713. The answer is (D). 3. Pick three consecutive digits for a, b, and c, such as 2, 3, and 4. Quantity A becomes 2 × 3 × 4, or 24, and Quantity B becomes 2 + 3 + 4, or 9. Quantity A is greater so eliminate (B) and (C). But if a, b, and c are −1, 0, and 1, respectively, then both quantities become 0 so eliminate (A). Therefore, the answer is (D). MORE ABOUT NUMBERS Prime Numbers Prime numbers are special numbers that are divisible by only two distinct factors: themselves and 1. Since neither 0 nor 1 is prime, the least prime number is 2. The rest, as you might guess, are odd, because all even numbers are divisible by two. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Note that not all odd numbers are prime; 15, for example, is not prime because it is divisible by 3 and 5. Said another way, 3 and 5 are factors of 15, because 3 and 5 divide evenly into 15. Let’s talk more about factors. Factors As we said, a prime number has only two distinct factors: itself and 1. But a number that isn’t prime—like 120, for example—has several factors. If you’re ever asked to list all the factors of a number, the best idea is to pair them up and work through the factors systematically, starting with 1 and itself. So, for 120, the factors are 1 and 120 2 and 60 3 and 40 4 and 30 5 and 24 6 and 20 8 and 15  10 and 12 Notice how the two numbers start out far apart (1 and 120) and gradually get closer together? When the factors can’t get any closer, you know you’re finished. The number 120 has 16 factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. Of these factors, three are prime (2, 3, and 5). That’s also an important point: Every number has a finite number of factors. Prime Factorization Sometimes the best way to analyze a number is to break it down to its most fundamental parts—its prime factors. To do this, we’ll break down a number into factors, and then continue breaking down those factors until we’re stuck with a prime number. For instance, to find the prime factors of 120, we could start with the most obvious factors of 120: 12 and 10. (Although 1 and 120 are also factors of 120, because 1 isn’t prime, and no two prime numbers can be multiplied to make 1, we’ll ignore it when we find prime factors.) Now that we have 12 and 10, we can break down each of those. What two numbers can we multiply to make 12? 3 and 4 work, and since 3 is prime, we can break down 4 to 2 and 2. 10 can be broken into 2 and 5, both of which are prime. Notice how we kept breaking down each factor into smaller and smaller pieces until we were stuck with prime numbers? It doesn’t matter which factors we used, because we’ll always end up with the same prime factors: 12 = 6 × 2 = 3 × 2 × 2, or 12 = 3 × 4 = 3 × 2 × 2. So the prime factor tree for 120 could look something like this: The prime factorization of 120 is 2 × 2 × 2 × 3 × 5, or 23 × 3 × 5. Note that these prime factors (2, 3, and 5) are the same ones we listed earlier. Multiples Since 12 is a factor of 120, it’s also true that 120 is a multiple of 12. It’s impossible to list all the multiples of a number, because multiples trail off into infinity. For example, the multiples of 12 are 12 (12 × 1), 24 (12 × 2), 36 (12 × 3), 48 (12 × 4), 60 (12 × 5), 72 (12 × 6), 84 (12 × 7), and so forth. If you ever have trouble distinguishing factors from multiples, remember this:  Factors are Few; Multiples are Many. Divisibility If a is a multiple of b, then a is divisible by b. This means that when you divide a by b, you get an integer. For example, 65 is divisible by 13 because 65 ÷ 13 = 5. Divisibility Rules The most reliable way to test for divisibility is to use the calculator that they give you. If a problem requires a lot of work with divisibility, however, there are several cool rules you can learn that can make the problem a lot easier to deal with. As you’ll see later in this chapter, these rules will also make the job of reducing fractions much easier. 1. All numbers are divisible by one. (Remember that if a number is prime, it is divisible by only itself and 1.) 2. A number is divisible by 2 if the last digit is even. 3. A number is divisible by 3 if the sum of the digits is a multiple of 3. For example, 13,248 is divisible by 3 because 1 + 3 + 2 + 4 + 8 = 18, and 18 is divisible by 3. 4. A number is divisible by 4 if the two digits at the end form a number that is divisible by 4. For example, 13,248 is divisible by 4 because 48 is divisible by 4. 5. A number is divisible by 5 if it ends in 5 or 0. 