GMAT sets, probability, and counting note: What is the probability of Julia winning if the jar contains 1 gray, 2 white and 4 green balls? Let's draw all possible outcomes and calculate all probabilities.


GMAT sets, probability, and counting note: Now, It is pretty obvious that the probability of Julia's win is:. C5 C8 C10 p = 1 − = 1 − = ∗C5 1 C8 C10 p = 1 ∗ ∗ = 8 P = + ∗ + ∗ = 4.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Symmetry Symmetry sometimes lets you solve seemingly complex probability problem in a few seconds.


GMAT word problem note: Bob and Rachel want to sit such that Bob is always left to Rachel. How many ways it can be done?


GMAT word problem note: Because of symmetry, the number of ways that Bob is left to Rachel is exactly 1/2 of all possible ways:.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Enumeration Enumeration is a method of counting all possible ways to arrange elements.


GMAT sets, probability, and counting note: Although it is the simplest method, it is often the fastest method to solve hard GMAT problems and is a pivotal principle for any other combinatorial method. In fact, combination and permutation is shortcuts for enumeration.


GMAT sets, probability, and counting note: The main idea of enumeration is writing down all possible ways and then count them. There are three marbles: 1 blue, 1 gray and 1 green.


GMAT sets, probability, and counting note: In how many ways is it possible to arrange marbles in a row? Let's write out all possible ways:.


GMAT sets, probability, and counting note: In how many ways is it possible.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. To arrange marbles in a row if blue and green marbles have to be next to each other?


GMAT statistics note: Let's write out all possible ways to arrange marbles in a row and then find only arrangements that satisfy question's condition:. There are three marbles: 1 blue, 1 gray and 1 green.


GMAT statistics note: In how many ways is it possible to arrange marbles in a row if gray marble have to be left to blue marble? Arrangements of n different objects Enumeration is a great way to count a small number of arrangements.


GMAT statistics note: But when the total number of arrangements is large, enumeration can't be very useful, especially taking into account GMAT time restriction. Fortunately, there are some methods that can speed up counting of all arrangements.


GMAT statistics note: The number of arrangements of n different objects in a row is a typical problem that can be solve this way: 1. How many objects we can put at 1st place?


GMAT statistics note: How many objects we can put at 2nd place? We can't put the object that already placed at 1st place.


GMAT statistics note: How nany objects we can put at n-th place? Therefore, the total number of arrangements of n different objects in a.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Combination A combination is an unordered collection of k objects taken from a set of n distinct objects.


GMAT sets, probability, and counting note: The number of ways how we can choose k objects out of n distinct objects is denoted as: knowing how to find the number of arrangements of n distinct objects we can easily find formula for combination: 1. The total number of arrangements of n distinct objects is n!


GMAT sets, probability, and counting note: Now we have to exclude all arrangements of k objects (k!) and remaining (n-k) objects ((n-k)!) as the order of chosen k objects and remained (n-k) objects doesn't matter. Permutation A permutation is an ordered collection of k objects taken from a set of n distinct objects.


GMAT sets, probability, and counting note: Now we have to exclude all arrangements of remaining (n-k) objects ((n-k)!) as the order of remained (n-k) objects doesn't matter. If we exclude order of chosen objects from permutation formula, we will get combination formula:.


GMAT statistics note: Circular arrangement note: for n distinct objects arranged in a circle, the number of different relative arrangements is (n - 1)!. Unlike a row, rotating every object by one position does not create a new circular arrangement because the relative order stays the same.


GMAT statistics note: On GMAT counting questions, define what counts as a different arrangement before choosing a permutations or combinations formula.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Html of 1 All times are UTC - 8 hours [DST].


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.


GMAT statistics note: Standard Deviation https://gmatclub.com/forum/standard-deviation-87905.html STANDARD DEVIATION Definition Standard Deviation (SD, or STD or) - a measure of the dispersion or variation in a distribution, equal to the square root of variance or the arithmetic mean (average) of squares of deviations from the arithmetic mean.


GMAT statistics note: In simple terms, it shows how much variation there is from the "average" (mean). It may be thought of as the average difference from the mean of distribution, how far data points are away from the mean.


GMAT statistics note: A low standard deviation indicates that data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.


GMAT statistics note: Standard deviation measures the spread of a set around its mean. Key GMAT properties: standard deviation is always non-negative, and it equals 0 only when all values in the set are equal.


GMAT statistics note: If every value in a set is multiplied by the same constant, the standard deviation changes by that same factor in absolute value. If the same constant is added to or subtracted from every value, the standard deviation does not change because the spread stays the same.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. If a new element is added to set and standard deviation of a new set is , then: 1) if 2) if 3) if 4) is the lowest if Tips and Tricks GMAC in majority of problems doesn't ask you to calculate standard deviation.


GMAT statistics note: Instead it tests your intuitive understanding of the concept. In 90% cases it is a faster way to use just average of instead of true formula for standard deviation, and treat standard deviation as " average difference between elements and mean".


GMAT statistics note: Therefore, before trying to calculate standard deviation, maybe you can solve a problem much faster by using just your intuition. Not all points contribute equally to standard deviation.


GMAT statistics note: Taking into account that standard deviation uses sum of squares of deviations from mean, the most remote points will essentially contribute to standard deviation. The point 10 gives in sum of squares but point 6 gives only.


GMAT statistics note: So, when you need to find what set has the largest standard deviation, always look for set with the largest range because remote points have a very significant contribution to standard deviation. If we create a new set that consists of all elements of the initial set but decreased by 17%, what is the chan.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Means that all elements strictly related to each other.


GMAT statistics note: If we shift the set by adding or subtracting any integer, does it change standard deviation (average deviation of elements from the mean)? One thing we should know is the number of elements in the set, because the more elements we have the broader they are distributed relative to the mean.


GMAT statistics note: Now, look at DS statements, all we need it is just first statement. Standard deviation of set is 18.3.


GMAT statistics note: How many elements are 1 standard deviation above the mean? Now, we need to count all numbers greater than 42+18.3 = 60.3.


GMAT statistics note: There is a set A of 19 integers with mean 4 and standard deviation of 3. Now we form a new set B by adding 2 more elements to the set A.


GMAT statistics note: What two elements will decrease the standard deviation the most? A) 9 and 3 B) -3 and 3 C) 6 and 1 D) 4 and 5 E) 5 and 5 Solution:.


GMAT statistics note: The closer to the mean, the greater decrease in standard deviation. D has 4 (equal our mean) and 5 (differs from mean only by 1).


GMAT statistics note: All other options have larger deviation from mean. Normal distribution It is a more advance concept that you will never see in GMAT but understanding statistic properties of standard deviation can help you to be m.


GMAT quant note: Percent problems usually relate part, whole, rate, and percent change.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Html of 1 All times are UTC - 8 hours [DST].


