The post Investing in Success: The Best In-Person or Online GMAT Tutors Can Make a Difference appeared first on Veritas Prep Blog.

]]>**Knowledge of All Aspects of the GMAT**

The best private GMAT tutor has more than just general advice regarding the GMAT. The person has thorough knowledge of the exam and its contents. There are several parts to the GMAT, including the Verbal, Quantitative, Integrated Reasoning, and Analytical Writing sections. A qualified tutor will have plenty of tips to share that can help you to navigate all of the sections on the GMAT. Plus, an experienced tutor will be able to evaluate the results of your practice GMAT to determine where you need to focus most of your study efforts. This puts the element of efficiency into your test prep.

The GMAT instructors at Veritas Prep achieved scores on the exam that placed them in the 99th percentile, so if you work with a Veritas Prep tutor, you know you’re studying with someone who has practical experience with the exam. Our tutors are experts at describing the subtle points of the GMAT to their students.

**Access to Quality Study Resources**

If you want to thoroughly prepare for the GMAT, you must use quality study materials. At Veritas Prep, we have a GMAT curriculum that guides you through each section of the test. Your instructor will show you the types of questions on the test and reveal proven strategies you can use to answer them correctly. Of course, our curriculum teaches you the facts you need to know for the test. But just as importantly, we show you how to apply those facts to the questions on the exam. We do this in an effort to help you think like a business executive as you complete the GMAT. Private tutoring services from Veritas Prep give you the tools you need to perform your best on the exam.

**Selecting Your Method of Learning**

The best GMAT tutors can offer you several options when it comes to preparing for the exam. Perhaps you work full-time as a business professional. You want to prepare for the GMAT but don’t have the time to attend traditional courses. In that case, you should search for an online GMAT tutor. As a result, you can prep for the GMAT without disrupting your busy work schedule. At Veritas Prep, we provide you with the option of online tutoring as well as in-person classes. We recognize that flexibility is important when it comes to preparing for the GMAT, and we want you to get the instruction you need to earn a high score on this important test.

**An Encouraging Instructor**

Naturally, when you take advantage of GMAT private tutoring services, you will learn information you need to know for the test. But a tutor should also take the time to encourage you as you progress in your studies. It’s likely that you’ll face some stumbling blocks as you prepare for the different sections of the GMAT. A good instructor must be ready with encouraging words when you’re trying to master difficult skills.

Encouraging words from a tutor can give you the push you need to conquer especially puzzling questions on the test. The understanding tutors at Veritas Prep have been through preparation for the GMAT as well as the actual test, so we understand the tremendous effort it takes to master all of its sections.

If you want to partner with the best GMAT tutor as you prep for the test, we have you covered at Veritas Prep! When you sign up to study for the GMAT with Veritas Prep, you are investing in your own success. Give us a call or write us an email today to let us know when you want to start gearing up for excellence on the GMAT!

*Getting ready to take the GMAT? We have **free online GMAT seminars** **running all the time. And, be sure to follow us on **Facebook**, **YouTube**, **Google+**, and **Twitter**!*

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]]>The post GMAT Writing Tips: Analytical Writing for the GMAT appeared first on Veritas Prep Blog.

]]>**Take a Few Minutes to Plan Your Essay**

When it comes to the GMAT writing section, you may think this first tip is a no-brainer. Unfortunately, some students become nervous or anxious about this part of the exam and forget to plan out their essay before diving into the task. This can result in a poorly organized essay or one that is missing important points.

Take the time to carefully read the directions and the argument. Then, create a rough outline of what points you want to include in the essay as well as where you want to include them. If you lose your train of thought while you’re writing, simply look at your outline to regain your focus.

**Determine the Flaws in the Argument**

Your essay’s plan should include the flaws in the author’s argument. Faulty comparisons and mistaken assumptions as well as vague words are all things to point out when critiquing the argument. Writing a quick note about each flaw you find can be helpful when it comes time to elaborate on them in your essay. Plus, making note of them helps you to remember to include all of them in the final piece.

**Use Specific Examples in Your Essay**

The use of specific examples is a key element for Analytical Writing. GMAT graders will be looking for specific examples as they score your essay. It’s not enough to state that a piece of the given argument is inaccurate – you have to use the information within the argument to prove your point. Also, using specific examples helps you to demonstrate that you understand the argument.