6. A number is divisible by 6 if it is divisible by both 2 and 3. Because 13,248 is even and divisible by 3, it must therefore be divisible by 6. 7. There is no easy rule for divisibility by 7. It’s easier to just try dividing by 7! 8. A number is divisible by 8 if the three digits at the end form a number that is divisible by 8. For example, 13,248 is divisible by 8 because 248 is divisible by 8. 9. A number is divisible by 9 if the sum of the digits is a multiple of 9. For example, 13,248 is divisible by 9 because 1 + 3 + 2 + 4 + 8 = 18, and 18 is divisible by 9. 10. A number is divisible by 10 if it ends in 0. Remainders If an integer is not evenly divisible by another integer, whatever integer is left over after division is called the remainder. You can find the remainder by finding the greatest multiple of the number you are dividing by that is still less than the number you are dividing into. The difference between that multiple and the number you are dividing into is the remainder. For example, when 19 is divided by 5, 15 is the greatest multiple of 5 that is still less than 19. The difference between 19 and 15 is 4, so the remainder when 19 is divided by 5 is 4. Working with Numbers A lot of your math calculation on the GRE will require you to know the rules for manipulating numbers using the usual mathematical operations: addition, subtraction, multiplication, and division. PEMDAS (Order of Operations) When simplifying an expression, you need to perform mathematical operations in a specific order. This order is easily identified by the mnemonic device that most of us come in contact with sooner or later at school—PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. (You might have remembered this as a kid by saying “Please Excuse My Dear Aunt Sally,” which is a perfect mnemonic because it’s just weird enough not to forget. What the heck did Aunt Sally do, anyway?) In order to simplify a mathematical term using several operations, perform the following steps: 1. Perform all operations that are in parentheses. 2. Simplify all terms that use exponents. 3. Perform all multiplication and division from left to right. Do not assume that all multiplication comes before all division, as the acronym suggests, because you could get a wrong answer. WRONG: 24 ÷ 4 × 6 = 24 ÷ (4 × 6) = 24 ÷ 24 = 1. RIGHT: 24 ÷ 4 × 6 = (24 ÷ 4) × 6 = 6 × 6 = 36. 4. Fourth, perform all addition and subtraction, also from left to right. It’s important to remember this order, because if you don’t follow it, your results will very likely turn out wrong. Try it out in a GRE example. (2 + 1)3 + 7 × 2 + 7 − 3 × 42 = ○ −16 ○ 0 ○ 288 ○ 576 ○ 1,152 Here’s How to Crack It Simplify (2 + 1)3 + 7 × 2 + 7 − 3 × 42 like this: Parentheses: (3)3 + 7 × 2 + 7 − 3 × 42 Exponents: 27 + 7 × 2 + 7 − 3 × 16 Multiply and Divide: 27 + 14 + 7 – 48 Add and Subtract: 41 + 7 − 48  48 − 48 = 0 The answer is (B). Working with Numbers Quick Quiz Question 1 of 7 Quantity A Quantity B The number of even multiples of 11 between 1 and 100 The number of odd multiples of 22 between 1 and 100 ○ Quantity A is greater. ○ Quantity B is greater. ○ The two quantities are equal. ○ The relationship cannot be determined from the information given. Question 2 of 7 Which of the following numbers has the same distinct prime factors as 42 ? ○ 63 ○ 98 ○ 210 ○ 252 ○ 296 Question 3 of 7 p and r are factors of 100 Quantity A Quantity B pr 100 ○ Quantity A is greater. ○ Quantity B is greater. ○ The two quantities are equal. ○ The relationship cannot be determined from the information given. Question 4 of 7 Quantity A Quantity B The remainder when 33 is divided by 12 The remainder when 200 is divided by 7 ○ Quantity A is greater. ○ Quantity B is greater.  ○ The two quantities are equal. ○ The relationship cannot be determined from the information given. Question 5 of 7 6(3 − 1)3 + 12 ÷ 2 + 32 = ○ 24 ○ 39 ○ 63 ○ 69 ○ 105 Question 6 of 7 If r, s, t, and u are distinct, consecutive prime integers less than 31, then which of the following could be the average (arithmetic mean) of r, s, t, and u ? Indicate all such numbers. 4 4.25 6 9 14 22 24 Question 7 of 7 The greatest prime number that is less than 36 is represented by x. y represents the least even number greater than 19 that is divisible by 3. What is the sum of 2 and the value of x divided by y ? Explanations for Working with Numbers Quick Quiz 1. The only even multiples of 11 between 1 and 100 are 22, 44, 66, and 88, so Quantity A equals 4. Quantity B is tricky, because if 22 is even, all multiples of 22 are also even. There are no odd multiples of 22, so Quantity B equals 0. The answer is (A). 