Strategies for the New GMAT Questions that You Need to Know!

MBA Interview QuestionsAbout a month ago, GMAC released the latest version of the GMAT Official Guide, 25% of which consisted of new questions. Though the GMAT tends not to change too drastically over time – how else could a school compare a score received by one candidate in 2015 to a score received by another candidate in 2010? – there can be subtle shifts of emphasis, and paying attention to the composition mix of the questions in the latest version of the Official Guide is a good way to ascertain if any such shift is in the offing.

My concern as an instructor is whether the philosophy I’m advocating and the techniques I’m teaching are as relevant for the newer questions as they have been for the older ones.

This philosophy can be summarized as follows: the GMAT is not, fundamentally, a content-based test, but rather, uses certain elements of our academic background to test how we think under pressure. Because the test is evaluating how we think, and not what we know, the cultivation of simple strategies, such as using the answer choices or picking easy numbers, is just as important as the re-mastery of the content you may have initially learned in eighth grade, but have subsequently forgotten.

Having thoroughly dissected the new questions in the latest version of the Official Guide, I can confidently report that this philosophy is more relevant than ever. Of the over 200 new quantitative questions, I didn’t do extensive calculations for a single problem. If anything, the kind of fluid logic-based approach that we preach at Veritas is more critical than ever.

Take this new question, for example:

Four extra-large sandwiches of exactly the same size were ordered for m students, where m > 4. Three of the sandwiches were evenly divided among the students. Since 4 students did not want any of the fourth sandwich, it was evenly divided among the remaining students. If Carol ate one piece from each of the four sandwiches, the amount of sandwich that she ate would be what fraction of a whole extra-large sandwich? 

A) (m+4)/[m(m-4)]
B) (2m-4)/[m(m-4)]
C) (4m-4)/[m(m-4)]
D) (4m-8)/[m(m-4)]
E) (4m-12)/[m(m-4)]

Of course, we could do this question algebraically. But if the GMAT is testing our ability to make good decisions under pressure, and if the algebra feels hard for you, then a better option is to make your life as easy as possible and select a simple number for m. If m is larger than 4, let’s say that m = 5. “m” represents the number of students, so now we have 5 students and, we’re told in the question stem, a total of 4 sandwiches. (The question of what kind of negligent, hard-hearted school knowingly packs only 4 sandwiches for all of its students to share will have to be addressed in another post. This question feels straight out of Oliver Twist.)

Okay. We’re told that 3 of the sandwiches are divided evenly among the 5 students. (3 sandwiches)/(5 students) means each student gets 3/5 of a sandwich.

Additionally, we’re told that 4 of the students don’t want any part of the remaining sandwich. Because we only have 5 students and 4 of them don’t want the remaining sandwich, the last student will get the entire fourth sandwich.

To summarize what we have so far: Each of the 5 students initially received 3/5 of a sandwich, and then one student received an entire additional sandwich, on top of that initial 3/5. The lucky fifth student received a total of 3/5 + 1 = 8/5 of a sandwich.

Last, we ‘re told that Carol ate a piece of each of the four sandwiches. But we established that only one student ate a piece of every sandwich, so Carol has to be that lucky student! Therefore, Carol ate 8/5 of a sandwich.

We’re asked what fraction of a sandwich Carol ate, so the answer is simply 8/5. Now all we have to do is plug ‘5’ in place of ‘m’ in each answer choice, and the one that gives us 8/5 will be our answer.

Most test-takers will simply start with A and work their way down until they find an option that works. The question-writer knows that this is how most test-takers proceed. Therefore, it’s a more challenging question if the correct answer is towards the bottom of our answer choices. So let’s use this logic to our advantage, start with E, and work our way up.

Answer choice E:  (4m-12)/[m(m-4)]

Substituting ‘5’ in place of ‘m,’ we get (4*5 – 12)/[5(5-4) = 8/5. That’s it! We’re done. The correct answer is E.

Takeaway: Keep reminding yourself that the GMAT (even with its new questions) is not designed to test what you know. While it is important to brush up on all of the fundamentals you acquired years before, the most successful test-takers will fluidly incorporate simple strategies when attacking complex questions, rather than simply grinding through longer calculations. Each new version of the Official Guide validates the wisdom of this approach.

*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 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.

How to Tackle Evenly Spaced Sets on the GMAT

MBA Applicant Evaluation WorkshopThere’s an amusing anecdote told about the great 18th century mathematician, Carl Friedrich Gauss. Apparently, when Gauss was young, he was something of a troublemaker in school, and as a punishment for one of his disruptive outbursts, his teacher ordered him to calculate the sum of all the numbers from 1 to 100 inclusive, thinking that such a calculation would be taxing and time-consuming. Gauss simply scratched his head, thought for a few seconds, and then astonished his teacher and classmates by spitting out the answer: 5,050. He was about seven years old when this happened.

So then, how is it possible for a child – a genius, perhaps, but still a child – to do such an extensive computation in his head? The answer involves exploiting certain properties of evenly spaced sets. An evenly spaced set, as the name implies, is one in which the gap between each successive element in the set is equal. So a set consisting of consecutive integers would be evenly spaced, as would a set consisting of consecutive multiples of 2 or consecutive multiples of 3, etc.

It is always true of evenly spaced sets that the median – the middle term of the set – is equal to the mean, or arithmetic average, of the set. Moreover, the mean can be calculated by adding the high and the low terms of the set, and then dividing by 2. We can use this property in conjunction with the equation: Average * Number of Terms = Sum to calculate the sum of any large evenly spaced set.

In the case of the set of the integers from 1 to 100 inclusive, it works like this:

Average = (High + Low)/2 = (100 + 1)/2 = 101/2 = 50.5.

The Number of Terms = 100. Technically, the equation for finding the number of terms in an evenly spaced set is [(High-Low)/increment] + 1, but clearly, there are 100 terms between 1 and 100. Just remember, when using this formula, we want to add one to make sure we’re not leaving off the last term.

Average * Number = 50.5 * 100 = 5050.

Not too bad, even for a seven-year-old. (Note to those curious about the history of mathematics, this isn’t exactly how Gauss did the calculation, but it’s close enough.)

Now let’s see this concept in action on the GMAT:

For any positive integer n, the sum of the first n positive integers equals (n(n+1))/2. What is the sum of all the even integers between 99 and 301? 
A) 10,000
B) 20,200
C) 22,650
D) 40,200
E) 45,150 

Notice that we don’t have to bother with the formula they give us. The set of all evens from 99 to 301 inclusive will really be from 100 to 300, as those are the lowest and highest even terms of the set, respectively.

Average = (High + Low)/2 = (300 + 100)/2 = 400/2 = 200.

Number of Terms = [(High-Low)/increment] + 1 = [(300-100)/2] + 1 = 101. (Note: we divide by ‘2’ here because we only want even numbers, or multiples of 2. Thus, there is an increment of 2 between each successive term in the set.)

Average * Number = 200 * 101 = 20,200. The answer is B. Not bad.

Great, you think. Now I can just go on autopilot and apply these formulas anytime I encounter a huge evenly spaced set. But the GMAT doesn’t work like that. Sometimes we use a formula, but just as often, we’ll use logic, or we’ll pick a number, or we’ll work with the answer choices.

This cannot be repeated enough: Quantitative Reasoning is not a math test. It’s a test that requires some mathematical knowledge in order to make good decisions under pressure. Sometimes the best decision is doing little or no math at all.

Consider the following question:

How many positive three-digit integers are divisible by both 3 and 4? 
A) 75
B) 128
C) 150
D) 225
E) 300 

First, note that any number that is divisible by both 3 and 4 will be divisible by 12, as 12 is the least common multiple of 3 and 4. Perhaps you also noted that we’re dealing with an evenly spaced set here, and that, if the set consists of multiples of 12, the increment is clearly 12. But this question is very different from the previous one because we’re not required to calculate a sum. We just need to know how many multiples of 12 exist between 100 and 999.

If my increment is 12, I know I’ll be dividing by 12 at some point. But I can see that my (High – Low) will be at most (999 – 100) = 899. (Technically, our highest multiple of 12 in this set is 996 and our low term is 108, but there’s actually no need to discern this.) Clearly 899/12 will be less than 100, so there have to be fewer than 100 terms in the set. Now look at the answer choices. Only A is less than 100, so I’m done. I don’t have to finish the calculation.

Takeaway: You will be learning many useful formulas for the GMAT, but make sure you don’t use them blindly. Expect to mix formal algebra with the well-worn strategies of picking numbers and working with the answer choices. On the GMAT, flexibility and mental agility will always take precedence over rote memorization.

*GMATPrep 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, 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.

Advanced Averages Concepts for the GMAT

Quarter Wit, Quarter WisdomLet’s discuss an advanced averages concept today.

Say, you have the following set of consecutive integers: 2, 3, 4, 5, 6, 7, 8

What is the average of this set? There are 7 consecutive integers here and the average is 5, the middle number.

Say the set is changed to: 2, 3, 4, 5, 6, 7, 8, 9 (another consecutive number is added to the extreme right). Now what is the average? It is the average of the two middle numbers (5+6)/2 = 5.5.

Let’s edit the set one more time: 1, 2, 3, 4, 5, 6, 7, 8, 9 (another consecutive number is added to the extreme left). The average now is 5 again.

Whenever you add a number on either side of a set of consecutive integers, the average changes by 0.5. This is obvious because odd number of consecutive integers have the middle number as the average and an even number of consecutive integers have the average of two middle numbers as the average. Since every time you add an integer, the number of integers changes from odd to even or from even to odd, the average changes by 0.5.

By the same logic, what happens when you remove an integer from either extreme?

Given a set 3, 4, 5, 6, 7, 8, 9, how will its average change if you remove 3?

The average of 3, 4, 5, 6, 7, 8, 9 is 6, and the average of 4, 5, 6, 7, 8, 9 is 6.5 — the average increases to 6.5 because you removed a small number.

Now how will the average change if you remove 9 instead of 3?

The average of 3, 4, 5, 6, 7, 8, 9 is 6, and the average of 3, 4, 5, 6, 7, 8 is 5.5 — here, the average decreases to 5.5 because you removed a large number.

So, every time you add or remove a number from one of the extremes, the average will move by 0.5.

What happens if you remove a number from somewhere in the middle?

The average changes but by how much? When you remove the greatest or the least number, the average changes by 0.5. So when you remove some other number, the average will change by something less than 0.5. For example, from the set 3, 4, 5, 6, 7, 8, 9, if you remove 8, the average changes from 6 to 5.667. If instead, you remove 7, the average changes to 5.833.

A few takeaways:

  1. When you remove an integer very close to the average, the average changes by very little. If you remove the average, the average doesn’t change (changes by 0). When you remove a number close to the extreme, the average changes by a larger number (up to a maximum of 0.5).
  2. When you remove a number less than the average, the average increases. When you remove a number more than the average, the average decreases.
  3. When you remove the smallest number, the average increases by 0.5. When you remove the greatest number, the average decreases by 0.5.

Now, a question based on this concept:

In a class, the teacher wrote a set of consecutive integers beginning with 1 on the blackboard. A student erased one number. The average of the remaining numbers was 29(14/19). What was the number that the student erased?

(A) 13

(B) 16

(C) 28

(D) 36

(E) 50

Solution:

The numbers on the board: 1, 2, 3, 4, …

The new average is 29(14/19). Since the average changes by not more than 0.5 when you remove an integer from a set of consecutive integers, the original average was either 29.5 or 30. So originally there were either 58 numbers (average 29.5) or 59 numbers (average 30).

When you remove a number, you are left with either 57 numbers or with 58 numbers. Now, the new average will tell you whether you are left with 57 numbers or 58 numbers. The denominator is 19 in the fraction, so when you divide the sum of all remaining integers by the number of integers, the number of integers (denominator) is 19 or a multiple of 19 — 57 is a multiple of 19, 58 is not. So you must have been left with 57 integers and the original number of integers must be 58. This means the original average must have been 29.5.

The original average of 29(1/2) increases to 29(14/19), i.e. an increase of 14/19 – 1/2 = 9/38.

When an integer was removed, the average increased by 9/38 so the integer must be less than the original average. Now use the concept of average that we have learned. One integer was bringing the rest of the numbers down by 9/38 each so the integer must have been (9/38)*57 = 13.5, which is less than the original average of 29.5.

This means the integer that was removed must have been (29.5 – 13.5) = 16, so the answer is B.

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

GMAT Tip of the Week: There’s a Hole in the Bucket… But Not in Your GMAT Score!

GMAT Tip of the WeekIf you’ve ever attended a summer camp or roasted marshmallows over a campfire, there’s a good chance you know the popular children’s singalong song “There’s a Hole in the Bucket.”  Sparing you the repeat lyrics, let’s take a look at the ridiculous (and GMAT-relevant) musical conversation between Dear Henry and Dear Liza:

Henry: There’s a hole in the bucket (dear Liza, dear Liza, dear Liza…)

Liza: Then fix it (dear Henry, dear Henry, dear Henry…)

Henry: With what shall I fix it?

Liza: With straw.

Henry: The straw is too long.

Liza: Well, cut it.

Henry: With what shall I cut it?

Liza: With an axe.

Henry: The axe is too dull.

Liza: Then sharpen it.

Henry: With what shall I sharpen it?

Liza: With a stone.

Henry: The stone is too dry.

Liza: Then wet it.

Henry: With what shall I wet it? (Editor’s note: really, Henry?)

Liza: With water.

Henry: With what shall I fetch it?

Liza: With a bucket.

Henry (and his redemption): There’s a hole in the bucket.

<Repeat over and over again>

Now, what makes that song such a children’s and family favorite?  In some part it’s popular because it repeats upon itself, but mostly it’s popular because even small children have to laugh at Henry’s heroic lack of critical thought.  Henry simply can’t function unless Liza directly hands him the specific next step.

…and Liza and Henry’s conversation is not all that much unlike many GMAT tutoring sessions.

Among the pool of GMAT test-takers, there are plenty of Henrys.  And as much as you may laugh at him, you’re playing the part of Henry just a little too much when you:

  • Stop working on a problem in less than 2 minutes and flip to the back of the book for the solution. (“With what shall I solve it, dear textbook, dear textbook…”)
  • Give up on the calculations without first checking the answer choices to see if they afford you a shortcut. (“The calculation is too long, dear GMAT, dear GMAT”)
  • Frustratedly ask “but how am I supposed to see that I should do that?”. (“But how should I know that, dear teacher, dear teacher…”)
  • Write off the question as flawed because you disagree with the correct answer. (“The solution is just wrong, dear answer key, dear answer key…”)

Eavesdrop on a GMAT tutoring session at your local library or coffee shop and there’s a good chance you’ll hear more Liza-and-Henry than you’d expect.  Students frequently ask for the rule but not the lesson, and tutors often simply oblige.  But to avoid Henrydom on test day (this conversation should last 3-5 seconds, not be a song that kids will sing for an entire field trip bus ride.  Figure it out, Henry!) you need to train yourself to ask and answer those questions for yourself.