**Read and Evaluate High-Scoring Analytical Essays**

When preparing for the GMAT Analytical Writing section, it’s a good idea to read and evaluate essays that received high scores. This can help you see what needs to be adjusted in your own writing to create an essay that earns a high score. In fact, you can break each essay down and highlight the individual elements that earned it a high score.

**Study the Scoring System for the GMAT Analytical Writing Section**

Studying the scoring rubric for the analytical essay is very helpful in your quest to craft a high-scoring piece. After writing a practice essay, you can compare its contents to the criteria on the rubric. If your essay is missing an element, you can go back and do a rewrite. This sort of practice takes a bit of time, but will prove beneficial on test day.

**Study with a GMAT Tutor**

A professional tutor can assist you in preparing for the section on Analytical Writing. GMAT tutors at Veritas Prep have taken the exam and earned a score in the 99th percentile. This means that when you prep for the Analytical Writing section with one of our tutors, you’re learning from a teacher with practical experience! Your tutor can help you boost your writing skills by reviewing the outline of your practice essay and giving you tips on how to improve it. Also, your tutor can provide strategies for what you can do to make your analytical essay more convincing.

We have a variety of tutoring options for those who want help preparing for the analytical essay section on the GMAT. At Veritas Prep, we know that you have a busy schedule, and we want to make it convenient to prep for this test. We also offer resources such as the opportunity for you to take a free GMAT test. This is an excellent way to find out how your skills measure up on each section of the exam. Call or contact us online today and let us give you a hand with your essay-writing skills!

*Getting ready to take the GMAT? We have **free online GMAT seminars** **running all the time. And, be sure to follow us on **Facebook**, **YouTube**, **Google+**, and **Twitter**!*

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]]>The post The Patterns to Solve GMAT Questions with Reversed-Digit Numbers – Part II appeared first on Veritas Prep Blog.

]]>The biggest takeaways from that post were:

- Anytime we add two two-digit numbers whose tens and units digits have been reversed, we will get a multiple of 11.
- Anytime we take the difference of two two-digit numbers whose tens and units digits have been reversed, we will get a multiple of 9.

For the hardest GMAT questions, we’re typically mixing and matching different types of number properties and strategies, so it can be instructive to see how the above axioms might be incorporated into such problems.

Take this challenging Data Sufficiency question, for instance:

*When the digits of two-digit, positive integer M are reversed, the result is the two-digit, positive integer N. If M > N, what is the value of M?*

*(1) The integer (M –N) has 12 unique factors.*

*(2) The integer (M –N) is a multiple of 9.*

The average test-taker looks at Statement 1, sees that it will be very difficult to simply pick numbers that satisfy this condition, and concludes that this can’t possibly be enough information. Well, the average test-taker also scores in the mid-500’s, so that’s not how we want to think.

First, let’s concede that Statement 1 is a challenging one to evaluate and look at Statement 2 first. Notice that Statement 2 tells us something we already know – as we saw above, *anytime* you have two two-digit numbers whose tens and units digits are reversed, the difference will be a multiple of 9. If Statement 2 is useless, we can immediately prune our decision tree of possible correct answers. Either Statement 1 alone is sufficient, or the statements together are not sufficient, as Statement 2 will contribute nothing. So right off the bat, the only possible correct answers are A and E.

If we had to guess, and we recognize that the average test-taker would likely conclude that Statement 1 couldn’t be sufficient, we’d want to go in the opposite direction – this question is significantly more difficult (and interesting) if it turns out that Statement 1 gives us considerably more information than it initially seems.

In order to evaluate Statement 1, it’s helpful to understand the following shortcut for how to determine the total number of factors for a given number. Say, for example, that we wished to determine how many factors 1000 has. We could, if we were sufficiently masochistic, simply list them out (1 and 1000, 2 and 500, etc.). But you can see that this process would be very difficult and time-consuming.

Alternatively, we could do the following. First, take the prime factorization of 1000. 1000 = 10^3, so the prime factorization is 2^3 * 5^3. Next, we take the exponent of each prime base and add one to it. Last, we multiply the results. (3+1)*(3+1) = 16, so 1000 has 16 total factors. More abstractly, if your number is x^a * y^b, where x and y are prime numbers, you can find the total number of factors by multiplying (a+1)(b+1).