2. The prime factorization of 42 is 2 × 3 × 7 so those are the distinct prime factors. Choice (A)  can be eliminated, because 63 is odd. The prime factorization of 252 is 2 × 2 × 3 × 3 × 7, so its distinct prime factors are the same. The answer is (D). 3. If p and r are factors of 100, then each must be one of these numbers: 1, 2, 4, 5, 10, 20, 25, 50, or 100. If you plug in p = 1 and r = 2, for example, then pr = 2 and Quantity B is greater so eliminate (A) and (C). If p = 50 and r = 100, however, then pr = 5,000, which is much greater than 100 so eliminate (B). Therefore, the answer is (D). 4. 12 goes into 33 two times. The remainder is 9. Since 9 is greater than 7, there is no need to calculate the remainder for Quantity B because it can’t possibly be greater than 6. Remember that the remainder is always less than the number you are dividing by. The answer is (A). 5. Follow PEMDAS and calculate the parentheses and exponents first: 6 × (3 − 1)3 + 12 ÷ 2 + 32 = 6 × 8 + 12 ÷ 2 + 9. Second, perform all multiplication and division: 6 × 8 + 12 ÷ 2 + 9 = 48 + 6 + 9. Now, it’s just a matter of addition: 48 + 6 + 9 = 63. The answer is (C). 6. This one can be tricky because of the math vocabulary; the question is really asking for the average of four prime numbers in a row. Start by making a list of all the consecutive prime numbers less than 31. Remember that 1 is not prime, and that 2 is the least prime number. Your list is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Starting with 2, 3, 5, 7, use the on-screen calculator to work out the different possible averages for four consecutive primes. Choices (B), (D), and (F) are the answers. 7. Work this question in bite-sized pieces. The greatest prime number less than 36 is 31, so x = 31. The least even number greater than 19 that is divisible by 3 is 24, so y = 24. The problem states that x is divided by y, so. The question asks for the sum of 2 plus the value of x divided by y, so 2 +. When adding a whole number to a fraction, set the denominators equal, so. PARTS OF THE WHOLE (FRACTIONS AND DECIMALS) It’s still necessary to be knowledgeable when it comes to fractions, decimals, and percents. Each of these types of numbers has an equivalent in the form of the other two, and fluency among the three of them can save you precious time on test day. For example, say you had to figure out 25% of 280. You could take a moment to realize that the fractional equivalent of 25% is. At this point, you might see that of 280 is 70, and your work would be done. If nothing else, memorizing the following table will increase your math IQ and give you a head start on your calculations. The Conversion Table Fractions Each fraction is made up of a numerator (the number on top) divided by a denominator (the number down below). In other words, the numerator is the part, and the denominator is the whole. By most accounts, the part is less than the whole, and that’s the way a fraction is “properly” written. Improper Fractions For a fraction, when the part is greater than the whole, the fraction is considered improper. The GRE won’t quiz you on the terminology, but it usually writes its multiple-choice answer choices in proper form. A proper fraction takes this form: Integer Converting from Improper to Proper To convert the improper fraction into proper form, find the remainder when 16 is divided by 3. Because 3 goes into 16 five times with 1 left over, rewrite the fraction by setting aside the 5 as an integer and putting the remainder over the number you divided by (in this case, 3). Therefore, is equivalent to 5 , because 5 is the integer, 1 is the remainder, and 3 is the divisor. The expression 5 is also referred to as a mixed number, because the number contains both an integer and a fraction. Converting from Proper to Improper Sometimes you’ll want to convert a mixed number into its improper format. Converting to an improper fraction from a mixed number is a little easier, because all you do is multiply the divisor (the denominator) by the integer and then add the remainder. Improper formats are much easier to work with when you have to add, subtract, multiply, divide, or compare fractions. The important thing to stress here is flexibility; you should be able to work with any fraction the GRE gives you, regardless of what form it’s