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. 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 word problem note: GMAT® is a registered trademark of the Graduate Management Admissions CouncilTM. Manhattan Prep is neither endorsed by nor affiliated with GMAC.


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT strategy note: Translate the wording into variables, relationships, and equations before calculating. 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:.


GMAT strategy note: Advanced Principles In This Chapter... Advanced Principles Principle #1:.


GMAT strategy note: Understand the Basics Principle #2:. Solve—and Put Pen to Paper Principle #4:.


GMAT strategy note: Strategies & Tactics In This Chapter... Principles In This Chapter...


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Never Rephrase Yes/No as Value Principle #3:.


GMAT algebra note: Work from Facts to Question Principle #4:. Be a Contrarian Principle #5:.


GMAT algebra note: Strategies & Tactics In This Chapter... Strategies & Tactics Advanced Strategies Advanced Guessing Tactics Summary Common Wrong Answers Problem Set Solutions Part 2:.


GMAT algebra note: Common Terms and Quadratic Templates In This Chapter... Visual Solutions In This Chapter...


GMAT sequences and patterns note: Translate the wording into variables, relationships, and equations before calculating. Hybrid Problems In This Chapter...


GMAT sequences and patterns note: Hybrid Problems Identify and Sequence the Parts Where to Start Minor Hybrids Problem Set Solutions Part 3:.


GMAT sequences and patterns note: 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:.


GMAT sequences and patterns note: Answer Key Workout Set 6 Solutions Workout Set 7 Workout Set 7 Answer Key.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Workout Set 7 Solutions Workout Set 8 Workout Set 8 Answer Key Workout Set 8 Solutions Workout Set 9 Workout Set 9:.


GMAT quant note: 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:.


GMAT quant note: Answers and Explanations mba Mission mba Mission Go Beyond Books.


GMAT coordinate geometry note: 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.


GMAT coordinate geometry note: 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.


GMAT coordinate geometry note: 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.


GMAT coordinate geometry note: 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.


GMAT coordinate geometry note: 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 permissio.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. 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 Advanced Quant GMAT Foundations of Verbal.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. I hope this book provides just the guidance you need to get the most out of your GMAT studies.


GMAT word problem note: 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.


GMAT word problem note: Email our Student Services team at gmat@manhattanprep.com or give us a shout at 212-721-7400 (or 800-576-4628 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 manhattanprep.com/GMAT/errata.


GMAT word problem note: 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.


GMAT word problem note: Project management and design were led by Matthew Callan and Mario Gambino.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Particular thanks to instructors Stacey Koprince and Ben Ku for their content contributions.


GMAT word problem note: 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.


GMAT word problem note: And now that you are one of our students too, please chime in! I look forward to hearing from you.


GMAT word problem note: Thanks again and best of luck preparing for the GMAT! Sincerely, Chris Ryan Executive Director, Product Strategy www.manhattanprep.com/gmat 138 West 25th Street, 7th Floor, New York, NY 10001 Tel: 212-721-7400 Fax: 646-514-7425.


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. 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.


GMAT coordinate geometry note: Introduction A Qualified Welcome Welcome to GMAT Advanced Quant! A er all, the point (0, 0) in the coordinate plane is called the origin, isn’t it?


GMAT coordinate geometry note: (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.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Who Should Use This Book You should use this book if you meet the following conditions:.


GMAT fractions, decimals, and percents note: If you match this description, then please turn the page. 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).


GMAT fractions, decimals, and percents note: 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:.


GMAT fractions, decimals, and percents note: 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.


GMAT fractions, decimals, and percents note: You want to become a significantly smarter test-taker.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. If you don’t match this description, then you will probably find this book too difficult at this stage of your preparation.


GMAT fractions, decimals, and percents note: 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.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. 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.


GMAT quant note: 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.


GMAT quant note: Go ahead and give these problems a try. You should not expect to solve any of them in two minutes.


GMAT quant note: In fact, you might find yourself completely stuck. If that’s the case, switch gears.


GMAT quant note: 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.


GMAT quant note: 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?


GMAT coordinate geometry note: 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.


GMAT coordinate geometry note: How many different arrows with these - properties can be constructed?


GMAT quant note: This problem does not require any non-GMAT math, such as trigonometry.).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The Purpose of This Book This book is designed to prepare you for the most difficult math problems on the GMAT.


GMAT number properties note: 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.


GMAT number properties note: 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?


GMAT number properties note: What characterizes these problems? There are two kinds of features:.


GMAT number properties note: Topical nuances or obscure principles 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.


GMAT number properties note: However, the exhaustive theory of divisibility and primes, for instance, is not repeated here. Connected to a particular topic Inherently hard to grasp or unfamiliar Easy to mix up.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Complex structures Complex structures are essentially disguises for simpler content.


GMAT word problem note: These disguises may be difficult to pierce. The path to the answer is twisted or clouded somehow.


GMAT word problem note: 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.


GMAT word problem note: The three problems on the previous page have complex structures. The solutions are a bit later in this chapter.


GMAT word problem note: In the meantime, take a look at another problem. 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.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. An Illustration Give this problem a whirl.


GMAT sequences and patterns note: 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?


GMAT sequences and patterns 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.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: 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.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. You might even say that you need two types of brain.


GMAT fractions, decimals, and percents note: The top-down brain is your conscious self. In cognitive science, the top-down brain is called the executive function.


GMAT fractions, decimals, and percents note: Top-down thinking and planning is indispensable to any problem-solving process. But the corporation in your head is a big place.


GMAT fractions, decimals, and percents note: For one thing, how does information get to the CEO? And how preprocessed is that information?


GMAT fractions, decimals, and percents note: The bottom-up brain is your preconscious processor. A er raw sensory input arrives, your bottom-up brain processes that input extensively before it reaches your top-down brain.


GMAT fractions, decimals, and percents note: 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.


GMAT fractions, decimals, and percents note: This all happens automatically and swi ly. In fact, it takes effort to interrupt this process.


GMAT fractions, decimals, and percents note: 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?


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Each of your brains needs the other one to solve difficult problems.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: To find all the clues in the woods, you need both a savvy detective and a sharp-nosed bloodhound. Your top-down brain is the detective.


GMAT sequences and patterns note: Be organized, fast, and flexible to crack the case. Your bottom-up brain is the bloodhound.


GMAT sequences and patterns note: To solve difficult GMAT problems, try to harmonize the activity of your two brains by following an organized, fast, and flexible problem-solving process. Organized You need a general step-by-step approach to guide you.


GMAT sequences and patterns note: One such approach, inspired by expert mathematician George Pólya, is Understand, Plan, Solve (UPS).


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. You may never have thought explicitly about steps 1 and 2 before.


GMAT word problem note: 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.


GMAT word problem note: Mathematicians know that the real math on hard problems is not Solve. The real math is Understand and Plan.