Now let’s apply this process to Statement 1. Imagine that the difference of M and N comes out to some two-digit number that can be expressed as x^a * y^b. If we have a total of 12 factors, then we know that (a+1)(b+1) = 12. So, for example, it would work if a = 3 and b = 2, as a + 1 = 4 and b + 1 = 3, and 4*3 =12. But it would also work if, say, a = 5 and b = 1, as a + 1 = 6 and b + 1 = 2, and 6*2 = 12. So, let’s list out some numbers that have 12 factors:

- 2^
**3*** 3^**2**(3+1)(2+1) = 12 - 2^
**5*** 3^**1**(5+1)(1+1) = 12 - 2^
**2*** 3^**3**(2+1)(3+1) = 12

Now remember that M – N, by definition, is a multiple of 9, which will have at least 3^2 in its prime factorization. So the second option is no longer a candidate, as its prime factorization contains only one 3. Also recall that we’re talking about the difference of two two-digit numbers. 2^2 * 3^3 is 4*27 or 108. But the difference between two positive two-digit numbers can’t possibly be a three-digit number! So the third option is also out.

The only possibility is the first option. If we know that the difference of the two numbers is 2^3 * 3^2, or 8*9 = 72, then only 91 and 19 will work. So Statement 1 alone is sufficient to answer this question, and the answer is A.

Algebraically, if M = 10x + y, then N = 10y + x.

M – N = (10x + y) – (10y + x) = 9x – 9y = 9(x – y).

If 9(x – y) = 72, then x – y = 8. If the difference between the tens and units digits is 8, the numbers must be 91 and 19.

Takeaway: the hardest GMAT questions will require a balance of strategy and knowledge. In this case, we want to remember the following:

- Anytime we take the difference of two two-digit numbers whose tens and units digits have been reversed, we will get a multiple of 9.

- If one statement is easier to evaluate than the other, tackle the easier one first. If it’s the case that one statement gives you absolutely nothing, and the other is complex, there is a general tendency for the complex statement alone to be sufficient.

- For the number x^a * y^b, where x and y are prime numbers, you can find the total number of factors by multiplying (a+1)(b+1).

*Plan on taking the GMAT soon? We have GMAT prep courses starting all the time. And be sure to follow us on Facebook, YouTube, Google+ and Twitter!*

*By David Goldstein, a Veritas Prep GMAT instructor based in Boston. You can find more articles written by him here.*

The post The Patterns to Solve GMAT Questions with Reversed-Digit Numbers – Part II appeared first on Veritas Prep Blog.

]]>The post GMAT Hacks, Tricks, and Tips to Make Studying and Preparing for the GMAT Simpler appeared first on Veritas Prep Blog.

]]>Preparing for this important exam may seem like a daunting task, but you can simplify the process with the help of some GMAT tips and tricks.

**Use Mnemonics to Learn Vocabulary Words**

Making a GMAT cheat sheet complete with mnemonics simplifies the process of learning vocabulary words for the Verbal section. Word pictures can help you to retain the words you’re learning. For instance, suppose you’re trying to learn the word “*extricate.” “Extricate”* means to free something or someone from a constraint or problem. You may pair the word with a mental picture of a group of people being freed from a stuck elevator by a technician. Creating mnemonics that relate to your life, family, or job can make them all the more memorable.

**Look for Vocabulary Words in Context**

Studying a GMAT cheat sheet full of words and mnemonics shouldn’t be the end of your vocabulary studies. It’s just as important to be able to recognize those words in context. If you’ve signed up to take the GMAT, there’s a good chance that you already read several business publications, so keep an eye out for the words used within those resources. Reading financial newspapers, magazines, and online articles that contain GMAT vocabulary words helps you become more familiar with them. After a while, you’ll know what the words mean without having to think about them.

**Learn the Test Instructions Before Test Day**

When you read the instructions for each section before test day arrives, you’ll know what to expect on the actual day. This can make you feel more relaxed about tackling each section. Also, you won’t have to use your test time reading instructions because you will already know what you’re doing.

**Always Keep Some Study Materials Close By**

When it comes to GMAT tips and strategies, the easiest ones can sometimes be the most effective. Even busy working professionals have free moments throughout the day. It’s a smart idea to use those moments for study and review. For instance, you can work on some practice math problems during a lunch or coffee break. If you have a dentist or doctor’s appointment, you can use virtual flashcards to quiz yourself on GMAT vocabulary words while you’re sitting in the waiting room. Taking a few minutes each day to review can add up to a lot of productive study time by the end of a week.