in. Adding and Subtracting Fractions There’s one basic rule for adding or subtracting fractions: You can’t do anything until all of the fractions have the same denominator. If that’s already the case, all you have to do is add or subtract the numerators, like this: When the fractions have different denominators, you must convert one or both of them first in order to find their common denominator. When you were a kid, you may have been trained to follow a bunch of complicated steps in order to find the “lowest common denominator.” It might be a convenient thing to learn in order to impress your math teacher, but on the GRE it’s way too much work. The Bowtie The Bowtie method has been a staple of The Princeton Review’s materials since the company began in a living room in New York City in 1981. It’s been around so long because it works so simply. To add and , for example, follow these three steps: Step One: Multiply the denominators together to form the new denominator. Step Two: Multiply the first denominator by the second numerator (5 × 4 = 20) and the second denominator by the first numerator (7 × 3 = 21) and place these numbers above the fractions, as shown below. See? A bowtie! Step Three: Add the products to form the new numerator. Subtraction works the same way. Note that with subtraction, the order of the numerators is important. The new numerator is 21 − 20, or 1. If you somehow get your numbers reversed and use 20 − 21, your answer will be − , which is incorrect. One way to keep your subtraction straight is to always multiply up from denominator to numerator when you use the Bowtie so the product will end up in the right place. Quantity A Quantity B ○ 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 First, eliminate (D) because both Quantity A and Quantity B contain numbers. Next, the Bowtie can also be used to compare fractions. Multiply the denominator of the fraction in Quantity B by the numerator of the fraction in Quantity A and write the product (40) over the fraction in Quantity A. Next, multiply the denominator of the fraction in Quantity A by the numerator of the fraction in Quantity B and write the product (39) over the fraction in Quantity B. Since 40 is greater than 39, the answer is (A). Multiplying Fractions Multiplying fractions isn’t nearly as complicated as adding or subtracting, because any two fractions can be multiplied by each other exactly as they are. In other words, the denominators don’t have to be the same. All you have to do is multiply all the numerators to find the new numerator, and multiply all the denominators to find the new denominator, like this: The great thing is that it doesn’t matter how many fractions you have; all you have to do is multiply across. Dividing Fractions Dividing fractions is almost exactly like multiplying them, except that you need to perform one extra step: When dividing fractions, don’t ask why; just flip the second fraction and multiply. Dividing takes two forms. When you’re given a division sign, flip the second fraction and multiply them, like this: Sometimes, you’ll be given a compound fraction, in which one fraction sits on top of another, like this: The fraction bar might look a little intimidating, but remember that a fraction bar is just another way of saying “divide.” In this case, flip the bottom and multiply. Reciprocals of Fractions When a fraction is multiplied by its reciprocal, the result is always 1. You can think of the reciprocal as being the value you get when the numerator and denominator of the fraction are “flipped.” The reciprocal of , for example, is. Knowing this will help you devise a nice shortcut for working with problems such as the following. Which of the following is the reciprocal of ?  ○ ○ ○ ○ ○ Here’s How to Crack It To solve this problem, multiply each answer choice by the original expression,. If the product is 1, you know you have a match. In this case, the only expression that works is , so the answer is (C). Reducing Fractions Are you scared of reducing, or canceling, fractions because you’re not sure what the rules are? If so, there’s only one rule to remember: You can do anything to a fraction as long as you do exactly the same thing to both the numerator and the denominator. When you reduce a fraction, you divide both the top and bottom by the same number. If you have the fraction , for example, you can divide both the numerator and denominator by a common factor, 3, like this: Be Careful If you are worried