We at Veritas Prep suggest the “toolkit” approach as opposed to a “if it’s this kind of problem I will steadfastly use this method without critical thought” mindset.  When the bucket has a hole or the straw is too long, ask yourself what other tools are in your toolkit.

For example, if you blank on a rule, try proving it with small numbers.  Unsure whether Even + Odd is Even or Odd?  Just try 2 + 1 (an even plus and odd) and recognize that the answer is 3 (Odd!).  Or if the algebra looks too messy, see if you can plug in an answer choice to get a better feel for the solutions’ relationship to the problem.

What makes “There’s a Hole in the Bucket” funny is what could ultimately make your own GMAT test experience miserable: you (and Henry) have to employ a combination of critical thinking, trial-and-error, and patience to solve problems. The exam simply isn’t testing your ability to memorize a “Liza List” of steps to solve each problem; many hard problems are designed specifically to reward those who overcome the adversity of the “obvious” method leading you down a rabbit hole of awful algebra or those who find a familiar theme in a completely unfamiliar setup.  So to train yourself to be an anti-Henry:

  • Force yourself to fight and struggle through hard practice problems. The written solution isn’t likely to be nearly as helpful as your having had to struggle to gain understanding.
  • Think in terms of your “toolkit” – if your first inclination doesn’t lead to success, rummage around your toolkit to see what other types of concepts might apply to that problem.
  • When you don’t know or can’t remember a rule, test the concept with small numbers to see if you can retrain your brain or prove the relationship to yourself.
  • Hold your tutor accountable – they should be asking you probing questions like Socrates, not handing you one-time solutions and steps like Liza (she’s not totally innocent in this either…she enables Henry way too much!)

The way the song goes, there will be a hole in Henry’s bucket forever, but if there’s a hole in your GMAT score you can fix it with a new study mindset (even if the straw is too long…).

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

By Brian Galvin.

Why is There “Math” in the GMAT Critical Reasoning Section?

No MathThe Critical Reasoning portion of the GMAT will sometimes test basic mathematical concepts. My more verbally-minded students sometimes complain that this tendency is unfair, as the test seems to have imported a question-type from the section of the test that they find less agreeable into the section they consider their strength. But the truth is that the “math” in Critical Reasoning is really about logic and intuition rather than higher-level abstraction.

Take percentages, for instance. We can understand percentage reasoning without doing much calculation. When I introduce this topic, I’ll offer a simple real-world example:

In the 2014 playoffs, Lebron James made roughly 56% of his field goal attempts. In the 2015 playoffs, he made roughly 42% of his attempts. Therefore, he made fewer field goals in 2015 than in 2014.

You don’t need to be an avid basketball fan to recognize the glaring logical flaw in this statement. To determine whether that percentage dip is meaningful, we have to know how many shot attempts he was taking. Because he took so many more shots in 2015 than in 2014, he ended up making more field goals in that year, when his field goal percentage was lower. The notion that a percentage isn’t terribly meaningful without knowing the percent of what is obvious to everyone.

What the GMAT will typically do, however, is to test the exact same concept using a scenario that we may not grasp quite as intuitively. Consider the following official argument:

In the United States, of the people who moved from one state to another when they retired, the percentage who retired to Florida has decreased by three percentage points over the last ten years.  Since many local businesses in Florida cater to retirees, these declines are likely to have a noticeably negative economic effect on these businesses and therefore on the economy of Florida.

Which of the following, if true, most seriously weakens the argument given?

  1. People who moved from one state to another when they retired moved a greater distance, on average, last year than such people did ten years ago.
  2. People were more likely to retire to North Carolina from another state last year than people were ten years ago.
  3. The number of people who moved from one state to another when they retired has increased significantly over the past ten years.
  4. The number of people who left Florida when they retired to live in another state was greater last year than it was 10 years ago.
  5. Florida attracts more people who move form one state to another when they retired than does any other state.

The logic here may not be as obvious as the Lebron example, but it is, in fact, identical. The argument’s conclusion is that Florida’s economy will suffer negative consequences. The central premise is that of the people moving from one state to another, a smaller percentage are going to Florida now than were going to Florida ten years ago. The assumption is that a smaller percentage moving to Florida means fewer people moving to Florida.

This line of reasoning is no more valid than asserting that Lebron shooting a lower percentage in 2015 than in 2014 means he made fewer shots in 2015. Just as we needed to know if there was a change in the total number of shots Lebron was taking in order to evaluate whether the change in percentage was meaningful, we need to know if there was a change in the total number of people moving from one state to another in order to properly assess whether it’s meaningful that a smaller percentage are moving to Florida.

Let’s evaluate the answer choices one by one:

  1. The distance people moved doesn’t matter. Out of Scope. A is out.
  2. North Carolina isn’t relevant to what’s happening in Florida. Out of Scope. B is out.
  3. This is the logical equivalent of pointing out that Lebron took many more shots in 2015 than in 2014. If far more people are moving from one state to another now than were moving from one state to another ten years ago, it’s possible that more total people are moving to Florida, even if a smaller percentage of movers are going to Florida. This looks good.
  4. First, the number of people leaving Florida has no bearing on whether a smaller percentage of people moving to Florida will have an impact on Florida’s economy. Moreover, we’re trying to weaken the idea that Florida’s economy will suffer. If more people are leaving Florida, it would strengthen the notion that Florida’s economy will endure negative consequences. That’s the opposite of what we want. D is out.
  5. Tempting perhaps, but ultimately, irrelevant. Just because Lebron led the league in field goals made in both 2015 and 2014 (he didn’t, but play along), doesn’t mean he didn’t make fewer field goals in 2015. E is out.

The answer is C.  If more people are moving from state to state, a lower percentage moving to Florida may not mean that fewer people are coming to Florida, just as Lebron’s dip in field goal percentage does not mean he was making fewer field goals if he was taking more shots.

Takeaway: The “math” concepts tested in Critical Reasoning are, in fact, logic concepts. By connecting the prompt to a more concrete real-world example, we make this logic far more intuitive and easily graspable when we encounter it on the test.

*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 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.

99th Percentile GMAT Score or Bust! Lesson 7: Read Like You Drive

raviVeritas Prep’s Ravi Sreerama is the #1-ranked GMAT instructor in the world (by GMATClub) and a fixture in the new Veritas Prep Live Online format as well as in Los Angeles-area classrooms. He’s beloved by his students for the philosophy “99th percentile or bust!”, a signal that all students can score in the elusive 99th percentile with the proper techniques and preparation. In this “9 for 99thvideo series, Ravi shares some of his favorite strategies to efficiently conquer the GMAT and enter that 99th percentile.

First, take a look at lessons 1, 2, 3, 4, 5, and 6!

Lesson Seven:

Read Like You Drive: very few GMAT examinees will make mistakes driving to the GMAT test center, but most test-takers will make several Reading Comprehension mistakes once they’re there. As Ravi will discuss in this video, however, the two activities are much more similar than you realize: your job is to follow the signs. Certain keywords in Reading Comprehension passages will tell you when to yield, stop, turn, and pass with care, and if you’re following those signs properly you can proceed much faster than your self-imposed “speed limit” (most people read the passages far too slowly – stay out of the left lane!) and save valuable time for the questions themselves.

Are you studying for the GMAT? We have free online GMAT seminars running all the time. And, be sure to find us on Facebook, YouTube and Google+, and follow us on Twitter!

Want to learn more from Ravi? He’s taking his show on the road for a one-week Immersion Course in New York this summer, and he teaches frequently in our new Live Online classroom.

By Brian Galvin

Is Positive Thinking Enough to Actually Succeed on the GMAT?

QuestioningAt some point during each course I teach, I’ll ask my students if they’re familiar with this famous quote from Henry Ford “Whether you think you can, or you think you can’t – you’re right.”  Of course, they always know it. It’s a quote so popular it’s become a pedagogical cliché. Next, I’ll ask them if they believe the quote is true. They usually do. I’ll follow up with a series of GMAT-related questions. “Who struggles with probability questions?” “Who sees Reading Comprehension as a weakness?” Different hands go up for different questions.

They realize immediately that there’s a disconnect here. Why would anyone maintain the belief that he or she struggles in a given area if he or she subscribes to the notion that the pessimistic belief is a self-fulfilling prophesy? My sense is that this disconnect is rooted in our tendency to nod politely when greeted with popular aphorisms we’d like to be true, while at some level, not really believing them.

We can pay lip service to Henry Ford all we want. Our actual belief is something more along the lines of: sure, it would be nice if you could improve your performance via thought alone, but that doesn’t actually work. It’s a fantasy, one that is so appealing that we’ll collectively agree to pretend that it’s true.

 Part of my job as an instructor is to get my students to move past the cliché and somehow internalize the truth of the sentiment that our beliefs do matter. This isn’t a New Age chimera that we’d like to be true. It’s an area of extensive scientific research. In 2007, researchers at Stanford University conducted a study in which they tracked the development of 7th grade students who believed that intelligence was innate vs. students who believed that intelligence is a fluid phenomenon, something that can be cultivated and improved through dedicated effort.

The students who believed that intelligence is innate were deemed the “fixed mindset” group, and the group who believed that intelligence could be improved were deemed the “growth mindset” group. Most importantly, at the start of the study, these groups had similar academic background. Sure enough, over the next couple of years, there was a marked divergence in performance – the growth mindset group outperformed their fixed mindset peers by a significant margin (take a look at this study here).

One component of the growth mindset is the belief that adversity isn’t evidence of an inherent shortcoming, but rather, an opportunity to learn and improve. This is absolutely essential on the GMAT. Students will, on average, take about a half-dozen practice tests. It is extremely rare that every one of those practice tests goes well.

At some point, during every class I teach, I’ll get a panicked email, the general gist of which is that things had been going well, but now, after a disappointing practice test, the student has significant doubts about whether the previous successes were real. I’m often asked if it will be necessary to push the test date back. The growth mindset compels us to see this setback as a positive. Isn’t it better to uncover the need for a strategic tweak on a low stakes practice test than on the official exam?

 Sure enough, once my students are able to re-frame their beliefs from, “I’m just not good at X,” to, “Maybe I’ve struggled with X in the past, but with a little practice I can actual convert this former liability into an asset,” they improve. The student who struggled with probability wasn’t inherently bad at probability, but had a less than stellar teacher in high school or college and never learned the underlying concepts properly. The student who struggled with Reading Comprehension simply wasn’t taking notes properly.

Most importantly, the students who believed that they just weren’t good at standardized tests realized that the ability to do well on standardized tests is a skill that they simply hadn’t acquired yet. In the past, when they were convinced that they couldn’t do well on, say, the SAT, they hadn’t bothered to study, because what was the point of expending any effort if the result was going to be disappointment? Once they see that they their past struggles weren’t functions of innate deficits, but rather, of self-limiting beliefs, a world of possibility opens up.

Takeaway: how we frame our thoughts with respect to academic performance is extraordinarily important. Unfortunately, our culture generally pays lip service to the growth mindset while perpetuating the notion of a fixed one. We’ll thoughtlessly spout that Henry Ford quote, all the while thinking of people as high IQ or low IQ, not realizing that IQ is itself malleable (take a look at this idea here).

Think of someone you knew in high school who did unusually well on the SAT’s. You probably thought, “That person is great at standardized tests,” rather than “That person has been successful at cultivating a particular skill set that translated well in the domain of this one particular exam.” But the latter is true. So don’t set arbitrary limits of yourself, because, contrary to some our deepest intuitions, belief and performance are inextricably linked.

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

2 Simple Calculation Tricks to Help You Overcome the GMAT Quant Section

tutoringMuch of the Quant section of the GMAT involves calculations, and as calculators are not allowed, we need to be able to add, subtract, multiply and divide numbers quickly and accurately. A surprising number of errors are made on easier problems simply because of computational error and, interestingly, many trap answers on the exam come from common errors made in computations!

American legend Wyatt Earp once said of gunfights, “Fast is fine, but accuracy is everything.” For the GMAT, if we can combine speed with accuracy, not only will you minimize silly mistakes but you can also pick up time that can be spent reasoning through more difficult problems or double-checking your work. Here are two simple math tricks that can help:

1) The Distributive Property is Your Friend

Many of us recall the basic properties of operations (associative property, commutative property, distributive property and identity property), but rarely do we consciously use them except in some algebraic manipulations. However the Distributive Property is also very useful in quickly and easily multiplying larger numbers.

For example, take the following: 163 x 30. Multiplying this out long-hand is not overly difficult, however, since we do not do this in our everyday lives, we are prone to errors. Using the Distributive Property here can help.  Intuitively, many of us would see that 163 x 30 can be broken up into 163 x 10 three times – THAT is the Distributive Property. We can then re-write the expression as 163 x (10 x 3), and then “distribute” the operands into (163 x 10) x 3

Now, couple the Distributive Property with some other tricks that many of you already use, and this tip becomes even more valuable.  Take, for instance, a more difficult calculation to “distribute” – say, 163 x 48.  You could “distribute” this into 163 x (40 + 8) and get 163 x 40 (an easy calculation) + 163 x 8.

A common trick that is used when multiplying by 8 or 9 is multiply the number by 10 and then subtract out “extra.” In this case, we would multiply 163 by 10 and then subtract out the 2 additional 163’s. If we combine these steps on the front end, we could say 163 x 48 can be re-written as 163 x (50 – 2) and get (163 x 50) – (163 x 2). 1633 x 50 can be further broken down as 163 x 10 x 5, 1630 x 5. Putting it all together, we get 163 x 50 = 163 x 10 x 5 = 1,630 x 5 = ½ of 16300 = 8,150.

8,150 – (163 x 2) = 8, 150 – 326 = 7, 824

This is a good trick to help make long-hand multiplication simpler. Be careful in breaking numbers down into too many pieces as this can overly complicate the process and lead to errors.

2) Estimation and Proportionality

The GMAT expects us to be able to move between fractions, decimals, percents and even ratios easily. Many times this is straightforward, but other times it can be quite vexing. For instance, a problem may tell you that the ratio of boys to girls in a particular class is 4:9 and ask what percentage of the class is boys. Obviously, we can see that the TRAP answer would be 44% (4/9 = 44%). But, how can we easily and accurately calculate, or estimate, the correct answer?

Here is a trick to help with that: because we have ratios down pat, we know that a ratio of 4:9 tells us that there are 4 boys out of a class of 13, so the proper fraction would be 4/13. Now the tough part is turning this into a percent. Since percents simply are fractions with a denominator of 100, we can set up an algebraic equation. In this case, we have 4/13 = x/100 and using cross-multiplication we see the answer is 400 ÷ 13, 30 and 10/13, or a little less than 31%.

