The post It’s All Greek to Me: How to Use Greek Concepts to Beat the GMAT appeared first on Veritas Prep Blog.

]]>A few months ago, I wrote about this difficult Data Sufficiency question.

When I first encountered this problem I couldn’t help but wonder what kind of mad scientist question-writer engineered it. Where would such an idea even come from? It turns out, it wasn’t a GMAC employee at all, but Archimedes, the famous Greek geometer and coiner of the phrase “Eureka!”

The question is based on his attempt to trisect an angle with only a straight edge and a compass. (Alas, Archimedes’ work, though ingenious, was not technically a correct solution to the problem, as it provides only an approximation.) The reader is hereby invited to contemplate the kind of person who encounters a proof by Archimedes and instinctively thinks, “This would make an excellent Data Sufficiency question on the GMAT!” We’d like to believe that the good folks at GMAC are just like you and me, but perhaps not.

So this got me thinking: what other interesting Greek contributions to mathematics might be helpful in analyzing GMAT questions? In Euclid’s work *Elements,* he offers a simple and elegant proof for why there is no largest prime number. The proof proceeds by positing a hypothetical largest prime number “p.” We can then construct a product that consists of every prime number 2*3*5*7….*p. We’ll call this product “q.”

The next consecutive number will be q + 1. Now, we know that “q” contains 2 as a factor, as “q,” supposedly, contains every prime as a factor. Therefore q +1 will *not* contain 2 as a factor. (The next number to contain 2 as a factor will be q + 2.) We know that “q” contains 3 as a factor. Therefore q + 1 will not contain 3 as a factor. (The next number to contain 3 as a factor will be q + 3.)

Uh oh. If “p” really is the largest prime number, we’ve got a problem, because q + 1 will not contain *any* of the primes between 2 and p as factors. So either q + 1 is itself prime, or there is some prime greater than p and less than q + 1 that we’ve failed to consider. Either way, we’ve proven that p can’t be the largest prime number – I told you the Greeks were neat.

One axiom that’s worth internalizing from Euclid’s proof is the notion that two consecutive numbers cannot have any factors in common aside from 1. When q contains every prime from 2 to p as a factor, q + 1 contains none of those primes. How would this be helpful on the GMAT? Glad you asked. Check out this question:

*x is the product of all even numbers from 2 to 50, inclusive. The smallest prime factor of x + 1 must be: *

*(**A) Between 1 and 10*

*(B) **Between 11 and 15*

*(C) **Between 15 and 20*

*(D) **Between 20 and 25*

*(E) **Greater than 25*

We’re given information about x, and we’re asked about x + 1. If x is the product of all even numbers from 2 to 50, we can write x = 2 * 4 * 6 …* 50. This is the same as (1*2) * (2*2) * (3*2)… (25*2), which means the product consists of all the integers from 1 to 25, inclusive, and a bunch of 2’s.

So now we know that every prime number between 2 and 25 will be a factor of x. What about x + 1? (Paging Euclid!) We know that 2 is not a factor of x + 1, as 2 is a factor of x, and so the next multiple of 2 would be x + 2. We know that 3 is also not a factor of x + 1, as 3 is a factor of x, and so the next multiple of 3 would be x + 3. And once we’ve internalized that two consecutive numbers cannot have any factors in common aside from 1, we know that if *all* the primes between 2 and 25 are factors of x, *none* of those primes can be factors of x + 1, meaning that the smallest prime of x, whatever is, will be greater than 25. The answer, therefore, is E.

**Takeaway:** One of the beautiful things about mathematics is that fundamental truths do not change over time. What worked for the Greeks will work for us. The same axioms that allowed ancient mathematicians to grapple with problems two millennia ago will allow us to unravel the toughest GMAT questions. Learning a few of these axioms is not only interesting – though I’d caution against bringing up Archimedes’ trisection proof at a dinner party – but also helpful on the GMAT.

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 by him here.*

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]]>The post Quarter Wit, Quarter Wisdom: Solving GMAT Critical Reasoning Questions Involving Rates appeared first on Veritas Prep Blog.

]]>Say my average driving speed is 60 miles/hr. Does it matter whether I drive for 2 hours or 4 hours? Will my average speed change if I drive more (theoretically speaking)? No, right? When I drive for more hours, the distance I cover is more. When I drive for fewer hours, the distance I cover is less. If I travel for a longer time, does it mean my average speed has decreased? No. For that, I need to know what happened to the distance covered. If the distance covered is the same while time taken has increased, only then can I say that my speed was reduced.