**Set a Timer for Practice Tests**

If you’re concerned about completing each section of the GMAT within the allotted number of minutes, one of our favorite GMAT hacks is to try setting a timer as you begin each section of a practice test. If the timer goes off before you’re finished with the section, you may be spending too much time on puzzling problems. Or perhaps you’re taking too much time to read the directions for each section rather than familiarizing yourself with them ahead of time.

Timing your practice tests helps you establish a rhythm that allows you to get through each section with a few minutes to spare for review. At Veritas Prep, we provide you with the opportunity to take a free exam. Taking this practice exam allows you to get a clear picture of what you’ll encounter on test day.

**Get Into the Habit of Eliminating Wrong Answer Options**

Another very effective GMAT strategy is to eliminate answer options that are clearly incorrect. With the exception of the analytical essay, this can be done on every portion of the test. Taking practice tests gives you the chance to establish this habit. By eliminating obviously incorrect answer options, you are making the most efficient use of your test time. Also, you are making the questions more manageable by giving yourself fewer answers to consider.

Here at Veritas Prep, our GMAT instructors follow a unique curriculum that shows you how to approach every problem on the test. We teach you how to strengthen your higher-order thinking skills so you’ll know how to use them to your advantage on the test. Contact our offices today to take advantage of our in-person prep courses or our private tutoring services. Learn GMAT hacks from professional instructors who’ve mastered the test!

*Getting ready to take the GMAT? We have **free online GMAT seminars** **running all the time. And, be sure to follow us on **Facebook**, **YouTube**, **Google+**, and **Twitter**!*

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]]>The post How to Solve “Hidden” Factor Problems on the GMAT appeared first on Veritas Prep Blog.

]]>But if tell you that I have a certain number of cupcakes and, were I so inclined, I could distribute the same number of cupcakes to each of 6 students with none left over or to each of 9 students with none left over, it’s the same concept, but I’m not telegraphing the subject in the same conspicuous manner as the previous question.

This kind of recognition comes in handy for questions like this one:

*All boxes in a certain warehouse were arranged in stacks of 12 boxes each, with no boxes left over. After 60 additional boxes arrived and no boxes were removed, all the boxes in the warehouse were arranged in stacks of 14 boxes each, with no boxes left over. How many boxes were in the warehouse before the 60 additional boxes arrived?*

(1) There were fewer than 110 boxes in the warehouse __before __the 60 additional arrived.

(2) There were fewer than 120 boxes in the warehouse __after__ the 60 additional arrived.

Initially, we have stacks of 12 boxes with no boxes left over, meaning we could have 12 boxes or 24 boxes or 36 boxes, etc. This is when you want to recognize that we’re dealing with a multiple/factor question. That first sentence tells you that the number of boxes is a multiple of 12. After 60 more boxes were added, the boxes were arranged in stacks of 14 with none left over – after this change, the number of boxes is a multiple of 14.

Because 60 is, itself, a multiple of 12, the new number must remain a multiple of 12, as well. [If we called the old number of boxes 12x, the new number would be 12x + 60. We could then factor out a 12 and call this number 12(x + 5.) This number is clearly a multiple of 12.] Therefore the new number, after 60 boxes are added, is a multiple of both 12 and 14. Now we can find the least common multiple of 12 and 14 to ensure that we don’t miss any possibilities.

The prime factorization of 12: 2^2 * 3

The prime factorization of 14: 2 * 7

The least common multiple of 12 and 14: 2^2 * 3 * 7 = 84.

We now know that, after 60 boxes were added, the total number of boxes was a multiple of 84. There could have been 84 boxes or 168 boxes, etc. And before the 60 boxes were added, there could have been 84-60 = 24 boxes or 168-60 = 108 boxes, etc.

A brief summary:

After 60 boxes were added: 84, 168, 252….

Before 60 boxes were added: 24, 108, 192….

That feels like a lot of work to do before even glancing at the statements, but now look at how much easier they are to evaluate!

Statement 1 tells us that there were fewer than 110 boxes before the 60 boxes were added, meaning there could have been 24 boxes to start (and 84 once 60 were added), or there could have been 108 boxes to start (and 168 once 60 were added). Because there are multiple potential solutions here, Statement 1 alone is not sufficient to answer the question.

