MANHATTAN PREP GMAT Advanced Quant GMAT STRATEGY GUIDE

MANHATTAN PREP GMAT Advanced Quant GMAT STRATEGY GUIDE This supplemental guide provides in-depth and comprehensive explanations of the advanced math skills necessary for the highest-level performance on the GMAT. GMAT® is a registered trademark of the Graduate Management Admissions council™. Manhattan Prep is neither endorsed by nor affiliated with GMAc. Table of Contents GMAT Advanced Quant Cover Title Page Copyright Instructional Guide Series Letter Introduction In This Chapter... A Qualified Welcome Who Should Use This Book Try It Yourself The Purpose of This Book An Illustration Learning How to Think Plan of This Book Solutions to Try-It Problems Part 1: Problem Solving and Data Sufficiency Strategies Chapter 1 Problem Solving: Advanced Principles In This Chapter... Chapter 1 Problem Solving: Advanced Principles Principle #1: Understand the Basics Principle #2: Build a Plan Principle #3: Solve—and Put Pen to Paper Principle #4: Review Your Work Problem Set Solutions Chapter 2: Problem Solving: Strategies & Tactics In This Chapter... Chapter 2 Problem Solving: Strategies & Tactics Advanced Strategies Advanced Guessing Tactics Problem Set Solutions Chapter 3: Data Sufficiency: Principles In This Chapter... Chapter 3 Data Sufficiency: Principles Principle #1: Follow a Consistent Process Principle #2: Never Rephrase Yes/No as Value Principle #3: Work from Facts to Question Principle #4: Be a Contrarian Principle #5: Assume Nothing Problem Set Solutions Chapter 4: Data Sufficiency: Strategies & Tactics In This Chapter... Chapter 4 Data Sufficiency: Strategies & Tactics Advanced Strategies Advanced Guessing Tactics Summary Common Wrong Answers Problem Set Solutions Part 2: Strategies for All Problem Types Chapter 5 Pattern Recognition In This Chapter... Pattern Recognition Problems Sequence Problems Units (Ones) Digit Problems Remainder Problems Other Pattern Problems Problem Set Solutions Chapter 6: Common Terms and Quadratic Templates In This Chapter... Chapter 6 Common Terms and Quadratic Templates Common Terms Quadratic Templates Quadratic Templates in Disguise Problem Set Solutions Chapter 7: Visual Solutions In This Chapter... Chapter 7 Visual Solutions Representing Objects with Pictures Rubber Band Geometry Baseline Calculations for Averages Number Line Techniques for Statistics Problems Problem Set Solutions Chapter 8: Hybrid Problems In This Chapter... Pop Quiz! Hybrid Problems Identify and Sequence the Parts Where to Start Minor Hybrids Problem Set Solutions Part 3: Practice Chapter 9 Workout Sets In This Chapter... Workout Set 1 Workout Set 1 Answer Key Workout Set 1 Solutions Workout Set 2 Workout Set 2 Answer Key Workout Set 2 Solutions Workout Set 3 Workout Set 3 Answer Key Workout Set 3 Solutions Workout Set 4 Workout Set 4 Answer Key Workout Set 4 Solutions Workout Set 5 Workout Set 5 Answer Key Workout Set 5 Solutions Workout Set 6 Workout Set 6: Answer Key Workout Set 6 Solutions Workout Set 7 Workout Set 7 Answer Key Workout Set 7 Solutions Workout Set 8 Workout Set 8 Answer Key Workout Set 8 Solutions Workout Set 9 Workout Set 9: Answer Key Workout Set 9 Solutions Workout Set 10 Workout Set 10 Answer Key Workout Set 10 Solutions Workout Set 11 Workout Set 11 Answer Key Workout Set 11 Solutions Workout Set 12 Workout Set 12 Answer Key Workout Set 12 Solutions Workout Set 13 Workout Set 13 Answer Key Workout Set 13 Solutions Workout Set 14 Workout Set 14 Answer Key Workout Set 14 Solutions Workout Set 15 Workout Set 15 Answer Key Workout Set 15 Solutions Workout Set 16 Workout Set 16 Answer Key Workout Set 16: Answers and Explanations mba Mission mba Mission Go Beyond Books. Try A Free Class Now. Prep Made Personal Acknowledgements A great number of people were involved in the creation of the book you are holding. Our Manhattan Prep resources are based on the continuing experiences of our instructors and students. The overall vision for this edition was developed by Chelsey Cooley, who determined what new areas to cover and who wrote all of the problems that are new to this edition. Chelsey served as the primary author of this edition and Emily Meredith Sledge was the primary editor; Emily also served as the primary author of the first edition of this guide. Mario Gambino managed production for the many—and quite complicated—images that appear in this guide. Matthew Callan coordinated the production work for this guide. Once the manuscript was done, Naomi Beesen and Ben Ku edited and Cheryl Duckler and Stacey Koprince proofread the entire guide from start to finish. Carly Schnur designed the covers. Retail ISBN: 978-1-5062-4993-3 Retail eISBN: 978-1-5062-4994-0 Course ISBN: 978-1-5062-4995-7 Course eISBN: 978-1-5062-4996-4 Copyright © 2020 Manhattan Prep, Inc. ALL RIGHTS RESERVED. No part of this work may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution—without the prior written permission of the publisher, MG Prep, Inc. GMAT® is a registered trademark of the Graduate Management Admission Council. Manhattan Prep is neither endorsed by nor affiliated with GMAC. GMAT® STRATEGY GUIDES GMAT All the Quant GMAT All the Verbal GMAT Integrated Reasoning and Essay STRATEGY GUIDE SUPPLEMENTS Math Verbal GMAT Foundations of Math GMAT Foundations of Verbal GMAT Advanced Quant January 7, 2020 Dear Student, Thank you for picking up a copy of Advanced Quant. I hope this book provides just the guidance you need to get the most out of your GMAT studies. At Manhattan Prep, we continually aspire to provide the best instructors and resources possible. If you have any questions or feedback, please do not hesitate to contact us. Email our Student Services team at or give us a shout at (or in the United States or Canada). We try to keep all our books free of errors, but if you think we’ve goofed, please visit Our Manhattan Prep Strategy Guides are based on the continuing experiences of both our instructors and our students. The primary author of the this edition of the Advanced Quant guide was Chelsey Cooley and the primary editor was Emily Meredith Sledge. Project management and design were led by Matthew Callan and Mario Gambino. I’d like to send particular thanks to instructors Stacey Koprince and Ben Ku for their content contributions. Finally, we are indebted to all of the Manhattan Prep students who have given us excellent feedback over the years. This book wouldn’t be half of what it is without their voice. And now that you are one of our students too, please chime in! I look forward to hearing from you. Thanks again and best of luck preparing for the GMAT! Sincerely, Chris Ryan Executive Director, Product Strategy Introduction In This chapter... Introduction A Qualified Welcome Welcome to GMAT Advanced Quant! In this venue, we decided to be a little nerdy and call the introduction Chapter 0. After all, the point (0, 0) in the coordinate plane is called the origin, isn’t it? (That’s the first and last math joke in this book.) Unfortunately, we have to qualify our welcome right away, because this book isn’t for everyone. At least, it’s not for everyone right away. Who Should Use This Book You should use this book if you meet the following conditions: You have achieved a scaled score of at least 47 (out of 51) on the Quant section of either the Manhattan Prep practice test or the official practice computer-adaptive test (cAT). You have worked through the Manhattan Prep All the Quant guide, which covers all of the topics and strategies you need for the Quant section, or you have worked through similar material from another company. This material should include the following: Algebra