Now we will look at an official question and hopefully convince you of the right answer:

*The faster a car is traveling, the less time the driver has to avoid a potential accident, and if a car does crash, higher speeds increase the risk of a fatality. Between 1995 and 2000, average highway speeds increased significantly in the United States, yet, over that time, there was a drop in the number of car-crash fatalities per highway mile driven by cars.*

*Which of the following, if true about the United States between 1995 and 2000, most helps to explain why the fatality rate decreased in spite of the increase in average highway speeds?*

*(A) The average number of passengers per car on highways increased.*

*(B) There were increases in both the proportion of people who wore seat belts and the proportion of cars that were equipped with airbags as safety devices.*

*(C) The increase in average highway speeds occurred as legal speed limits were raised on one highway after another.*

*(D) The average mileage driven on highways per car increased.*

*(E) In most locations on the highways, the density of vehicles on the highway did not decrease, although individual vehicles, on average, made their trips more quickly.*

Let’s break down the given argument:

- The faster a car, the higher the risk of fatality.
- In a span of 5 years, the average highway speed has increased.
- In the same time, the number of car crash fatalities per highway mile driven by cars has reduced.

This is a paradox question. In last 5 years, the average highway speed has increased. This would have increased the risk of fatality, so we would expect the number of car crash fatalities per highway mile to go up. Instead, it actually goes down. We need to find an answer choice that explains why this happened.

*(A) The average number of passengers per car on highways increased.*

If there are more people in each car, the risk of fatality increases, if anything. More people are exposed to the possibility of a crash, and if a vehicle is in fact involved in an accident, more people are at risk. It certainly doesn’t explain why the rate of fatality actually decreases.

*(B) There were increases in both the proportion of people who wore seat belts and the proportion of cars that were equipped with airbags as safety devices.*

This option tells us that the safety features in the cars have been enhanced. That certainly explains why the fatality rate has gone down. If the cars are safer now, the risk of fatality would have reduced, hence this option does help us in explaining the paradox. This is the answer, but let’s double-check by looking at the other options too.

*(C) The increase in average highway speeds occurred as legal speed limits were raised on one highway after another.*

This option is irrelevant – why the average speed increased is not our concern at all. Our only concern is that average speed has, in fact, increased. This should logically increase the risk of fatality, and hence, our paradox still stands.

*(D) The average mileage driven on highways per car increased.*

This is the answer choice that troubles us the most. The rate we are concerned about is number of fatalities/highway mile driven, and this option tells us that mileage driven by cars has increased.

Now, let’s consider the parallel with our previous distance-rate-time example:

Rate = Distance/Time

We know that if I drive for more time, it doesn’t mean that my rate changes. Here, however:

Rate = Number of fatalities/highway mile driven

In this case, if more highway miles are driven, it doesn’t mean that the rate will change. It actually has no impact on the rate; we would need to know what happened to the number of fatalities to find out what happened to the rate. Hence this option does not explain the paradox.

*(E) In most locations on the highways, the density of vehicles on the highway did not decrease, although individual vehicles, on average, made their trips more quickly.*

This answer choice tells us that on average, the trips were made more quickly, i.e. the speed increased. The given argument already tells us that, so this option does not help resolve the paradox.

Our answer is, therefore, (B).

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**!

*Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the **GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

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]]>The post GMAT Tip of the Week: Kanye, Wiz Khalifa, Twitter Beef…and GMAT Variables appeared first on Veritas Prep Blog.

]]>**Kanye didn’t consider all the possibilities when he saw variables.**

A brief history of the beef: there was musical origin, as Wiz wanted a bit of credit for his young/wild/free friends for the term “Wave,” as Kanye changed his upcoming album title from *Swish* to *Waves*. But where things escalated quickly all stemmed from Wiz’s use of variables in the following tweet:

*Hit this kk and become yourself.*

Kanye, whose wife bears those exact initials, K.K., immediately interpreted those variables as a reference to Kim and lost his mind. But Wiz had intended those variables kk to mean something entirely different, a reference to his favorite drug of choice. And then…well let’s just say that things got out of hand.

So back to the GMAT: Kanye’s main mistake was that he didn’t consider alternate possibilities for the variables he saw in the tweet, and quickly built in some incorrect assumptions that led to disastrous results. Do not let this happen to you on the GMAT! Here’s how it could happen:

**1) Forgetting about not-obvious numbers.**

If a problem, for example, defines k as 10 < k < 12, you can’t just think “k = 11” because you don’t know that k has to be an integer. 11.9 or 10.1 are also possibilities. Similarly if k^2 = 121, you have to consider that k could be -11 as well as it could be 11.

Ultimately, that was Yeezy’s mistake: he saw KK and with tunnel vision saw the most obvious possibility. But why couldn’t “KK” have been Krispy Kreme or Kyle Korver or Kato Kaelin? Before you leap to conclusions on a GMAT variable, see if there’s anything else it could be.

**2) Assuming that each variable must represent a different number.**

This one is a bit more nuanced. Suppose you were asked:

*For positive integers a and b, is the product ab > 1?*

*(1) a = 1*

With that statement, you might start thinking, “Well if a is 1, b has to be something else…” but all the variable b really means is “a number we don’t know.” Just because a problem assigns two different variables does not mean that they represent two different numbers! B could also be 1…we just don’t know yet.