about when you can cancel terms in a fraction, here’s an important rule to remember. If you have more than one term in the numerator of a fraction but only a single term in the denominator, you can’t divide into one of the terms and not the other. The only way you can cancel something out is if you can factor out the same number from both terms in the numerator and then divide. Decimals Decimals are just fractions with a hidden denominator: Each place to the right of the decimal point represents a fraction. Comparing Decimals To compare decimals, you have to look at the decimals place by place, from left to right. As soon as the digit in a specific place of one number is greater than its counterpart in the other number, you know which is bigger. For example, 15.345 and 15.3045 are very close in value because they have the same digits in their tens, units, and tenths places. But the hundredths digit of 15.345 is 4 and the hundredths digit of 15.304 is 0, so 15.345 is greater. Rounding Decimals In order to round a decimal, you have to know how many decimal places the final answer is supposed to have (which the GRE will usually specify) and then base your work on the decimal place immediately to the right. If that digit is 5 or higher, round up; if it’s 4 or lower, round down. For example, if you had to round 56.729 to the tenths place, you’d look at the 2 in the hundredths place, see that it was less than 5, and round down to 56.7. If you rounded to the hundredths place, however, you’d consider the 9 in the thousandths place and round up to 56.73. Fractions and Decimals Quick Quiz Question 1 of 8 The fraction is equivalent to what percent? ○ 30% ○ 33 % ○ 35% ○ 37.5% ○ 62.5% Question 2 of 8 What is the sum of and ? ○ ○ ○ ○ ○ Question 3 of 8 Indicate all such values. 59,049 310 Question 4 of 8 If x is the 32nd digit to the right of the decimal point when is expressed as a decimal, and y is the 19th digit to the right of the decimal point when is expressed as a decimal, what is the value of xy ? Question 5 of 8 If two-thirds of 42 equals four-fifths of x, what is the value of x ? ○ 28 ○ 35 ○ 42 ○ 63 ○ 84 Question 6 of 8 Quantity A Quantity B ○ Quantity A is greater. ○ Quantity B is greater. ○ The two quantities are equal. ○ The relationship cannot be determined from the information given. Question 7 of 8 How many digits are there between the decimal point and the first even digit after the decimal point in the decimal equivalent of ? Question 8 of 8 A set of digits under a bar indicates that those digits repeat infinitely. What is the value of (0.000023)(106 − 104) ? ○ 0.023 ○ 0.23 ○ 22.77 ○ 23  ○ 23.23 Explanations for Fractions and Decimals Quick Quiz 1. Reduce to and remember the common conversion table to convert the fraction to 37.5%. The answer is (D). 2. Use the Bowtie method. First, multiply the denominators: 5 × 12 = 60. When you multiply diagonally, as in the diagram given in the text, the numerator becomes 35 + 24, or 59. The new fraction is , and the answer is (D). 3. Simplify the negative exponents by taking the reciprocal of the corresponding positive exponent, which gives you. Now you have three reciprocals, so flip them over and calculate: = 59,049. You can also express each number as a power of 3, which gives you (33)(34)(33) = 310, which makes (B) correct. 310 can also be expressed as. Thus, the correct answer is (A), (B), and (C). 4. Divide to convert the fractions into decimals. First, = 0.2727. This is really a pattern question: The odd numbered terms are 2 and the even numbered terms are 7. The 32nd digit to the right of the decimal is an even term, so x = 7. Next, = 0.6363. This time, the odd numbered terms are 6 and the even numbered terms are 3. The 19th digit on the right side of the decimal place is an odd term, so y = 6. Lastly, xy = 7 × 6 = 42. 5. One-third of 42 is 14, so two-thirds is 28. 28 equals of x translates to 28 = x. Multiply both sides by the reciprocal, , to eliminate the fraction and isolate the x. The fractions on the  right cancel out and on the left you have (28) = 35, so 35 = x. The answer is (B). 6. To compare two fractions, just cross-multiply and compare the products. 7 × 8 = 56 and 11 × 5 = 55, so Quantity A is greater. 7. The answer is 0. First, divide to convert into a decimal. The on-screen calculator doesn’t do exponents, so you may want to factor the denominator: = 0.00003125. Since 0 is even, there are no digits between the decimal point and the first even digit after the decimal point, and the correct answer is 0. In fact, if you noticed that any decimal starting with a 0 would have the same answer, you only needed to make sure the denominator was larger than 10. 