An easier way, might be to use proportionality to estimate the answer. Here is how that would work:  proportionally, our fraction of 4/13 needs to be converted to a fraction over 100. To increase our denominator of 13 to 100, we need to multiply it by a bit over 7 ½. To keep the fraction in its original proportion, we will also need to multiply the numerator by the same (a bit over 7 ½), giving us a value of a little over 30%.

The takeaway from this is that by sharpening your computational skills, you can save time, improve accuracy and even minimize your effort on the exam. This can translate to higher scores by eliminating silly mistakes and saving brain power for the more difficult questions.

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 Dennis Cashion, a Veritas Prep instructor based in Denver.

Why You Shouldn’t Rely on Your Ear for GMAT Sentence Correction Questions

Phone InterviewThe other night, when we were reviewing Sentence Correction strategies in class, a student asked if it was acceptable to rely on his ear to find the correct answer. This was what he’d done when he’d taken his diagnostic test, and he’d performed quite well on this section, so he figured it just made more sense to devote his study time to other areas. It’s a common question. After all, if you’re naturally good at something, does it really make sense to make an investment of time and energy just to tamper with an approach that’s been effective?

Whenever I get this question, I always take pains to give a nuanced response. My goal, when I’m teaching, isn’t to indoctrinate anyone or impose a given philosophical approach to a problem. The last thing any of us should be doing when we take the GMAT is wringing our hands over whether our instinct for how to tackle the problem is the “right” one. However, some approaches have potential shortcomings that we need to be mindful of, and using your ear alone to solve Sentence Correction questions is no exception.

The first problem with using your ear alone is that while a good instinct for syntax and grammar is immensely helpful for writers, on the GMAT, this instinct will often cause us to reject sentences that are technically correct but are specifically engineered to sound a little off. If you were a question-writer for the GMAT, and your goal was to make a given question as challenging as possible, wouldn’t you make some correct answers sound a little strange to amplify the difficulty of the question?

In these cases, we simply have to use a blend of logic and grammar rules to rule out the four definitively wrong answer choices. The remaining answer, which sounds strange, but has no glaring errors, will have to be correct. Take this official question, for example:

For many revisionist historians, Christopher Columbus has come to personify devastation and enslavement in the name of progress that has decimated native people of the Western Hemisphere.

A) devastation and enslavement in the name of progress that has decimated native peoples of the Western Hemisphere.
B) devastation and enslavement in the name of progress by which native peoples of the Western Hemisphere have been decimated.
C) devastating and enslaving in the name of progress those native peoples of the Western Hemisphere that have been decimated.
D) devastating and enslaving those native peoples of the Western hemisphere which in the name of progress are decimated.
E) the devastation and enslavement in the name of progress that have decimated the native peoples of the Western Hemisphere.

I like to pride myself on having a good ear when it comes to Sentence Correction, but none of these options strike me as terribly appealing. Let’s evaluate them one by one:

A, in its entirety, reads as follows: Christopher Columbus has come to personify devastation and enslavement in the name of progress that has decimated native peoples of the Western Hemisphere.

Notice, the relative clause beginning with “that.” “That” has a singular verb “has,” meaning that the antecedent for “that” should be the closest singular noun. Here, the closest singular noun is “progress.” Read literally, the sentence is saying that progress has decimated native peoples! That makes no sense. Eliminate A.

B: Christopher Columbus has come to personify devastation and enslavement in the name of progress by which native peoples of the Western Hemisphere have been decimated.

Again, it sounds like “progress” is responsible for the decimation of native peoples. No good.

C: Christopher Columbus has come to personify devastating and enslaving in the name of progress those native peoples of the Western Hemisphere that have been decimated. 

Here “that” seems to refer to “native peoples.” The GMAT prefers “who” when referring to people. Moreover, the phrase “those native peoples that have been decimated” makes it sound as though there were some native peoples who were devastated and others who weren’t. This is not the intended meaning of the sentence. Eliminate C.

D: Christopher Columbus has come to personify devastating and enslaving those native peoples of the Western hemisphere which in the name of progress are decimated. 

This one is riddled with problems. Again the phrase “those native peoples” is problematic. “Which” appears to refer to people, when the GMAT would prefer “who.” And last the verb “are” implies that the action is happening in the present tense. Clearly incorrect.

That leaves us with E: Christopher Columbus has come to personify the devastation and enslavement in the name of progress that have decimated the native peoples of the Western Hemisphere. 

It still sounds off to my ear, but if we read the sentence without the prepositional phrase “in the name of progress,” we get: Christopher Columbus has come to personify the devastation and enslavement that have decimated the native peoples of the Western Hemisphere. This makes perfect sense. Notice that because “that” has the plural verb “have,” it must have a plural antecedent, so “that” refers to “devastation and enslavement.” Not the world’s prettiest sentence, but far superior to the other four options, each of which have glaring mistakes.

Takeaway: No single strategy will allow you to answer every question within a given category correctly. Because some correct Sentence Correction answers are engineered to sound strange, it’s important to keep logic and grammar in mind as we’re justifying our decisions to eliminate the incorrect answers.

*GMAT Prep 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

A Closer Look at Set and Ratio GMAT Quant Questions

Quarter Wit, Quarter WisdomWriting this post on Teacher’s Day made me dedicate this post to questions on teachers! Considering that all GMAT questions are written by teachers, oddly enough, I found very few questions actually involving them. Looks like we are a humble bunch! Today, we will discuss two GMAT Quant questions on two different topics of discussion – sets and ratios. Both questions are official and of higher difficulty.

Question 1: Of the 1400 college teachers surveyed, 42% said they considered engaging in research an essential goal. How many of the college teacher surveyed were women?

Statement I: In the survey 36% of men and 50% of women said that they consider engaging in research activity an essential goal.

Statement II: In the survey 288 men said that they consider engaging in research activity an essential goal.

Solution:

On reading the question stem we realise that this question involves two variables:

Research Essential – Not Essential

Men – Women

This should immediately make us think about a matrix. Not that we cannot solve the question without one, but you know that I am a huge proponent of visual approaches.

We are given that 42% of total teachers (1400) considered research essential. So this means that 58% did not consider it essential. No need to actually calculate the number right now, let’s wait and see what else we know (anyway, we love to procrastinate calculations in Data Sufficiency questions).

Statement I: In the survey 36% of men and 50% of women said that they consider engaging in research activity an essential goal.

Say the number of women is W. We need the value of W. The number of men must be ‘Total – W’ = 1400 – W. 36% of men and 50% of women consider research essential. Knowing this, we see that we get:

TeachersDay

36% * (1400 – W) + 50% * W = 42% * 1400

This is a linear equation in W so we can solve it to get the value of W. Therefore, this statement alone is sufficient.

Statement II: In the survey 288 men said that they consider engaging in research activity an essential goal.

This statement doesn’t tell us the number of women who consider research essential, so it is not sufficient alone, therefore the answer is A, Statement I alone is sufficient but Statement II is not.

Question 2: If the ratio of the number of teachers to the number of students is the same in School District A and School District B, what is the ratio of the number of students in School District A to the number of students in School District B?

Statement I: There are 10,000 more students in School District A than there are in School District B.

Statement II: The ratio of the number of teachers to the number of students in School District A is 1 to 20.

Solution:

In both schools, the ratio of the number of teachers : the number of students is the same.

Statement I: There are 10,000 more students in School District A than there are in School District B.

We don’t know the number of students in either school district, so it is not informative enough to know that School District A has 10,000 more students. Therefore, this statement alone is not sufficient.

Statement II: The ratio of the number of teachers to the number of students in School District A is 1 to 20.

With this statement, we know that the ratio of the number of teachers : the number of students in School District A = 1:20.

Say the number of teachers in A = a; the number of students in A = 20a. We also know the ratio of the number of teachers : the number of students in School District B = 1:20.

Say the number of teachers in B = b; the number of students in B = 20b. Mind you, we don’t know the value of a and b. All we know is that the teacher student ratio is 1:20 in both.

The ratio of the number of students in A: the number of students in B = 20a : 20b = a:b. With this ratio, we don’t know a:b (even using both statements, we just know that a – b = 10,000). Therefore, the answer is E, Statements 1 and 2 together are not sufficient.

Were you able to solve both questions effortlessly? No? Don’t worry, that’s what we are here for! (Ignore the preposition at the end. It sounds most natural this way.)

Not so humble anymore, eh? :)

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

Master the GMAT by Applying Jedi-like Skills

Yoda ForceOnce you begin studying for the GMAT, you’ll realize quickly that there are different levels of mastery. There’s that initial level of competence in which you learn, or relearn, many of the foundational concepts that you learned in middle school and have since forgotten. There’s a more intermediate level of mastery in which you’re able to blend strategic thinking with foundational concepts.

Then there’s the highest level in which you achieve a kind of trance-like, fugue state that allows you to incorporate multiple strategies to break down a single complex problem and then seamlessly shift to a fresh set of strategies on the next problem, which, of course, will be testing slightly different concepts from the previous one.

It’s the GMAT equivalent of becoming a Jedi who can anticipate his opponent’s next light saber strike several moves in advance or becoming Neo in the Matrix, finally deciphering the structure of the streaming code that animates his synthetic world. Pick whatever sci-fi analogy you like – it’s this kind of expertise that we’re shooting for when we prepare for the test. The pertinent questions are then the following: how do we accomplish this level of expertise, and what does it look like once we’re finally there?

Fortunately for you, dear student, our books are organized with this philosophy in mind. Once you’ve worked through the skill-builders and the lessons, you’ll likely be at the intermediate level of competence. Then it will be through drilling with homework problems and taking practice tests that you’ll achieve the level of mastery we seek. But let’s take a look at a Sentence Correction question to get a sense of how our thought processes might unfold, once we’re functioning in full Jedi-mode.

Unlike most severance packages, which require workers to stay until the last day scheduled to collect, workers at the automobile company are eligible for its severance package even if they find a new job before they are terminated. 

(A) the last day scheduled to collect, workers at the automobile company are eligible for its severance package

(B) the last day they are scheduled to collect, workers are eligible for the automobile company’s severance package

(C) their last scheduled day to collect, the automobile company offers its severance package to workers

(D) their last scheduled day in order to collect, the automobile company’s severance package is available to workers

(E) the last day that they are scheduled to collect, the automobile company’s severance package is available to workers

Having done hundreds of questions, you’ll notice one structural clue leap immediately: “unlike.” When you see words such as “like” or “unlike” you know that you’re dealing with a comparison, so your first task is to make sure you’re comparing appropriate items. You’ll also note that the clause beginning with “which require” modifies “severance packages,” so whatever is compared to these severance packages will come after the modifier.

In A, you’re comparing “severance packages” to “workers.” We’d rather compare severance packages to severance packages or workers to workers. No good.

In B, again, you’re comparing “severance packages” to “workers.”

In C, you’re comparing “severance packages” to “the automobile company.” Nope.

That leaves us with D and E, both of which compare “severance packages” to “automobile’s company severance package.” Here, you’re comparing one group of severance packages to another, so this is logical. But now you have to switch gears – the comparison issue allowed you to eliminate some incorrect answer choices, but you’ll have to use another issue to differentiate between your remaining options.

Once we’re down to two options, you can simply read the two sentences and look for differences. One difference is that E contains the word “that” in the phrase “the last day that they are scheduled to collect.” Perhaps it sounds okay to your ear, but you’ll recall that when “that” is used as a relative pronoun, it should touch the noun it modifies. In this case, it touches, “last day.” Read literally, the phrase, “the last day that they are scheduled to collect,” makes it sound as though “they” are collecting the “last day.” Surely this isn’t what the sentence intends to convey, so we’re then left with ‘D,’ which is the correct answer.

Takeaway:

Notice how many disparate concepts you had to juggle here: You had to recognize the structural clue indicating that “unlike” signifies a comparison; recognize that temporarily skipping over a longer modifying phrase is an effective way to get a sense of the core clause you’re evaluating; recall that once you’re down to two answer choices, you can simply zero in on differences between your options; remember the rule stipulating that relative pronouns must touch what they modify; and last, you had to recognize that Sentence Correction is not only about grammar but also about logic and meaning, and all in under a minute and a half. I’d say that’s pretty Jedi-like.

*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

GMAT Tip of the Week: Test Day Should Not Be Labor Day

GMAT Tip of the WeekAs we head into the Labor Day weekend here in the U.S., it seems a fitting time to talk about labor.  Precious few people consider the GMAT to be a labor of love; to most aspiring (and perspiring?) MBAs, the GMAT is a lot of hard work.  And while, to earn the score that you’re hoping for, it’s likely that you’ll have to put in a good amount of sweat and a few tears (but hopefully no blood…), it’s important to recognize that test day itself should not be a Labor Day!

Your hard work should take place well before you get to the test center, so that on test day you’re not overworking yourself.  Working too hard on test day takes time (which is a precious resource on the exam), saps your mental energy (which also tends to be in short supply as you get later into the test with only two 8-minute breaks to recharge), and leads to errors.  Accordingly, here are a few tips to help you take the heavy labor out of your test day:

1. Only do the math you absolutely have to do.

The GMAT rewards efficiency and ingenuity, and has been known to set up problems that can be awful if done “by the book” but relatively smooth if you recognize common patterns.  For example:

  • Answers are assets! If the math looks like it’s going to get messy, look at the answer choices.  If they’re really far apart, you may be able to estimate after just a step or two.  Or if the answer choices are really “clean” numbers (0, 1, 10…these are really easy numbers with which to perform calculations) you may be able to plug them into the problem and backsolve without any algebra.
  • Don’t multiply until you’ve divided. Working step by step through a problem, you may see that you have to multiply, say, 51 by 18.  Which is an ugly thing to have to do for two reasons: that calculation will take time by hand, and it will leave you with a new number that will be hard to work with for the following step.  But the next step might be to divide by 34.  If you save the multiplication (just call it (51)(18) and don’t actually perform the step), then you can divide by 2 and 17.  Which works out pretty cleanly: 51/17 is 3 and 18/2 is 9, so now you’re just multiplying 3 by 9 and the answer has to be 27.   The GMAT goes heavy on divisibility, so keep in mind that you’ll do a lot of division on this test…meaning that it usually makes sense to wait to multiply until after you’ve seen what you’ll have to divide by.
  • Think in terms of number properties. Often you can determine quickly whether the answer has to be even or odd, or whether it has to be positive or negative, or what the first or last digit will be.  If you’ve made those determinations, quickly scan the answer choices and see how many fit those criteria.  If only one does, you’re done.  And if 2-3 do but they’re easier to plug in to the problem or to estimate between, then you can avoid doing the actual math.