Where this manifests itself as a problem most often is on function problems. When people see the setup, for example:

*The function f is defined for all values x as f(x) = x^2 – x – 1*

They’ll often be confused when that’s paired with a question like, “Is f(a) > 1?” and a statement like:

*(1) -2 < a < 2*

“I know about f(x) but I don’t know anything about f(a),” they might say, but the way these variables work, f(x) means “the function of any number…we just don’t know which number” so when you then see f(a), a becomes that number you don’t know. You’ll do the same thing for a: f(a) = a^2 – a – 1. What goes in the parentheses is just “the number you perform the function on” – the function doesn’t just apply to the variable in the definition, but to any number, variable, or combination that is then put in the parentheses.

The real lesson here is this: variables on the GMAT are a lot like variables in Wiz Khalifa’s Twitter feed. You might think you know what they mean, but before you stake your reputation (or score) on your response to those variables, consider all the options. Hit this GMAT and become yourself.

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**!

*By Brian Galvin.*

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]]>The post Why Logic is More Important Than Algebra on the GMAT appeared first on Veritas Prep Blog.

]]>If you’ve followed this blog for any length of time, you know that one of the themes we stress is that Quantitative Reasoning is not, primarily, a math test. Though math is certainly involved – How could it not be? – logic and reasoning are far more important factors than conventional mathematical facility. I stress this in every class I teach. So why the misconception that we need to hone our algebra chops?

I suspect that the culprit here is the explanations that often accompany official GMAC questions. On the whole, they tend to be biased in favor of purely algebraic solutions. They’re always* technically* correct, but often suboptimal for the test-taker who needs to arrive at a solution within two minutes. Consequently, many students, after reviewing these solutions and arriving at the conclusion that they would not have been capable of the hairy algebra proffered in the official solution, think they need to work on this aspect of their prep. And for the most part it isn’t true.

Here’s a good example:

*If x, y, and k are positive numbers such that [x/(x+y)]*10 + [y/(x+y)]*20 = k and if x < y, which of the following could be the value of k? *

A) 10

B) 12

C) 15

D) 18

E) 30

A large percentage of test-takers see this question, rub their hands together, and dive into the algebra. The solution offered in the Official Guide does the same – it is about fifteen steps, few of them intuitive. If you were fortunate enough to possess the algebraic virtuosity to solve the question in this manner, you’d likely chew up 5 or 6 minutes, a disastrous scenario on a test that requires you to average 2 minutes per problem.

The upshot is that it’s important for test-takers, when they peruse the official solution, not to arrive at the conclusion that they need to solve this question the same way the solution-writer did. Instead, we can use the same simple strategies we’re always preaching on this blog: pick some simple numbers.

We’re told that x<y, but for my first set of numbers, I like to make x and y the same value – this way, I can see what effect the restriction has on the problem. So let’s say x = 1 and y = 1. Plugging those values into the equation, we get:

(1/2) * 10 + (1/2) * 20 = k

5 + 10 = k

15 = k

Well, we know this isn’t the answer, because x should be less than y. So scratch off C. And now let’s see what the effect is when x is, in fact, less than y. Say x = 1 and y = 2. Now we get:

(1/3) * 10 + (2/3) * 20 = k

10/3 + 40/3 = k

50/3 = k

50/3 is about 17. So when we honor the restriction, k becomes larger than 15. The answer therefore must be D or E. Now we could pick another set of numbers and pay attention to the trend, or we can employ a bit of logic and common sense. The first term in the equation x/(x+y)*10 is some fraction multiplied by 10. So this term, logically, is some value that’s less than 10.

The second term in the equation is y/(x+y)*20, is some fraction multiplied by 20, this term must be less than 20. If we add a number that’s less than 10 to a number that’s less than 20, we’re pretty clearly not going to get a sum of 30. That leaves us with an answer of 18, or D.

(Note that if you’re really savvy, you’ll recognize that the equation is a weighted average. The coefficients in the weighted average are 10 and 20. If x and y were equal, we’d end up at the midway point, 15. Because 20 is multiplied by y, and y is greater than x, we’ll be pulled towards the high end of the range, leading to a k that must fall between 15 and 20 – only 18 is in that range.)

Takeaway: Never take a formal solution to a problem at face value. All you’re seeing is one way to solve a given question. If that approach doesn’t resonate for you, or seems so challenging that your conclusion is that you must purchase a host of textbooks in order to improve your formal math skills, then you haven’t absorbed what the GMAT is really about. Often, the relevant question isn’t, “Can you do the math?” It’s, “Can you reason your way to the answer without actually doing the math?”

*Official Guide 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 follow us on **Facebook**, **YouTube**, **Google+ **and **Twitter**!