8. Before you break out the calculator, distribute the (0.000023). This gives you (0.000023) (106) – (0.000023)(104). Now move the decimal point 6 places to the right for the first term and 4 places to the right for the second term, which gives you 23.23 −.23 = 23. The correct answer is (D). PARTS OF A WHOLE (PERCENTS) As you may have noted from the conversion chart, decimals and percents look an awful lot alike. In fact, all you have to do to convert a decimal to a percent is to move the decimal point two places to the right and add the percent sign: 0.25 becomes 25%, 0.01 becomes 1%, and so forth. This is because they’re both based on multiples of 10. Percents also represent division with a denominator that is always 100. Calculating Percents Now, let’s review four different ways to calculate percents: translating, conversion, proportions, and tip calculations. Some people find that one way makes more sense than others. There is no best way to find percents. Try out all four, and figure out which one seems most natural to you. You will probably find that some methods work best for some problems, and other methods work best for others. Translating Translating is one of the more straightforward and versatile methods of calculating percents. Each word in a percent problem is directly translated into a mathematical term, according to the following chart: Term Math equivalent  what x (variable) is = of × (multiply) percent ÷ 100 This table can be a great help. For example, take a look at this Quant Comp problem. Quantity A Quantity B 24% of 15% of 400 52% of 5% of 600 ○ 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 it’s not immediately obvious which quantity is larger, you’re going to have to do some actual calculation. Start with Quantity A. 24% of 15% of 400 can be translated, piece by piece, into math. Remember that % means to divide by 100 and of means to multiply. As always, write down everything on your scratch paper. After translation, we get. Cancel out one of the 100s in the denominator with the 400 in the numerator of the last fraction to get. Now feel free to use the calculator. Multiply the top of the fractions first and then the bottom, resulting in = 14.4. Now do the same with Quantity B. 52% of 5% of 600 becomes. Cancel out the first 100 with the 600 and you’ll have = 15.6. Quantity B is therefore bigger, and the answer is (B). Word Problems and Percents Word problems are also far less onerous when you apply the math translation table. Thirty percent of the graduate students at Hardcastle State University are from outside the state, and 75% of out-of-state students receive some sort of financial aid. If there are 3,120 graduate students at Hardcastle State, how many out-of-state students do NOT receive financial aid? ○ 234 ○ 468 ○ 702 ○ 936 ○ 1,638 Here’s How to Crack It Note the range in answer choices. This is another problem that is ripe for Ballparking. Estimate that there are 3,000 graduate students. 10% of 3,000 is 300, so 30% is 900. 25% do not receive financial aid, which is what you’re looking for, so of 900 is 225. Since there are a few more than 3,000 graduate students, the answer must be a few more than 225. Only (A) is even close. Note that (C) represents the 75% who do receive financial aid. Make sure you read slowly and carefully and take all word problems in bite-sized pieces. Conversion The second way to deal with percentage questions is to use the chart on this page. This will allow you to quickly change each percentage into a fraction or a decimal. This method works well in tandem with Translating, as reduced fractions are often easier to work with when you are setting up a calculation. Proportions You can also set up a percent question as a proportion by matching up the part and whole. Look at a couple quick examples: 1) 60 is what percent of 200? Because 60 is some part of 200, you can set up the following proportion:  Notice that each fraction is simply the part divided by the whole. On the left side, 60 is part of 200. You want to know what that part is in terms of a percent, so on the right side set up x (the percent you want to find), divided by 100 (the total percent). 