2. Don’t take too many notes.

Particularly with Reading Comprehension passages, GMAT test-takers on average take far too many notes.  This hurts you for two reasons: first, it’s time consuming, and on a question type that’s already time consuming by nature.  And second, very few of the notes that people take are useful. People tend to take notes on details – you generally write down what you don’t think you’ll remember – but the test will typically only ask you about one detail per passage.  And the passage stays on the screen the whole time, so if you need to find a detail it’s just as easy to find it on the screen as it is in your notes (plus you’ll want to read the exact way that it was written, which your notes won’t necessarily have).  So use your time wisely: use your initial read of the passage to get a feel for the general direction of the passage, and then you’ll know which area/paragraph to go back to if and when you do need to find the details.

3. Stay flexible.

The GMAT is a test that rewards “mental agility,” meaning that it often designs problems that look like they should be solved one way (say, algebra) but quickly become labor-intensive that way and then reward those who are able to quickly change approaches (maybe to backsolving or picking numbers).  When it looks like you’ve just set yourself up for a massive amount of work, take a quick step back and re-analyze.  At this point are the answer choices more helpful?  Should you abandon your number-picking and go back to doing the algebra?  Does re-reading the question allow you to set it up differently?  Generally speaking, if the math starts to get labor-intensive you’re missing a better method.  So let that be your catalyst for re-assessing.

As you sit down to take the GMAT (to get into a great business school to become a more valuable member of the labor force), those 4 hours you spend at the test center probably won’t be a labor of love.  But they shouldn’t be full of labor, anyway.  Heed this advice to lighten your labor and the GMAT just might feel like more of a day off than anything (like, you know, Labor Day).

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

By Brian Galvin

99th Percentile GMAT Score or Bust! Lesson 6: Practice Tests Aren’t Real Tests

Veritas Prep’s Ravi Sreerama is the #1-ranked GMAT instructor in the world (by GMATClub) and a fixture in the new Veritas Prep Live Online format as well as in Los Angeles-area classrooms.  He’s beloved by his students for the philosophy “99th percentile or bust!”, a signal that all students can score in the elusive 99th percentile with the proper techniques and preparation.   In this “9 for 99thvideo series, Ravi shares some of his favorite strategies to efficiently conquer the GMAT and enter that 99th percentile.

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.

Are you studying for the GMAT? We have free online GMAT seminars running all the time. And, be sure to find us on Facebook, YouTube and Google+, and follow us on Twitter!

Want to learn more from Ravi? He’s taking his show on the road for a one-week Immersion Course in New York this summer, and he teaches frequently in our new Live Online classroom.

By Brian Galvin

Think Like Einstein to Answer GMAT Data Sufficiency Questions

I recently read Manjit Kumar’s, Quantum, which is about the philosophical disagreement between Niels Bohr and Albert Einstein with respect to the nature of reality.  In high school physics, we learned about Heisenberg’s Uncertainty Principle, which posits that we can never know both the position and the momentum of an electron with absolute certainty. The more precisely we measure an electron’s position, the less we know about its momentum, and vice versa.

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

Catching Sneaky Remainder Questions on the GMAT

One of my favorite topics to teach is remainders. We learn about remainders in grade school and when I introduce the topic in class, the response is often amused incredulity. It isn’t hard to see that when 16 is divided by 7, the remainder is 2. How can it possibly be the case that something we learned in fifth grade is included on a test that helps determine where we go to graduate school?

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

 

How to Compare Effectively During the GMAT

Quarter Wit, Quarter WisdomA lot of GMAT test takers complain about insufficient time. This is understandable as far as the Verbal section is concerned. We all have different reading speeds and that itself accounts for a lot of time issues in the Verbal section. Obviously then there are other factors – your comfort with the language, your comprehension skills, your conceptual understanding of the Verbal question types, etc.

However, timing issues should not arise in the Quant section. Your reading speed has very little effect on the overall timing scheme because most of the time during the Quant section is spent in solving the question. So if you are falling short on time, it means the methods you are using are not appropriate. We have said it before and will say it again – most GMAT Quant questions can be done in under one minute if you just look for the right thing.

For example, of the four listed numbers below, which number is the greatest and which is the least?

2/3

2^2/3^2

2^3/3^3

Sqrt(2)/Sqrt(3)

Now, how much time you take to solve this depends on how you approach this problem. If you get into ugly calculations, you will end up wasting a ton of time.

2/3 = .667

2^2/3^2 = 4/9 = .444

2^3/3^3 = 8/27 = .296

Sqrt(2)/Sqrt(3) = 1.414/1.732 = .816

So we know that the greatest is Sqrt(2)/Sqrt(3) and the least is 2^3/3^3. We got the answer but we wasted at least 2-3 mins in getting it.

We can do the same thing very quickly. We know that the squares/cubes/roots etc of numbers vary according to where the numbers lie on the number line.

2/3 lies in between 0 and 1, as does 1/4.

The Sqrt(1/4) = 1/2, which is greater than 1/4, so we know that the Sqrt(2/3) will be greater than 2/3 as well.

Also, the square and cube of 1/4 is less than 1/4, so the square and cube of 2/3 will also be less than 2/3. So the comparison will look like this:

(2/3)^3 < (2/3)^2 < 2/3 < Sqrt(2/3)

That is all you need to do! We arrived at the same answer using less than 30 secs.

Using this technique, let’s solve a question:

Which of the following represents the greatest value?

(A) Sqrt(3)/Sqrt(5) + Sqrt(5)/Sqrt(7) + Sqrt(7)/Sqrt(9)

(B) 3/5 + 5/7 + 7/9

(C) 3^2/5^2 + 5^2/7^2 + 7^2/9^2

(D) 3^3/5^3 + 5^3/7^3 + 7^3/9^3

(E) 3/5 + 1 – 5/7 + 7/9

Such a question can baffle someone who believes in calculating everything. We know better than that!

Note that the base values in all the options are 3/5, 5/7 and 7/9. This should hint that we need to compare term to term and not the entire expressions. Also, all values lie between 0 and 1 so they will behave the same way.

Sqrt(3)/Sqrt(5) is the same as Sqrt(3/5). The square root of a number between 0 and 1 is greater than the number itself.

3^2/5^2 is the same as (3/5)^2. The square (and cube) of a number between 0 and 1 is less than the number itself.

So, the comparison will look like this:

(3/5)^3 < (3/5)^2 < 3/5 < Sqrt(3/5)

(5/7)^3 < (5/7)^2 < 5/7 < Sqrt(5/7)

(7/9)^3 < (7/9)^2 < 7/9 < Sqrt(7/9)

This means that out of (A), (B), (C) and (D), the greatest one is (A).

Now we just need to analyse (E) and compare it with (B).

The first term is the same, 3/5.

The last term is the same, 7/9.

The only difference is that (B) has 5/7 in the middle and (E) has 1 – 5/7 = 2/7 in the middle. So (E) is certainly less than (B).

We already know that (A) is greater than (B), so we can say that (A) must be the greatest value.

A quick recap of important number properties:

Case 1: N > 1

N^2, N^3, etc. will be greater than N.

The Sqrt(N) and the CubeRoot(N) will be less than N.

The relation will look like this:

… CubeRoot(N) < Sqrt(N) < N < N^2 < N^3 …

Case II: 0 < N < 1

N^2, N^3 etc will be less than N.

The Sqrt(N) and the CubeRoot(N) will be greater than N.

The relation will look like this:

… N^3 < N^2 < N < Sqrt(N) < CubeRoot(N)  …

Case III: -1 < N < 0

Even powers will be greater than N and positive; Odd powers will be greater than N but negative.

The square root will not be defined, and the cube root of N will be less than N.

CubeRoot(N) < N < N^3 < 0 < N^2

Case IV: N < -1

Even powers will be greater than N and positive; Odd powers will be less than N.

The square root will not be defined, and the cube root of N will be greater than N.

N^3 < N < CubeRoot(N) < 0 < N^2

Note that you don’t need to actually remember these relations, just take a value in each range and you will know how all the numbers in that range behave.

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

GMAT Tip of the Week: 10 Must-Know Divisibility Rules For the GMAT (#3 Will Blow Your Mind!)

GMAT Tip of the WeekYou clicked, didn’t you? You’re helpless when presented with an enumerated list and a teaser that at least one of the items is advertised to be – but probably won’t be – mind-blowing. (In this case it kind of is…if not mind-blowing, it’s at least very powerful). So in this case, let’s use click bait for good and enumerated lists to talk about numbers. Here are 10 important (and “BuzzFeedy”) divisibility rules you should know heading into the GMAT:

 

 

1) 1

1 may be the loneliest number but it’s also a very important number for divisibility! Every integer is divisible by 1, and the result of any integer x divided by 1 is just x (when you divide an integer by 1, it stays the same). On the GMAT, the fact that every integer is divisible by 1 can be quite important. For example, a question might ask (as at least one official problem does): Does integer x have any factors y such that 1 < y < x?

Because every integer is divisible by itself and 1, that question is really just asking, “Is x prime or not?” because if there is a factor y that’s between 1 and x, x is not prime, and there is not such a factor, then x is prime. That “> 1” caveat in the problem may seem obtuse, but when you understand divisibility by 1, you can see that the abstract question stem is really just asking you about prime vs. not-prime as a number property. The concept that all integers are divisible by 1 may seem basic, but keeping it top of mind on the GMAT can be extremely helpful.

2) 2

It takes 2 to make a thing go right…in relationships and on the GMAT! A number is divisible by 2 if that number is even (and a number is even if it’s divisible by 2). That means that if an integer ends in 0, 2, 4, 6, or 8, you know that it’s divisible by 2. And here’s a somewhat-surprising fact: the number 0 is even! 0 is divisible by 2 with no remainder (0/2 = 0), so although 0 is neither positive nor negative it fits the definition of even and should therefore be something you keep in mind because 0 is such a unique number.

The GMAT frequently tests even/odd number properties, so you should make a point to get to know them. Because any even number is divisible by 2 (which also means that it can be written as 2 times an integer), an even number multiplied by any integer will keep 2 as a factor and remain even. So even x even = even and even x odd = even.

3) 3

It’s been said that good things come in 3s, and divisibility rules are no exception! The divisibility rule for 3 works much like a magic trick and is one that you should make sure is top of mind on test day to save you time and help you unravel tricky numbers. The rule: if you sum the digits of an integer and that sum is divisible by 3, then that integer is divisible by 3. For example, consider the integer 219. 2 + 1 + 9 = 12 which is divisible by 3, so you know that 219 is divisible by 3 (it’s 3 x 73).

This rule can help you in many ways. If you were asked to determine whether a number is prime, for example, and you can see that the sum of the digits is a multiple of 3, you know immediately that it’s not prime without having to do the long division to prove it. Or if you had a messy fraction to reduce and noticed that both the numerator and denominator are divisible by 3, you can use that rule to begin reducing the fraction quickly. The GMAT tests factors, multiples, and divisibility quite a bit, so this is a critical rule to have at your disposal to quickly assess divisibility. And since 1 out of every 3 integers is divisible by 3, this rule will help you out frequently!

4) 4

Presidential Election and Summer Olympics enthusiasts, be four-warned! You already know the divisibility rule for 4: take the last two digits of an integer and treat them as a two-digit number, and if that’s divisible by 4 so is the whole number. So for 2016 – next year and that of the next presidential election and Brazil Olympics – the last two-digit number, 16, is divisible by 4, so you know that 2016 is also divisible by 4.

If you fail to see immediately that a number is divisible by 4 given that rule, fear not! Being divisible by 4 just means that a number is divisible by 2 twice. So if you didn’t immediately see that you could factor a 4 out of 2016 (it’s 504 x 4), you could divide by 2 (2 x 1008) and then divide by 2 again (2 x 2 x 504) and end up in the same place without too much more work.

5) 5

Who needs only 5 fingers to divide by 5? All of us – divisibility by 5 is so easy you should be able to do it with one hand tied behind your back! If an integer ends in 5 or 0 you know that it’s divisible by 5 (and we’ll talk more about what extra fact 0 tells you in just a bit…).

6) 6

Your favorite character from the hit 1990’s NBC sitcom “Blossom” is also an easy-to-use divisibility rule! Since 6 is just the product of 2 and 3 (2 x 3 = 6), if a number meets the divisibility rules for both 2 (it’s even) and 3 (the sum of the digits is divisible by 3) it’s divisible by 6. So if you need to reduce a number like 324, you might want to start by dividing by 6, instead of by 2 or 3, so that you can factor it in fewer steps.

7) 7

Ah, magnificent 7. While there is a “trick” for divisibility by 7, 7 occurs much less frequently in divisibility-based problems (as do other primes like 11, 13, 17, etc.), so 7 is a good place to begin to think about a strategy that works for all numbers, rather than memorizing limited-use tricks for each number. To test whether a large number, such as 231, is divisible by 7, find an obvious multiple of 7 nearby and then add or subtract multiples of 7 to see whether doing so will land on that number. For 231, you should recognize that a nearby multiple of 7 is 210 (you know 21 is 7 x 3, so putting a 0 on the end of it just means that 210 is 7 x 30). Then as you add 7s to get there, you go to 217, then to 224, then to 231. So in your head you can see that 231 is 3 more 7s than 7 x 30 (which you know is 210), so 231 = 7 x 33.

8) 8

8 is enough! As you saw above with 4s and 6s, when you start working with non-prime factors it’s often easier to just divide out the smaller prime factors one at a time than to try to determine divisibility by a larger composite number in one fell swoop. Since 8 = 2 x 2 x 2, you’ll likely find more success testing for divisibility by 8 by just dividing by 2, then dividing by 2 again, then dividing by 2 a third time. So for a number like 312, rather than working through long division to divide by 8, just divide it in half (156) then in half again (78) then in half again (39), and you’ll know that 312 = 39 x 8.

9) 9

While “nein” may be German for “no,” you should be saying “yes” to divisibility by nine! 9 shares a big similarity with 3 in that a sum-of-the-digits rule applies here too. If you sum the digits of an integer and that sum is a multiple of 9, the integer is also divisible by 9. So, for example, with the number 729, because 7 + 2 + 9 = 18, you know that 729 is divisible by 9 (it’s 81 x 9, which actually is 9 to the 3rd power).

10) 10

We’ve saved the best for last! If a number ends in 0, it’s divisible by 10, giving you a great opportunity to make the math easy. For example, a number like 210 (which you saw above) lets you pull the 0 aside and say that it’s 21 x 10, which means that it’s 3 x 7 x 10.