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

The post Why Logic is More Important Than Algebra on the GMAT appeared first on Veritas Prep Blog.

]]>The post Quarter Wit, Quarter Wisdom: Should You Use the Permutation or Combination Formula? appeared first on Veritas Prep Blog.

]]>People have tried to answer this question in various ways, but some students still remain unsure. So we will give you a rule of thumb to follow in all permutation/combination questions:

**You never NEED to use the permutation formula!** You can always use the combination formula quite conveniently. First let’s look at what these formulas do:

**Permutation:** nPr = n!/(n-r)!

Out of n items, select r and arrange them in r! ways.

**Combination:** nCr = n!/[(n-r)!*r!]

Out of n items, select r.

So the only difference between the two formulas is that nCr has an additional r! in the denominator (that is the number of ways in which you can arrange r elements in a row). So you can very well use the combinations formula in place of the permutation formula like this:

nPr = nCr * r!

The nCr formula is far more versatile than nPr, so if the two formulas confuse you, just forget about nPr.

Whenever you need to “select,” “pick,” or “choose” r things/people/letters… out of n, it’s straightaway nCr. What you do next depends on what the question asks of you. Do you need to arrange the r people in a row? Multiply by r!. Do you need to arrange them in a circle? Multiply by (r-1)!. Do you need to distribute them among m groups? Do that! You don’t need to think about whether it is a permutation problem or a combination problem at all. Let’s look at this concept more in depth with the use of a few examples.

*There are 8 teachers in a school of which 3 need to give a presentation each. In how many ways can the presenters be chosen?*

In this question, you simply have to choose 3 of the 8 teachers, and you know that you can do that in 8C3 ways. That is all that is required.

8C3 = 8*7*6/3*2*1 = 56 ways

Not too bad, right? Let’s look at another question:

*There are 8 teachers in a school of which 3 need to give a presentation each. In how many ways can all three presentations be done?*

This question is a little different. You need to find the ways in which the presentations can be done. Here the presentations will be different if the same three teachers give presentations in different order. Say Teacher 1 presents, then Teacher 2 and finally Teacher 3 — this will be different from Teacher 2 presenting first, then Teacher 3 and finally Teacher 1. So, not only do we need to select the three teachers, but we also need to arrange them in an order. Select 3 teachers out of 8 in 8C3 ways and then arrange them in 3! ways:

We get 8C3 * 3! = 56 * 6 = 336 ways

Let’s try another one:

*Alex took a trip with his three best friends and there he clicked 7 photographs. He wants to put 3 of the 7 photographs on Facebook. How many groups of photographs are possible?*

For this problem, out of 7 photographs, we just have to select 3 to make a group. This can be done in 7C3 ways:

7C3 = 7*6*5/3*2*1 = 35 ways

Here’s another variation:

*Alex took a trip with his three best friends and there he clicked 7 photographs. He wants to put 3 of the 7 photographs on Facebook, 1 each on the walls of his three best friends. In how many ways can he do that?*

Here, out of 7 photographs, we have to first select 3 photographs. This can be done in 7C3 ways. Thereafter, we need to put the photographs on the walls of his three chosen friends. In how many ways can he do that? Now there are three distinct spots in which he will put up the photographs, so basically, he needs to arrange the 3 photographs in 3 distinct spots, which that can be done in 3! ways:

Total number of ways = 7C3 * 3! = (7*6*5/3*2*1) * 6= 35 * 6 = 210 ways

Finally, our last problem:

*12 athletes will run in a race. In how many ways can the gold, silver and bronze medals be awarded at the end of the race?*

We will start with selecting 3 of the 12 athletes who will win some position in the race. This can be done in 12C3 ways. But just selecting 3 athletes is not enough — they will be awarded 3 distinct medals of gold, silver, and bronze. Athlete 1 getting gold, Athlete 2 getting silver, and Athlete 3 getting bronze is not the same as Athlete 1 getting silver, Athlete 2 getting gold and Athlete 3 getting bronze. So, the three athletes need to be arranged in 3 distinct spots (first, second and third) in 3! ways:

Total number of ways = 12C3 * 3! ways

Note that some of the questions above were permutation questions and some were combination questions, but remember, we don’t need to worry about which is which. All we need to think about is how to solve the question, which is usually by starting with nCr and then doing any other required steps. Break the question down — select people and then arrange if required. This will help you get rid of the “permutation or combination” puzzle once and for all.

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**!

*Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the **GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

The post Quarter Wit, Quarter Wisdom: Should You Use the Permutation or Combination Formula? appeared first on Veritas Prep Blog.

]]>The post GMAT Tip of the Week: Stay In Your Lane (In The Snow And On Sentence Correction) appeared first on Veritas Prep Blog.