Fractions, Decimals, Percents, and Ratios Geometry Number Properties Word Problems You are already comfortable with the core principles in these topics. You want to raise your performance to a scaled score of 49 or higher. You want to become a significantly smarter test-taker. If you match this description, then please turn the page. If you don’t match this description, then you will probably find this book too difficult at this stage of your preparation. For now, you are better off working on topic-focused material, as found in the All the Quant guide, and ensuring that you have mastered that material before you return to this book. Try It Yourself Throughout the chapters of this guide, you’ll see Try-It problems— problems designed to test your skills on certain aspects of GMAT problems. Take a look at the following three Try-It problems, which are very difficult. They are at least as hard as any real GMAT problem—probably even harder. Go ahead and give these problems a try. You should not expect to solve any of them in two minutes. In fact, you might find yourself completely stuck. If that’s the case, switch gears. Do your best to eliminate some incorrect answer choices and take an educated guess. Try-It #0-1 A jar is filled with red, white, and blue tokens that are equivalent except for their color. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token. If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar? (A) 9 (B) 12 (C) 15 (D) 18 (E) 21 Try-It #0-2 Arrow , which is a line segment exactly 5 units long with an arrowhead at A, is constructed in the xy-plane. The x- and y- coordinates of A and B are integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these - properties can be constructed? (A) 50 (B) 168 (C) 200 (D) 368 (E) 536 Try-It #0-3 In the diagram to the right, the value of x is closest to which of the following? (A) (B) 2 (C) (D) (E) 1 (Note: This problem does not require any non-GMAT math, such as trigonometry.) The Purpose of This Book This book is designed to prepare you for the most difficult math problems on the GMAT. So…what is a difficult math problem, from the point of view of the GMAT? A difficult math problem is one that most GMAT test-takers get wrong under exam conditions. In fact, this is essentially how the GMAT measures difficulty: by the percent of test-takers who get the problem wrong. So what kinds of math questions do most test-takers get wrong? What characterizes these problems? There are two kinds of features: 1. Topical nuances or obscure principles Connected to a particular topic Inherently hard to grasp or unfamiliar Easy to mix up These topical nuances are largely covered in the Extra sections of the Manhattan Prep All the Quant guide. This book includes many problems that involve topical nuances. However, the exhaustive theory of divisibility and primes, for instance, is not repeated here. 2. Complex structures May use simple principles in ways that aren’t obvious May require multiple steps May make you consider many cases May combine more than one topic May need a flash of real insight to complete May make you change direction or switch strategies along the way Complex structures are essentially disguises for simpler content. These disguises may be difficult to pierce. The path to the answer is twisted or clouded somehow. To solve problems that have simple content but complex structures, you need approaches that are both more general and more creative. This book focuses on these more general and more creative approaches. The three problems on the previous page have complex structures; the solutions are a bit later in this chapter. In the meantime, take a look at another problem. An Illustration Give this problem a whirl. Don’t go on until you have spent a few minutes on it—or until you have figured it out. Try-It #0-4 What should the next number in this sequence be? 1 2 9 64 Note: This problem is not exactly GMAT-like, because there is no mathematically definite rule. However, you’ll know when you’ve solved the problem. The answer will be elegant. This problem has very simple content but a complex structure. Researchers in cognitive science have used sequence-completion problems such as this one to develop realistic models of human thought. Here is one such model, simplified but practical. Top-Down Brain and Bottom-Up Brain To solve the sequence-completion problem above, you need two kinds of thinking: You might even say that you need two types of brain. The top-down brain is your conscious self. If you imagine the contents of your head as a big corporation, then your top-down brain is the cEO, responding to input, making decisions, and issuing orders. In cognitive science, the top-down brain is called the executive function. Top-down thinking and planning is indispensable to any problem-solving process. But the corporation in your head is a big place. For one thing, how does information get to the cEO? And how preprocessed is that information? The bottom-up brain is your preconscious processor. After raw sensory input arrives, your bottom-up brain processes that input extensively before it reaches your top-down brain. For instance, to your optic nerve, every word on this page is just a lot of black squiggles. Your bottom-up brain immediately turns these squiggles into letters, joins the letters into words, summons relevant images and concepts, and finally serves these images and concepts to your top-down brain. This all happens automatically and swiftly. In fact, it takes effort to interrupt this process. Also, unlike your top-down brain, which does things one at a time, your bottom-up brain can easily do many things at once. How does all this relate to solving the sequence problem above? Each of your brains needs the other one to solve difficult problems. Your top-down brain needs your bottom-up brain to notice patterns, sniff out valuable leads, and make quick, intuitive leaps and connections. But your bottom-up brain is inarticulate and distractible. Only your top- down brain can build plans, pose explicit questions, follow procedures, and state findings. Imagine that you are trying to solve a tough murder case. To find all the clues in the woods, you need both a savvy detective and a sharp-nosed bloodhound. To solve difficult GMAT problems, try to harmonize the activity of your two brains by following an organized, fast, and flexible problem-solving process. You need a general step-by-step approach to guide you. One such approach, inspired by expert mathematician George Pólya, is Understand, Plan, Solve (UPS): 1. Understand the problem first. 2. Plan your attack by adapting known techniques in new ways. 3. Solve by executing your plan. You may never have thought explicitly about steps 1 and 2 before. It may have been easy or even automatic for you to Understand easier problems and to Plan your approach to them. As a result, you may tend to dive right into the Solve stage. This is a bad strategy. Mathematicians know that the real math on hard problems is not Solve; the real math is Understand and Plan. Speed is important for its own sake on the GMAT, of course. What you may not have thought as much about is that being fast can also lower your stress level and promote good process. If you know you can solve quickly, then you can take more time to comprehend the question, consider the given information, and select a strategy. To this end, make sure that you can complete calculations and manipulations fairly rapidly so that you can afford to spend some time on the Understand and Plan stages of your problem-solving process. A little extra time invested up front can pay off handsomely later. To succeed against difficult problems, you sometimes have to “unstick” yourself. Expect to run into brick walls and encounter dead ends. Returning to first principles and to the general process (e.g., making sure that you fully Understand the problem) can help you back up out of the mud. Let’s return to the sequence problem and play out a sample interaction between the two brains. The path is not linear; there are several dead ends, as you would expect. This dialog will lead to the answer, so don’t start reading until you’ve given the problem a final shot (if you haven’t already solved it). The top-down brain is labeled TD; the bottom-up brain is labeled BU. 1 2 9 64 TD: “Okay, let’s Understand this thing. At a glance, they’ve given me an increasing list of numbers, and they want me to find the number that “should” go in the blank, whatever “should” means. What’s a good Plan? Hmm. No idea. Stare at the numbers given?” BU notices that 9 = 32 and 64 = 82. Likes the two squares. 