Working with 10s makes mental (or pencil-and-paper) math quick and convenient, so you should seek out opportunities to use such numbers in your calculations. For example, look at 693: If you add 7, you get to a number that ends in two 0s (so it’s 7 x 10 x 10), meaning that you know that 693 is divisible by 7 (it’s 7 away from an easy multiple of 7) *and* that it’s 7 x 99 because it’s one less 7 than 7 x 100. Because the GMAT rewards quick mental math, it’s a good idea to quickly check for, “If I have to add x to get to the nearest 0, then does that give me a multiple of x?” (297 is 3 away from 300, so you know that 297 = 99 x 3). And since 10 = 2 x 5, it’s also helpful sometimes to double a number that ends in 5 (try 215, which times 2 = 430) to see how many 10s you have (43). That tells you that 215 = 43 x 5 because 215 x 2 = 43 x (2 x 5). Working with 10s can make mental math extremely quick – we’d rate numbers that end in 0 a perfect 10!

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

By Brian Galvin

Finding the Product of Factors on GMAT Questions

Quarter Wit, Quarter WisdomWe have discussed how to find the factors of a number and their properties in these two posts:

Writing Factors of an Ugly Number

Factors of Perfect Squares

Today let’s discuss the concept of ‘product of the factors of a number’.

From the two posts above, we know that the factors equidistant from the centre multiply to give the number. We also know that the behaviour is a little different for perfect squares. Let’s take two examples to understand this.

Example 1: Say N = 6

Factors of 6 are 1, 2, 3, 6

1*6 = 6 (first factor * last factor)

2*3 = 6 (second factor and second last factor)

Product of the four factors of 6 is given by 1*6 * 2*3 = 6*6 = 6^2 = [Sqrt(N)]^4

Example 2: Say N = 25 (a perfect square)

Factors of 25 are 1, 5, 25

1*25 = 25 (first factor * last factor)

5*5 = 25 (middle factor multiplied by itself)

Product of the three factors of 25 is given by 1*25 * 5 = 5^3 = [Sqrt(N)]^3

If a number, N, can be expressed as: 2^a * 3^b * 5^c *…

The total number of factors f = (a+1)*(b+1)*(c+1)…

The product of all factors of N is given by [Sqrt(N)]^f i.e. N^(f/2)

Let’s look at a couple of questions based on this principle:

Question 1: If the product of all the factors of a positive integer, N, is

2^(18) * 3^(12), how many values can N take?

(A) None

(B) 1

(C) 2

(D) 3

(E) 4

Solution: Since the product of all factors of N has only 2 and 3 as prime factors, N must have two prime factors only: 2 and 3.

Let N = 2^a * 3^b

Given that N^(f/2) = 2^(18) * 3^(12)

(2^a * 3^b)^[(a+1)(b+1)/2] = 2^(18) * 3^(12)

a*(a+1)*(b+1)/2 = 18

b*(a+1)*(b+1)/2 = 12

Dividing the two equations, we get a/b = 3/2

Smallest values: a = 3, b = 2. It satisfies our two equations.

Can we have more values for a and b? Can a = 6 and b = 4? No. Then the product a*(a+1)*(b+1)/2 would be much larger than 18.

So N = 2^3 * 3^2

There is only one such value of N.

Answer (B)

Question 2: If the product of all the factors of a positive integer, N, is 2^9 * 3^9, how many values can N take?

(A) None

(B) 1

(C) 2

(D) 3

(E) 4

Solution: Since the product of all factors of N has only 2 and 3 as prime factors, N must have two prime factors only: 2 and 3.

Let N = 2^a * 3^b

Given that N^(f/2) = 2^(9) * 3^(9)

(2^a * 3^b)^[(a+1)(b+1)/2] = 2^(9) * 3^(9)

a*(a+1)*(b+1)/2 = 9

b*(a+1)*(b+1)/2 = 9

Dividing the two equations, we get a/b = 1/1

Smallest values: a = 1, b = 1 – Does not satisfy our equation

Next set of values: a = 2, b = 2 – Satisfies our equations

All larger values will not satisfy our equations.

Answer (B)

Note that we can easily use hit and trial in these questions without actually working through the equations.

This is how we will do it:

N^(f/2) = 2^(18) * 3^(12)

Case 1: Assume values of f/2 from common factors of 18 and 12 – say 2

[2^9 * 3^6]^2

Can f/2 = 2 i.e. can f = 4?

If N = 2^9 * 3^6, total number of factors f = (9+1)*(6+1) = 70

This doesn’t work.

Case 2: Assume f/2 is 6

[2^3 * 3^2]^6

Can f/2 = 6 i.e. can f = 12?

If N = 2^3 * 3^2, total number of factors f = (3+1)*(2+1) = 12

This works.

The reason hit and trial isn’t a bad idea is that there will be only one such set of values. If we can quickly find it, we are done.

Why should we then bother to find it at all. Shouldn’t we just answer with option ‘B’ in both cases? Think of a case in which the product of all factors is given as 2^(16) * 3^(14). Will there be any value of N in such a case?

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

GMAT Tip of the Week: Small Numbers Lead to Big Scores

GMAT Tip of the WeekThe last thing you want to see on your score report at the end of the GMAT is a small number. Whether that number is in the 300s (total score) or in the single-digits (percentile), your nightmares leading up to the test probably include lots of small numbers flashing on the screen as you finish the test. So what’s one of the most helpful tools you have to keep small numbers from appearing on the screen?

Small numbers on your noteboard.

Have you found yourself on a homework problem or practice test asking yourself “can I just multiply these?”? Have you forgotten a rule and wondered whether you could trust your memory? Small numbers can be hugely valuable in these situations. Consider this example:

For integers x and y, 2^x + 2^y = 2^30. What is the sum x + y?

(A) 30
(B) 40
(C) 50
(D) 58
(E) 64

Every fiber of your being might be saying “can I just add x + y and set that equal to 30?” but you’re probably at least unsure whether you can do that. How do you definitively tell whether you can do that? Test the relationship with small numbers. 2^30 is far too big a number to fathom, but 2^6 is much more convenient. That’s 64, and if you wanted to set the problem up that way:

2^x + 2^y = 2^6

You can see that using combinations of x and y that add to 6 won’t work. 2^3 + 2^3 is 8 + 8 = 16 (so not 64). 2^5 + 2^1 is 32 + 2 = 34, which doesn’t work either. 2^4 + 2^2 is 16 + 4 = 20, so that doesn’t work. And 2^0 + 2^6 is 1 + 64 = 65, which is closer but still doesn’t work. Using small numbers you can prove that that step you’re wondering about – just adding the exponents – isn’t valid math, so you can avoid doing it. Small numbers help you test a rule that you aren’t sure about!

That’s one of two major themes with testing small numbers. 1) Small numbers are great for testing rules. And 2) Small numbers are great for finding patterns that you can apply to bigger numbers. To demonstrate that second point about small numbers, let’s return to the problem. 2^30 again is a number that’s too big to deal with or “play with,” but 2^6 is substantially more manageable. If you want to get to:

2^x + 2^y = 2^6, think about the powers of 2 that are less than 2^6:

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64

Here you can just choose numbers from the list. The only two that you can use to sum to 64 are 32 and 32. So the pairing that works here is 2^5 + 2^5 = 2^6. Try that again with another number (what about getting 2^x + 2^y = 2^5? Add 16 + 16 = 32, so 2^4 + 2^4), and you should start to see the pattern. To get 2^x + 2^y to equal 2^z, x and y should each be one integer less than z. So to get back to the bigger numbers in the problem, you should now see that to get 2^x + 2^y to equal 2^30, you need 2^29 + 2^29 = 2^30. So x + y = 29 + 29 = 58, answer choice D.

The lesson? When problems deal with unfathomably large numbers, it can often be quite helpful to test the relationship using small numbers. That way you can see how the pieces of the puzzle relate to each other, and then apply that knowledge to the larger numbers in the problem. The GMAT thrives on abstraction, presenting you with lots of variables and large numbers (often exponents or factorials), but you can counter that abstraction by using small numbers to make relationships and concepts concrete.

So make sure that small numbers are a part of your toolkit. When you’re unsure about a rule, test it with small numbers; if small numbers don’t spit out the result you’re looking for, then that rule isn’t true. But if multiple sets of small numbers do produce the desired result, you can proceed confidently with that rule. And when you’re presented with a relationship between massive numbers and variables, test that relationship using small numbers so that you can teach yourself more concretely what the concept looks like.

The best way to make sure that your GMAT score report contains big numbers? Use lots of small numbers in your scratchwork.

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

By Brian Galvin

5 Reasons That Studying for the GMAT Sucks

GMATLet’s face it. Except for the folks who write the test and prepare you for the test, no one really loves the GMAT. Any anyone who tells you otherwise either scored an 800 with no prep or is lying.

But self-inflicted misery loves company, so in no particular order, let’s take a look at some of the things that suck and more importantly, how to cope:

 

  • Integrated Reasoning (IR) : It was introduced a few years ago, and even though multiple surveys and studies show it does correlate well with skills needed to succeed in business and the corporate world, schools still seem to have varying opinions on its value and how best to use it in the admissions process. For now, think of IR as the appetizer or warm-up. It’s tough, but it’s 30 minutes and can serve as a solid warmup before tackling the tougher ‘main course’ of quant and verbal. You wouldn’t start sprinting out of the gates in a race; treat the GMAT the same way, and if you bank some early points, that can’t hurt either.
  • AWA: Similar to IR, it doesn’t factor into your Total score, and schools differ on how they evaluate the essay. That being said, consider it a pre-pre-warm up, and more importantly, remember that schools can download a copy of your essay when they view your scores. So it’s important to put forth your best effort (now is NOT the time to challenge authority and write what you truly think of the GMAT or B school admissions process) and treat it as another writing sample that schools can use to evaluate your brilliance and creativity under pressure. Also, if English isn’t your first language, it’s absolutely going to be leveraged as an additional writing sample.
  • Data Sufficiency: This isn’t math, at least not in the sense that you’re used to seeing. What happened to the two trains leaving from separate stations and determining where they’ll meet? While that’s more problem solving, data sufficiency is important for schools to gauge your decision making abilities when you have limited or inaccurate information In a perfect world, you could make informed decisions with an infinite amount of time and all of the necessary details. But the world isn’t ideal, and like the cliché says, time is money. So data sufficiency quantifies what schools want to see: can you discern at what point do you have enough information to make an informed decision or at what point do you not have enough information and need to walk away.
  • Getting up early/Staying up late/Giving up Happy Hour aka Time Suck: We’ve all heard of FOMO, or “fear of missing out.” You’re likely going to have to make FOMO your new BFF while you’re preparing. In order to get the score you want, it’s important to put forth the effort. Just like training for a marathon or triathlon, you can’t take shortcuts or it’ll show on race day, and only you truly know the full measure of the effort you’re putting forth. So before you even start studying, make sure you’re mapping out a 3-4 month window where you know you can truly carve out time on a daily (regular!) basis to prepare, and more importantly, dedicate quality time to preparation.
  • Expenses!: The GMAT is expensive! And so is preparation! But if you think about it compared to the investment you’re about to make in your future and your long-term earnings potential, $250 for the test, $20 in bus fare/gas/transportation, and $50 for a celebratory steak after you crush it is a drop in the bucket. In life, there are absolutely times you should clip coupons, look for a better value and skimp on the extras. This is not one of them. Consider the GMAT the first step in a much larger investment in yourself.

It’s not rocket science (if it was, that might be the MCAT, not the GMAT), but it is important to recognize and embrace the challenges of this process. If it was easy, there would be far more individuals taking the GMAT every year (though nearly 250,000 is some decently sized competition). And one day while you’re studying, you’ll realize that while you don’t necessarily love it, the “studying for the GMAT sucks” factor is not quite as strong as it once was.   Take that as your reminder to keep your eye on the end game and keep plugging away. Your former self will thank you down the road.

Are you studying for the GMAT? We have free online GMAT seminars running all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!

By Joanna Graham

How to Solve Relative Rate of Work Questions on the GMAT

Quarter Wit, Quarter WisdomToday, we look at the relative rate concept  of work, rate and time – the parallel of relative speed of distance, speed and time.

But before we do that, we will first look at one fundamental principle of work, rate and time (which has a parallel in distance, speed and time).

Say, there is a straight long track with a red flag at one end. Mr A is standing on the track 100 feet away from the flag and Mr B is standing on the track at a distance 700 feet away from the flag. So they have a distance of 600 feet between them. They start walking towards each other. Where will they meet? Is it necessary that they will meet at 400 feet from the red flag – the mid point of the distance between them? Think about it – say Mr A walks very slowly and Mr B is super fast. Of the 600 feet between them, Mr A will cover very little distance and Mr B will cover most of the distance. So where they meet depends on their rate of walking. They will not necessarily meet at the mid point. When do they meet at the mid point? When their rate of walking is the same. When they both cover equal distance.

Now imagine that you have two pools of water. Pool A has 100 gallons of water in it and the Pool B has 700 gallons. Say, water is being pumped into pool A and water is being pumped out of pool B. When will the two pools have equal water level? Is it necessary that they both have to hit the 400 gallons mark to have equal amount of water? Again, it depends on the rate of work on the two pools. If water is being pumped into pool A very slowly but water is being pumped out of pool B very fast, at some point, they both might have 200 gallons of water in them. They will both have 400 gallons at the same time only when their rate of pumping is the same. This case is exactly like the case above.

Now let’s go on to the question from the GMAT Club tests which tests this understanding and the concept of  relative rate of work:

Question: Tanks X and Y contain 500 and 200 gallons of water respectively. If water is being pumped out of tank X at a rate of K gallons per minute and water is being added to tank Y at a rate of M gallons per minute, how many hours will elapse before the two tanks contain equal amounts of water?

(A) 5/(M+K) hours

(B) 6/(M+K) hours

(C) 300/(M+K) hours

(D) 300/(M−K) hours

(E) 60/(M−K) hours

Solution: There are two tanks with different water levels. Note that the rate of pumping is given as K gallons per min and M gallons per min i.e. they are different. So we cannot say that they both will have equal amount of water when they have 350 gallons. They could very well have equal amount of water at 300 gallons or 400 gallons etc. So when one expects that water in both tanks will be at 350 gallon level, one is making a mistake. The two tanks are working for the same time to get their level equal but their rates are different. So the work done is different. Note here that equal level does not imply equal work done. The equal level could be achieved at 300 gallons when work done would be different – 200 gallons removed from tank X and 100 gallons added to tank Y. The equal level could be achieved at 400 gallons when work done would be different again – 100 gallons removed from tank X and 200 gallons added to tank Y.

To achieve the ‘equal level,’ tank Y needs to gain water and tank X needs to lose water. Total 300 gallons (500 gallons – 200 gallons) of work needs to be done. Which tank will do how much depends on their respective rates.

Work to be done together = 300 gallons

Relative rate of work = (K + M) gallons/minute

The rates get added because they are working in opposite directions – one is removing water and the other is adding water. So we get relative rate (which is same as relative speed) by adding the individual rates.