2) What is 30% of 200? Now you don’t know how much of 200 you’re dealing with, but you know the percentage. So put the unknown, x, over the whole, 200, and the percentage on the right: 3) 60 is 30% of what number? You know the percentage, and you know the part, but you don’t know the whole, so that’s the unknown: Notice that the setup for each problem (they’re actually just variations of the same problem: 30% of 200 is 60) is essentially the same. You had one unknown, either the part, the whole, or the percentage, and you wrote down everything you knew. Every proportion will therefore look like this: Once you’ve set up the proportion, cross-multiply to solve. Ben purchased a computer and paid 45% of the price immediately. If he paid $810 immediately, what is the total price of the computer? Here’s How to Crack It Find the parts of the proportion that you know. You know he paid 45%, so you know the percentage, and you know the part he paid: $810. You’re missing the whole price. On your scratch paper, set up the proportion. Cross-multiply to get 45x = 81,000. Divide both sides by 45 (feel free to use the calculator here) to get x = $1,800. Tip Calculation The last method for calculating percentages is a variation on a method many people use to calculate the tip for a meal. To find 10% of any number, simply move the decimal one place to the left. 10% of 100 = 10.0 10% of 30 = 3.0 10% of 75 = 7.5 10% of 128 = 12.8 10% of 87.9 = 8.79 To find 1% of any number, move the decimal two places to the left. 1% of 100 = 1.00 1% of 70 = 0.70 1% of 5 = 0.05 1% of 2,145 = 21.45 You can then find the value of any percentage by breaking the percentage into 1%, 10%, and 100% pieces. Remember that 5% is half of 10%, and 50% is half of 100%. 5% of 60 = half of 10% = half of 6 = 3 20% of 35 = 10% + 10% = 3.5 + 3.5 = 7 52% of 210 = 50% + 1% + 1% = 105 + 2.1 + 2.1 = 109.2 40% of 70 = 10% + 10% + 10% + 10% = 7 + 7 + 7 + 7 = 28 Generally, we’ll use this most often with Ballparking, especially on Charts and Graphs questions, which are covered in Chapter 6. Last month Dave spent at least 20% but no more than 25% of his monthly income on groceries. If his monthly income was $2,080.67, which of the following could be the amount Dave spent on groceries? Indicate all such values.  $391.92 $432.88 $456.02 $497.13 $530.17 $545.60 $592.43 Here’s How to Crack It Ugh. Those are some ugly numbers. Ballpark a little bit, and use some quick tip calculations to simplify. First, write down A B C D E F G vertically on your scratch paper. Look for a number greater than 20% of $2,080.67. Ignore the 67 cents for now. If any answers are only a couple pennies away from the answer, then you can go back and use more exact numbers, but that’s fairly unlikely. 10% of $2,080 is $208, which means that 20% is $208 + $208 = $416. Dave must have spent at least $416 on groceries, so cross off (A). You know he spent no more than 25% of his income. 5% of $2,080 is half of 10%, which means that 5% of $2,080 is $104, and 25% of $2,080 is 10% + 10% + 5% = $208 + $208 + $104 = $520. He couldn’t have spent more than $520 on groceries, so cross off (E), (F), and (G). The answers are everything between $416 and $520: (B), (C), and (D). Percent Change Percent change is based on two quantities: the change and the original amount. %change = × 100 Quantity A Quantity B The percent change from 10 to 11 The percent change from 11 to 10 ○ 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 At first glance, you might assume that the answer is (C), because the numbers in each quantity look so similar. However, even though both quantities changed by a value of 1, the original amounts are different. When you follow the formula, you find that Quantity A equals × 100, or 10%, while Quantity B equals × 100, or 9%. The answer is (A). Percentage change also factors into charts and graphs questions, so we’ll talk about it more in Chapter 6. From here, we head to a powerful little device that helps us convey very big and very small numbers with very little effort: exponents. Before we get into exponents, though, practice those percents. Percents Quick Quiz Question 1 of 6 What is 26% of 3,750 ? ○ 753 ○ 975 ○ 1,005 ○ 2,775 ○ 9,750 Question 2 of 6 Quantity A Quantity B 200% of 20% of 300 120% of 25% of 400 ○ Quantity A is greater. ○ Quantity B is greater. ○ The two quantities are equal. ○ The relationship cannot be determined from the information given. Question 3 of 6 A think tank projects that people over the age of 65 will comprise 25% of the U.S. population by 2010. If the current population of 300 million is expected to grow by 8% by 2010, how many people over age 65, in millions, will there be in 2010 ? Question 4 of 6  Marat sold his condo at a price that was 18% more than the price he paid for it. If he bought his condo at a price that was 42% less than the buyer’s $175,000 asking price, which of the following must be true? Indicate all such values. The person who sold Marat the condo lost money. Marat bought the condo for $101,500. 