1 2 32 82 TD: “Write in the two squares.” 1 2 32 82 BU notices that 1 is a square, too. sq no sq sq TD: “Are they all perfect squares? No, since 2 isn’t.” BU doesn’t like this break in the pattern. 1 2 32 26 — TD: “Wait, back up. What about primes, factoring all the way. 8 = 23, and so 82 = (23)2 = 26.” BU notices 6 = 2 × 3, but so what? 1 2 32 (23)2 TD: “Let’s write 26 as (23)2. Anything there?” BU notices lots of 2’s and 3’s, but so what? TD: “Okay, keep looking at this. Are the 2’s and 3’s stacked somehow?” BU notices no real pattern. There’s 2−3−2 twice as you go across, but so what? And the 1 is weird by itself. 1 1 2 7 9 55 64 TD: “No good leads there. Hmm...time to go back to the original and try taking differences.” BU notices no pattern. The numbers look even uglier. 1 2 9 64 TD: “Hmm. No good. Go back to original numbers again. What’s going on there?” BU notices that the numbers are growing quickly, like squares or exponentials. 12 2 32 82 TD: “Must have something to do with those squares. I should look at those again.” BU notices a gap on the left, among the powers. 1? 21 32 82 TD: “How about looking at 2. Write it with exponents: 2 = 21. - Actually, 1 doesn’t have to be 12. One can be to any power and still be 1. The power is a question mark.” BU notices 21 then 32. Likes the counting numbers. BU really wants 1, 2, 3, 4 somehow. 1? 21 32 4?? TD: “Try 4 in that last position. Could the last term be 4 somehow?” BU likes the look of this. 8 and 4 are related. 1? 21 32 43 TD: “64 is 4 to the what? 42 = 16, times another 4 equals 64, so it’s 4 to the third power. That fits.” BU is thrilled: 1, 2, 3, 4 below and 1, 2, 3 up top. 10 21 32 43 TD: “Extend left. It’s 10. confirmed. The bases are 1, 2, 3, 4, etc., and the powers are 0, 1, 2, 3, etc.” BU is content. 10 21 32 43 54 TD: “So the answer is 54, which is 252, or 625.” Your own process was almost certainly different in the details. Also, your internal dialog was very rapid—parts of it probably only took fractions of a second to transpire. After all, you think at the speed of thought. The important thing is to recognize how the bottom-up bloodhound and the top-down detective worked together in the case above. The TD detective set the overall agenda and then pointed the BU bloodhound at the clues. The bloodhound did practically all the “noticing,” which in some sense is where all the magic happened. But sometimes the bloodhound got stuck, so the detective had to intervene, consciously trying a new path. For instance, 64 reads so strongly as 82 that the detective had to actively give up on that reading. There are so many possible meaningful sequences that it wouldn’t have made sense to apply a strict recipe from the outset: “Try X first, then Y, then Z…” Such an algorithm would require hundreds of possibilities. Should you always look for 1, 2, 3, 4? Should you never find differences or prime factors because they weren’t that useful here? Of course not! A computer can rapidly and easily apply a complicated algorithm with hundreds of steps, but humans can’t. (If you are an engineer or programmer, maybe you wish you could program your own brain, but so far, that’s not possible!) What humans are good at, though, is noticing patterns. The bottom-up brain is extremely powerful—far more powerful than any computer yet built. As you gather problem-solving tools, the task becomes knowing when to apply which tool. This task becomes harder as problem structures become more complex. But if you deploy your bottom-up bloodhound according to a general problem-solving process such as Understand, Plan, Solve, then you can count on the bloodhound to notice the relevant aspects of the problem—the aspects that tell you which tool to use. You can break down Understand, Plan, Solve into several discrete steps: Understand Glance at the problem briefly: does anything stand out? Read the problem. Jot down any obvious formulas or numbers. Plan Reflect on what you were given: what clues might help tell you how to approach this problem? Organize your approach: choose a solution path. Solve Work the problem! You’ll get lots of practice using the UPS process throughout this guide. Learning How to Think This book is intended to make you smarter. It is also intended to make you scrappier. That description encompasses two main ideas: employing GMAT strategies as well as textbook solution methods and knowing when to let go. If you have traditionally been good at paper-based standardized tests, then you may be used to solving practically every problem the “textbook” way. Problems that forced you to get down and dirty—to work backwards from the choices, to estimate and eliminate—may have annoyed you. A major purpose of this book is to help you learn to choose the best GMAT approach. On the hardest Quant problems, the textbook approach is often not the best GMAT approach. Unfortunately, advanced test-takers are sometimes very stubborn. Sometimes they feel they should solve a problem according to some theoretical approach. Or they fail to move to Plan B or C rapidly enough, so they don’t have enough time left to execute that plan. In the end, they might wind up guessing purely at random—and that’s a shame. GMAT problems often have back doors—ways to solve that don’t involve crazy computation or genius-level insights. Remember that in theory, GMAT problems can all be solved in two minutes. By searching for the back door, you might avoid all the bear traps that the problem writer set out by the front door! In addition to learning alternative solution methods, you also need to learn when to let go. As you know, the GMAT is an adaptive test. If you keep getting questions correct, the test will keep getting harder…and harder… and harder… At some point, there will appear a monster problem, one that announces “I must break you.” In your battle with this problem, you could actually lose the bigger war—even if you ultimately conquer this particular problem. Maybe it takes you eight minutes, or it beats you up so badly that your head starts pounding. This will take its toll on your score. This will happen to everyone, no matter how good you are at the GMAT. Why? The GMAT is not an academic test, though it certainly appears to be. Business schools are primarily interested in whether you’re going to be an effective businessperson. Good businesspeople are able to assess a situation rapidly, manage scarce resources, distinguish between good opportunities and bad ones, and make decisions accordingly. The GMAT wants to put you in a situation where the best decision is, in fact, to guess and move on, because business schools are interested in learning whether you have