First, take a look at Lessons** 1, 2, 3, 4 **and** 5**!

**Lesson Six: **

Practice Tests Aren’t Real Tests: read the popular GMAT forums and you’ll see lots of handwringing and bellyaching about practice tests scores…but not very much analysis beyond the scores themselves. In this video, Ravi (along with his alter ego Allen Iverson) talks about practice, stressing the importance of using the tests to increase your score more than to merely try to predict it. Pacing is paramount and diagnosis is divine; as Ravi will explain, practice tests are critical for learning how you would perform if that were the real thing, with the added bonus of having the opportunity to fix those things that you don’t like about that practice performance.

Click **HERE** to check out Ravi’s latest video on this subject.

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*By Brian Galvin*

There are two ways to interpret this phenomenon. Einstein thought that an electron had a defined position and momentum. We simply weren’t capable of documenting both at the same time due to the clumsiness of our measuring instruments. Bohr, on the other hand, believed that an electron didn’t have a position or momentum until we measured it. In other words, the electron doesn’t exist before it’s observed (which, of course, raises knotty metaphysical questions about how the observer exists, if the observer is herself made of sub-atomic particles, none of which exist before they’re observed. But this one is a little harder to connect to the GMAT, so the reader is invited to contemplate such a conundrum in his or her own time, once the test is in the rear view mirror).

Though physicists, by and large, are more likely to accept Bohr’s interpretation than Einstein’s, on the GMAT we’ll want to reason more like Einstein, particularly when it comes to Data Sufficiency. In almost every class I teach, a student will ask a question along the lines of, “Is it possible that, in a value question, Statement 1 will tell you definitively that x equals 8, and that Statement 2 will tell you definitively that x equals some other number?” The answer is a resounding “No” – x has a unique value, the question is whether we can definitively divine what that value is. If Statement 1 tells us decisively that x = 8, Statement 2 cannot tell us that x equals, say, 10.

Let’s see how this principle can be helpful in action:

*If a certain positive integer is divided by 9, the remainder is 3. What is the remainder when the integer is divided by 5?*

*1) **If the integer is divided by 45, the remainder is 30.*

*2) **The integer is divisible by 2*

Statement 1 tells me that when I divide an integer by 45, I get a remainder of 30. So I could test 75, because that will give a remainder of 30 when divided by 45 (And, just as importantly, it gives a remainder of 3 when divided by 9 – I have to satisfy the conditions embedded in the question stem too!). The question asks me for the remainder when the integer is divided by 5. Well, 75/5 will give no remainder, so the remainder, in this case, is 0.

Let’s see if that will always be the case. Next, we’ll test 105, which gives a remainder of 30 when divided by 45, and gives a remainder of 3 when divided by 9 [note: I can generate fresh numbers to test by simply adding the divisor, 30, to the previous number I test (75 + 30 = 105)]. Clearly 105/5 will give a remainder of 0, as any number that ends in 5 will be divisible by 5. The same will be true of 145, or 175, or 205. The remainder, when the integer in question is divided by 5 will always be 0, so Statement 1 is sufficient.

Now let’s reason like Einstein. We know that the answer to the question has a definitive value of 0. That can’t change. The only way Statement 2 can be sufficient is if it gives us that *same value*. So let’s pick a number that is divisible by 2 but gives a remainder of 3 when divided by 9. 12 will work. The remainder, when 12 is divided by 5, is 2. All we need to see is that we *did not get 0.*

We don’t have to test another number. Statement 2 cannot, alone, be sufficient, because we already know – the Einsteins that we are – that the value in question is 0. Statement 2 cannot tell us that the value is definitively 2 (if we continued to test, we’d eventually find values that gave us a remainder of 0 when we divided by 5, but because there are other possibilities, Statement 2 doesn’t give us enough information to determine, without a doubt, that the value is 0). We’re done. Statement 2 is insufficient. The answer is A: Statement 1 alone is sufficient.

Note that this same logic will work on “YES/NO” questions as well. If Statement 1 tells us that the answer to the question is definitively “YES”, Statement 2 cannot tell us that the answer is definitively “NO”, and vice versa. Recognizing this can save us valuable time.

Takeaway: Although Niels Bohr might say that there is no answer to a Data Sufficiency question until we evaluate a statement, for these questions we want to think more like Einstein and recognize that, in the mind of the question-writer, there is an objective answer – the question is whether we have enough information to definitely deduce what that answer is. There may be no objective reality in the quantum world, but on the GMAT, there most certainly is.

*GMATPrep question courtesy of the Graduate Management Admissions Council.