GMAT word problem note: Fast 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.


GMAT word problem note: 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.


GMAT word problem note: A little extra time invested up front can pay off handsomely later. Flexible To succeed against difficult problems, you sometimes have to “unstick” yourself.


GMAT word problem note: 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.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The top-down brain is labeled TD.


GMAT number properties note: The bottom-up brain is labeled BU. 1 2 9 64 __ TD: “Okay, let’s Understand this thing.


GMAT number properties note: 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. Stare at the numbers given?” BU notices that 9 = 32 and 64 = 82.


GMAT number properties note: 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?


GMAT number properties note: No, since 2 isn’t.” BU doesn’t like this break in the pattern. 1 2 32 26 — TD: “Wait, back up.


GMAT number properties note: What about primes, factoring all the way. 8 = 23, and so 82 = (23)2 = 26.” BU notices 6 = 2 × 3, but so what?


GMAT number properties note: 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?


GMAT number properties note: TD: “Okay, keep looking at this. Are the 2’s and 3’s stacked somehow?”.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. There’s 2−3−2 twice as you go across, but so what?


GMAT sequences and patterns note: And the 1 is weird by itself. Original and try taking differences.” BU notices no pattern.


GMAT sequences and patterns note: The numbers look even uglier. Go back to original numbers again.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: I should look at those again.” BU notices a gap on the le, among the powers. 21 32 82 __ TD: “How about looking at 2.


GMAT sequences and patterns note: Write it with exponents: 2 = 21. Actually, 1 doesn’t have to be 12.


GMAT sequences and patterns note: One can be to any power and still be 1. The power is a question mark.” BU notices 21 then 32.


GMAT sequences and patterns note: BU really wants 1, 2, 3, 4 somehow. __ TD: “Try 4 in that last position.


GMAT sequences and patterns note: Could the last term be 4 somehow?” BU likes the look of this. 21 32 43 __ TD: “64 is 4 to the what?


GMAT sequences and patterns note: 42 = 16, times another 4 equals 64, so.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. That fits.” BU is thrilled: 1, 2, 3, 4 below and 1, 2, 3 up top.


GMAT number properties note: 10 21 32 43 __ TD: “Extend le. The bases are 1, 2, 3, 4, etc., and the powers are 0, 1, 2, 3, etc.” BU is content.


GMAT number properties note: 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.


GMAT number properties note: A er 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.


GMAT number properties note: 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.


GMAT number properties note: 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.


GMAT number properties note: 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?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Factors because they weren’t that useful here?


GMAT number properties note: 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.


GMAT number properties note: 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.


GMAT number properties note: 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.


GMAT number properties note: You can break down Understand, Plan, Solve into several discrete steps:. Understand Glance at the problem briefly: does anything stand out?


GMAT number properties note: 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?


GMAT quant note: Focus on equations, expressions, variables, and step-by-step manipulation. Organize your approach: choose a solution path.


GMAT quant note: You’ll get lots of practice using the UPS process throughout this guide.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Learning How to Think This book is intended to make you smarter.


GMAT word problem note: 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.


GMAT word problem note: 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.


GMAT word problem note: 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 o en not the best GMAT approach.


GMAT word problem note: Unfortunately, advanced test-takers are sometimes very stubborn. Sometimes they feel they should solve a problem according to some theoretical approach.


GMAT word problem note: Or they fail to move to Plan B or C rapidly enough, so they don’t have enough time le to execute that plan. In the end, they might wind up guessing purely at random—and that’s a shame.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. GMAT problems o en have back doors—ways to solve that don’t involve crazy computation or genius-level insights.


GMAT word problem note: 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!


GMAT word problem note: 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.


GMAT word problem note: If you keep getting questions correct, the test will keep getting 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.


GMAT word problem note: 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.


GMAT word problem note: This will happen to everyone, no matter how good you are at the GMAT. The GMAT is not an academic test, though it certainly appears to be.


GMAT word problem note: 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 de.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Whether you have the presence of mind to recognize a bad opportunity and the discipline to let it go.


GMAT word problem note: 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.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Plan of This Book The rest of this book has three parts:.


GMAT word problem note: Strategies & Tactics Part Two:. 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:.


GMAT word problem note: The next four chapters, in Part II, focus on techniques.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. That apply across several topics but are more specific than the approaches in Part I.


GMAT statistics note: Each of the eight chapters in Part I and Part II contains the following:. Many of these problems will be GMAT-like in format, but many will not.


GMAT statistics note: 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.


GMAT statistics note: Other Workout problem sets are mixed by both approach and topic. Note that these problems are not arranged in order of difficulty.


GMAT statistics note: Also, you should know that some of these problems draw on advanced content covered in the Manhattan Prep All the Quant guide. Try-It Problems embedded throughout the text Problem Sets at the end of the chapter.


GMAT quant note: 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.


GMAT quant note: 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.


GMAT quant note: 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.


GMAT quant note: 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?


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Jar is filled with red, white, and blue tokens...


GMAT sets, probability, and counting note: Chance of randomly selecting... TD: “I need to Understand this problem first.


GMAT sets, probability, and counting note: 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.


GMAT sets, probability, and counting note: 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.


GMAT sets, probability, and counting note: In simpler words, if I pick a red, then a white, that’s the same chance as if I pick a blue. Okay, what else?” “multiple of 3” thing yet.


GMAT sets, probability, and counting note: Smallest possible total number of tokens in the jar? They’re small, but not tiny.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. TD: “Let’s Reflect for a moment to figure out a Plan.


GMAT algebra note: How about algebra— if I name the number of each color, then I can represent each fact and also what I’m looking for. Multiply red and white fractions.


GMAT algebra note: Simplify algebraically.” BU is now unsure. 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...


GMAT algebra note: TD: “Let’s start over conceptually. Can I learn anything interesting?” BU notices that blues are different.


GMAT algebra note: TD: “How are blues different? Picking a red, then a white is as likely as picking a blue.


GMAT algebra note: 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.


GMAT algebra note: Fewer blues than reds or whites B < R and B < W TD: “Are there fewer blues? Focus on the algebraic setup.”.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In the very first equation above, each fraction on the le is less than 1, so their product is even smaller.


GMAT algebra note: The denominators of the three fractions are all the same. BU notices fractions less than 1.


GMAT algebra note: 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?


GMAT algebra note: Neither R nor W can equal 3 (since B is smaller than either). The smallest possible total is now 15.


GMAT algebra note: Algebra by itself isn’t getting me there. Try the most constrained variable:.


GMAT algebra note: Since it’s the smallest quantity, but still positive, pretend B is 3. Divide by RW.” BU likes having only two variables.


GMAT algebra note: TD: “Need to test other numbers. Apply constraints I know—B is the smallest number.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Try the next possibility.” BU doesn’t like breaking the symmetry between R and W.