the presence of mind to recognize a bad opportunity and the discipline to let it go. Show the GMAT that you know how to manage your scarce resources (time and mental energy) and that you can recognize and cut off a bad opportunity. Plan of This Book The rest of this book has three parts: Part One: Problem Solving and Data Sufficiency Strategies Chapter 1: Problem Solving: Advanced Principles Chapter 2: Problem Solving: Strategies & Tactics Chapter 3: Data Sufficiency: Principles Chapter 4: Data Sufficiency: Strategies & Tactics Part Two: Strategies for All Problem Types Chapter 5: Pattern Recognition Chapter 6: Common Terms & Quadratic Templates Chapter 7: Visual Solutions Chapter 8: Hybrid Problems Part Three: Practice Workouts 1−16: Sixteen sets of 10 problems each The four chapters in Part I focus on principles, strategies, and tactics related to the two types of GMAT math problems: Problem Solving (PS) and Data Sufficiency (DS). The next four chapters, in Part II, focus on techniques that apply across several topics but are more specific than the approaches in Part I. Each of the eight chapters in Part I and Part II contains the following: Try-It Problems embedded throughout the text Problem Sets at the end of the chapter Many of these problems will be GMAT-like in format, but many will not. Part III contains sets of GMAT-like Workout problems, designed to exercise your skills as if you were taking the GMAT and seeing its hardest problems. Several of these sets contain clusters of problems relating to the chapters in Parts I and II, although the problems within each set do not all resemble each other in obvious ways. Other Workout problem sets are mixed by both approach and topic. Note that these problems are not arranged in order of difficulty. Also, you should know that some of these problems draw on advanced content covered in the Manhattan Prep All the Quant guide. Solutions to Try-It Problems If you haven’t tried to solve the first three Try-It problems in the Try It Yourself section at the beginning of this chapter, then go back and try them now. Think about how to get your top-down brain and your bottom-up brain to work together like a detective and a bloodhound. Come back when you’ve tackled the problems, even if you don’t get to an answer (in this case, do make a guess). In these solutions, we’ll outline sample dialogs between the top-down detective and the bottom-up bloodhound. Try-It #0-1 A jar is filled with red, white, and blue tokens that are equivalent except for their color. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token. If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar? (A) 9 (B) 12 (C) 15 (D) 18 (E) 21 SOLUTION TO TRY-IT #0-1 … jar is filled with red, white, and blue tokens … chance of randomly selecting … TD: “I need to Understand this problem first. There’s a jar, and it’s got red, white, and blue tokens in it.” BU notices “chance” and “randomly.” That’s probability. TD: “All right, this is a probability problem. Now, what’s the situation?” BU notices that there are two situations. … chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token. TD: “Let’s rephrase. In simpler words, if I pick a red, then a white, that’s the same chance as if I pick a blue. Jot that down. Okay, what else?” … number of tokens of every color is a multiple of 3 … BU doesn’t want to deal with this “multiple of 3” thing yet. … smallest possible total number of tokens in the jar? TD: “Okay, what are they asking me?” BU notices “smallest possible total number.” Glances at answer choices. They’re small, but not tiny. Hmm. TD: “Let’s Reflect for a moment to figure out a Plan. How can I approach this? How about algebra— if I name the number of each color, then I can represent each fact and also what I’m looking for. Okay, I use R, W, and B. Make probability fractions. Multiply red and white fractions. Simplify algebraically.” BU is now unsure. No obvious path forward. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token … TD: “Let’s start over conceptually. Reread the problem. can I learn anything interesting?” BU notices that blues are different. TD: “How are blues different? Hmm. Picking a red, then a white is as likely as picking a blue. What does that mean?” BU notices that it’s unlikely to pick a blue. So there aren’t many blues compared to reds or whites. Fewer blues than reds or whites B R and B W TD: “Are there fewer blues? Yes. Justify this. Focus on the algebraic setup.” In the very first equation above, each fraction on the left is less than 1, so their product is even smaller. The denominators of the three fractions are all the same. So the numerator of the product (B) must be smaller than either of the other numerators (R and W ). BU notices fractions less than 1. All positive. TD: “Two positive fractions less than 1 multiplied together give an even smaller number.” TD: “Yes, there are fewer blues.” BU is quiet. If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar? Neither R nor W can equal 3 (since B is smaller than either). Let R = W = 6. (A) and (B) are out now. The smallest possible total is now 15. TD: “Time to go back and reread the rest of the problem.” BU again notices “multiple of 3,” also in answer choices. Small multiples. TD: “change of Plan: Algebra by itself isn’t getting me there. What about plugging in a number? Try the most constrained variable: B. Since it’s the smallest quantity, but still positive, pretend B is 3. Execute this algebraically. Divide by RW.” BU likes having only two variables. TD: “Need to test other numbers. Apply constraints I know—B is the smallest number. Rule out answer choices as I go.” TD: “6 and 6 don’t work, because the right side adds up to larger than 1. The correct answer is (D). Let’s look at another pathway—one that moves more quickly to the back door. ALTERNATIVE SOLUTION TO TRY-IT #0-1 … chance of randomly selecting … BU notices “chance.” BU doesn’t like probability. TD: “Oh man, probability. Okay, let’s make sense of this and see whether there are any back doors. That’s the Plan.” … the number of tokens of every color is a multiple of 3 … BU notices that there are only limited possibilities for each number. TD: “Okay, every quantity is a multiple of 3. That simplifies things. There are 3, 6, 9, etc., of each color.” A jar is filled with red, white, and blue tokens … BU is alert—what about 0? TD: “What about 0? Hmm…the wording at the beginning assumes that there actually are tokens of each color. So there can’t be 0 tokens of any kind.” (A) 9 (B) 12 (c) 15 (D) 18 (E) 21 TD: “Now let’s look at the answer choices.” BU notices that they’re small. (A) 9 TD: “Try plugging in the choices. Let’s start at the easy end—in this case, the smallest number.” BU notices 9 = 3 + 3 + 3. Select a red: Select a white: TD: “The only possible way to have 9 total tokens is to have 3 reds, 3 whites, and 3 blues. So…does that work? Plug into probability formula.” , which is not "select a blue" (A) 9 (B) 12 (c) 15 (D) 18 (E) 21 (B) 12 Select a red: TD: “No, that doesn’t work. This is good. Knock out (A). Let’s keep going. Try (B).” BU notices 12 = 3 + 3 + 6. TD: “Only way to have 12 total is 3, 3, and 6. Which one’s which? Picking a red and then a white is the same as picking a blue, so the blue should be one of the 3’s. Let’s say red is 3 and white is 6.” Select a white: , which is not “select a blue” (A) 9 (B) 12 (c) 15 (D) 18 (E) 21 (c) 15 TD: “That doesn’t work either. Knock out (B). Keep going.” BU notices 15 has a few options. Select a red: Select a white: TD: “I can make 15 by 3, 6, and 6 or by 3, 3, and 9. Try 3−6−6; make blue the 3.” TD: “Nope. What about 3−3−9.” blue" , which