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

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

But in mathematics, seemingly basic topics often have broader applications. So let’s consider both simple and complex applications of remainders on the GMAT. The most straightforward scenario is for the question to ask what the remainder is in a given context. We’ll start by looking at an official Data Sufficiency question of moderate difficulty:

*What is the remainder when x is divided by 3?*

*1) The sum of the digits of x is 5*

*2) When x is divided by 9, the remainder is 2*

Pretty straightforward question. In Statement 1, we could approach by simply picking numbers. If the sum of the digits of x is 5, x could be 14. When 14 is divided by 3, the remainder is 2. Similarly, x could be 32. When 32 is divided by 3, the remainder will again be 2. Or x could be 50, and still, the remainder when x is divided by 3 will be 2. So no matter what number we pick, the remainder will always be 2. Statement 1 alone is sufficient.

Note that if we know the rule for divisibility by 3 – if the digits of a number sum to a multiple of 3, the number itself is a multiple of 3 – we can reason this out without picking numbers. If the sum of the digits of x were exactly 3, the remainder would be 0. If the sum of the digits of x were 4, then logically, the remainder would be 1. Consequently, if the sum of the digits of x were 5, the remainder would have to be 2.

Again, in Statement 2, we can pick numbers. We’re told that when x is divided by 9, the remainder is 2. To quickly generate a list of numbers that we might test, we can start with multiples of 9: 9, 18, 27, 36, etc. Then, we can add two to each of those multiples of 9 to get the following list of numbers: 11, 20, 29, 38, etc. All of these numbers will give us a remainder of 2 when divided by 9. Now we can test them. If x is 11, when x is divided by 3, the remainder will be 2. If x is 20, when x is divided by 3, the remainder will be 2. We’ll quickly see that the remainder will always be 2, so Statement 2 is also sufficient on its own. The answer to this question is D, either statement alone is sufficient. That’s not too bad.

But the GMAT won’t always be so conspicuous about what category of math it’s testing. Take this more challenging question, for example:

* **June 25, 1982 fell on a Friday. On which day of the week did June 25, 1987 fall. (Note: 1984 was a leap year.)*

* **A) **Sunday*

*B) **Monday*

*C) **Tuesday*

*D) **Wednesday*

*E) **Thursday*

If you’re anything like my students, it’s not blindingly obvious that this is a remainder question in disguise. But that is precisely what we’re dealing with. Consider a very simple case. Say that June 1 is a Monday, and I want to know what day of the week it will be 14 days later. Clearly, that would also be a Monday. And if I asked you what day of the week it would be 16 days later, you’d know that it would be a Wednesday – two days after Monday. Put another way – because we’re dealing with weeks, or increments of 7 – all we need to do is divide the number of days elapsed by 7 and then find the remainder in order to determine the day of the week. 16 divided by 7 gives a remainder of 2, so if June 1 is a Monday, 16 days later must be 2 days after Monday.

Suddenly the aforementioned question is considerably more approachable. From June 25, 1982 to June 25, 1983 a total of 365 days will pass. 365/7 gives a remainder of 1, so if June 25, 1982 was a Friday, June 25 1983 will be a Saturday. From June 25, 1983 to June 25, 1984, 366 days will pass because 1984 is a leap year. 366/7 gives a remainder of 2, so if June 25, 1983 was a Saturday, June 25, 1984 will be 2 days later, or Monday. We already know that in a typical 365 day year, the remainder will be 1, so June 25, 1985 will be Tuesday, June 25, 1986 will be Wednesday and June 25, 1987 will be Thursday, which is our answer.

Takeaway: the challenge of the GMAT isn’t necessarily that questions are asking you to do difficult math, but that it can be hard to figure out what the questions are asking you to do. When you encounter something that seems unfamiliar or strange, remind yourself that virtually every problem you encounter will involve the application of a concept considerably simpler than the nebulous wording the question might suggest.

*Official Guide questions courtesy of the Graduate Management Admissions Council.

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

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

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I didn’t make up that 1.3/3.2 calculation. It comes directly from an official question, and it’s quite clearly designed to elicit the panicked response it usually gets when I ask it in class. Here is the full question:

*The age of the Earth is approximately 1.3 * 10^17 seconds, and one year is approximately 3.2 * 10^7 seconds. Which of the following is closest to the age of the Earth in years?*

*5 * 10^9**1 * 10^9**9 * 10^10**5 * 10^11**1 * 10^11*

Most test-takers quickly see that in order to convert from seconds to years, we have to perform the following calculation: 1.3 * 10^17 seconds * 1 year/3.2 * 10^ 7 seconds or (1.3 * 10^17)/(3.2 * 10^ 7.)

It’s here when many test-takers freeze. So let’s estimate. We’ll round 1.3 down to 1, and we’ll round 3.2 down to 3. Now we’re calculating or (1* 10^17)/(3 * 10^ 7.) We can rewrite this expression as (1/3) * (10^17)/(10^7.) This becomes .333 * 10^10. If we borrow a 10 from 10^10, we’ll get 3.33 * 10^9. We know that this number is a little smaller than the correct answer, because we rounded the numerator down from 1.3 to 1, and this was a larger change than the adjustment we made to the denominator. If 3.33 * 10^9 is a little smaller than the correct answer, the answer must be B. (Similarly, if we were to estimate 13/3, we’d see that the number is a little bigger than 4.)