GMAT sets, probability, and counting note: TD: “Does it matter whether R = 6 and W = 9 or the other way around? Plug in and go.” TD: “This works.


GMAT sets, probability, and counting note: Let’s look at another pathway—one that moves more quickly to the back door. ALTERNATIVE SOLUTION TO TRY-IT #0-1 ...


GMAT sets, probability, and counting note: Chance of randomly selecting... BU notices “chance.” BU doesn’t like probability.


GMAT sets, probability, and counting note: Okay, let’s make sense of this and see whether there are any back doors. The number of tokens of every color is BU notices that there are only limited possibilities for each number.


GMAT sets, probability, and counting note: TD: “Okay, every quantity is a multiple of 3. There are 3, 6, 9, etc., of each color.”.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. A jar is filled with red, white, and blue tokens...


GMAT sets, probability, and counting note: Beginning assumes that there actually are tokens of each color. Let’s start at the easy end—in this case, the smallest number.” BU notices 9 = 3 + 3 + 3.


GMAT sets, probability, and counting note: Select a white: , which is not "select a blue" TD: “The only possible way to have 9 total tokens that work? TD: “Only way to have 12 total is 3, 3, and 6.


GMAT sets, probability, and counting note: 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.”.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Keep going.” BU notices 15 has a few options.


GMAT quant note: Make blue the 3.” TD: “Nope. Make blue the 3.” TD: “That’s it!


GMAT coordinate geometry note: 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.


GMAT coordinate geometry note: 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.


GMAT coordinate geometry note: 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?


GMAT coordinate geometry note: SOLUTION TO TRY-IT #0-2 BU notices “xy-plane.” TD: “Let’s Understand first. This is a coordinate- plane problem.


GMAT coordinate geometry note: Put in boundaries as necessary.” BU wonders where this is going. How many different arrows with these TD: “What is the question asking for again?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Properties can be constructed?


GMAT number properties note: Reread the question.” BU wonders which properties. The x- and y-coordinates of A and B are to be integers that satisfy the inequalities 0 ≤ x ≤ 9 and 0 ≤ y ≤ 9.


GMAT number properties note: 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.


GMAT number properties note: The tip and the end of the arrow have to touch holes in the pegboard exactly. The Plan is to start counting.


GMAT number properties note: How to Organize?” BU imagines many possible arrows. Brute force can’t be the right way forward.


GMAT number properties note: The arrows can point in all sorts of different ways. TD: “Let’s simplify the Plan.


GMAT number properties note: Let’s focus on just one orientation of arrows—pointing straight up. How many places can the arrow be?” BU wants to go up & down, then right & le.


GMAT number properties note: 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.


GMAT number properties note: TD: “Count the positions in one column, then multiply by the number of columns. Be careful to count endpoints.”.


GMAT geometry note: There are 10 identical columns: x = 0 through x = 9. 5 × 10 = 50 possible positions for the arrow pointing straight up.


GMAT geometry note: 50 × 2 = 100 possible positions for the arrow if it points straight up or to the right. Other possibilities?” BU notices the square is the same vertically as horizontally.


GMAT geometry note: TD: “I get the same result for arrows pointing right. Am I done?” 50 × 4 = 200 possible positions BU wonders about “down” and “le.” TD: “These arrows can point straight down or straight le, too.


GMAT geometry note: Those would have the same result. So there are 50 positions in each of the four directions.


GMAT geometry note: Calculate at this point and evaluate answers. BU is suspicious: somehow too easy.


GMAT geometry note: TD: “Could the arrows be at an angle?” BU notices that the arrow is 5 units long, associated with 3−4−5 triangles.


GMAT geometry note: Put the arrow as the hypotenuse of a 3−4−5 triangle. Looks like there are four ways if I go 3 up and 4 across: up right, up le, down right, down le.” BU is happy.


GMAT geometry note: TD: “Likewise, there must be four ways if I go 4 up and 3 across: again, up right, up le, down right, and down le. 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.


GMAT geometry note: 8 × 42 = 336 possible positions at an angle. In total, there are 200 + 336 = 536 positions.


GMAT geometry note: TD: “Now count just one of these ways. Be sure to include endpoints.” BU notices the symmetry.


GMAT geometry note: 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.


GMAT geometry note: There are 8 ways to point the arrow at an angle. There isn’t much of an alternative to the approach above.


GMAT geometry note: SOLUTION TO TRY-IT #0-3 TD: “Okay, let’s Understand this. The problem wants the value of x.


GMAT geometry note: How about a Plan?” BU notices this is an isosceles triangle, because there are two sides labeled x. How about the two equal angles?


GMAT geometry note: Divide 135° equally across the two missing angles. TD: “Figure out the two missing angles.


GMAT geometry note: Use the 180° rule.” BU doesn’t recognize this triangle.


GMAT geometry note: TD: “Hmm...here’s a Plan: add a perpendicular line to make right triangles. Drop the line from the top point.


GMAT geometry note: I'll label corners while I'm at it. Now fill in angles.” BU notices 45−45−90 and is happy.


GMAT geometry note: 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.


GMAT geometry note: Write the Pythagorean theorem for the small triangle on the le, using the as the hypotenuse.” BU thinks this equation is really ugly. Expand the quadratic and simplify.” BU doesn’t like the square root on the bottom.


GMAT geometry note: TD: “Multiply by to get rid of it on the - bottom of the fraction.”.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. BU has no idea how to take the square root of this.


GMAT algebra note: 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). That matches the squared value, not the square root.


GMAT algebra note: 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.


GMAT algebra note: 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.


GMAT algebra note: In this regard, the solution above works perfectly well.


GMAT geometry note: 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.


GMAT geometry note: 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.


GMAT geometry note: It equals 1.2 multiplied by 1.4, or approximately 1.7. How can you estimate that math quickly?


GMAT geometry note: Unfortunately, you might guess wrong at this point. But the odds are much better than they were at the outset.


GMAT geometry note: 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.


GMAT geometry note: That was a substantial introduction.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.


GMAT strategy note: Focus on equations, expressions, variables, and step-by-step manipulation. Understand the Basics Principle #2:.


GMAT strategy note: Solve—and Put Pen to Paper Principle #4:.


GMAT strategy note: Track mean, median, range, distribution, and how operations change spread. Advanced Principles Chapters 1 and 2 of this book focus on the more fundamental of the two types of GMAT math questions:.


GMAT strategy note: However, Chapters 3 and 4 deal specifically with DS issues. 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.


GMAT strategy note: 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.


GMAT strategy note: His little book How to Solve It has been in print since 1945—it’s worth getting a copy. In the meantime, keep reading.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Understand the Basics Take time to think and plan before you start solving a difficult problem.


GMAT word problem note: 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:.


GMAT word problem note: 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?