is not "select a , TD: “Not this one either.” which is not "select a blue" (A) 9 (B) 12 (c) 15 (D) 18 (E) 21 (D) 18 TD: “Knock out (c). Try (D).” TD: “Maybe 3-6-9 first. Make blue the 3.” TD: “That’s it! Answer’s (D).” , which IS "select a blue" Many people find this second approach less stressful and more efficient than the textbook approach. In fact, there is no way to find the correct answer by pure algebra. Ultimately, you have to test suitable numbers. Try-It #0-2 Arrow , which is a line segment exactly 5 units long with an arrowhead at A, is constructed in the xy-plane. The x- and y- coordinates of A and B are integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9. How many different arrows with these properties can be constructed? (A) 50 (B) 168 (C) 200 (D) 368 (E) 536 SOLUTION TO TRY-IT #0-2 properties can be constructed? Reread the question.” BU wonders which properties. … exactly 5 units long with an arrowhead at A … the x- and y-coordinates of A and B are to be integers that satisfy the inequalities 0 c x c 9 and 0 c y c 9. TD: “What are the properties of the arrows supposed to be again? Each arrow is 5 units long.” BU notices “integers” and “coordinates” and pictures a pegboard. TD: “Reflect. The tip and the end of the arrow have to touch holes in the pegboard exactly. Okay. The Plan is to start counting. How to Organize?” BU imagines many possible arrows. Brute force can’t be the right way forward. The arrows can point in all sorts of different ways. TD: “Let’s simplify the Plan. Let’s focus on just one orientation of arrows—pointing straight up. Draw this situation. How many places can the arrow be?” BU wants to go up & down, then right & left. In one column, there are 5 positions for the arrowhead: y = 5, 6, 7, 8, or 9. That’s the same as 9 − 5 + 1, by the way. TD: “count the positions in one column, then multiply by the number of columns. Be careful to count endpoints.” There are 10 identical columns: x = 0 through x = 9. 5 × 10 = 50 possible positions for the arrow pointing straight up. 50 × 2 = 100 possible positions for the arrow if it points straight up or to the right. TD: “Great. I’ve Solved one part. Other possibilities?” BU notices the square is the same vertically as horizontally. Go right. TD: “I get the same result for arrows pointing right. 50 more positions. Is that it? Am I done?” 50 × 4 = 200 possible positions BU wonders about “down” and “left.” TD: “These arrows can point straight down or straight left, too. Those would have the same result. So there are 50 positions in each of the four directions. Calculate at this point and evaluate answers. Eliminate (A) and (B).” Answer seems to be (C). BU is suspicious: somehow too easy. TD: “Tentative answer is (C), but I'm not done.” BU wonders about still other ways for the arrows to point. Three up, four across: TD: “Could the arrows be at an angle?” BU notices that the arrow is 5 units long, associated with 3−4−5 triangles. TD: “3−4−5 triangles. Yes. Put the arrow as the hypotenuse of a 3−4−5 triangle. How can this be done? Try to place the arrow. Remember the reversal. Looks like there are four ways if I go 3 up and 4 across: up right, up left, down right, down left.” BU is happy. This is the trick. Four up, three across: TD: “Likewise, there must be four ways if I go 4 up and 3 across: again, up right, up left, down right, and down left. By the way, the answer must be (D) or (E).” Three up, four across, pointing up to the right: There are 7 positions vertically for the arrowhead (9 − 3 + 1) and 6 positions horizontally (9 − 4 + 1), for a total of 7 × 6 = 42 positions. 8 × 42 = 336 possible positions at an angle. In total, there are 200 + 336 = 536 positions. TD: “Now count just one of these ways. Same ideas as before. Be sure to include endpoints.” BU notices the symmetry. The 3 up, 4 across is the same as the 4 up, 3 across, if you turn the square. TD: “Each of these angled ways will be the same. There are 8 ways to point the arrow at an angle. Finish the calculation and confirm the answer.” The correct answer is (E). There isn’t much of an alternative to the approach above. With counting problems, it can often be very difficult to estimate the answer or work backwards from the answer choices. Try-It #0-3 In the diagram to the right, the value of x is closest to which of the following? (A) (B) 2 (C) (D) (E) 1 SOLUTION TO TRY-IT #0-3 TD: “Hmm...here’s a Plan: add a perpendicular line to make right triangles. Drop the line from the top point. I'll label corners while I'm at it. Now fill in angles.” BU notices 45−45−90 and is happy. TD: “Use the 45−45−90 to write expressions for its sides. Then can be split up into two pieces, and I can set up the Pythagorean theorem.” BU feels that this process is kind of ugly. TD: “Let’s push through. Write the Pythagorean theorem for the small triangle on the left, using the as the hypotenuse.” BU thinks this equation is really ugly. TD: “Push through. Expand the quadratic and simplify.” BU doesn’t like the square root on the bottom. TD: “Multiply by to get rid of it on the - bottom of the fraction.” BU has no idea how to take the square root of this. TD: “Neither do I. Let’s try estimating. If x2 is about 3.5, then the square root must be a bit less than 2 (since the square root of 4 is 2). 182 is 324 and 192 is 361, so the answer is around 1.8 or 1.9.” TD: “Answer (A) is about 3.5; that matches the squared value, not the square root. The answer needs to be less than 2, so (B) is also wrong. is about 1.7. Answers (D) and (E) are too small, so the answer is (c).” (A) 2 + (B) 2 (c) (D) (E) 1 The correct answer is (c). The method you just saw is algebraically intensive, and so your bottom-up bloodhound might have kicked up a fuss along the way. Sometimes, your top-down brain needs to ignore the bottom-up brain. Remember, when you’re actually taking the GMAT, you have to solve problems quickly—and you don’t need to publish your solutions in a mathematics journal. What you want is to get the correct answer as quickly and as easily as possible. In this regard, the solution above works perfectly well. Alternatively, the question stem asks for an approximate answer, so you can also try estimating from the start. Draw the triangle carefully and start with the same perpendicular line as before. This line is a little shorter than the side of length (which is about 1.4). call the two shorter legs 1.2 and calculate the hypotenuse. It equals 1.2 multiplied by 1.4, or approximately 1.7. (Bonus question: How can you estimate that math quickly? Answer below.) Now, examine the answer choices using 1.4 for and 1.7 for : (A) (B) (c) (D) (E) 3.4 2 1.7 1.4 1 They’re all close, but you can pretty confidently eliminate answers (A) and (E). Furthermore, the answer needs to be less than 2, so (B) can’t be it. Answer (c) is closer than (D), so (c) is probably it. Unfortunately, you might guess wrong at this point. But the odds are much better than they were at the outset. It is worthwhile to look for multiple solution paths as you practice. Your top-down brain will become faster, more organized, and more flexible, enabling your bottom-up brain to have more flashes of insight. That was a substantial introduction. Now, on to chapter 1! PART ONE Problem Solving and Data Sufficiency Strategies CHAPTER 1 Problem Solving: Advanced Principles In This chapter... Chapter 1 Problem Solving: Advanced Principles Chapters 1 and 2 of this book focus on the more fundamental of the two types of GMAT math questions: Problem Solving (PS). Some of the content applies to any kind of math problem, including Data Sufficiency (DS). However, Chapters 3 and 4 deal specifically with DS issues. This chapter outlines broad principles for solving advanced PS problems. You’ve already seen very basic versions of the first three principles in the introduction, in the dialogs between the top-down and the bottom-up brain. As mentioned earlier, these principles draw on the work of George Pólya, who was a brilliant mathematician and teacher of mathematics. Pólya was teaching future mathematicians, not GMAT test-takers, but what he said still applies. His little book How to Solve It has been in print since 1945—it’s worth getting a copy. In the meantime, keep reading. Principle #1: Understand the Basics Take time to think and plan before you start solving a difficult problem. If Quant is your strength, you may want to dive straight into every problem as soon as you see it, without pausing to consider all of the angles. There are two good reasons to slow down: 1. You need to manage your time and mental energy across the entire GMAT. If you pause briefly to find a more efficient solution, you’ll save time and energy for other problems. 