Statement 2 tells us that there were fewer than 120 boxes after 60 boxes were added. This means there could have been 84 boxes – that’s the only possibility, as the next number, 168, already exceeds 120. So we know for a fact that there are 84 boxes after 60 were added, and 24 boxes before they were added. Statement 2 alone is sufficient, and the answer is B.

Takeaway: questions that look strange or funky are always testing concepts that have been tested in the past – otherwise, the exam wouldn’t be standardized. By making these connections, and recognizing that a verbal clue such as “none left over” really means that we’re talking about multiples and factors, we can recognize even the most abstract patterns on the toughest of GMAT questions.

*Plan on taking the GMAT soon? We have GMAT prep courses starting all the time. And be sure to follow us on Facebook, YouTube, Google+ and Twitter!*

*By David Goldstein, a Veritas Prep GMAT instructor based in Boston. You can find more articles written by him here.*

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]]>The post GMAT Probability Practice: Questions and Answers appeared first on Veritas Prep Blog.

]]>You may already know that there are certain formulas that can help solve GMAT probability questions, but there is more to these problems than teasing out the right answers. Take a look at some advice on how to tackle GMAT probability questions to calm your fears about the test:

**Probability Formulas**

As you work through GMAT probability practice questions, you will need to know a few formulas. One key formula to remember is that the probability equals the number of desired outcomes divided by the number of possible outcomes. Another formula deals with discrete events and probability – that formula is P(A and B) = P(A)*P(B). Figuring out the probability of an event not occurring is one minus the probability that the event will occur. Putting these formulas into practice is the most effective way to remember them.

**Is it Enough to Know the Basic Formulas for Probability?**

Some test-takers believe that once you know the formulas related to probability for GMAT questions, then you have the keys to success on this portion of the test. Unfortunately, that is not always the case. The creators of the GMAT are not just looking at your ability to plug numbers into formulas – you must understand what each question is asking and why you arrived at a particular answer. Successful business executives use reason and logic to arrive at the decisions they make. The creators of the GMAT want to see how good you are at using these same tools to solve problems.

**The Value of Practice Exams**

Taking a practice GMAT can help you determine your skill level when it comes to probability questions and problems on every other section of the test. Also, a practice exam gives you the chance to become accustomed to the amount of time you’ll have to finish the various sections of the test.

At Veritas Prep, we have one free GMAT practice test available to anyone who wants to get an idea of how prepared they are for the test. After you take the practice test, you will receive a score report and thorough performance analysis that lets you know how you fared on each section. Your performance analysis can prove to be one of the most valuable resources you have when starting to prepare for the GMAT. Follow-up practice tests can be just as valuable as the first one you take. These tests reveal your progress on probability problems and other skills on the GMAT. The results can guide you on how to adjust your study schedule to focus more time on the subjects that need it.

**Getting the Right Kind of Instruction**

When it comes to probability questions, GMAT creators have been known to set subtle traps for test-takers. In some cases, you may happen upon a question with an answer option that jumps out at you as the right choice. This could be a trap.

If you study for the GMAT with Veritas Prep, we can teach you how to spot and avoid those sorts of traps. Our talented instructors have not only taken the GMAT; they have mastered it. Each of our tutors received a score that placed them in the 99th percentile. Consequently, if you study with Veritas Prep, you’ll benefit from the experience and knowledge of tutors who have conquered the GMAT. When it comes to probability questions, GMAT tutors at Veritas Prep have you covered!

In addition to providing you with effective GMAT strategies, tips, and top-quality instruction, we also give you choices regarding the format of your courses. We have prep classes that are given online and in person – learn your lessons where you want, and when you want. You may want to go with our private tutoring option and get a GMAT study plan that is tailored to your needs. Contact Veritas Prep today and dive into your GMAT studies!

*Getting ready to take the GMAT? We have **free online GMAT seminars** **running all the time. And, be sure to follow us on **Facebook**, **YouTube**, **Google+**, and **Twitter**!*

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]]>The post When to Pick Your Own Numbers on GMAT Quant Questions appeared first on Veritas Prep Blog.

]]>As luck would have it, on her previous practice exam she’d received the following problem, which both illustrates the value of picking numbers and demonstrates why this approach works so well.