This strategy will work just as well on tough Data Sufficiency questions:

*If it took Carlos ½ hour to cycle from his house to the library yesterday, was the distance that he cycled greater than 6 miles? (1 mile = 5280 feet.)*

*The average speed at which Carlos cycled from his house to the library yesterday was greater than 16 feet per second.**The average speed at which Carlos cycled from his house to the library yesterday was less than 18 feet per second.*

The fact that we’re given the conversion from miles to feet is a dead-giveaway that we’ll need to do some unit conversions to solve this question. So we know that the time is ½ hour, or 30 minutes. We want to know if the distance is greater than 6 miles. We’ll call the rate ‘r.’ If we put this question into the form of Rate * Time = Distance, we can rephrase the question as:

*Is r * 30 minutes > 6 miles?*

We can simplify further to get: *Is r > 6 miles/30 minutes* or *Is r > 1 mile/5 minutes?*

A quick glance at the statements reveals that, ultimately, I want to convert into feet per second. I know that 1 mile is 5280 feet and that 5 minutes is 5 *60, or 300 seconds.

Now *Is r > 1 mile/5 minutes?* becomes *Is r > 5280 feet/ 300 seconds.* Divide both by 10 to get Is r > 528 feet/30 seconds. Now, let’s estimate. 528 is pretty close to 510. I know that 510/30 is the same as 51/3, or 17. Of course, I rounded down by 18 from 528 to 510, and 18/30 is about .5, so I’ll call the original question:

*Is r > 17.5 feet/second?*

If we get to this rephrase, the statements become a lot easier to test. Statement 1 tells me that Carlos cycled at a speed greater than 16 feet/second. Well, that could mean he went 16.1 feet/second, which would give me a NO to the original question, or he could have gone 30 feet/second, so I can get a YES to the original question. Not Sufficient.

Statement 2 tells me that his average speed was less than 18 feet/second. That could mean he went 17.9 feet/second, which would give me a YES. Or he could have gone 2 feet/second, which would give me a NO.

Together, I know he went faster than 16 feet/second and slower than 18 feet/second. So he could have gone 16.1 feet/second, which would give a NO, and he could have gone 17.9, which would give a YES, so even together, the statements are not sufficient, and the answer is E.

The takeaway: estimation isn’t simply a luxury on the GMAT; on certain questions, it’s a necessity. If you find yourself grinding through a host of ungainly arithmetical calculations, stop, and remind yourself that there has to be a better, more time-efficient approach.

*GMATPrep questions courtesy of the Graduate Management Admissions Council.

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There are many ways the GMAT test makers ensure that you’re thinking logically about the solution of the question. One common example is that the question will give you a story that you have to translate into an equation. Anyone with a calculator can do 15 * 6 * 2 but it’s another skill entirely to translate that a car dealership that’s open every day but Sunday sells 3 SUVs, 5 trucks and 7 sedans per day for a sale that lasts a fortnight (sadly, the word fortnight is somewhat rare on the GMAT). Which skill is more important in business, crunching arbitrary numbers or deciphering which numbers to crunch? (Trick question: they’re both important!) The difference is a computer will calculate numbers much faster than a human ever will, but being able to determine what equation to set up is the more important skill.

This distinction is rather ironic, because the GMAT often provides questions that are simply equations to be solved. If the thought process is so important, why provide questions that are so straight forward? Precisely because you don’t have a calculator to solve them and you still need to use reasoning to get to the correct answer. An arbitrarily difficult question like 987 x 123 is trivial with a calculator and provides no educational benefit, simply an opportunity to exercise your fingers (and they want to look good for summer!) But without a calculator, you can start looking at interesting concepts like unit digits and order of magnitude in order to determine the correct answer. For business students, this is worth much more than a rote calculation or a mindless computation.

Let’s look at an example that’s just an equation but requires some analysis to solve quickly:

*(36^3 + 36) / 36 =*

*A) 216
*

This question has no hidden meaning and no interpretation issues. It is as straight forward as 2+2, but much harder because the numbers given are unwieldy. This is, of course, not an accident. A significant number of people will not answer this question correctly, and even more will get it but only after a lengthy process. Let’s see how we can strategically approach a question like this on test day.

Firstly, there’s nothing more to be done here than multiplying a couple of 2-digit numbers, then performing an addition, then performing a division. In theory, each of these operations is completely feasible, so some people will start by trying to solve 36^3 and go from there. However, this is a lengthy process, and at the end, you get an unwieldy number (46,656 to be precise). From there, you need to add 36, and then divide by 36. This will be a very difficult calculation, but if you think of the process we’re doing, you might notice that you just multiplied by 36, and now you’ll have to divide by 36. You can’t exactly shortcut this problem because of the stingy addition, but perhaps we can account for it in some manner.