Note here that rate is given in gallons per minute. But the options have hours so we must convert the rate to gallons per hour.

Relative rate of work = (K + M) gallons/minute = (K + M) gallons/(1/60) hour = 60*(K + M)  gallons/hour

Time taken to complete the work = 300/60(K+M) hours = 5/(K+M) hours

Answer (A)

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!

GMAT Tip of the Week: Eazy E Shows You How To Take Your Quant Score Straight Outta Compton And Straight To Cambridge

GMAT Tip of the WeekIf you listened to any hip hop themed radio today, the day of the Straight Outta Compton movie premiere, you may have heard interviews with Dr. Dre. You almost certainly heard interviews with Ice Cube. And depending on how old school the station is there’s even a chance you heard from DJ Yella or MC Ren.

But on the radio this morning – just like on your GMAT exam – there was no Eazy-E. Logistically that’s because – as the Bone Thugs & Harmony classic “Tha Crossroads” commemorated – Eazy passed away about 20 years ago. But in GMAT strategy form, Eazy’s absence speaks even louder than his vocals on his NWA and solo tracks. “No Eazy-E” should be a mantra at the top of your mind when you take the GMAT, because on Data Sufficiency questions, choice E – the statements together are not sufficient to solve the problem – will not be given to you all that easily (Data Sufficiency “E” answers, like the Boyz in the Hood, are always hard).

Think about what answer choice E really means: it means “this problem cannot be solved.” But all too often, examinees choose the “Eazy-E,” meaning they pick E when “I can’t do it.” And there’s a big chasm. “It cannot be solved” means you’ve exhausted the options and you’re maybe one piece of information (“I just can’t get rid of that variable”) or one exception to the rule (“but if x is a fraction between 0 and 1…”) that stands as an obstacle to directly answering the question. Very rarely on problems that are above average difficulty is the lack of sufficiency a wide gap, meaning that if E seems easy, you’re probably missing an application of the given information that would make one or both of the statements sufficient. The GMAT just doesn’t have an incentive to reward you for shrugging your shoulders and saying “I can’t do it;” it does, however, have an incentive to reward those people who can conclusively prove that seemingly insufficient information can actually be packaged to solve the problem (what looks like E is actually A, B, C, or D) and those people who can look at seemingly sufficient information and prove why it’s not actually quite enough to solve it (the “clever” E).

So as a general rule, you should always be skeptical of Eazy-E.

Consider this example:

A shelf contains only Eazy-E solo albums and NWA group albums, either on CD or on cassette tape. How many albums are on the shelf?

(1) 2/3 of the albums are on CD and 1/4 of the albums are Eazy-E solo albums.

(2) Fewer than 30 albums are NWA group albums and more than 10 albums are on cassette tape.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
(C) Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient to answer the question asked
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed

Statistically on this problem (the live Veritas Prep practice test version uses hardcover and paperback books of fiction or nonfiction, but hey it’s Straight Outta Compton day so let’s get thematic!), almost 60% of all test-takers take the Eazy-E here, presuming that the wide ranges in statement 2 and the ratios in statement 1 won’t get the job done. But a more astute examinee is skeptical of Eazy-E and knows to put in work! Statement 1 actually tells you more than meets the eye, as it also tells you that:

  • 1/3 of the albums are on cassette tape
  • 3/4 of the albums are NWA albums
  • The total number of albums must be a multiple of 12, because that number needs to be divisible by 3 and by 4 in order to create the fractions in statement 1

So when you then add statement 2, you know that since there are more than 10 albums total (because at least 11 are cassette alone) so the total number could be 12, 24, 36, 48, etc. And then when you apply the ratios you realize that since the number of NWA albums is less than 30 and that number is 3/4 of the total, the total must be less than 40. So only 12, 24, and 36 are possible. And since the number of cassettes has to be greater than 10, and equate to 1/3 of the total, the total must then be more than 30. So the only plausible number is 36, and the answer is, indeed, C.

Strategically, being wary of Eazy-E tells you where to invest your time. If E seems too easy, that means that you should spend the extra 30-45 seconds seeing if you can get started using the statements in a different way. So learn from hip hop’s first billionaire, Dr. Dre, who split with Eazy long ago and has since seen his business success soar. Avoid Eazy-E and as you drive home from the GMAT test center you can bask in the glow of those famous Ice Cube lyrics, “I gotta say, today was a good day.”

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

By Brian Galvin

The Importance of Estimation on the GMAT

SciNotationIn the first session of every new class I teach, I try to emphasize the power and effectiveness of estimating when dealing with potentially complex calculations. No one ever disputes that this is a good approach, but an unspoken assumption is that while you may save a bit of time by estimating, it isn’t absolutely crucial to do so. After all, how long does it take to do a little arithmetic? The problem is that, under pressure, hard arithmetic can cause us to freeze. To illustrate this, I’ll ask, “quick, what’s 1.3 divided by 3.2?” This is usually greeted by blank stares or nervous laughter. But when I ask “okay, what’s 1 divided by 3?” they see the point: trying to solve 1.3/3.2 won’t just be time-consuming, but can easily lead to a careless mistake prompted by arithmetical paralysis.

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?

  1. 5 * 10^9
  2. 1 * 10^9
  3. 9 * 10^10
  4. 5 * 10^11
  5. 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.)

  1. The average speed at which Carlos cycled from his house to the library yesterday was greater than 16 feet per second.
  2. 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.

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

Identifying the Paradox on GMAT Critical Reasoning Questions

Quarter Wit, Quarter WisdomLet’s take a look at a very tricky GMAT Prep critical reasoning problem today. Problems such as these make CR more attractive than RC and SC to people who have a Quantitative bent of mind. It’s one of the “explain the paradox” problems, which usually tend to be easy if you know exactly how to tackle them, but the issue here is that it is hard to put your finger on the paradox.

Once you do, then the problem is quite easy.

 

Question: Technological improvements and reduced equipment costs have made converting solar energy directly into electricity far more cost-efficient in the last decade. However, the threshold of economic viability for solar power (that is, the price per barrel to which oil would have to rise in order for new solar power plants to be more economical than new oil-fired power plants) is unchanged at thirty-five dollars.

Which of the following, if true, does most to help explain why the increased cost-efficiency of solar power has not decreased its threshold of economic viability?

(A) The cost of oil has fallen dramatically.

(B) The reduction in the cost of solar-power equipment has occurred despite increased raw material costs for that equipment.

(C) Technological changes have increased the efficiency of oil-fired power plants.

(D) Most electricity is generated by coal-fired or nuclear, rather than oil-fired, power plants.

(E) When the price of oil increases, reserves of oil not previously worth exploiting become economically viable.

Solution: We really need to understand this $35 figure that is given. The argument calls it “the threshold of economic viability for solar plant.” It is further explained as price per barrel to which oil would have to rise in order for new solar power plants to be more economical than new oil-fired power plants.

Note the exact meaning of this “threshold of economic viability”. It is the price TO WHICH oil would have to rise to make solar power more economical i.e. the price to which oil would have to rise to make electricity generated out of oil power plants more expensive than electricity generated out of solar power plants. So this is a hypothetical price of oil. It is not the price BY WHICH oil would have to rise. So this number 35 has nothing to do with the actual price of oil right now – it could be $10 or $15. The threshold of economic viability will remain 35.

So what the argument tells us is that tech improvements have made solar power cheaper but the price to which oil should rise has stayed the same. If you are not sure where the paradox is, let’s take some numbers to understand:

Previous Situation:

– Sunlight is free. Infrastructure needed to convert it to electricity is expensive. Say for every one unit of electricity, you need to spend $50 in a solar power plant.

– Oil is expensive. Infrastructure needed to convert it to electricity, not so much. Say for every one unit of electricity, the oil needed costs $25 and cost of infrastructure to produce a unit of electricity is $15. So total you spend $40 for a unit of electricity in an oil fired plant.

Oil based electricity is cheaper. If the cost of oil rises by $10 and becomes $35 from $25 assumed above, solar power will become viable. Electricity produced from both sources will cost the same.

Again, note properly what the $35 implies.

Raw material cost in solar plant + Infrastructure cost in solar plant = Raw material cost in oil plant + Infrastructure cost in oil plant

0 + 50 = Hypothetical cost of oil + 15

Hypothetical cost of oil = 50 – 15

That is, this $35 = Infra price per unit in solar plant – Infra price per unit in oil plant

This threshold of economic viability for solar power is the hypothetical price per barrel to which oil would have to rise (mind you, this isn’t the actual price of oil) to make solar power viable.

What happens if you need to spend only $45 in a solar power plant for a unit of electricity? Now, for solar viability, ‘cost of oil + cost of infrastructure in oil power plant’ should be only $45. If ‘cost of infrastructure in oil power plant’ = 15, we need the oil to go up to $30 only. That will make solar power plants viable. So the threshold of economic viability will be expected to decrease.

Now here lies the paradox – The argument tells you that even though the cost of production in solar power plant has come down, the threshold of economic viability for solar power is still $35! It doesn’t decrease. How can this be possible? How can you resolve it?

One way of doing it is by saying that ‘Cost of infrastructure in oil power plant’ has also gone down by $5.

Raw material cost in solar plant + Infrastructure cost in solar plant = Raw material cost in oil plant + Infrastructure cost in oil plant

0 + $45 = $35 + Infrastructure cost in oil plant

Infrastructure cost in oil plant = $10

Current Situation:

– Sunlight is free. Infrastructure needed to convert it to electricity is expensive. For every one unit of electricity, you need to spend $45 in a solar power plant.

– Oil is expensive. Infrastructure needed to convert it to electricity, not so much. For every one unit of electricity, you need to spend $25 + $10 = $35 in an oil fired power plant.

You still need the oil price to go up to $35 so that cost of electricity generation in oil power plant is $45.

So you explained the paradox by saying that “Technological changes have increased the efficiency of oil-fired power plants.” i.e. price of infrastructure in oil power plant has also decreased.

Hence, option (C) is correct.

The other option which seems viable to many people is (A). But think about it, the actual price of the oil has nothing to do with ‘the threshold of economic viability for solar power’. This threshold is $35 so you need the oil to go up to $35. Whether the actual price of oil is $10 or $15 or $20, it doesn’t matter. It still needs to go up to $35 for solar viability. So option (A) is irrelevant.

We hope the paradox and its solution make sense.

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!

GMAT Tip of the Week: Jim Harbaugh Says Milk Does A GMAT Score Good

GMAT Tip of the WeekSomeday when he’s not coaching football, playing with the Oakland Athletics, visiting with the Supreme Court, or Tweeting back and forth with Lil Wayne and Nicki Minaj, Jim Harbaugh should sit down and take the GMAT.

Because if his interaction with a young, milk-drinking fan is any indication, Harbaugh understands one of the key secrets to success on the GMAT Quant section:

 

Harbaugh, who told HBO Real Sports this summer about his childhood plan to grow to over 6 feet tall – great height for a quarterback – by drinking as much milk as humanly possible – is a fan of all kinds of milk: chocolate, 2%… But as he tells the young man, the ideal situation for growing into a Michigan quarterback is drinking whole milk, just as the ideal way to attend the Ross School of Business a few blocks from Harbaugh’s State Street office is to use whole numbers on the GMAT.

The main reason? You can’t use a calculator on the GMAT, so while your Excel-and-calculator-trained mind wants to say calculate “75% of 64” as 0.75 * 64, the key is to think in terms of whole numbers whenever possible. In this case, that means calling “75%” 3/4, because that allows you to do all of your calculations with the whole numbers 3 and 4, and not have to set up decimal math with 0.75. Since 64/4 is cleanly 16 – a whole number – you can calculate 75% by dividing by 4 first, then multiplying by 3: 64/4 is 16, then 16 * 3 = 48, and you have your answer without ever having to deal with messier decimal calculations.

This concept manifests itself in all kinds of problems for which your mind would typically want to think in terms of decimal math. For example:

With percentages, 25% and 75% can be seen as 1/4 and 3/4, respectively. Want to take 20%? Just divide by 5, because 0.2 = 1/5.

If you’re told that the result of a division operation is X.4, keep in mind that the decimal .4 can be expressed as 2/5, meaning that the divisor has to be a multiple of 5 and the remainder has to be even.

If at some point in a calculation it looks like you need to divide, say, 10 by 4 or 15 by 8 or any other type of operation that would result in a decimal, wait! Leaving division problems as improper fractions just means that you’re keeping two whole numbers handy, and knowing the GMAT at some point you’ll end up having to multiply or divide by a number that lets you avoid the decimal math altogether.

So learn from Jim Harbaugh and his obsession with whole milk. Whole milk may be the reasons that his football dreams came true; whole numbers could be a major reason that your business school dreams come true, too.

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

By Brian Galvin

A Secret Shortcut to Increase Your GMAT Score

Ron Point_GMAT TipsThe GMAT is an exam that aims to test how you think about things. Many people have heard this mantra when studying for the GMAT, but it’s not always clear what it means. While there are many formulae and concepts to know ahead of taking the exam, you will be constantly thinking throughout the exam about how to solve the question in front of you. The GMAT specializes in asking questions that require you to think about the solution, not just to plug in numbers mindlessly and return whatever your calculator tells you (including typos and misplaced decimals).

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
B) 1216
C) 1297
D) 1333
E) 1512

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.

Dr. Larry Rudner Endorses Veritas Prep’s GMAT Practice Tests

GMACThree years ago this month, the team here at Veritas Prep launched a new project to completely reinvent how we build and administer GMAT practice tests for our students. A home-built system that started with the GMAT Question Bank (launched in October, 2012) soon grew into a whole computer-adaptive testing system containing thousands of questions and employing Item Response Theory to produce some of the most authentic practice tests in the industry. We launched our new practice test in May, 2013, and five months later we made five tests available to everyone. We later added two more tests, bringing the total number to seven that anyone could get. (Veritas Prep students get five additional computer-adaptive tests, for a total of 12.)

The whole time, we kept evaluating the current bank questions (aka “items” in testing parlance), adding new ones, and measuring the ability levels of tens thousands of GMAT students. To date, we have gathered more than 12 million responses from students, and put all of that data to work to keep making our tests better and better. And we keep doing this every week.

Earlier this year, we embarked on a new chapter in the development of our computer-adaptive testing system: We began working with Dr. Larry Rudner, the former Chief Psychometrician at the Graduate Management Admission Council (GMAC), and the definitive authority on the GMAT examination. Dr. Rudner took a look at every aspect of our system, from how we manage our items, to how good each item is at helping our system measure ability levels, to how we employ Item Response Theory to produce an accurate ability level for each test taker. In the end, not only did Dr. Rudner provide us with a roadmap for how to make our tests even better, but he also gave us a great deal of praise for the system that we have now.