2. If you start doing math without thinking first, you might have to change your approach later in the problem, which takes time that you don’t have. You also risk falling for traps. To remind yourself to slow down and plan, understand each problem by taking three steps: Glance at the entire problem: is it PS or DS? If it’s PS, glance at the answer choices. If it’s DS, glance at the statements. Knowing what type of problem you’re dealing with will help you read more effectively. Pólya recommended that you ask yourself simple questions as you read a problem. Here are some great Pólya-style questions that can help you understand: What exactly is the problem asking for? What information would I need in order to find the answer? What information do I already have? What information don’t I have? Sometimes you care about something you don’t know. This could be an intermediate unknown quantity that you didn’t think of earlier. Other times, you don’t know something, and you don’t care. For instance, if a problem includes the quantity 11! (11 factorial), you will practically never need to know the exact value of that quantity. What is this problem testing? In other words, why is this problem on the GMAT? What aspect of math are they testing? What kind of reasoning do they want me to demonstrate? Have I seen a problem like this before? Have you already solved a similar problem? What approach worked best? You don’t need to meticulously go through every one of these questions whenever you solve a problem. (However, that’s a good thing to do when you review a problem!) They’re here to help you consider how you might read more productively. As you read the problem, jot down any given numbers or formulas on your scrap paper. That doesn’t mean you should start doing math while you’re trying to read. If you start trying to solve the problem when you haven’t even finished reading it, you’re getting ahead of yourself. Start the following problem by taking these three steps: Glance, Read, and Jot. Try-It #1-1 x = 910 − 317 and is an integer. If n is a positive integer that has exactly two factors, how many different values for n are possible? (A) One (B) Two (C) Three (D) Four (E) Five Glance. This is a PS problem. The answers are numbers but in written form; this format is reserved for problems that ask for the number of numbers or number of possibilities for something. The numbers are small. Read. Dive into the text. Here are some possible answers to the Pólya questions: What exactly is the problem asking for? The number of possible values for n. This means that n might have multiple possible values. In fact, it probably can take on more than one value. I may not need these actual values. I just need to count them. What are the quantities I care about? I’m given x and n as variables. These are the quantities I care about. What do I know? x = 910 − 317 That is, x = a specific large integer, expressed in terms of powers of 9 and 3. is an integer. That is, x is divisible by n, or n is a factor of x. Finally, n is a positive integer that has exactly two factors. Prime numbers have exactly two factors. So I can rephrase the information: n is prime. (Primes are always positive.) What don’t I know? Here’s something I don’t know: I don’t know the value of x as a series of digits. Using a calculator or Excel, I could find out that x equals 3,357,644,238. But I don’t know this number at the outset. Moreover, because this calculation is far too cumbersome, it must be the case that I don’t need to find this number. What is this problem testing? From the foregoing, I can infer that this problem is testing Divisibility & Primes. I will probably also need to manipulate exponents, since I see them in the expression for x. You can ask these questions in whatever order is most helpful for the problem. For instance, you might not look at what the problem is asking for until you’ve understood the given information. Jot. As you decide that a piece of information is important, jot it down on your scrap paper. At this point, your scrap paper might look something like this: Principle #2: Build a Plan Next, think about how you will solve the problem: Reflect. Here are some Pólya questions that help you think about what you know and come up with a plan: Is a good From your answers above, you may already see a way to reach the answer. If approach you can envision the rough outlines of the correct path, then go ahead and already get started. obvious? If not, what in If you are stuck, look for particular clues to tell you what to do next. Revisit the problem your answers to the basic questions. What do those answers mean? can you can help me rephrase or reword them? can you combine two pieces of information in any figure out a way, or can you rephrase the question, given everything you know? good approach? can I Try relating the problem to other problems you’ve faced. This can help you remember a categorize the problem or recall a solution process. similar problem? Organize. For the Try-It #1-1 problem, some of the information is already rephrased (reorganized). Go further now, combining information and simplifying the question: Given: n is a prime number AND n is a factor of x. combined: n is a prime factor of x. Question: How many different values for n are possible? combined: How many different values for n, a prime factor of x, are possible? Rephrased: How many distinct prime factors does x have? You need the prime factorization of x. Notice that n is not even in the question anymore. The variable n just gave you a way to ask this underlying question. consider the other given fact: x = 910 − 317. It can be helpful initially to put certain complicated facts to the side. At this stage, however, you know that you need the prime factors of x. So now you have the beginning of a plan: factor this expression into its primes. Principle #3: Solve—and Put Pen to Paper The third step is to do the work: solve. You’ll want to execute that solution in an error-free way—it would be terrible to get all the thinking correct, then make a careless computational mistake. That’s why we say you should put pen to paper. In the expression 910 − 317, the 3 is prime but the 9 is not. Since the problem is asking about prime factors, rewrite the equation in terms of prime numbers: Next, pull out a common factor from both terms. The largest common factor is 317: Now, you have what you need: the prime factorization of x. The number x has three distinct prime factors: 2, 3, and 13. The correct answer is (c). The idea of putting pen to paper also applies when