Multiplying 36 by itself twice will be tedious, but since you’re dividing by 36 afterwards, perhaps you can omit the final multiplication as it will essentially cancel out with the division. The only caveat is that we have to add 36 in between multiplying and dividing, but logically we’re adding 36 and then dividing the sum by 36, which means that this is tantamount to just adding 1. As such, this problem kind of breaks down to just 36 * 36, and then you add 1. If you were willing to multiply 36^3, then 36^2 becomes a much simpler calculation. This operation will yield the correct answer (we’ll see shortly that we don’t even need to execute it), and you can get there entirely by reasoning and logic.

Moreover, you can solve this question using (our friendly neighbour) algebra. When you’re facing a problem with addition of exponents, you always want to turn that problem into multiplication if at all possible. This is because there are no good rules for addition and subtraction with exponents, but the rules for multiplication and division are clear and precise. Taking just the numerator, if you have 36^3 + 36, you can factor out the 36 from both terms. This will leave you with 36 *(36^2 + 1). Considering the denominator again, we end up with (36 *(36^2 + 1)) / 36. This means we can eliminate both the 36 in the numerator and the 36 in the denominator and end up with just (36^2 + 1), which is the same thing we found above.

Now, 36*36 is certainly solvable given a piece of paper and a minute or so, but you can tell a lot from the answer by the answer choices that are given to you. If you square a number with a units digit of 6, the result will always end with 6 as well (this rule applies to all numbers ending in 0, 1, 5 and 6). The result will therefore be some number that ends in 6, to which you must add 1. The final result must thus end with a 7. Perusing the answer choices, only answer choice C satisfies that criterion. The answer must necessarily be C, 1297, even if we don’t spend time confirming that 36^2 is indeed 1,296.

In the quantitative section of the GMAT, you have an average of 2 minutes per question to get the answer. However, this is simply an average over the entire section; you don’t have to spend 2 minutes if you can shortcut the answer in 30 seconds. Similarly, some questions might take you 3 minutes to solve, and as long as you’re making up time on other questions, there’s no problem taking a little longer. However, if you can solve a question in 30 seconds that your peers spend 2 or 3 minutes solving, you just used the secret shortcut that the exam hopes you will use.

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

*Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.*

This means that we can soon begin to discuss Barack Obama’s legacy. As with any legacy, it’s important to look at the terms globally, and not necessarily get bogged down by one or two memorable moments. A legacy is a summary of the major points and the minor points of one’s tenure. As such, it’s difficult to sum up a presidency that spanned nearly a decade and filter it down to simply “Obamacare” or “Killing Bin Laden” or “Relations with Cuba”. Not everyone will agree on what the exact highlights were, but we must be able to consider all the elements holistically.

On the GMAT, Sentence Correction is often the exact same way. If only a few words are highlighted, then your task is to make sure those few words make sense and flow properly with the non-underlined portion. If, however, the entire sentence is underlined, you have “carte blanche” (or Cate Blanchett) to make changes to any part of the sentence. The overarching theme is that the whole sentence has to make sense. This means that you can’t get bogged down in one portion of the text, you have to evaluate the entire thing. If some portion of the phrasing is good but another contains an error, then you must eliminate that choice and find and answer that works from start to finish.

Let’s look at a topical Sentence Correction problem and look for how to approach entire sentences:

*Selling two hundred thousand copies in its first month, the publication of The Audacity of Hope in 2006 was an instant hit, helping to establish Barack Obama as a viable candidate for president.*

*A) Selling two hundred thousand copies in its first month, the publication of The Audacity of Hope in 2006 was an instant hit, helping to establish Barack Obama as a viable candidate for president.
*

An excellent strategy in Sentence Correction is to look for decision points, significant differences between one answer choice and another, and then make decisions based on which statements contain concrete errors. However, when the whole sentence is underlined, this becomes much harder to do because there might be five decision points between statements, and each one is phrased a little differently. You can still use decision points, but it might be simpler to look through the choices for obvious errors and then see if the next answer choice repeats that same gaffe (not a giraffe).

Looking at the original sentence (answer choice A), we see a clear modifier error at the beginning. Once the sentence begins with “Selling two hundred thousand copies in its first month,…” the very next word after the comma must be the noun that has sold 200,000 copies. Anything else is a modifier error, whether it be “Barack Obama wrote a book that sold” or “the publication of the book” or any other variation thereof. We don’t even need to read any further to know that it can’t be answer choice A. We’ll also pay special attention to modifier errors because if it happened once it can easily happen again in this sentence.

Answer choice B, unsurprisingly, contains a very similar modifier error. The sentence begins with: “The publication in 2006 of the Audacity of Hope was an instant hit:…”. This means that the publication was a hit, whereas logically the book was the hit. This is an incorrect answer choice again, and so far we haven’t even had to venture beyond the first sentence, so don’t let the length of the answer choices daunt you.