GMAT word problem note: If it’s DS, glance at the statements. Knowing what type of problem you’re dealing with will help you read more effectively.


GMAT word problem note: 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:.


GMAT word problem note: 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.


GMAT word problem note: 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.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. You don’t need to meticulously go through every one of these questions whenever you solve a problem.


GMAT word problem note: (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.


GMAT word problem note: 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.


GMAT word problem note: Start the following problem by taking these three steps:. What exactly is the problem asking for?


GMAT word problem note: What information would I need in order to find the answer? What information do I already have?


GMAT word problem note: What information don’t I have? Sometimes you care about something you don’t know.


GMAT word problem note: 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.


GMAT word problem note: For instance, if a problem includes the quantity 11! (11 factorial), you will practically never need to know the exact value of that quantity.


GMAT word problem note: What is this problem testing? In other words, why is this problem on the GMAT?


GMAT word problem note: What aspect of math are they testing? What kind of reasoning do they want me to demonstrate?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #1-1 x = 910 − 317 and is an integer.


GMAT number properties note: If n is a positive integer that has exactly two factors, how many different values for n are possible? The answers are numbers but in written form.


GMAT number properties note: This format is reserved for problems that ask for the number of numbers or number of possibilities for something. Here are some possible answers to the Pólya questions:.


GMAT number properties note: What exactly is the problem asking for? The number of possible values for n.


GMAT number properties note: This means that n might have multiple possible values. In fact, it probably can take on more than one value.


GMAT number properties note: I may not need these actual values. What are the quantities I care about?


GMAT number properties note: I’m given x and n as variables. These are the quantities I care about.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. X = 910 − 317 That is, x = a specific large integer, expressed in terms of powers of 9 and 3.


GMAT number properties note: 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.


GMAT number properties note: Prime numbers have exactly two factors. So I can rephrase the information: n is prime.


GMAT number properties note: (Primes are always positive.) What don’t I know? Here’s something I don’t know:.


GMAT number properties note: 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.


GMAT number properties note: 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.


GMAT number properties note: What is this problem testing? From the foregoing, I can infer that this problem is testing Divisibility & Primes.


GMAT number properties note: 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.


GMAT number properties note: For instance, you might not look at what the problem is asking for until you’ve understood the given information. As you decide that a piece of information is important, jot it down on your scrap paper.


GMAT number properties note: At this point, your scrap paper might look something like this.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Build a Plan Next, think about how you will solve the problem:.


GMAT statistics note: Here are some Pólya questions that help you think about what you know and come up with a plan:. Is a good approach already obvious?


GMAT statistics note: From your answers above, you may already see a way to reach the answer. If you can envision the rough outlines of the correct path, then go ahead and get started.


GMAT statistics note: If not, what in the problem can help me figure out a good approach? If you are stuck, look for particular clues to tell you what to do next.


GMAT statistics note: Revisit your answers to the basic questions. Can you rephrase or reword them?


GMAT statistics note: Can you combine two pieces of information in any way, or can you rephrase the question, given everything you know? Can I remember a similar problem?


GMAT statistics note: Try relating the problem to other problems you’ve faced. This can help you categorize the problem or recall a solution process.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. For the Try-It #1-1 problem, some of the information is already rephrased (reorganized).


GMAT number properties note: Go further now, combining information and simplifying the question:. Given: n is a prime number AND n is a factor of x.


GMAT number properties note: Combined: n is a prime factor of x. How many different values for n are possible?


GMAT number properties note: How many different values for n, a prime factor of x, are possible? How many distinct prime factors does x have?


GMAT number properties note: You need the prime factorization of x. Notice that n is not even in the question anymore.


GMAT number properties note: The variable n just gave you a way to ask this underlying question. It can be helpful initially to put certain complicated facts to the side.


GMAT number properties note: 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.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Solve—and Put Pen to Paper The third step is to do the work: solve.


GMAT number properties note: 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.


GMAT number properties note: 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:.


GMAT number properties note: Next, pull out a common factor from both terms. The largest common factor is Now, you have what you need: the prime factorization of x.


GMAT number properties note: The number x has three distinct prime factors: 2, 3, and 13. The idea of putting pen to paper also applies when you get stuck anywhere along the way on a monster problem.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Think back to those killer Try-It problems in the introduction.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: Do not just stare, hoping that you suddenly get it. Instead, ask yourself the Pólya questions again and write down whatever you can:.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: Think back to the sequence problem in the introduction. You’ll keep seeing 64 as 82 unless you try writing it in another way.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: LOOK FOR PATTERNS Reinterpretations of given information or of the question Intermediate results, whether specific or general Avenues or approaches that didn’t work.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Every GMAT Quant problem has a two-minutes-or-faster solution path, which may depend upon a pattern that you’ll need to extrapolate.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: 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?


GMAT sequences and patterns note: Nobody is going to write out all 61 terms and then add them up in two minutes. The recursive definition of Sn doesn’t yield any secrets upon first glance.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. The terms of the sequence are ....


GMAT sequences and patterns note: Three terms repeat in this cyclical pattern forever. Every third term is the same.


GMAT sequences and patterns 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: .


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. It is almost impossible to stare at the recursive definition of this sequence and discern the resulting pattern.


GMAT sequences and patterns note: The best way to identify the pattern is to calculate a few values of the sequence and look for the pattern. DRAW IT OUT Some problems are much easier to solve if you draw out what’s happening in the problem.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: 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?


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Represent Truck A and Truck B as of 1:00 p.m.


GMAT word problem note: How does the distance between Truck A and Truck B change as time goes by? Since the answers are all a matter of minutes a er 1:00 p.m., try a convenient increment of a few minutes.


GMAT word problem note: A er 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?


GMAT word problem note: Because Truck A is traveling due south and Truck B is traveling due east, the triangle must be a right triangle. At this point, you could solve the problem in one of two ways.


GMAT word problem note: 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.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Therefore, y could equal 6 or 8 miles.


GMAT algebra note: In other words, the trucks will be exactly 10 miles apart at 1:12 p.m. However, many other kinds of problems can benefit from visual thinking.


GMAT algebra note: 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.


GMAT algebra note: Once you understand how the math works, return to the more complex problem and apply the same solution path.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. 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?


GMAT number properties note: 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:.


GMAT number properties note: What does the square of an odd integer look like?


GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. Are there any patterns or commonalities?


GMAT algebra note: All of the numbers are perfect squares. Therefore, is an odd perfect square.


GMAT algebra note: The question asks for the smallest possible value of xy. What do you need to figure out in order to find that?


GMAT algebra note: If the expression is distracting you, try figuring out what this would mean for a simpler version of the expression. What if is an odd perfect square?


GMAT algebra note: In order for the number to be odd, you have to get rid of the even number 20 (because an even number times any number equals an even number). The only way to get rid of the even part is to divide it out by y.