What exactly did he say about our GMAT practice tests? See for yourself:

After months spent evaluating every aspect of their GMAT practice exams, it’s clear that Veritas Prep has mastered the science of test simulation. They offer thousands of realistic questions that have been validated using Item Response Theory and a powerful computer adaptive testing algorithm that closely matches that of the real GMAT® exam. Simply stated, Veritas Prep gives students a remarkably accurate measure of how they will perform on the Official GMAT.”

– Lawrence M. Rudner, PhD, MBA. Former Chief Psychometrician at GMAC and the definitive authority on the GMAT exam

Our work on our practice tests will never stop — after all, every month we add new items to our GMAT Question Bank, and many of these questions eventually make it into our computer-adaptive tests — but Dr. Rudner’s endorsement is particularly satisfying given the thousands of hours that have gone into building a testing system as robust as ours. When you take this or any practice test (even the official ones from GMAC), keep in mind that it never can perfectly predict how you will perform on test day. But, with Veritas Prep’s own practice tests, you have the confidence of knowing that more than three years of hard work and over 12 million responses from other students have gone into giving you as authentic a practice experience as possible.

We plan on putting this system to use in even more places, and helping even more students prepare for a wide variety of exams… That’s how powerful Item Response Theory is. Stay tuned!

Finally, we love talking and writing about this stuff. If you’re relatively new to studying for the GMAT or understanding how these tests work, check out some of our previous articles on computer-adaptive testing:

By Scott Shrum

3 Important Concepts for Statistics Questions on the GMAT

Quarter Wit, Quarter WisdomWe have discussed these three concepts of statistics in detail:

– Arithmetic mean is the number that can represent/replace all the numbers of the sequence. It lies somewhere in between the smallest and the largest values.

– Median is the middle number (in case the total number of numbers is odd) or the average of two middle numbers (in case the total number of numbers is even).

– Standard deviation is a measure of the dispersion of the values around the mean.

A conceptual question is how these three measures change when all the numbers of the set are varied is a similar fashion.

For example, how does the mean of a set change when all the numbers are increased by say, 10? How does the median change? And what about the standard deviation? What happens when you multiply each element of a set by the same number?

Let’s discuss all these cases in detail but before we start, we would like to point out that the discussion will be conceptual. We will not get into formulas though you can arrive at the answer by manipulating the respective formulas.

When you talk about mean or median or standard deviation of a list of numbers, imagine the numbers lying on the number line. They would be spread on the number line in a certain way. For example,

——0—a———b—c———————d———e————————f—g———————

Case I:

When you add the same positive number (say x) to all the elements, the entire bunch of numbers moves ahead together on the number line. The new numbers a’, b’, c’, d’, e’, f’ and g’ would look like this

——0——————a’———b’—c’———————d’———e’————————f’—g’——————

The relative placement of the numbers does not change. They are still at the same distance from each other. Note that the numbers have moved further to the right of 0 now to show that they have moved ahead on the number line.

The mean lies somewhere in the middle of the bunch and will move forward by the added number. Say, if the mean was d, the new mean will be d’ = d + x.

So when you add the same number to each element of a list, 

New mean = Old mean + Added number.

On similar lines, the median is the middle number (d in this case) and will move ahead by the added number. The new median will be d’ = d + x

So when you add the same number to each element of a list, 

New median = Old median + Added number

Standard deviation is a measure of dispersion of the numbers around the mean and this dispersion does not change when the whole bunch moves ahead as it is. Standard deviation does not depend on where the numbers lie on the number line. It depends on how far the numbers are from the mean. So standard deviation of 3, 5, 7 and 9 is the same as the standard deviation of 13, 15, 17 and 19. The relative placement of the numbers in both the cases will be the same. Hence, if you add the same number to each element of a list, the standard deviation will stay the same.

Case II:

Let’s now move on to the discussion of multiplying each element by the same positive number.

The original placing of the numbers on the number line looked like this:

——0—a———b—c———————d———e————————f—g———————

The new placing of the numbers on the  number line will look something like this:

——0———a’——————b’———c’————————————d’—————————e—- etc

The numbers spread out. To understand this, take an example. Say, the initial numbers were 10, 20 and 30. If you multiply each number by 2, the new numbers are 20, 40 and 60. The difference between them has increased from 10 to 20.

If you multiply each number by x, the mean also gets multiplied by x. So, if d was the mean initially, d’ will be the new mean which is x*d.

New mean = Old mean * Multiplied number

Similarly, the median will also get multiplied by x.

New median = Old median * Multiplied number

What happens to standard deviation in this case? It changes! Since the numbers are now further apart from the mean, their dispersion increases and hence the standard deviation also increases. The new standard deviation will be x times the old standard deviation. You can also establish this using the standard deviation formula.

New standard deviation = Old standard deviation * Multiplied number

The same concept is applicable when you increase each number by the same percentage. It is akin to multiplying each element by the same number. Say, if you increase each number by 20%, you are, in effect, multiplying each number by 1.2. So our case II applies here.

Now, think about what happens when you subtract/divide each element by the same number.

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!

GMAT Tip of the Week: Behind the Scenes of Your GMAT Score

GMAT Tip of the WeekAmong the most frequent questions we receive here at Veritas Prep headquarters (sadly, “How much am I allowed to tip my instructor?” is not one of them!) is the genre of “On my most recent practice test, I got X right and Y wrong and only Z wrong in a row… Why was my score higher/lower than my other test with A right and B wrong and C wrong in a row?” inquiries from students desperately trying to understand the GMAT scoring algorithm. We’ve talked previously in this space about why simply counting rights and wrongs isn’t all that great a predictor of your score. And perhaps the best advice possible relates to our Sentence Correction advice here a few months ago: Accept that there are some things you can’t change and focus on making a difference where you can.

But we also support everyone’s desire to leave no stone unturned in pursuit of a high GMAT score and everyone’s intellectual curiosity with regard to computer-adaptive testing. So with the full disclosure that these items won’t help you game the system and that your best move is to turn that intellectual curiosity toward mastering GMAT concepts and strategies, here are four major reasons that your response pattern — did you miss more questions early in the test vs. late in the test; did you miss consecutive questions or more sporadic questions, etc. — won’t help you predict your score:

1) The all-important A-parameter.

Item Response Theory incorporates three metrics for each “item” (or “question” or “problem”): the B parameter is the closest measurement to pure “difficulty”. The C parameter is essentially a measure of likelihood that a correct answer can be guessed. And the A parameter tells the scoring system how much to weight that item. Yes, some problems “count” more than others do (and not because of position on the test).

Why is that? Think of your own life; if you were going to, say, buy a condo in your city, you’d probably ask several people for their opinion on things like the real estate market in that area, mortgage rates, the additional costs of home ownership, the potential for renting it if you were to move, etc. And you’d value each opinion differently. Your very risk-loving friend may not have the opinion to value highest on “Will I be able to sell this at a profit if I get transferred to a new city?” (his answer is “The market always goes up!”) whereas his opinion on the neighborhood itself might be very valuable (“Don’t underestimate how nice it will be to live within a block of the blue line train”). Well, GMAT questions are similar: Some are extremely predictive (e.g. 90% of those scoring over 700 get it right, and only 10% of those scoring 690 or worse do) and others are only somewhat predictive (60% of those 700+ get this right, but only 45% of those below 700 do; here getting it right whispers “above 700” whereas before it screams it).

So while you may want to look at your practice test and try to determine where it’s better to position your “misses,” you’ll never know the A-values of any of the questions, so you just can’t tell which problems impacted your score the most.

2) Content balancing.

OK, you might then say, the test should theoretically always be trying to serve the highest value questions, so shouldn’t the larger A-parameters come out first? Not necessarily. The GMAT values balanced content to a very high degree: It’s not fair if you see a dozen geometry problems and your friend only sees two, or if you see the less time-consuming Data Sufficiency questions early in the test while someone else budgets their early time on problem solving and gets a break when the last ten are all shorter problems. So the test forces certain content to be delivered at certain times, regardless of whether the A-parameter for those problems is high or low. By the end of the test you’ll have seen various content areas and A-parameters… You just won’t know where the highest value questions took place.

3) Experimental items.

In order to know what those A, B, and C parameters are, the GMAT has to test its questions on a variety of users. So on each section, several problems just won’t count — they’re only there for research. And this can be true of practice tests, too (the Veritas Prep tests, for example, do contain experimental questions). So although your analysis of your response pattern may say that you missed three in a row on this test and gotten eight right in a row on the other, in reality those streaks could be a lot shorter if one or more of those questions didn’t count. And, again, you just won’t know whether a problem counted or not, so you can’t fully read into your response pattern to determine how the test should have been scored.

4) Item delivery vs. Score calculation.

One common prediction people make about GMAT scoring is that missing multiple problems in a row hurts your score substantially more than missing problems scattered throughout the test. The thinking goes that after one question wrong the system has to reconsider how smart it thought you were; then after two it knows for sure that you’re not as smart as advertised; and by the third it’s in just asking “How bad is he?” In reality, however, as you’ve read above, the “get it right –> harder question; get it wrong –> easier question” delivery system is a bit more nuanced and inclusive of experimentals and content balancing than people think. So it doesn’t work quite like the conventional wisdom suggests.

What’s more, even when the test delivers you an easier question and then an even easier question, it’s not directly calculating your score question by question. It’s estimating your score question-by-question in order to serve you the most meaningful questions it can, but it calculates your score by running its algorithm across all questions you’ve seen. So while missing three questions in a row might lower the current estimate of your ability and mean that you’ll get served a slightly easier question next, you can also recover over the next handful of questions. And then when the system runs your score factoring in the A, B, and C parameters of all of your responses to “live” (not experimental) questions, it doesn’t factor in the order in which those questions were presented — it only cares about the statistics. So while it’s certainly a good idea to get off to a good start in the first handful of problems and to avoid streaks of several consecutive misses, the rationale for that is more that avoiding early or prolonged droughts just raises your degree of difficulty. If you get 5 in a row wrong, you need to get several in a row right to even that out, and you can’t afford the kinds of mental errors that tend to be common and natural on a high-stakes exam. If you do manage to get the next several right, however, you can certainly overcome that dry spell.

In summary, it’s only natural to look at your practice tests and try to determine how the score was calculated and how you can use that system to your advantage. In reality, however, there are several unseen factors that affect your score that you just won’t ever see or know, so the best use of that curiosity and energy is learning from your mistakes so that the computer — however it’s programmed — has no choice but to give you the score that you want.

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

By Brian Galvin

How to Evaluate the Entire Sentence on Sentence Correction GMAT Questions

Ron Point_GMAT TipsAs the Donald Trump sideshow continues to dominate American news, politics is again being pushed to the forefront as the country gears up for an election in 15 months. The nominees are not yet confirmed, but many candidates are jockeying for position, trying to get their names to resonate with the American population. This election will necessarily have a new candidate for both parties, as Barack Obama will have completed the maximum of two elected terms allowed by the Constitution (via the 22nd amendment).

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.
B) The publication in 2006 of the Audacity of Hope was an instant hit: in two months it sold two hundred thousand copies and helped establish Barack Obama as a viable candidate for president.
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.
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.
E) 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.

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.

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.

1 Strategy That Will Lead You to Better Pacing on the GMAT

Keep the PaceLet’s look at a vastly important testing issue that is largely misunderstood and its seriousness under-appreciated.  Throughout multiple years of tutoring, this has been one of the most common and detrimental problems that I have had to work to correct in my students.  It pertains to the entire GMAT exam, but is typically more relevant to the quant section as students often struggle more with pacing during quant.

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!

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!

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.

What to Do When Math Fails You on the GMAT!

Quarter Wit, Quarter WisdomThey say Mathematics is a perfect Science. There is a debate over this among scientists but we can definitely say that Mathematical methods are not perfect so we cannot use them blindly. We could very well use the standard method for some given numbers and get stranded with “no solution.” The issue is what do we do when that happens?

For example, review this post on averages.

Here we saw that:

Average Speed = 2ab/(a + b)

Applicable when one travels at speed a for half the distance and speed b for other half of the distance. In this case, average speed is the harmonic mean of the two speeds.

So now, say if we have a question which looks like this:

Question: In the morning, Chris drives from Toronto to Oakville and in the evening he drives back from Oakville to Toronto on the same road. Was his average speed for the entire round trip less than 100 miles per hour?

Statement 1: In the morning, Chris drove at an average speed of at least 10 miles per hour while travelling from Toronto to Oakville.

Statement 2: In the evening, Chris drove at an average speed which was no more than 50 miles per hour while travelling from Oakville to Toronto.

Solution: We know that the question involves average speed. The case involves travelling at a particular average speed for one half of the journey and at another average speed for the other half of the journey.

So average speed of the entire trip will be given by 2ab/(a+b)

But the first problem is that we are given a range of speeds. How do we handle ‘at least 10’ and ‘no more than 50’ in equation form? We have learnt that we should focus on the extremities so let’s analyse the problem by taking the numbers are the extremities:10 and 50

Statement 1: In the morning, Chris drove at an average speed of at least 10 miles per hour while travelling from Toronto to Oakville.

What if Chris drives at an average speed of 10 mph in the morning and averages 100 mph for the entire journey? What will be his average speed in the evening? Perhaps around 200, right? Let’s see.

100 = 2*10*b/(10 + b)

1000 + 100b = 20b

1000 = -80b

b = – 1000/80

How can speed be negative?

Let’s hold on here and try the same calculation for statement 2 too.

Statement 2: In the evening, Chris drove at an average speed which no more than 50 miles per hour while travelling from Oakville to Toronto.

If Chris drives at an average speed of 50 mph in the evening, and averages 100 mph, let’s find his average speed in the morning.

100 = 2a*50/(a + 50)

100a + 5000 = 100a

5000 = 0

This doesn’t make any sense either!

What is going wrong? Look at it conceptually:

Say, Toronto is 100 miles away from Oakville. If Chris wants his average speed to be 100 mph over the entire trip, he should cover 100+100 = 200 miles in 2 hrs.

What happens when he travels at 10 mph in the morning? He takes 100/10 = 10 hrs to reach Oakville in the morning. He has already taken more time than what he had allotted for the entire round trip. Now, no matter what his speed in the evening, his average speed cannot be 100mph. Even if he reaches Oakville to Toronto in the blink of an eye, he would have taken 10 hours and then some time to cover the total 200 miles distance. So his average speed cannot be equal to or more than 200/10 = 20 mph.

Similarly, if he travels at 50 mph in the evening, he takes 2 full hours to travel 100 miles (one side distance). In the morning, he would have taken some time to travel 100 miles from Toronto to Oakville. Even if that time is just a few seconds, his average speed cannot be 100 mph under any circumstances.