Answer choice C, “*helping to establish Barack Obama as a viable candidate for president was the publication of The Audacity of Hope in 2006, which was an instant hit: it sold two hundred thousand copies in its first month” *contains another fairly glaring error. On the GMAT, the relative pronoun “which” must refer to the word right before the comma. In this case, that would be the year 2006, instead of the actual book. Similarly to the first two choices, this answer also contains a pronoun error because the “it” after the colon would logically refer back to the publication instead of the book as well. One error is enough, and we’ve already got two, so answer choice C is definitely not the correct selection.

Answer choice D, “The *Audacity of Hope was an instant hit: it helped establish Barack Obama as a viable candidate for president, selling two hundred thousand copies in its first month and published in 2006” *sounds pretty good until you get to the very end. The “published in 2006” is a textbook dangling modifier, and would have been fine had it been placed at the beginning of the sentence. Unfortunately, as it is written, this is not a viable answer choice (you are the weakest link).

By process of elimination, it must be answer choice E. Nonetheless, if we read through it, we’ll find that it doesn’t contain any glaring errors: “The *Audacity of Hope, published in 2006, was an instant hit: in two months, it sold two hundred thousand copies and helped establish its author, Barack Obama, as a viable candidate for president.” *The title of the book is mentioned initially, a modifier is correctly placed and everything after the colon describes why it was regarded as a hit. Holistically, there’s nothing wrong with this answer choice, and that’s why E must be the correct answer.

Overall, it’s easy to get caught up in one moment or another, but it’s important to look at things globally. A 30-word passage entirely underlined can cause anxiety in many students because there are suddenly many things to consider at the same time. There’s no reason to panic. Just review each statement holistically, looking for any error that doesn’t make sense. If everything looks good, even if it wasn’t always ideal, then the answer choice is fine. It’s important to think of your legacy, and on the GMAT, that means getting a score that lets you achieve your goals.

**No single question matters unless you let it.**

Reflect on that for a second, because it’s super important, weird, true, and again…important. The GMAT exam is not testing your ability to get as many questions right as you can. You can get the exact same percentage of questions right on two different exams and end up getting very different scores as a result of the complicated scoring algorithm. Mistakes that will crush your score are a large string of consecutive incorrect answers, unanswered questions remaining at the end of the section (these hurt your score even more than answering them incorrectly would), and a very low hit rate for the last 5 or 10 questions. These are all problems that are likely to arise if you spend way too much time on one/several questions.

Each individual question is actually pretty insignificant. The GMAT has 37 quantitative questions to gauge your ability level (currently ignoring the issue of experimental questions), so whether you get a certain question right or wrong doesn’t matter much. Let’s look at a hypothetical example and pick on question #17 for a second (just because it looked at me wrong!). If you start question 17, realize that it is not going your way, and ultimately make an educated guess after about 2 minutes and get it wrong…that doesn’t hurt you a lot. You missed the question, but you didn’t let it burn a bunch of your time and you live to fight another day (or in this case question).

Now let’s look at question 17 again, but from the perspective of being stubborn. If you start the question and are struggling with it but refuse to quit, thinking something like “this is geometry, I am so good at geometry, I have to get this right!”, then it will become very significant. In a bad way. In this example you spend 6 minutes on the question and you get it right. Congratulations! Except…you are now statistically not even going to get to attempt to answer two other questions because of the time that you just committed to it (with an average of 2 minutes per question on the quant section, you just allocated 3 questions’ worth of time to one question).

So your victory over infamous question 17 just got you 2 questions wrong! That’s a net negative. Loop in the concept of experimental questions, the fact that approximately one-fourth of quant questions don’t count, and therefore it is entirely possible that #17 isn’t even a real question, and the situation is pretty depressing.

Pacing is critical, and your pacing on quant questions should very rarely ever go above 3 minutes. Spending an excess amount of time on a question but getting it right is not a success; it is a bad strategic move. I challenge you to look at any practice tests that you have taken and decide whether you let this happen. Were there a few questions that you spent way over 2 minutes on and got right, but then later in the test a bunch of questions that you had to rush on and ended up missing, even though they may not have been that difficult? If that’s the case, then your timing is doing some serious damage. Work to correct this fatal error ASAP!

*Brandon Pierpont is a GMAT instructor for Veritas Prep. He studied finance at Notre Dame and went on to work in private equity and investment banking. When he’s not teaching the GMAT, he enjoys long-distance running, wakeboarding, and attending comedy shows.*

As an example, consider a simple question that asks you how many even numbers there are between 1 and 100. Of course, you could write out all 100 terms and identify which ones are even, say by circling them, and then sum up all the circled terms. This strategy would work, but it is completely inefficient and anyone who’s successfully passed the fourth grade would be able to see that you can get the answer faster than this. If every second number is even, then you just have to take the number of terms and divide by 2. The only difficulty you could face would be the endpoints (say 0 to 100 instead), but you can adjust for these easily. The next question might be count from 1 to 1,000, and you definitely don’t want to be doing that manually.