GMAT algebra note: If y is 4, then the expression would become . As long as x is an odd number, 5x will be odd, too.


GMAT algebra note: 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.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. In other words, y2 must cancel out all the even factors in 1,620.


GMAT number properties note: 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.


GMAT number properties note: If so, then the expression becomes Now, what about x? If you’re not sure, return to your simpler problem thinking.


GMAT number properties note: In the last step, in order to make 5x odd, x has to be odd. 5x also has to be a perfect square.


GMAT number properties note: If you make x = 5, then 5x = 25, an odd perfect square. The expression 5x, therefore, needs a second 5 to make this a perfect square.


GMAT number properties note: Make sure that 405x has two pairs of every factor. 405 contains only one 5, so x must contain another 5.


GMAT number properties note: 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.


GMAT number properties note: 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.


GMAT number properties note: If a problem has many complexities, you can attack it by ignoring some of the complexities at first.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Then, see whether you can adjust the solution to the simpler problem in order to solve the original.


GMAT sequences and patterns note: To recap, put your work on paper. Don’t try to solve hard problems in your head.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: For instance, make tables to keep track of cases. The more organized you are, the more insights you will have into difficult problems.


GMAT sequences and patterns note: Write out the first few cases. Solve an easier problem, then apply your method to the harder problem.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Review Your Work When you are done with a test or practice set, you are not really done.


GMAT word problem note: 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 a er you’ve finished it and picked your answer, when you look at it with a clear head and no timer.


GMAT word problem note: 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.


GMAT word problem note: Most of these questions are useful even if you got the problem correct. Don’t restrict yourself to reviewing problems you got wrong.


GMAT word problem note: Review any problem you might learn something from and ask yourself:. What are all of the pathways to the answer?


GMAT word problem note: 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?


GMAT word problem note: If I see one of those clues in a different problem, what should I do? What traps or tricks are built into this problem?


GMAT word problem note: Where could I have made a mistake? If I did make a mistake, what went wrong in my problem-solving process?


GMAT word problem note: Do I need to change how I approach similar problems?


GMAT quant note: Focus on equations, expressions, variables, and step-by-step manipulation. What could I take from this problem to help me solve other problems in the future?


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Understand, Plan, and Solve.


GMAT sets, probability, and counting note: Then, review each problem in depth. As you review, do two things:.


GMAT sets, probability, and counting note: The solutions include our own responses to these two tasks. Yours might look different, and that's fine.


GMAT sets, probability, and counting note: Problem Set Identify exactly what the problem is asking for and what that means in the simplest possible terms. Note at least one general takeaway that might be useful on other problems in the future.


GMAT sets, probability, and counting note: 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.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. If x is a positive integer, what is the units digit of (24)5 + 2x(36)6(17)3?


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: If he makes exactly 95 cookies for the bake sale, what is the minimum number of chocolate chip cookies that he could make?


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. 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.


GMAT algebra note: 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?


GMAT algebra note: The sum of all distinct solutions for x in the equation x2 − 8x + 21 = |x − 4| + 5 is equal to which of the following?


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Solutions Each solution addresses the two steps from the instructions:.


GMAT number properties note: Identify exactly what the problem is asking for, and what that means in the simplest possible terms. Note at least one general takeaway that might useful on other problems in the future.


GMAT number properties note: 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.


GMAT number properties note: There are many balls, each with a different factor of 210. Each factor of 210 is represented.


GMAT number properties note: One ball is selected randomly. Some balls have a multiple of 42 (e.g., 42 itself).


GMAT number properties note: How many factors of 210 there are How many of these factors are multiples of 42 What the problem is testing:.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Plan: 210 to primes → build full list of factors from prime components → distinguish between multiples of 42 and non-multiples → count factors → compute probability.


GMAT number properties note: Alternatively, you could list all the factors of 210 using factor pairs.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. There are 16 factors of 210, and two of them (42 and 210) are multiples of 42.


GMAT number properties note: You can also count the factors using 210’s prime factorization: (2)(3)(5) Here’s a shortcut to determine the number of distinct factors of 210. Add 1 to the power of each prime factor and multiply:.


GMAT number properties note: 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?


GMAT number properties note: 42 itself is a multiple of 42, of course. To find any others, divide 210 by 42 to get 5.


GMAT number properties note: This number is a prime, so the only other possible factor is 42 × 5, or 210. The problem is straightforward in one sense: it says the word factor explicitly.


GMAT number properties note: 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.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The factor-counting shortcut, but only if you do know how to deal with the multiples of 42.


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: Furthermore, the problem provides crazy numbers. You are absolutely not going to multiply these out.


GMAT sequences and patterns note: There must be some kind of pattern at work. Use the Last Digit Shortcut (discussed in the All the Quant guide).


GMAT sequences and patterns note: 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.


GMAT sequences and patterns note: The pattern for the units digit of 4integer = [4, 6]. Units digit of (36)6 must be 6, as every power of 6 ends in 6.


GMAT sequences and patterns note: Units digit of (17)3 = units digit of (7)3. The pattern for the units digit of 7integer = [7, 9, 3, 1].


GMAT sequences and patterns note: The product of the units digits is (4)(6)(3) = 72, which has a units digit of 2. Patterns were very important on this one!


GMAT sequences and patterns note: 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.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The problem asks for the minimum number of chocolate chip cookies.


GMAT number properties note: 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.


GMAT number properties note: 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.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. 7C 6P = 95 − 7C Is 6P a multiple of 6?


GMAT algebra note: The two competing constraints made testing choices the most efficient method. There are three equations and three variables.


GMAT algebra note: What is the easiest way to solve for the volume? The question asks for the volume of the box.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The width, w, appears in all three constraint equations, so solve for the other variables in terms of w and substitute into the longest constraint:.


GMAT algebra note: Since w can’t be zero, you can divide it out safely. The question is complex enough that you could check your work at the end by also calculating the volume of the new solid (w + 1)(l + 1)(h − 9).


GMAT algebra note: As always, you have to decide whether to spend that time here versus elsewhere.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Notice that, though the initial volume formula seemed long and annoying, the calculations canceled out nicely in the end.


GMAT algebra note: This is common on the GMAT—common enough, in fact, to suspect that you may be doing something wrong if the algebra becomes very messy. The question implies that there may be multiple solutions, as does the nonlinear given equation.


GMAT algebra note: What is the most efficient way to find those solutions? There are actually two good approaches.


GMAT algebra note: Choose the one that is easier for you. Split it into two equations, the “positive” version and the “negative” version:.


GMAT algebra note: The question asks for the sum of the distinct solutions. In other words, if the number 2 were to show up twice as a solution, you would count it only once.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Sum of the different solutions: 5 + 4 + 3 = 12.