]]>When driving in the snow:

- Don’t brake until you have to.
- Don’t make sudden turns or lane changes, and only turn if you have to.
- Stay calm and leave yourself space and time to make decisions.

And those same lessons apply to GMAT Sentence Correction. Approach these questions like you would approach driving in a blizzard, and you may very well earn that opportunity to drive through blustery New England storms as you pursue your MBA. What does that mean?

**1) Stay In Your Lane**

Just as quick, sudden jerks of the steering wheel will doom you on snowy/icy roads, sudden and unexpected decisions on GMAT Sentence Correction will get you in trouble. Your “lane” consists of the decisions that you’ve studied and practiced and can calmly execute: Modifiers, Verbs (tense and agreement), Pronouns, Comparisons, Parallelism in a Series, etc. It’s when you get out of that lane that you’re prone to skidding well off track. For example, on this problem (courtesy the Official Guide for GMAT Review):

*While Jackie Robinson was a Brooklyn Dodger, his courage in the face of physical threats and verbal attacks was not unlike that of Rosa Parks, who refused to move to the back of a bus in Montgomery, Alabama.*

*(A) not unlike that of Rosa Parks, who refused*

* (B) not unlike Rosa Parks, who refused*

* (C) like Rosa Parks and her refusal*

* (D) like that of Rosa Parks for refusing*

* (E) as that of Rosa Parks, who refused*

Your “lane” here is to check for Modifiers (Is “who refused” correct? Is it required?) and for logical, clear meaning (it is required, because otherwise you aren’t sure who refused to move to the back of the bus). But examinees are routinely baited into “jerking the wheel” and turning against the strange-but-correct structure of “not unlike.” When you’re taken off of your game, you often eliminate the correct answer (A) because you’re turning into a decision you’re just not great at making.

**2) Don’t Turn or Brake Until You Have To**

The GMAT *does* test Redundancy and Pronoun Reference (among other things), but those are error types that are dangerous to prioritize – much like it’s dangerous while driving in snow to decide quickly that you need to turn or hit the brakes. Too often, test-takers will slam on the Sentence Correction brakes at their first hint of, “That’s redundant!” (like they would for “not unlike” above) or “There are multiple nouns – that pronoun is unclear!” and steer away from that answer choice.

The problem, as you saw above, is that often this means you’re turning away from the proper path. “Not unlike” may scream “double-negative” or “redundant” to many, but it’s a perfectly valid way to express the idea that the two things aren’t close to identical, but they’re not as different as you might think. And you don’t need to know THAT, as much as you need to know that you shouldn’t ever make redundancy your first decision, because if you’re like most examinees you’re probably not that great at you…AND you don’t have to be, because the path toward your strengths will get you to your destination.

Similarly, this week the Veritas Prep Homework Help service got into an interesting email thread about why this sentence:

*Based on his experience in law school, John recommended that his friend take the GMAT instead of the LSAT.*

has a pronoun reference error, but this sentence:

*Mothers expect unconditional love from their children, and they are rarely disappointed.*

does not. And while there likely exists a technical, grammatical reason why, the GMAT reason really comes down to this: Does the problem make you address the pronoun reference? If not, don’t worry about it. In other words, don’t brake or turn until you have to. If you look at those sentences in GMAT problem form, you might have:

__Based on his experience in law school, John__ recommended that his friend take the GMAT instead of the LSAT.

*(A) Based on his experience in law school, John*

*(B) Having had a disappointing experience in law school, John*

*(C) Given his experience in law school, John*

Here, the question forces you to deal with the pronoun problem. The major differences between the choices are that A and C involve a pronoun, and B doesn’t. Here, you have to deal with that issue. But for the other sentence, you might see:

__Mothers expect unconditional love from their children, and they are__ rarely disappointed.

*(A) Mothers expect unconditional love from their children, and they are*

*(B) The average mother expects unconditional love from their children, and are*

*(C) The average mother expects unconditional love from their children, and they are*

*(D) Mothers, expecting unconditional love from their children, they are*

Here, the only choice that doesn’t include the pronoun “they” is choice B, but that choice commits a glaring pronoun (and verb) agreement error (“the average mother” is singular, but “their children” is plural…and the verb “are” is, too). So you don’t need to worry about the “they” (which clearly refers to “mothers” and not “children,” even though there happen to be two plural nouns in the sentence).

Grammatically, the presence of multiple nouns doesn’t alone make the pronoun itself ambiguous, but strategically for the GMAT, what you really need to know is that you don’t have to hit the brakes at the first sign of “unclear reference.” Wait and see if the answer choices give you a chance to address that, and if they do, then make sure that those choices are free of other, more binary errors first. Don’t turn or brake unless you have to.