Other questions might not be as straight forward, but can be solved using similar mathematical properties. It’s important to note that you don’t have a calculator on the GMAT, but you will have one handy for the rest of your life (even in a no-WiFi zone!). This means that the goal of the test is not to waste your time executing calculations you would execute on your calculator in real life, but rather to evaluate how you think and whether you can find a logical shortcut that will yield the correct answer quickly.

Let’s look at an example that can waste a lot of time if you’re not careful:

*Brian plays a game in which he rolls two die. For each die, an even number means he wins that amount of money and an odd number means he loses that amount of money. What is the probability that he loses money if he plays the game once?*

*A) 11/12
*

First, it’s important to interpret the question properly. Brian will roll two die, independently of one another. For each even number rolled, he will win that amount of money, so any given die is 50/50. If both end up even, he’s definitely winning some money, but if one ends up even and the other odd, he may win or lose money depending on the values. The probability should thus be close to being 50/50, but a 5 with a 4 will result in a net loss of 1$, whereas a 5 with a 6 will result in a net gain of 1$. Clearly, we need to consider the actual values of each die in some of our calculations.

Let’s start with the brute force approach (similar to writing out 1-100 above). There are 6 sides to a die, and we’re rolling 2 dice, so there are 6^2 or 36 possibilities. We could write them all out, sum up the dollar amounts won or lost, and circle each one that loses money. However, it is essentially impossible to do this in less than 2 minutes (or even 3-4 minutes), so we shouldn’t use this as our base approach. We may have to write out a few possibilities, but ideally not all 36.

If both numbers are even, say 2 and 2, then Brian will definitely win some money. The only variable is how much money, but that is irrelevant in this problem. Similarly, if he rolls two odd numbers, say 3 and 3, then he’s definitely losing money. We don’t need to calculate each value; we simply need to know they will result in net gains or net losses. For two even numbers, in which we definitely win money, this will happen if the first die is a 2, a 4 or a 6, and the second die is a 2, a 4 or a 6. That would leave us with 9 possibilities out of the 36 total outcomes. You can also calculate this by doing the probability of even and even, which is 3/6 * 3/6 or 9/36. Similarly, odd and odd will also yield 9/36 as the possibilities are 1, 3, and 5 with 1, 3, and 5. Beyond this, we don’t need to consider even/even or odd/odd outcomes at all.

The interesting part is when we come to odds and evens together. One die will make Brian win money and the other will make him lose money. The issue is in the amplitude. Since we’ve eliminated 18 possibilities that are all entirely odd or even, we only need to consider the 18 remaining mixed possibilities. There is a logical way to solve this issue, but let’s cover the brute force approach since it’s reasonable at this point. The 18 possibilities are:

Odd then even: Even then odd:

1, 2 3, 2 5, 2 2, 1 4, 1 6, 1

1, 4 3, 4 5, 4 2, 3 4, 3 6, 3

1, 6 3, 6 5,6 2, 5 4, 5 6, 5

Looking at these numbers, it becomes apparent that each combination is there twice ((2,1) or (1,2)). The order may matter when considering 36 possibilities, but it doesn’t matter when considering the sums of the die rolls. (2,1) and (1,2) both yield the same result (net gain of 1), so the order doesn’t change anything to the result. We can simplify our 18 cases into 9 outcomes and recall that each one weighs 1/18 of the total:

(1,2) or (2,1): Net gain of 1$

(1,4) or (4,1): Net gain of 3$

(1,6) or (6,1): Net gain of 5$

Indeed, no matter what even number we roll with a 1, we definitely make money. This is because 1 is the smallest possible number. Next up:

(3,2) or (2,3): Net loss of 1$

(3,4) or (4,3): Net gain of 1$

(3,6) or (6,3): Net gain of 3$

For 3, one of the outcomes is a loss whereas the other two are gains. Since 3 is bigger than 2, it will lead to a loss. Finally:

(5,2) or (2,5): Net loss of 3$

(5,4) or (4,5): Net loss of 1$

(5,6) or (6,5): Net gain of 1$

For 5, we tend to lose money, because 2/3 of the possibilities are smaller than 5. Only a 6 paired with the 5 would result in a net gain. Indeed, all numbers paired with 6 will result in a net gain, which is the same principle as always losing with a 1.

Summing up our 9 possibilities, 3 led to losses while 6 led to gains. The probability is thus not evenly distributed as we might have guessed up front. Indeed, the fact that any 6 rolled with an odd number always leads to a gain whereas any 1 rolled with an even number always leads to a loss helps explain this discrepancy.

To find the total probability of losing money, we need to find the probability of reaching one of these three odd-even outcomes. The chance of the dice being odd and even (in any order) is ½, and within that the chances of losing money are 3/9: (3, 2), (5, 2), and (5, 4). Thus we have 3/9 * ½ = 3/18 or 1/6 chance of losing money if it’s odd/even. Similarly, if it ends up odd/odd, then we always lose money, and that’s 3/9 * 3/9 = 9/36 or ¼. We have to add the two possibilities since any of them is possible, and we get ¼ + 1/6, if we put them on 12 we get 3/12 + 2/12 which equals 5/12. This is answer choice D.