you get stuck anywhere along the way on a monster problem. Think back to those killer Try-It problems in the introduction. Those are not the kinds of problems you can figure out just by looking at them. When you get stuck on a tough problem, take action. Do not just stare, hoping that you suddenly get it. Instead, ask yourself the Pólya questions again and write down whatever you can: Reinterpretations of given information or of the question Intermediate results, whether specific or general Avenues or approaches that didn’t work This way, your top-down brain can help your bottom-up brain find the correct leads—or help it let go. In particular, it’s almost impossible to abandon an unpromising line of thinking without writing something down. Think back to the sequence problem in the introduction. You’ll keep seeing 64 as 82 unless you try writing it in another way. Do not try to juggle everything in your head. Your working memory has limited capacity, and your bottom-up brain needs that space to work. A multistep problem cannot be solved in your brain as quickly, easily, and accurately as it can be on paper. As you put pen to paper, there are three themes you'll want to keep in mind. 1. LOOK FOR PATTERNS Every GMAT Quant problem has a two-minutes-or-faster solution path, which may depend upon a pattern that you’ll need to extrapolate. You’ll know a pattern is needed when a problem asks something that would be impossible to calculate (without a calculator) in two minutes. When this happens, write out the first five to eight items in the sequence or list in order to try to spot the pattern. Try-It #1-2 for all integer values of n greater than 1. If S1 = 1, what is the sum of the first 61 terms in the sequence? (A) −48 (B) −31 (C) −29 (D) 1 (E) 30 Nobody is going to write out all 61 terms and then add them up in two minutes. There must be a pattern. The recursive definition of Sn doesn’t yield any secrets upon first glance. So write out the early cases in the sequence, starting at n = 1 and looking for a pattern: etc. The terms of the sequence are . . . . Three terms repeat in this cyclical pattern forever; every third term is the same. Note: If you don’t spot a pattern within the first five to eight terms, stop using this approach and see whether there’s another way (including guessing!). The problem asks for the sum, so find the sum of each group of three consecutive terms: . There are 20 groups in the first 61 terms, and one additional term that hasn’t been counted yet. So the sum of the first 61 terms is as follows: The correct answer is (c). It is almost impossible to stare at the recursive definition of this sequence and discern the resulting pattern. The best way to identify the pattern is to calculate a few values of the sequence and look for the pattern. You will learn more about pattern recognition in chapter 5. 2. DRAW IT OUT Some problems are much easier to solve if you draw out what’s happening in the problem. Whenever a story problem describes something that could actually happen in the real world, you could try to draw out the solution. For instance, if a problem involves motion, you can draw snapshots representing the problem at different points in time. Try-It #1-3 Truck A is on a straight highway heading due south at the same time Truck B is on a different straight highway heading due east. At 1:00 p.m., Truck A is exactly 14 miles north of Truck B. If both trucks are traveling at a constant speed of 30 miles per hour, at which of the following times will they be exactly 10 miles apart? (A) 1:10 p.m. (B) 1:12 p.m. (C) 1:14 p.m. (D) 1:15 p.m. (E) 1:20 p.m. Represent Truck A and Truck B as of 1:00 p.m. How does the distance between Truck A and Truck B change as time goes by? Try another point in time. Since the answers are all a matter of minutes after 1:00 p.m., try a convenient increment of a few minutes. After 10 minutes, each truck will have traveled 5 miles (30 miles per 60 minutes = 5 miles in 10 minutes). How far apart will the trucks be then? On the diagram to the right, the distance is represented by x. Because Truck A is traveling due south and Truck B is traveling due east, the triangle must be a right triangle. Therefore, x2 = 92 + 52. At this point, you could solve the problem in one of two ways. The first is to notice that once both trucks travel 6 miles, the diagram will contain a 6 : 8 : 10 triangle. Therefore, of an hour later, at 1:12 p.m., the trucks will be exactly 10 miles apart. Alternatively, you could set up an algebraic equation and solve for the unknown number of miles traveled, such that the distance between the trucks is 10. call that distance y: Therefore, y could equal 6 or 8 miles. In other words, the trucks will be exactly 10 miles apart at 1:12 p.m. and at 1:16 p.m. Either way, the correct answer is (B). Notice how instrumental these diagrams were for the solution process. You may already accept that Geometry problems require diagrams. However, many other kinds of problems can benefit from visual thinking. You will learn more about advanced visualization techniques in Chapter 7. 3. SOLVE AN EASIER PROBLEM A problem may contain large numbers or complicated expressions that actually distract you from the task at hand: finding a solution path. When this happens, one tactic is to simplify part of the problem and solve that. Once you understand how the math works, return to the more complex problem and apply the same solution path. Try-It #1-4 If x and y are positive integers and is the square of an odd integer, what is the smallest possible value of xy ? (A) 1 (B) 8 (C) 10 (D) 15 (E) 28 As you read, jot down the given information. Note that you might not immediately write down the square of an odd integer info if you still have to puzzle out what it means: What does the square of an odd integer look like? List out a few examples, on paper or in your head: Are there any patterns or commonalities? All of the numbers are odd. All of the numbers are perfect squares. Therefore, is an odd perfect square. Add that to your notes. The question asks for the smallest possible value of xy. What do you need to figure out in order to find that? If the expression is distracting you, try figuring out what this would mean for a simpler version of the expression. Interesting. How can you apply that thinking to the real problem? It’s still true that, in order for to be odd, you have to get rid of the even factors in the numerator. In other words, y2 must cancel out all the even factors in 1,620. The y2 must contain at least two 2’s, so y itself has to contain at least one 2. Okay, that takes care of y: at minimum, y must be 2. If so, then the expression becomes . Now, what about x? If you’re not sure, return to your simpler problem thinking. Back to the real problem. Make sure that 405x has two pairs of every factor. 