*A total of 30 percent of the geese included in a certain migration study were male. If some of the geese migrated during the study and 20 percent of the migrating geese were male, what was the ratio of the migration rate for the male geese to the migration rate for the female geese? *

*[Migration rate for geese of a certain sex = (number of geese of that sex migrating) / (total number of geese of that sex)] *

A) 1/4

B) 7/12

C) 2/3

D) 7/8

E) 8/7

This is a perfect opportunity to break out two of my favorite GMAT tools: picking numbers and making charts. So, let’s say there are 100 geese in our population. That means that if 30% are male, we’ll have 30 male geese and 70 females geese, giving us the following chart:

Male | Female | Total | |

Migrating | |||

Not-Migrating | |||

Total | 30 | 70 | 100 |

Now, let’s say 10 geese were migrating. That means that 90 were not migrating. Moreover, if 20 percent of the migrating geese were male, we know that we’ll have 2 migrating males and 8 migrating females, giving us the following:

Male | Female | Total | |

Migrating | 2 | 8 | 10 |

Not-Migrating | |||

Total | 30 | 70 | 100 |

(Note that if we wanted to, we could fill out the rest of the chart, but there’s no reason to, especially when we’re trying to save as much time as possible.)

Our migration rate for the male geese is 2/30 or 1/15. Our migration rate for the female geese is 8/70 or 4/35. Ultimately, we want the ratio of the male migration rate (1/15) to the female migration rate (4/35), so we need to simplify (1/15)/(4/35), or (1*35)/(15*4) = 35/60 = 7/12. And we’re done – B is our answer.

My student was skeptical. How did we know that 10 geese were migrating? What if 20 geese were migrating? Or 50? Shouldn’t that change the result? This is the beauty of picking numbers – it doesn’t matter what number we pick (so long as we don’t end up with an illogical scenario in which, say, the number of migrating male geese is greater than the number of total male geese). To see why, watch what happens when we do this algebraically:

Say that we have a total of “t” geese. If 30% are male, we’ll have 0.30t male geese and 0.70t females geese. Now, let’s call the migrating geese “m.” If 20% are male, we’ll have 0.20m migrating males and 0.80m migrating females. Now our chart will look like this:

Male | Female | Total | |

Migrating | 0.20m | 0.80m | m |

Not-Migrating | |||

Total | 0.30t | 0.70t | t |

The migration rate for the male geese is 0.20m/0.30t or 2m/3t. The migration rate for the female geese is 0.80m/0.70t or 8m/7t. We want the ratio of the male migration rate (2m/3t) to the female migration rate (8m/7t), so we need to simplify (2m/3t)/(8m/7t) = (2m*7t)/(3t * 8m) = 14mt/24mt = 7mt/12mt = 7/12. It’s clear now why the numbers we picked for m and t don’t matter – they cancel out in the end.

Takeaway: We cannot say this enough: the GMAT is not testing your ability to do formal algebra. It’s testing your ability to make good decisions in a stressful environment. So your goal, when preparing for this test, isn’t to become a virtuoso mathematician, even for the toughest questions. It’s to practice the kind of simple creative thinking that will get you to your answer with the smallest investment of your time.

*Plan on taking the GMAT soon? We have GMAT prep courses starting all the time. And be sure to follow us on Facebook, YouTube, Google+ and Twitter!*

*By David Goldstein, a Veritas Prep GMAT instructor based in Boston. You can find more articles written by him here.*

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]]>The post How to Use Units Digits to Avoid Doing Painful Calculations on the GMAT appeared first on Veritas Prep Blog.

]]>The units digit of 130,467 * 367,569 would be the same as the units digit of 7*9, as only the units digits of the larger numbers are relevant in such a calculation. 7*9 = 63, so the units digit of 130,467 * 367,569 is 3. This is one of those concepts that is so simple and elegant that it seems too good to be true.

And yet, this simple, elegant rule comes into play on the GMAT with surprising frequency.

Take this question for example:

*If n is a positive integer, how many of the ten digits from 0 through 9 could be the units digit of n^3?*

*A) three*

*B) four*

*C) six*

*D) nine*

*E) ten*

Surely, you think, the solution to this question can’t be as simple as cubing the easiest possible numbers to see how many different units digits result. And yet that’s exactly what we’d do here.