**3) Stay calm and leave yourself space to make decisions.**

Just like a driver in the snow, as a GMAT test-taker you’ll be nervous and antsy. But don’t let that force you into rash decisions! Assess the answer choices before you try to determine whether something outside your 100% confidence interval is right or wrong in the original. You don’t need to make a decision on Choice A right away, just like you don’t need to change lanes simply for the sake of doing so. Have a plan and stick to it, both on the GMAT and on those snowy roads this weekend.

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**!

*By Brian Galvin.*

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]]>The post Quarter Wit, Quarter Wisdom: Keeping an Open Mind in Critical Reasoning appeared first on Veritas Prep Blog.

]]>We have discussed necessary and sufficient conditions before; we have also discussed assumptions before. This question from our own curriculum is an innovative take on both of these concepts. Let’s take a look.

*All of the athletes who will win a medal in competition have spent many hours training under an elite coach. Michael is coached by one of the world’s elite coaches; therefore it follows logically that Michael will win a medal in competition.*

*The argument above logically depends on which of the following assumptions?*

*(A) Michael has not suffered any major injuries in the past year.*

*(B) Michael’s competitors did not spend as much time in training as Michael did.*

*(C) Michael’s coach trained him for many hours.*

*(D) Most of the time Michael spent in training was productive.*

*(E) Michael performs as well in competition as he does in training.*

First we must break down the argument into premises and conclusions:

**Premises:**

- All of the athletes who will win a medal in competition have spent many hours training under an elite coach.
- Michael is coached by one of the world’s elite coaches.

**Conclusion:** Michael will win a medal in competition.

Read the argument carefully:

*All of the athletes who will win a medal in competition have spent many hours training under an elite coach.*

Are you wondering, “How does one know that all athletes who will win (in the future) would have spent many hours training under an elite coach?”

The answer to this is that it doesn’t matter how one knows – it is a premise and it has to be taken as the truth. How the truth was established is none of our business and that is that. If we try to snoop around too much, we will waste precious time. Also, what may seem improbable may have a perfectly rational explanation. Perhaps all athletes who are competing have spent many hours under an elite coach – we don’t know.

Basically, what this statement tells us is that spending many hours under an elite coach is a NECESSARY condition for winning. What you need to take away from this statement is that “many hours training under an elite coach” is a necessary condition to win a medal. Don’t worry about the rest.

*Michael is coached by one of the world’s elite coaches. *

It seems that Michael satisfies one necessary condition: he is coached by an elite coach.

**Conclusion:** Michael will win a medal in competition.

Now this looks like our standard “gap in logic”. To get this conclusion, the necessary condition has been taken to be sufficient. So if we are asked for the flaw in the argument, we know what to say.

Anyway, let’s check out the question (this is usually our first step):

*The argument above logically depends on which of the following assumptions?*

Note the question carefully – it is asking for an assumption, or a necessary premise for the conclusion to hold.

We know that “many hours training under an elite coach” is a necessary condition to win a medal. We also know that Michael has been trained by an elite coach. Note that we don’t know whether he has spent “many hours” under his elite coach. The necessary condition requires “many hours” under an elite coach.

If Michael has spent many hours under the elite coach then he satisfies the necessary condition to win a medal. It is still not sufficient for him to win the medal, but our question only asks for an assumption – a necessary premise for the conclusion to hold. It does not ask for the flaw in the logic.

Focus on what you are asked and look at answer choice (C):

*(C) Michael’s coach trained him for many hours.*

This is a necessary condition for Michael to win a medal. Hence, it is an assumption and therefore, (C) is the correct answer.

Don’t worry that the argument is flawed. There could be another question on this argument which asks you to find the flaw in it, however this particular question asks you for the assumption and nothing more.

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**!

*Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the **GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

The post Quarter Wit, Quarter Wisdom: Keeping an Open Mind in Critical Reasoning appeared first on Veritas Prep Blog.

]]>The post GMAT Tip of the Week: Your MLK Study Challenge (Remove Your Biases) appeared first on Veritas Prep Blog.

]]>If Dr. King were alive today, he would certainly be proud of the legislation he inspired to end much of the explicit bias – you can’t eat here, vote there, etc. – that was part of the American legal code until the 1960s. But he would undoubtedly be dismayed by the implicit bias that still runs rampant across society.

This implicit bias is harder to detect and even harder to “fix.” It’s the kind of bias that, for example, the movie *Freaknomics* shows; often when the name at the top of a resume connotes some sort of stereotype, it subconsciously colors the way that the reader of that resume processes the rest of the information on it.