GMAT algebra note: Approach #2: simplify the equation and use theory to finish it off. You square a number and get the absolute value of that same number (not squared!).


GMAT algebra note: Only a few numbers can make that true: 1 squared equals |1|, 0 squared equals |0|, −1 squared equals |−1|. When a problem asks for distinct solutions, count each unique solution.


GMAT algebra note: Ignore multiple instances of the same value. Both are valid solution methods.


GMAT algebra note: Which one do you think you will be able to remember and use more easily?


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Advanced Strategies Advanced Guessing Tactics.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Strategies & Tactics Sometimes you will encounter a Problem Solving (PS) problem that you can’t answer—either because its content is difficult or obscure or because you don’t have enough time to solve completely in two minutes.


GMAT word problem note: Here, we make the distinction between solution strategies and guessing tactics. Solution strategies are broad: they apply to a wide variety of problems, they provide a complete approach, and they can be used safely in most circumstances.


GMAT word problem note: Moreover, a particular tactic may only be useful in special situations or for parts of a problem.


GMAT strategy note: Translate the wording into variables, relationships, and equations before calculating. One of the most productive strategies on the GMAT is to pick good numbers and plug them into unknowns.


GMAT strategy note: Try this when the concepts are especially complex or when conditions are placed on key inputs that are otherwise unspecified (e.g., n is a prime number). Doing so can o en help you avoid demanding calculations or the need to set up and solve complicated algebraic expressions.


GMAT strategy note: Test Cases In certain circumstances, a problem allows multiple possible scenarios, or cases. On PS questions, the problem usually asks you to find something that must be true or could be true.


GMAT strategy note: On these problems, you can test different numbers to eliminate answers until only one remains. Avoid Needless Computation The GMAT rarely requires you to carry out intensive calculations to arrive at an answer.


GMAT strategy note: Look for opportunities to avoid tedious computation by factoring, simplifying, or estimating.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The correct answer may be part of one of these pairs.


GMAT word problem note: In such cases, look for clues as to the correct sign of the correct answer.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. CHOOSE SMART NUMBERS Some types of problems allow you to pick real numbers and solve the problem arithmetically rather than algebraically.


GMAT algebra note: Likewise, you can o en pick a smart number for a fraction or percent problem without specified absolute value amounts. Other problem types allow this strategy as well.


GMAT algebra note: For instance, a problem may put specific conditions on the inputs but not give you exact numbers. In this case, you can go ahead and just pick inputs that fit the conditions.


GMAT algebra note: If a problem specifies that “x must be a positive even integer” but does not specify the value of x, picking 2 for x will probably get you to a solution quickly and easily. The GMAT Official Guide contains many problems that are difficult to solve algebraically but much easier to solve with real numbers.


GMAT algebra note: However, as an advanced test-taker, you might consider it a point of pride not to plug in a number. You might want to prove a “theoretically correct” answer.


GMAT algebra note: The GMAT tests you on your flexibility of thinking and your ability to manage a very limited amount of time. Use the easiest and most efficient solution path, not the te.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Andra, Elif, and Grady each invested in a certain stock.


GMAT algebra note: Andra invested q dollars, which was 40% more than Elif invested. If Elif invested 25% less than Grady invested, what was the total amount invested by all three, in terms of q?


GMAT algebra note: This problem can be solved algebraically: write a couple of equations and solve for all three variables, then add them up. A glance at the answers, though, indicates that the algebra is likely to get messy.


GMAT algebra note: Instead, choose a real number and solve the problem arithmetically. If you’ve made it to the GMAT Advanced Quant book, then you have likely used this strategy before (or at least learned about it).


GMAT algebra note: At times, you may have been frustrated because this technique didn’t actually seem easier than doing the math algebraically. If so, here’s the missing piece: you need to learn how to choose smart numbers in the best possible way.


GMAT algebra note: Most of the time, you’re going to choose for the variable given (in this case, q). In some cases, though, starting with the given variable doesn’t make your life any easier.


GMAT algebra note: The problem above actually has three unknowns: one for Andra, one for Elif, and one for Grady. Take a look at the relationship between those unknowns before you decide which one is the best starting point.


GMAT strategy note: Percent problems usually relate part, whole, rate, and percent change. Andra invests 40% more than Elif.


GMAT strategy note: For just these two, it would be easier to pick a number for Elif and then calculate Andra’s amount. Elif invests 25% less than Grady.


GMAT strategy note: For these two, it is easier to start with Grady and then calculate Elif. As a result, start with Grady, then find Elif, then find Andra.


GMAT strategy note: If Grady invests $100, then Elif invests 25% less, or $75. Andra invests 40% more than Elif, or $75 + $30 = $105.


GMAT strategy note: Make sure to note on your scrap paper that q = 105. Collectively, the three invest $100 + $75 + $105 = $280.


GMAT strategy note: Note a few important aspects that will help you to choose smart numbers efficiently and effectively. First, if there are multiple unknowns and you have to choose where to start, pause to think about how to make the math as easy as possible.


GMAT strategy note: In the case of the problem above, if you had picked q = 100 for Andra, your next step would 2q = 2(105) = $210.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Have been to figure out Elif’s amount.


GMAT word problem note: It is not the case that Elif would be $60, or 40% less than Andra. Rather, Andra is 40% more than Elif: 1.4e = $100, so .


GMAT word problem note: Elif would actually equal approximately $71.42857. Nobody’s going to want to go down that path!


GMAT word problem note: At this stage, you have two choices: you can go back and pick for someone else or you can think about what numbers would make this particular path easier. (In this case, that would still result in a messy number for the next calculation.


GMAT word problem note: You can stop and eliminate a choice whenever you can tell that it will not equal $280. Third, practice this strategy extensively in order to expose yourself to these little variations and possible sticking points.


GMAT word problem note: As you become proficient with the strategy, you’ll be amazed at how much time and mental effort it can save.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Rewards finding the correct answer as efficiently as possible or guessing when needed.


GMAT number properties note: Try-It #2-2 If , which of these integers could be the value of z? Look at all of those fractions!


GMAT number properties note: Solving for z algebraically in this problem would not be easy. The numerators are always 2, so what needs to happen to bring the two halves of the equation together?


GMAT number properties note: The le side needs to be made smaller, so the denominator needs to be bigger.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Try-It #2-3 A certain college party is attended by both male and female students.


GMAT algebra note: The ratio of male-to-female students is 3 to 5. If 5 of the male students were to leave the party, the ratio would change to 1 to 2.


GMAT algebra note: How many total students are at the party? Of course, you could set up equations for the unknowns in the problem and solve them algebraically.


GMAT algebra note: However, the numbers in the answers are pretty straightforward integers.


GMAT fractions, decimals, and percents note: Translate the wording into variables, relationships, and equations before calculating. What characteristic must be true of the correct answer?