It’s convenient to shortcut this problem somewhat by identifying that it cannot end up at 50/50 (answer choice C) because of the added weight of even numbers. Since 6 will win over anything, you start getting the feeling that your probability of losing will be lower than ½. From there, your choices are D or E, 15/36 or 12/36. Short of taking a guess, you could start writing out a few possibilities without having to consider all 36 outcomes, and determine that all odd/odd combinations will work. After that, you look at the few possibilities that could work ((5,4), (4,5), etc) and determine that there are more than 12 total possibilities, locking you in to answer choice D.

Many students struggle with problems such as these because they appear to be simple if you just write out all the possibilities. Especially when your brain is already feeling fatigued, you may be tempted to try and save mental energy by using brute force to solve problems. Beware, the exam wants you to do this (It’s a trap!) and waste precious time. If you need to write out some possibilities, that’s perfectly fine, but try and avoid writing them all out by using logic and deduction. On test day, if you use logic to save time on possible outcomes, you won’t lose.

This statement “No outside knowledge is required on the GMAT” is true in spirit, but a fundamental understanding of certain basic concepts is sometimes required. The exam won’t expect you to know the distance between New York and Los Angeles (19,600 furlongs or so), but you should know that both cities exist. The exam will always give you conversions when it comes to distances (miles to feet, for example), temperatures (Fahrenheit to Celsius) or anything else that can be measured in different systems, but the basic concepts that any human should know are fair game on the exam.

If you think about the underlying logic, it makes sense that a business person needs to be able to reason things out, but the reasoning must also be based on tenets that people can agree on. You won’t need to know something like all the variables involved in a carbon tax or on the electoral process of Angola, but you should know that Saturday comes after Friday (and Sunday comes afterwards).

Let’s look at a relatively simple question that highlights the need to think critically about outside knowledge that may be important:

*Tom was born on October 28 ^{th}. On what day of the week was he born?*

*1) In the year of Tom’s birth, January 20 ^{th} was a Sunday.
*

*A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
*

Since this is a data sufficiency question, it’s important to note that we must only determine whether or not the information is sufficient, we do not actually need to figure out which day of the week it is. Once we know that the information is knowable, we don’t need to proceed any further.

In this case, we are trying to determine Tom’s birthday with 100% certainty. There are only 7 days in a week, but we need a reference point somewhere to determine which year it is or what day of the year another day of that same year falls (ideally October 27^{th}!).

Statement 1 gives us a date for that same year. This should be enough to solve the problem, except for one small detail: the day given is in January. Since the Earth’s revolution around the sun is not an exact multiple of its rotation around itself, some years contain one extra day on February 29^{th}, and are identified as leap years. The day of January 20^{th} gives us a fixed point in that year, but since it is before February 28^{th}, we don’t know if March 1^{st} will be 40 days or 41 days away from January 20^{th}. Since this is the case, October 28^{th} could be one of two different days of the week, depending on whether we are in a leap year, and so this statement is insufficient.

Statement 2, on the other hand, gives us a date in July. Since July is after the possible leap day, this means that the statement must be sufficient. Specifically, if July 17^{th} was a Wednesday, then October 28^{th} would have to be a Monday. You could do the calculations if you wanted to: there are 14 more days in July, 31 in August, 30 in September and 28 in October, for a total of 103 days, or 14 weeks and 5 days. The 14 weeks don’t change anything to the day of the week, so we must advance 5 days from Wednesday, taking us to the following Monday. Statement 2 must be sufficient, even if we don’t need to execute the calculations to be sure.

Interestingly, if you consider January 20^{th} to be a Sunday, then you could get a year like 2013 in which the 28^{th} of October is a Monday. 2013 is not a leap year, so July 17^{th} is also a Wednesday and either statement would lead to the same answer. However, if you consider January 20^{th} to be a Sunday, you could also get a year like 2008, which was a leap year, and then October 28^{th} was a Tuesday. July 17^{th} would no longer be a Wednesday, which is why the second statement is consistently correct whereas the first statement could lead to one of two possibilities. Some students erroneously select answer choice D, that both statements together solve this issue. While the combination of statements does guarantee one specific answer, you’re overpaying for information because statement 2 does it alone. The answer you should pick is B.

On the GMAT, it’s important that outside knowledge not be tested explicitly because it’s a test of how you think, not of what you know. However, some basic concepts may come up that require you to use logic based on things you know to be true. You will never be undone on a GMAT question because “I didn’t know that,” but rather because “Oh, I forgot to take that into account.” The GMAT is primarily a test of thinking, and it’s important to keep in mind little pieces of knowledge that could have big implications on a question. As they say, knowing is half the battle (G.I. Joe!).