405 contains only one 5, so x must contain another 5. Also contained in 405 is 81, which is 92. That set of factors already represents a perfect square, so the minimum requirement is that x equals 5. If y must be 2, at minimum, and x must be 5, at minimum, then the smallest - possible value of xy is 10. You can generalize this approach. If a problem has many complexities, you can attack it by ignoring some of the complexities at first. Solve a simpler problem. Then, see whether you can adjust the solution to the simpler problem in order to solve the original. To recap, put your work on paper. Don’t try to solve hard problems in your head. Instead, do the following: Find a pattern: Write out the first few cases. Visualize a scene: Draw it out! Solve an easier problem, then apply your method to the harder problem. In general, jot down intermediate results as you go. You may see them in a new light and consider how they fit into the solution. Also, try to be organized. For instance, make tables to keep track of cases. The more organized you are, the more insights you will have into difficult problems. Principle #4: Review Your Work When you are done with a test or practice set, you are not really done. When you first do a problem under timed conditions, your brain is too busy solving the problem to effectively learn and remember. What you learn from a new problem comes after you’ve finished it and picked your answer, when you look at it with a clear head and no timer. Give yourself twice as much time to review each problem as you spent doing the problem in the first place. Here are some things you might consider as you review a problem. Most of these questions are useful even if you got the problem correct. Don’t restrict yourself to reviewing problems you got wrong. Review any problem you might learn something from and ask yourself: What are all of the pathways to the answer? Which is the best? What is the easiest and fastest way to implement it? What clues in the problem told me to use a certain approach or take a certain step? If I see one of those clues in a different problem, what should I do? What traps or tricks are built into this problem? Where could I have made a mistake? If I did make a mistake, what went wrong in my problem-solving process? Do I need to change how I approach similar problems? What could I take from this problem to help me solve other problems in the future? When you do the following problem set, apply the first three principles from this chapter to each problem: Understand, Plan, and Solve. Then, review each problem in depth. As you review, do two things: 1. Identify exactly what the problem is asking for and what that means in the simplest possible terms. 2. Note at least one general takeaway that might be useful on other problems in the future. The solutions include our own responses to these two tasks. Yours might look different, and that's fine. Problem Set 1. Each factor of 210 is inscribed on its own plastic ball and all of the balls are placed in a jar. If a ball is randomly selected from the jar, what is the probability that the ball is inscribed with a multiple of 42 ? (A) (B) (c) (D) (E) 2. If x is a positive integer, what is the units digit of (24)5 + 2x(36)6(17)3 ? (A) 2 (B) 3 (c) 4 (D) 6 (E) 8 3. A baker makes a combination of chocolate chip cookies and peanut butter cookies for a school bake sale. His recipes only allow him to make chocolate chip cookies in batches of 7 and peanut butter cookies in batches of 6. If he makes exactly 95 cookies for the bake sale, what is the minimum number of chocolate chip cookies that he could make? (A) 7 (B) 14 (c) 21 (D) 28 (E) 35 4. A rectangular solid is changed such that the width and length are each increased by 1 inch and the height is decreased by 9 inches. Despite these changes, the new rectangular solid has the same volume as the original rectangular solid. If the width and length of the original rectangular solid are equal and the height of the new rectangular solid is 4 times the width of the original rectangular solid, what is the volume of the rectangular solid? (A) (B) (c) (D) (E) 18 50 100 200 400 5. The sum of all distinct solutions for x in the equation x2 − 8x + 21 = |x – 4| + 5 is equal to which of the following? (A) −7 (B) 7 (c) 10 (D) 12 (E) 14 Solutions Each solution addresses the two steps from the instructions: 1) Identify exactly what the problem is asking for, and what that means in the simplest possible terms. 2) Note at least one general takeaway that might useful on other problems in the future. 1. (c) 1. What it’s asking: The problem is asking for the probability that the selected ball is a multiple of 42. The quantities you care about are the factors of 210. What you know: There are many balls, each with a different factor of 210. Each factor of 210 is represented. One ball is selected randomly. Some balls have a multiple of 42 (e.g., 42 itself ); some do not (e.g., 1). What you don’t know: How many factors of 210 there are How many of these factors are multiples of 42 What the problem is testing: Probability; Divisibility & Primes The real question: Plan: 210 to primes → build full list of factors from prime components → distinguish between multiples of 42 and non-multiples → count factors → compute probability. Alternatively, you could list all the factors of 210 using factor pairs. 1 210 2 105 3 70 5 42 6 35 7 30 10 21 14 15 There are 16 factors of 210, and two of them (42 and 210) are multiples of 42. You can also count the factors using 210’s prime factorization: (2)(3)(5) (7) = (21)(31)(51)(71). Here’s a shortcut to determine the number of distinct factors of 210. Add 1 to the power of each prime factor and multiply: There are 16 different factors of 210: 2 x 2 x 2 x 2 = 16. How many of these 16 factors are multiples of 42? 42 itself is a multiple of 42, of course. To find any others, divide 210 by 42 to get 5. This number is a prime, so the only other possible factor is 42 × 5, or 210. There are two multiples of 42 out of a total of 16 factors, so the probability is . The correct answer is (c). 2. At least one takeaway: The problem is straightforward in one sense: it says the word factor explicitly. Listing all the factors is feasible in two minutes, but you do need to be going down that solution path fairly quickly because it will take some time. It may be slightly faster to use the factor-counting shortcut, but only if you do know how to deal with the multiples of 42. 2. (A) 2: 1. What it’s asking: The problem asks for the units digit. Because the problem talks about a product, you care only about the units digits, not the overall values. Furthermore, the problem provides crazy numbers; you are absolutely not going to multiply these out. There must be some kind of pattern at work. Use the Last Digit Shortcut (discussed in the All the Quant guide). What jumps out? If x is a positive integer, then 2x must be even and 5 + 2x must be odd. Units digit of (24)5 + 2x = units digit of (4)odd. The pattern for the units digit of 4integer = [4, 6]. Thus, the units digit is 4. Units digit of (36)6 must be 6, as every power of 6 ends in 6. Units digit of (17)3 = units digit of (7)3. The pattern for the units digit of 7integer = [7, 9, 3, 1]. Thus, the units digit is 3. The product of the units digits is (4)(6)(3) = 72, which has a units digit of 2. The correct answer is (A). 2. At least one takeaway: Patterns were very important on this one! If you forget any of the units digit patterns, start listing out the early cases. At most, you’ll need to list four cases to find the pattern. 3. (E) 35: 1. What it’s asking: The problem asks for the minimum number of chocolate chip cookies. Given: The baker only makes chocolate chip (c) or peanut butter (P) cookies. He can only make chocolate chip cookies in batches of 7 and peanut butter cookies in batches of 6. He makes exactly 95 cookies total. What jumps out? c and P must be integers. Therefore: The answer choices are small multiples of 7, so work backwards from the answers on this problem. Because the problem asks you to minimize the number of chocolate chip cookies, start with the smallest answer choice. Make a chart: 7c 6P = 95 − 7c Is 6P a multiple of 6? (i.e., Is P an integer?) 7 88 N 14 81 N 21 74 N 7c 6P = 95 − 7c Is 6P a multiple of 6? (i.e., Is P an integer?) 28 67 N 35 60 Y Use the answer choices to calculate the value of 6P. cross off an answer choice if 6P is not a multiple of 6. The first ans