GMAT fractions, decimals, and percents note: Once you have found the answer using this technique, you can stop. You don’t need to test any remaining answers.


GMAT fractions, decimals, and percents note: Also, you don’t have to translate equations or figure out how to eliminate variables and solve. You just work each number through the problem until you’re done.


GMAT fractions, decimals, and percents note: As you work through an answer, think about how the math is playing out. Again, you will usually be able to eliminate some answers without actually having to try them.


GMAT fractions, decimals, and percents note: If there are 30 students total, then the unknown multiplier is = 3.75, leaving you with a non-integer number of students. This is impossible, so this answer must be incorrect.


GMAT fractions, decimals, and percents note: If 5 males leave, there will be 4 le, leaving a ratio of 4: 15. This ratio is also pretty far away from the correct ratio of 1: 2, so consider trying the larger remaining answer, 80, next.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Some problems allow you to choose real numbers to solve, but you can’t choose just one set of numbers as you do when you choose smart numbers.


GMAT number properties note: Rather, you have to test multiple scenarios to get yourself to the one correct answer. On Problem Solving problems, this tends to occur with must be true or could be true problems.


GMAT number properties note: Try It #2-4 If n is a positive integer, what must be true of n3 − n? This problem is asking about a theoretical concept: what mathematical characteristic must be true of this expression?


GMAT number properties note: In order to solve, you can test allowable cases to narrow down the answers until only one remains. Note that in this PS problem, one of the answers must be correct.


GMAT number properties note: In this case, any positive integer n will help you get to the answer, so start with the simplest possible positive integer: 1. In general, when Testing Cases on PS or DS, start with the simplest number that fits the problem’s parameters.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Because one of the statements talks about having more than a certain number of prime factors, try a larger number next: n n3 − n Yes, 0 is divisible by 4.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. You could save yourself some time on this one by recognizing that the expression n3 − n can be rewritten as:.


GMAT number properties note: In other words, the expression represents three consecutive integers. Test some cases to discover what must be true about the product of three consecutive integers.


GMAT number properties note: If n = 1, then the three consecutive integers are 0, 1, 2. If n = 2, then the three consecutive integers are 1, 2, and 3.


GMAT number properties note: This product does not contain two 4’s, so it is not a multiple of 4. If n = 3, then the three consecutive integers are 3, 4, and 5.


GMAT number properties note: (The prime factors are 2, 3, and 5.) Eliminate.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. In this case, you’re trying to find factors, and the three consecutive integers tell you that directly.


GMAT number properties note: You don’t need to multiply them out. (You’ll learn more about avoiding unnecessary computation in the next section.) Instead, if possible, try to notice a pattern.


GMAT number properties note: In every set of three consecutive numbers, you will always have at least one even number. You will also always have a multiple of 3.


GMAT number properties note: (In Case 1, 0 is a multiple of 3.) As a result, the product will always be a multiple of 6. You may also notice that the problem contains no upper limit.


GMAT number properties note: By choosing a large enough number, you’re going to be able to create a number that contains more than two distinct prime numbers. You can test cases directly to eliminate the four incorrect answers, or you can use a few cases to help you figure out the theory underlying the problem, which will also get you to the correct answer.


GMAT number properties note: AVOID NEEDLESS COMPUTATION You won’t see many GMAT problems that require substantial calculation to arrive at a precise answer. Rather, correct answers on difficult problems will generally be relatively easy to compute once the difficult concept or trick in the problem has been correctly identified and addressed.


GMAT number properties note: On several types of GMAT problems, a significant amount of computation can be a.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Estimation Intelligent estimation can save you time and effort on many problems.


GMAT fractions, decimals, and percents note: Try-It #2-5 The percent change from 29 to 43 is approximately what percent of the percent change from 43 to 57? In this case, the question stem straight up tells you that you can estimate.


GMAT fractions, decimals, and percents note: Any time you see the word approximately (or a synonym), definitely do not try to solve for the exact answer. The question stem is pretty complex.


GMAT fractions, decimals, and percents note: Use the percent change formula:.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. You can simplify things by solving the question in parts.


GMAT fractions, decimals, and percents note: First, it talks about the percent change from 29 to 43. The percent change for the top half of the fraction, is , but that number is cumbersome.


GMAT fractions, decimals, and percents note: This is approximately The percent change from 43 to 57 is . The question is really asking:.


GMAT fractions, decimals, and percents note: The fraction is equivalent to 150%. The wording of the question can sometimes provide a strong clue that estimation should be used.


GMAT fractions, decimals, and percents note: Look for phrases such as these:. The test doesn’t have to tell you that you can estimate, though.


GMAT fractions, decimals, and percents note: Heavy Long Division is what percent of ... The number is approximately equal to which of the following?


GMAT fractions, decimals, and percents note: This result is closest to which of the following? Which of the following is most nearly equal to...?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Very few problems on the GMAT truly require long division, even though it might appear otherwise.


GMAT number properties note: You can almost always approximate the answer or reduce the division by taking out common factors. Try-It #2-6 is equivalent to which of the following?


GMAT number properties note: At first glance, it appears that precise long division is necessary. However, with some manipulation and factoring, the solution is much more straightforward.


GMAT number properties note: The key to factoring this fraction is to move the decimals of both the numerator and denominator three places to the right so that you’re dealing with integers. Then you might notice that 3,507 is divisible by 7 (3,500 and 7 are both divisible by 7, providing a clue that you may be able to factor out 7).


GMAT number properties note: Moreover, 10,020 is divisible by 10 and by 2 (10,020 ends in a 0, and 1,002 is even):.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Alternatively, you might observe that 10.02 is very slightly larger than 10.


GMAT algebra note: Try-It #2-7 What is the value of Since 102 is extremely small compared to 105, and the choices are somewhat spread out, estimate:. Notice that by ignoring the 102 term, you made the denominator slightly larger than it originally was.


GMAT algebra note: Therefore, 0.81918 is slightly smaller than the correct answer, 0.82. Quadratic Expressions in Word Problems Some Word Problems result in a quadratic equation.


GMAT algebra note: You are probably pretty good at solving quadratic equations, so your natural bias would be to set up and solve the equation.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. May be very difficult to solve.


GMAT algebra note: Try-It #2-8 A shoe cobbler charges n dollars to repair a single pair of loafers. Tomorrow, he intends to earn $240 repairing loafers.


GMAT algebra note: If he were to reduce his fee per pair by $20, he would have to repair an additional pair of the loafers to earn the same amount of revenue. How many pairs of loafers does he intend to repair tomorrow?


GMAT algebra note: The problem may not seem too bad... Until you try to set it up algebraically.


GMAT algebra note: Assign x to represent the number of pairs of loafers the cobbler intends to repair tomorrow. Using the equation for revenue gives you the following:.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. From here, the algebra gets complicated very quickly.