1^3 = 1

2^3 = 8

3^3 = 27 à units 7

4^3 = 64 à units 4

5^3 = ends in 5 (Fun fact: 5 raised to any positive integer will end in 5.)

6^3 = ends in 6 (Fun fact: 6 raised to any positive integer will end in 6.)

7^3 = ends in 3 (Well 7*7 = 49. 49*7 isn’t that hard to calculate, but only the units digit matters, and 9*7 is 63, so 7^3 will end in 3.)

8^3 = ends in 2 (Well, 8*8 = 64, and 4*8 = 32, so 8^3 will end in 2.)

9^3 = ends in 9 (9*9 = 81 and 1 * 9 = 9, so 9^3 will end in 9.)

10^3 = ends in 0

Amazingly, when I cube all the integers from 1 to 10 inclusive, I get 10 different units digits. Pretty neat. The answer is E.

Of course, this question specifically invoked the term “units digit.” What are the odds of that happening? Maybe not terribly high, but any time there’s a painful calculation, you’d want to consider thinking about the units digits.

Take this question, for example:

*A certain stock exchange designates each stock with a one, two or three letter code, where each letter is selected from the 26 letters of the alphabet. If the letters may be replaced and if the same letters used in a different order constitute a different code, how many different stocks is it possible to uniquely designate with these codes? *

*A) 2,951
B) 8,125
C) 15,600
D) 16,302
E) 18,278 *

Conceptually, this one doesn’t seem that bad.

If I wanted to make a one-letter code, there’d be 26 ways I could do so.

If I wanted to make a two-letter code, there’d be 26*26 or 26^2 ways I could do so.

If I wanted to make a three-letter code, there’d be 26*26*26, or 26^3 ways I could so.

So the total number of codes I could make, given the conditions of the problem, would be 26 + 26^2 + 26^3. Hopefully, at this point, you notice two things. First, this arithmetic will be deeply unpleasant to do. Second, all of the answer choices have different units digits!

Now remember that 6 raised to any positive integer will always end in 6. So the units digit of 26 is 6, and the units digit of 26^2 is 6 and the units digit of 26^3 is also 6. Therefore, the units digit of 26 + 26^2 + 26^3 will be the same as the units digit of 6 + 6 + 6. Because 6 + 6 + 6 = 18, our answer will end in an 8. The only possibility here is E. Pretty nifty.

Takeaway: Painful arithmetic can always be avoided on the GMAT. When calculating large numbers, note that we can quickly find the units digit with minimal effort. If all the answer choices have different units digits, the question writer is blatantly telegraphing how to approach this problem.

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]]>One of my main goals in teaching a class is to persuade everyone that this is not, in fact, how hard questions work on this test. Hard questions don’t ask you do to something you don’t know how to do. Rather, they’re cleverly designed to provoke an anxiety response that makes it difficult to realize that you do know exactly how to solve the problem.

Take this official question, for example:

*Let a, b, c and d be nonzero real numbers. If the quadratic equation ax(cx + d) = -b(cx +d) is solved for x, which of the following is a possible ratio of the 2 solutions?*

*A) **–ab/cd*

*B) –ac/bd*

*C) –ad/bc*

*D) ab/cd*

*E) ad/bc*

Most students see this and panic. Often, they’ll start by multiplying out the left side of the equation, see that the expression is horrible (acx^2 + adx), and take this as evidence that this question is beyond their skill level. And, of course, the question was designed to elicit precisely this response. So when I do this problem in class, I always start by telling my students, much to their surprise, that every one of them already knows how to do this. They’ve just succumbed to the question writer’s attempt to convince them otherwise.

So let’s start simple. I’ll write the following on the board: xy = 0. Then I’ll ask what we know about x or y. And my students shrug and say x or y (or both) is equal to 0. They’ll also wonder what on earth such a simple identity has to do with the algebraic mess of the question they’d been struggling with.

I’ll then write this: zx + zy = 0. Again, I’ll ask what we know about the variables. Most will quickly see that we can factor out a “z” and get z(x+y) = 0. And again, applying the same logic, we see that one of the two components of the product must equal zero – either z = 0 or x + y = 0.

Next, I’ll ask if they would approach the problem any differently if I’d given them zx = -zy – they wouldn’t.