While that kind of subconscious bias is a topic for a different blog to cover, it has an incredible degree of relevance to the way that you attack GMAT Data Sufficiency problems. If you’re serious about studying for the GMAT, you’ll probably have long enacted your own versions of the Voting Rights Act and Civil Rights Act well before you get to test day – that is to say, you’ll have figured out how to eliminate the kind of explicit bias that comes from reading a question like:

*If y is an odd integer and the product of x and y equals 222, what is the value of x?*

*1) x > 0*

*2) y is a 3 digit number*

Here, you’ll likely see very quickly that Statement 1 is not sufficient, and come back to Statement 2 with fresh eyes. You don’t know that x is positive, so you’ll quickly see that y could be 111 and x could be 2, or that y could be -111 and x could be -2, so Statement 2 is clearly also not sufficient. The explicit bias that came from seeing “x is positive” is relatively easy to avoid – you know not to carry over that explicit information from Statement 1 to Statement 2.

But you also need to be just as aware of implicit bias. Try this question, as it is more likely to appear on the actual GMAT:

*If y is an odd integer and the product of x and y equals 222, what is the value of x?*

*1) x is a prime number*

*2) y is a 3 digit number*

On this version of the problem, people become extremely susceptible to implicit bias. You no longer get to quickly rule out the obvious “x is positive.” Here, the first statement serves to pollute your mind – it is, on its own merit, sufficient (if y is odd and the product of x and y is even, the only prime number x could be is 2, the only even prime), but it also serves to get you thinking about positive numbers (only positive numbers can be prime) and integers (only integers are prime). But those aren’t explicitly stated; they’re just inferences that your mind quickly makes, and then has trouble getting rid of. So as you assess Statement 2, it’s harder for you to even think of the possibilities that:

x could be -2 and y could be -111: You’re not thinking about negatives!

x could be 2/3 and y could be 333: You’re not thinking about non-integers!

On this problem, over 50% of users say that Statement 2 is sufficient (and less than 25% correctly answer A, that Statement 1 alone is sufficient), because they fall victim to that implicit bias that comes from Statement 1 whispering – not shouting – “positive integers.”

Harder problems will generally prey on your more subtle bias, so you need to make sure you’re giving each statement a fresh set of available options. So this Martin Luther King, Jr. weekend, applaud the progress that you have made in removing explicit bias from your Data Sufficiency regimen – you now know not to include Statement 1 directly in your assessment of Statement 2 ALONE – but remember that implicit bias is just as dangerous to your score. Pay attention to the times that implicit bias draws you to a poor decision, and be steadfast in your mission to give each statement its deserved, unbiased attention.

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**!

*By Brian Galvin.*

The post GMAT Tip of the Week: Your MLK Study Challenge (Remove Your Biases) appeared first on Veritas Prep Blog.

]]>The post Quarter Wit, Quarter Wisdom: An Interesting Property of Exponents appeared first on Veritas Prep Blog.

]]>*For integers x and y, 2^x + 2^y = 2^(36). What is the value of x + y?*

Think about it for a few seconds – could you come up with the answer in the blink of an eye? If yes, great! Close this window and wait for the next week’s post. If no, then read on. There is much to learn today and it is an eye-opener!

Let’s start by jotting down some powers of numbers:

**Power of 2:** 1, 2, 4, 8, 16, 32 …

**Power of 3:** 1, 3, 9, 27, 81, 243 …

**Power of 4:** 1, 4, 16, 64, 256, 1024 …

**Power of 5:** 1, 5, 25, 125, 625, 3125 …

and so on.

Obviously, for every power of 2, when you multiply the previous power by 2, you get the next power (4*2 = 8).

For every power of 3, when you multiply the previous power by 3, you get the next power (27*3 = 81), and so on.

Also, let’s recall that multiplication is basically repeated addition, so 4*2 is basically 4 + 4.

This leads us to the following conclusion using the power of 2:

4 * 2 = 8

4 + 4 = 8

2^2 + 2^2 = 2^3

(2 times 2^2 gives 2^3)

Similarly, for the power of 3:

27 * 3 = 81

27 + 27 + 27 = 81

3^3 + 3^3 + 3^3 = 3^4

(3 times 3^3 gives 3^4)

And for the power of 4:

4 * 4 = 16

4 + 4 + 4 + 4 = 16

4^1 + 4^1 + 4^1 + 4^1 = 4^2

(4 times 4^1 gives 4^2)

Finally, for the power of 5:

125 * 5 = 625

125 + 125 + 125 + 125 + 125 = 625

5^3 + 5^3 + 5^3 + 5^3 + 5^3 = 5^4

(5 times 5^3 gives 5^4)

Quite natural and intuitive, isn’t it? Take a look at the previous question again now.

*For integers x and y, 2^x + 2^y = 2^(36). What is the value of x + y?*

*A) 18*

*(B) 32*

*(C) 35*

*(D) 64*

*(E) 70*

Which two powers when added will give 2^(36)?

From our discussion above, we know they are 2^(35) and 2^(35).

2^(35) + 2^(35) = 2^(36)

So x = 35 and y = 35 will satisfy this equation.

x + y = 35 + 35 = 70

Therefore, our answer is E.

One question arises here: Is this the only possible sum of x and y? Can x and y take some other integer values such that the sum of 2^x and 2^y will be 2^(36)?