There are different ways of asking the same thing on the GMAT. Sometimes, the question is simply: Find the value of x. Other times, you get a convoluted story that summarizes to: Find the value of x. While these two questions are essentially the same, and both have the same answer, the first scenario is easier for most students to understand than the second scenario. This is because the second question is exactly the first question but with an extra step at the beginning (watch your step!), and if you don’t solve the first step, you never even get to the crux of the question.

Consider the following two problems. The first one simply asks you to divide 96 by 6. Even without a calculator, this question should take no more than 30 seconds to solve. Now consider a similar prompt: “Sally goes to the store to buy 7 dozen eggs. When she leaves the store, she accidentally drops one carton containing 12 eggs. Unable to salvage any, she goes back into the store and buys two more cartons of 12 eggs each. Once home, she separates the eggs into bags of 6, in order to save space in the fridge. How many bags of eggs does Sally make?”

The second prompt is exactly the same as the first question, but takes much longer to read through, execute rudimentary math of (7 x 12 – 12 + 24) / 6, and yield a final answer of 16. Anyone who can solve the first question should be able to solve the second question, but fewer students answer the second question correctly. Between the two is the fine art of translating GMATese (patent pending) to a simple mathematical formula. Even for native English speakers, this can be difficult, and is often the difference between getting the correct answer and getting the right answer to a different question.

Let’s look at such a question that looks like it needs to be deciphered by a team of translators:

*“X and Y are both integers. If X / Y = 59.32, then what is the sum of all the possible two digit remainders of X / Y?”*

*A) 560
*

While this question may appear to be giving you a simple formula, it’s not that easy to interpret what is being asked. One integer is being divided by another, and the result is a quotient and a remainder. The remainder is then only one of multiple possible remainders, and all these possible remainders must be summed up to give a single value. The GMAT isn’t giving us a story on this question, but there’s a lot to chew on.

First off, the quotient doesn’t actually matter in this equation. X / Y = 59.32, but it could have been 29.32 or 7.32 or any other integer quotient, the only thing we care about is the remainder. This means that essentially X/Y is 0.32, and we must find possible values for that. Clearly, X could be 32 and Y could be 100, thus leaving a remainder of 32 and the equivalent of the fractional component of 0.32 in the quotient. This could work, and is two digits, which means that it’s one possible remainder on the list that we must sum up.

What could we do next? Well if 32/100 works, then all other fractional values that can be simplified from that proportion should work as well. This means that 16/50, which is half of the original fraction, should work as well. If we divide by 2 again, we get 8/25. This value satisfies the fraction of the quotient, but not the requirement that it must be two digits. We cannot count 8 as a possible remainder, but this does help open up the pattern of the remainders.

The fraction 8/25 is the key to solving all the other fractions, because it cannot be reduced any further. From 8/25, every time we increase the numerator by 8, we can increase the denominator by 25, and we will maintain the same fractional value. As such, we can have 16/50, 24/75, 32/100, 40/125, etc, without changing the value of the fraction. How far do we need to go? Well the question is asking for 2-digit remainders, so we only need to increase the numerator by 8 until it is no longer 2-digits. The denominator can be truncated, because when it comes to 40/125, all the question wants is 40.

Once we understand what this question is really asking for, it just wants the sum of all the 2-digit multiples of 8. There aren’t that many, so you can write them all down if you want to: 16, 24, 32, 40, 48, 56, 64, 72, 80, 88 and 96. Outside this range, the numbers are no longer 2-digits. This whole question could have been rewritten as: “Sum up the 2-digit multiples of 8” and we would have saved a lot of time (more than last month’s leap second brouhaha).

Solving for our summation is simple when we have a calculator, but there is a handy shortcut for these kinds of calculations. Since the numbers are consecutive multiples of 8, all we need to do is find the average and multiply by the number of terms. The average is the (biggest + smallest) / 2, which becomes (96 + 16) / 2 = 56. From there, we wrote out 11 terms, so it’s just 56 x 11 = 616, answer choice B.

It’s worth mentioning that there’s a formula for the number of terms as well: Take the biggest number, subtract the smallest number, divide by the frequency, and then add back 1 to account for the endpoints. This becomes ((96 – 16) /8) + 1, or (80 / 8) + 1 or 10 + 1, which is just 11. If you only have about a dozen terms to sum up, it’s not hard to consider writing each one down, but if you had to sum up the 3-digit multiples of 8, you wouldn’t spend hours writing out all the different values (hint: there are 112). It’s always better to know the formula, just in case.

On the GMAT, you’re often faced with questions that end up throwing curveballs at you. Interpreting what the question is looking for is half the difficulty, and solving the equations in a relatively short amount of time is the other half. If all the questions were written in straight forward mathematical terms, the exam would be significantly easier. As it is, you want to make sure that you don’t give away easy points on questions that you know how to solve. On test day, the exam will ask you: “¿Habla GMAT?” and your answer should be a resounding “¡si!”