Now it clicks. We can take our initial equation in the aforementioned problem: ax(cx +d) = -b(cx+d), and see that we have a ‘cx + d’ on both sides of the equation, just as we’d had a “z” on both sides of the previous example. If I’m able to get everything on one side of the equation, I can factor out the common term.

Now ax(cx +d) = -b(cx+d) becomes ax(cx +d) + b(cx+d) = 0.

Just as we factored out a “z” in the previous example, we can factor out “cx + d” in this one.

Now we have (cx + d)(ax + b) = 0.

Again, if we multiply two expressions to get a product of zero, we know that at least one of those expressions must equal 0. Either cx + d = 0 or ax + b = 0.

If cx + d = 0, then x = -d/c.

If ax + b = 0, then x = -b/a.

Therefore, our two possible solutions for x are –d/c and –b/a. So, the ratio of the two would simply be (-d/c)/(-b/a). Recall that dividing by a fraction is the equivalent of multiplying by the reciprocal, so we’re ultimately solving for (-d/c)(-a/b). Multiplying two negatives gives us a positive, and we end up with da/cb, which is equivalent to answer choice E.

Takeaway: Anytime you see something on the GMAT that you think you don’t know how to do, remind yourself that the question was designed to create this false impression. You know how to do it – don’t hesitate to dive in and search for how to apply this knowledge.

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]]>His story is a fairly typical one: like the vast majority of GMAT test-takers, he enrolled in the class looking to hit a 700. His scores improved steadily throughout the course, and when he took the test the first time, he’d received a 720, which was in line with his last two practice exams. After he finished the official test, he called me – both because he was feeling pretty good about his score but also because a part of him was sure he could do better.

My feeling at the time was that there really wasn’t any pressing need for a retake: a 720 is a fantastic score, and once you hit that level of success, the incremental gains of an improvement begin to suffer from the law of diminishing returns. Still, when you’re talking about the most competitive MBA programs, you want any edge you can get. Moreover, he’d already made up his mind. He wanted to retake.

Part of his decision was rooted in principle. He was sure he could hit the 99^{th} percentile, and he wanted to prove it to himself. The problem, he noted, was that he’d already mastered the test’s content. So if there was nothing left for him to learn, how did he jump to the 99^{th} percentile?

The answer can be found in the vast body of literature enumerating the psychological variables that influence test scores. We like to think of tests as detached analytic tools that measure how well we’ve mastered a given topic. In reality, our mastery of the content is one small aspect of performance.

Many of us know this from experience – we’ve all had the experience of studying hard for a test, feeling as though we know everything cold, and then ending up with a score that didn’t seem to reflect how well we’d learned the material. After I looked at the research, it was clear that the two most important psychological variables were 1) confidence and 2) how well test-takers managed test anxiety. (And there’s every reason to believe that those two variables are interconnected.)

I’ve written in the past about how a mindfulness meditation practice can boost test day performance. I’ve also written about how perceiving anxiety as excitement, rather than as a nefarious force that needs to be conquered, has a similarly salutary effect. Recently I came across a pair of newer studies.

In one, researchers found that when students wrote in their journals for 10 minutes about their test-taking anxiety the morning of their exams, their scores went up substantially. In another, the social psychologist Amy Cuddy found that body language had a profound impact on performance in all sorts of domains. For example, her research has revealed that subjects who assumed “power poses” for two minutes before a job interview projected more confidence during the interview and were better able to solve problems than a control group that assumed more lethargic postures. (To see what these power poses look like, check out Cuddy’s fascinating Ted talk here.) Moreover, doing power poses actually created a physiological change, boosting testosterone and reducing the stress hormone Cortisol.

Though her research wasn’t targeted specifically at test-takers, there’s every reason to believe that there would be a beneficial effect for students who practiced power poses before an exam. Many teachers acquainted with Cuddy’s research now recommend that their students do this before tests.

So the missing piece of the puzzle for my student was simply confidence. His strategies hadn’t changed. His knowledge of the core concepts was the same. The only difference was his psychological approach. So now I’m recommending that all of my students do the following to cultivate an ideal mindset for producing their best possible test scores:

- Perform mindfulness meditation for the two weeks leading up to the exam.
- Reframe test-day anxiety as excitement.
- Spend 10 minutes the morning of the test writing in a journal.
- Practice two minutes of power poses in the waiting room before sitting for the exam and between the Quant and Verbal section.

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