36 percent of the price at which Marat sold the condo is $43,117.20. Question 5 of 6 If 25% of p equals 65% of 80 and if q is 50% of p, which of the following must be true? Indicate all such values. 65 is 62.5% of q q is 130% of 80 p is 200% of q = 50% of 1 Question 6 of 6 When John withdraws x% of his $13,900 balance from his checking account, his new balance is less than $10,000. Quantity A Quantity B x 25 ○ Quantity A is greater. ○ Quantity B is greater. ○ The two quantities are equal. ○ The relationship cannot be determined from the information given. Explanations for Percents Quick Quiz 1. Convert. 26% is close to 25%, so what is of 3,750? Only (B) is close. 2. In Quantity A, 20% or of 300 is 60. 100% of 60 is 60, so 200% must be 120. In Quantity B, 25% or of 400 is 100. 20% of 100 is 20, so 120% is 120. The two quantities are equal, and the answer is (C). 3. 1% of 300 million is 3 million, so 8% is 24 million. If the population grew by 8%, that means that it added another 24 million people, so the new total is 324 million. 25% or of 324 million is 81. So there will be 81 million people over 65 in the United States in 2010; enter  81 in the box. 4. Take bite-sized pieces. Marat bought his condo for 58% of $175,000 = $101,500. (Alternatively, you could take 42% of $175,000 and subtract that from $175,000 to get $101,500.) Choice (B) is correct. Marat later sold his condo for 18% more than he paid for it: 1.18 × $101,500 = $119,770. Calculate 36% of that selling price: 0.36 × $119,700 = $43,117.20. Choice (C) is correct. Because you know only the asking price of the person who sold Marat the condo, and not how much he or she paid for it, (A) may or may not be true. Eliminate it. The correct answers are (B) and (C). 5. Start by translating: 0.25p = 0.65(80), so 0.25p = 52 and p = 208. That means q = 0.50(208) = 104. Now replace p and q in the answer choices with the appropriate values. For (A), you have 65 = × 104, which is true. Choice (A) is correct. For (B), 104 = × 80. This equation is also true, so keep (B). Choice (C) is also correct because 208 = × 104. Finally, (D) is correct because = , which is 50% of 1. The correct answers are (A), (B), (C), and (D). 6. Whenever you see a variable in one column and an actual number in the other column, try plugging the number in for the variable. In this case, if John withdrew 25% of his savings, that would be $3,475, and his balance would still be $10,425. Therefore, he must withdraw more than 25% in order for his balance to dip below $10,000. The answer is (A). EXPONENTS The superscripted number to the upper right corner of an integer or other math term is called an exponent, and it tells you the number of times that number or variable is multiplied by itself. For example, 54 = 5 × 5 × 5 × 5. The exponent is 4, and the base is 5. You can add or subtract two exponential terms as long as both the base and the exponents are the same. 5x5 + x5 = 6x5 6b3 − 4b3 = 2b3 15ab2c3 − 9ab2c3 + 2ab2c3 = 8ab2c3 Multiplying and Dividing Exponential Terms Let’s say we’re multiplying a2 × a3. If we expand out those terms, then we get (a × a)(a × a × a) = a5 = a(2+3). So when multiplying terms that have the same base, add the exponents. Note that the terms have to have the same base. If we’re presented with something like a2 × b3, then we can’t simplify it any further than that. Now let’s deal with division. Let’s start by expanding out an exponent problem using division, to see what we can eliminate. Notice that the three a terms on the bottom canceled out with three a terms on top? We ended up with a2, which is the same as a(5−3). So when dividing terms that have the same base, subtract the exponents. Parentheses with exponents work exactly as they do with normal multiplication. (ab)3 = (ab)(ab)(ab) = a3b3. When a term inside parentheses is raised to a power, the exponent is applied to each individual term within the parentheses. What if a term inside the parentheses already has an exponent? For example, what if we have (a2)3 ? Well, that’s (a2)(a2)(a2) =