Well, we know that no matter which integer values x and y take, 2^x and 2^y will always be positive, which means both x and y must be less than 36. Now note that no matter which two powers of 2 you add, their sum will always be less than 2^(36). For example:

2^(35) + 2^(34) < 2^(35) + 2^(35)

2^(2) + 2^(35) < 2^(35) + 2^(35)

etc.

So if x and y are both integers, the only possible values that they can take are 35 and 35.

How about something like this: 2^x + 2^y + 2^z = 2^36? What integer values can x, y and z take here?

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

*GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

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]]>The post GMAT Tip of the Week: Make 2016 The Year Of Number Fluency appeared first on Veritas Prep Blog.

]]>- Oh, that’s divisible by 9
- Well, obviously that’s divisible by 4
- Huh, 20 and 16 are consecutive multiples of 4
- 2, 0, 1, 6 – that’s three evens and an odd
- I wonder what the prime factors of 2016 are…

Why? Because the GMAT – and its no-calculator-permitted format for the Quant Section – is a test that highly values and rewards mathematical fluency. The GMAT tests patterns in, and properties of, numbers quite a bit. Whenever you see a number flash before your eyes, you should be thinking about even vs. odd, prime vs. composite, positive vs. negative, “Is that number a square or not?” etc. And, mathematically speaking, the GMAT is a multiplication/division test more than a test of anything else, so as you process numbers you should be ready to factor and divide them at a moment’s notice.

Those who quickly see relationships between numbers are at a huge advantage: they’re not just ready to operate on them when they have to, they’re also anticipating what that operation might be so that they don’t have to start from scratch wondering how and where to get started.

With 2016, for example:

The last two digits are divisible by 4, so you know it’s divisible by 4.

The sum of the digits (2 + 0 + 1 + 6) is 9, a multiple of 9, so you know it’s divisible by 9 (and also by 3).

So without much thinking or prompting, you should already have that number broken down in your head. 16 divided by 4 is 4 and 2000 divided by 4 is 500, so you should be hoping that the number 504 (also divisible by 9) shows up somewhere in a denominator or division operation (or that 4 or 9 does).

So, for example, if you were given a problem:

*In honor of the year 2016, a donor has purchased 2016 books to be distributed evenly among the elementary schools in a certain school district. If each school must receive the same number of books, and there are to be no books remaining, which of the following is NOT a number of books that each school could receive?*

*(A) 18*

*(B) 36*

*(C) 42*

*(D) 54*

*(E) 56*

You shouldn’t have to spend any time thinking about choices A and B, because you know that 2016 is divisible by 4 and by 9, so it’s definitely divisible by 36 which means it’s also divisible by every factor of 36 (including 18). You don’t need to do long division on each answer choice – your number fluency has taken care of that for you.

From there, you should look at the other numbers and get a quick sense of their prime factors:

42 = 2 * 3 * 7 – You know that 2016 is divisible by 2 and 3, but what about 7?

54 = 2 * 3 * 3 * 3 – You know that 2016 is divisible by that 2 and that it’s divisible by 9, so you can cover two of the 3s. But is 2016 divisible by three 3s?

56 = 2 * 2 * 2 * 7 – You know that two of the 2s are covered, and it’s quick math to divide 2016 by 4 (as you saw above, it’s 504). Since 504 is still even, you know that you can cover all three 2s, but what about 7?

Here’s where good test-taking strategy can give you a quick leg up: to this point, a savvy 700-scorer shouldn’t have had to do any real “work,” but testing all three remaining answer choices could now get a bit labor intensive. Unless you recognize this: for C and E, the only real question to be asked is “Is 2016 divisible by 7?” After all, you’re already accounted for the 2 and 3 out of 42, and you’ve already accounted for the three 2s out of 56.

7 is the only one you haven’t checked for. And since there can only be one correct answer, 2016 must be divisible by 7…otherwise you’d have to say that C and E are both correct.

But even if you’re not willing to take that leap, you may still have the hunch that 7 is probably a factor of 2016, so you can start with choice D. Once you’ve divided 2016 by 9 (here you may have to go long division, or you can factor it out), you’re left with 224. And that’s not divisible by 3. Therefore, you know that 2016 cannot be divided evenly into sets of 54, so answer choice D must be correct. And more importantly, good number fluency should have allowed you to do that relatively quickly without the need for much (if any) long division.

So if you didn’t immediately think “divisible by 4 and 9!” when you saw the year 2016 pop up, make it your New Year’s resolution to start thinking that way. When you see numbers this year, start seeing them like a GMAT expert, taking note of clear factors and properties and being ready to quickly operate on that number.

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**!

*By Brian Galvin.*

The post GMAT Tip of the Week: Make 2016 The Year Of Number Fluency appeared first on Veritas Prep Blog.

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