But statement 1 says that his speed in morning was at least 10 mph which means that he could have traveled at 10 mph in the morning or at 100 mph. In one case, his average speed for the round trip cannot be 100 mph and in the other case, it can very well be. Hence statement 1 alone is not sufficient.

On the other hand, statement 2 says that his speed in the evening was 50 mph or less. This means he would have taken AT LEAST 2 hours in the morning. So his average speed for the round trip cannot be 100 mph under any circumstances. So statement 2 alone is sufficient to answer this question with ‘No’.

Answer (B)

Takeaway: If your average speed is s for a certain trip, your average speed for half the distance must be more than s/2.

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!

GMAT Tip of the Week: Kanye West’s Everything I Am Teaches Critical Reasoning

GMAT Tip of the Week“Everything I’m not made me everything I am,” says Kanye West in his surprisingly-humble track Everything I Am. And while, unsurprisingly, much of what he’s talking about is silencing his critics, he might as well be rapping about making you an elite critic on Critical Reasoning problems. Because when it comes to some of the most challenging Critical Reasoning problems on the GMAT, everything they’re not makes them everything they are. Which is a convoluted way of saying this:

On challenging Strengthen and Assumption questions, the correct answer often tells you that a potential flaw with the argument is not true.

Everything that’s not true in that answer choice, then, makes the conclusion substantially more valid.

Consider this argument, for example:

Kanye received the most votes for the “Best Hip Hop Artist” award at the upcoming MTV Video Music Awards, so Kanye will be awarded the trophy for Best Hip Hop Artist.

If this were the prompt for a question that asked “Which of the following is an assumption required by the argument above?” a correct answer might read:

A) The Video Music Award for “Best Hip Hop Artist” is not decided by a method other than voting.

And the function of that answer choice is to tell you what’s not true (“everything I’m not”), removing a flaw that allows the conclusion to be much more logically sound (“…made me everything I am.”) These answer choices can be challenging in context, largely because:

1) Answer choices that remove a flaw can be difficult to anticipate, because those flaws are usually subtle.

2) Answer choices that remove a flaw tend to include a good amount of negation, making them a bit more convoluted.

In order to counteract these difficulties, it can be helpful to use “Everything I’m not made me everything I am” to your advantage. If what’s NOT true is essential to the conclusion’s truth, then if you consider the opposite – what if it WERE true – you can turn that question into a Weaken question. For example, if you took the opposite of the choice above, it would read:

The VMA for “Best Hip Hop Artist” is decided by a method other than voting.

If that were true, the conclusion is then wholly unsupported. So what if Kanye got the most votes, if votes aren’t how the award is determined? At that point the argument has no leg to stand on, so since the opposite of the answer directly weakens the argument, then you know that the answer itself strengthens it. And since we’re typically all much more effective as critics than we are as defenders, taking the opposite helps you to do what you’re best at. So consider the full-length problem:

Editor of an automobile magazine: The materials used to make older model cars (those built before 1980) are clearly superior to those used to make late model cars (those built since 1980). For instance, all the 1960’s and 1970’s cars that I routinely inspect are in surprisingly good condition: they run well, all components work perfectly, and they have very little rust, even though many are over 50 years old. However, almost all of the late model cars I inspect that are over 10 years old run poorly, have lots of rust, and are barely fit to be on the road.

Which of the following is an assumption required by the argument above?

A) The quality of materials used in older model cars is not superior to those used to make other types of vehicles produced in the same time period.

B) Cars built before 1980 are not used for shorter trips than cars built since then.

C) Manufacturing techniques used in modern automobile plants are not superior to those used in plants before 1980.

D) Well-maintained and seldom-used older model vehicles are not the only ones still on the road.

E) Owners of older model vehicles take particularly good care of those vehicles.

First notice that several of the answer choices (A, B, C, and D) include “is not” or “are not” and that the question stem asks for an assumption. These are clues that you’re dealing with a “removes the flaw” kind of problem, in which what is not true (in the answer choices) is essential to making the conclusion of the argument true. Because of that, it’s a good idea to take the opposites of those answer choices so that instead of removing the flaw in a Strengthen/Assumption question, you’re introducing the flaw and making it a Weaken. When you do that, you should see that choice D becomes:

D) Well-maintained and seldom-used older model vehicles ARE the only ones still on the road.

If that’s the case, the conclusion – “the materials used to make older cars are clearly superior to those used in newer ones” – is proven to be flawed. All the junkers are now off the road, so the evidence no longer holds up; you’re only seeing well-working old cars because they’re the most cared-for, not because they were better made in the first place.

And in a larger context, look at what D does ‘reading forward’: if it’s not only well-maintained and seldom-driven older cars on the road, then you have a better comparison point. So what’s not true here makes the argument everything it is. But dealing in “what’s not true” can be a challenge, so remember that you can take the opposite of each answer choice and make this “Everything I’m Not” assumption question into a much-clearer “Everything I Am” Weaken question.

Are you studying for the GMAT? We have free online GMAT seminars running all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!

By Brian Galvin

Find Time-Saving Strategies for GMAT Test Day

Ron Point_GMAT TipsI’ve often heard from people studying for the GMAT that they would score much higher on the test if there were no time limit to each section. The material covered on the exam is not inherently complicated, but the combination of subtle wordplay and constant stress about time management creates an environment where test takers often rush through prompts and misinterpret questions. Unfortunately, time management and stress management are two of the major skills being tested on the GMAT, so the time limit isn’t going away any time soon (despite my frequent letters to the GMAC). Instead, it’s worth mastering simple techniques to save time and extrapolate patterns based on smaller samples.

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
B) 7/12
C) 1/2
D) 5/12
E) 1/3

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.

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.

Expression vs Equation on GMAT

Quarter Wit, Quarter WisdomToday, we want to take up a conceptual discussion on expressions and equations and the differences between them. The concept is quite simple but a discussion on these is warranted because of the similarity between the two.

An expression contains numbers, variables and operators.

For example

x + 4

2x – 4x^2

5x^2 + 4x -18

and so on…

These are all expressions. We CANNOT equate these expressions to 0 by default. We cannot solve for x in these cases. As the value of x changes, the value of the expression changes.

For example, given x + 4, if x is 1, value of the expression is 5. If x is 2, value of the expression is 6. If value of the expression is given to be 10, x is 6 and so on.

We cannot say, “Solve x + 4.”

If we set an algebraic expression equal to something, with an “=“ sign, we have an equation.

So here are some ways of converting the above expressions into equations:

I. x + 4 = -3

II. 2x – 4x^2 = 0

III. 5x^2 + 4x -18 = 3x

Now the equation can be solved. Note that the right hand side of the equation needn’t always be 0. It might be something other than 0 and you might need to make it 0 by bringing whatever is on the right hand side to the left hand side or by segregating the variable if possible:

I. x + 4 = -3

x + 7 = 0

x = -7

II. 2x – 4x^2 = 0

2x(1 – 2x) = 0

x = 0 or 1/2

III. 5x^2 + 4x -18 – 3x = 0

5x^2 + x – 18 = 0

5x^2 + 10x – 9x – 18 = 0

5x(x + 2) -9(x + 2) = 0

(x + 2)(5x – 9) = 0

x = -2, 9/5

In each of these cases, we get only a few values for x because we were given equations.

Think about what you mean by “solving an equation”. Let’s take a particular type of equation – a quadratic.

This is how you usually depict a quadratic:

f(x) = ax^2 + bx + c

or

y = ax^2 + bx + c

This is a parabola – upward facing if a is positive and downward facing if a is negative.

When we solve ax^2 + bx + c = 0 for x, it means, when y = 0, what is the value of x? So you are looking for x intercepts.

When we solve ax^2 + bx + c = d for x, it means, when y = d, what is the value of x? Depending on the values of a, b, c and d, you may or may not get values for x.

Let’s take an example:

x^2 – 2x – 3 = 0

(x + 1)(x – 3) = 0

x = -1 or 3

This is what it looks like:

images

When y is 0, x can take two values: -1 and 3.

So what do we do when we have x^2 – 2x -3 = -3?

We solve it in the same way:

x^2 – 2x -3 + 3 = 0

x(x – 2) = 0

x = 0 or 2

So when y is -3, x is 0 or 2. It has 2 values for y = -3 as is apparent from the graph too.

Similarly, you can solve for it when y = 5 and get two values for x.

What happens when you put y = -5? x will have no value for y = -5 so the equation x^2 – 2x – 3 = – 5 has no real solutions (so ‘no solutions’ as far as we are concerned).

We hope you understand the difference between an expression and an equation now and also that you cannot equate any given expression to 0 and solve it.

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!

GMAT Tip of the Week: Shake & Bake Your Way To Function Success

GMAT Tip of the WeekFor many of us, cooking a delicious homemade meal and solving a challenge-level GMAT math problem are equally daunting challenges. So many steps, so many places to make a mistake…why can’t there be an easier way? Well, the fine folks at Kraft foods solved your first problem years ago with a product called “Shake and Bake.” You take a piece of chicken (your input), stick in the bag of seasoning, shake it up, bake it, and voila – you have yourself a delicious meal with minimal effort. So gourmet level cooking is now nothing to fear…but what about those challenging GMAT quant problems?

You’re in luck. Function problems on the GMAT are essentially Shake and Bake recipes. Consider the example:

f(x) = x^2 – 80

If that gets your heart rate and stress level up, you’re not alone. Function notation just looks challenging. But it’s essentially Shake and Bake if you dissect what a function looks like.

The f(x) portion tells you about your input. f(x) = ________ means that, for the rest of that problem, whatever you see in parentheses is your “input” (just like the chicken in your Shake and Bake).

What comes after the equals sign is the recipe. It tells you what to do to your input to get the result. Here f(x) = x^2 – 80 is telling you that whatever your input, you square it and then subtract 80 and that’s your output. …And that’s it.

So if they ask you for:

What is f(9)? (your input is 9), then f(9) = 9^2 – 80, so you square 9 to get 81, then you subtract 80 and you have your answer: 1.

what is f(-4)? (your input is -4), then f(4) = 4^2 – 80, which is 16 – 80 = -64.

What is f(y^2)? (your input is y^2), then f(y^2) = (y^2)^2 – 80, which is y^4 – 80.

What is f(Rick Astley), then your input is Rick Astley and f(Rick Astley) = (Rick Astley)^2 – 80.

It really doesn’t matter what your input is. Whatever the test puts in the parentheses, you just use that as your input and do whatever the recipe says to do with it. So for example:

f(x) = x^2 – x. For which of the following values of a is f(a) > f(8)?

I. a = -8
II. a = -9
III a = 9

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

While this may look fairly abstract, just consider the inputs they’ve given you. For f(8), just put 8 wherever that x goes in the “recipe” f(x) = x^2 – x:

f(8) = 8^2 – 8 = 64 – 8 = 56

And then do the same for the three other possible values:

I. f(-8) means put -8 wherever you see the x: (-8)^2 – (-8) = 64 + 8 = 72, so f(-8) > f(8).
II. f(-9) means put -9 wherever you see the x: (-9)^2 – (-9) = 81 + 9 = 90, so f(-9) > f(8)
III. f(9) means put a 9 wherever you see the x: 9^2 – 9 = 81 – 9 = 72, so f(9) > f(8), and the answer is E.

Ultimately with functions, the notation (like the Shake and Bake ingredients) is messy, but with practice the recipes become easy to follow. What goes in the parentheses is your input, and what comes after the equals sign is your recipe. Follow the steps, and you’ll end up with a delicious GMAT quant score.

Are you studying for the GMAT? We have free online GMAT seminars running all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!

By Brian Galvin

When You’ll Need to Bring Outside Knowledge to the GMAT

Ron Point_GMAT TipsIt is often said that outside knowledge is not required on the GMAT. The idea is that everyone should be on relatively equal footing when starting to prepare for this exam, minimizing the advantage that someone with a B.Comm might have over someone with an engineering or philosophy degree. Of course, it’s difficult to determine at what point does outside knowledge begin and end. Knowing that there are 26 letters in the (English) alphabet or that blue and red are different colors is never explicitly mentioned in the GMAT preparation, but the concepts certainly can come up in GMAT questions.

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 28th. On what day of the week was he born?

1) In the year of Tom’s birth, January 20th was a Sunday.
2) In the year of Tom’s birth, July 17th was a Wednesday.

A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
D) Each statement alone is sufficient to answer the question.
E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

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 27th!).

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 29th, and are identified as leap years. The day of January 20th gives us a fixed point in that year, but since it is before February 28th, we don’t know if March 1st will be 40 days or 41 days away from January 20th.   Since this is the case, October 28th 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 17th was a Wednesday, then October 28th 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 20th to be a Sunday, then you could get a year like 2013 in which the 28th of October is a Monday. 2013 is not a leap year, so July 17th is also a Wednesday and either statement would lead to the same answer. However, if you consider January 20th to be a Sunday, you could also get a year like 2008, which was a leap year, and then October 28th was a Tuesday. July 17th 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!).

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.

When Not to Use Number Plugging on the GMAT

Quarter Wit, Quarter WisdomA few weeks back we discussed the kind of questions which beg you to think of the process of elimination – a strategy probably next only to number plugging in popularity.

Today we discuss the kind of questions which beg you to stay away from number plugging (but somehow, people still insist on using it because they see variables).

Not every question with variables is suitable for number plugging. If there are too many variables, it can be confusing and error prone. Then there are some other cases where number plugging is not suitable. Today we discuss an official question where you face two of these problems.

Question: If m, p , s and v are positive, and m/p < s/v, which of the following must be between m/p and s/v?

I. (m+s)/(p+v)

II. ms/pv

III. s/v – m/p

(A) None

(B) I only

(C) II only

(D) III only

(E) I and II both

Solution: The moment people see m, p, s and v variables, they jump to m = 1, p = 2 etc.

But two things should put you off number plugging here:

– There are four variables – just too many to plug in and manage.

– The question is a “must be true” question. Plugging in numbers is not the best strategy for ‘must be true’ questions. If you know that say, statement 1 holds for some particular values of m, p, s and v (say, 1, 2, 3 and 4), that’s fine but how do you know that it will be true for every set of valid values of m, p, s and v? You cannot try every set because the variables can take an infinite variety of values. If you find a set of values for the variables such that statement 1 does not hold, then you know for sure that it may not be true. In this case, number plugging does have some use but it may be a while before you can arrive at values which do not satisfy the conditions. In such questions, it is far better to take the conceptual approach.

We can solve this question using some number line and averaging concepts.

We are given that m/p < s/v

This means, this is how they look on the number line:

…………. 0 ……………….. m/p …………………… s/v ……………..

(since m, p, s and v are all positive (not necessarily integers though) so m/p and s/v are to the right of 0)

Let’s look at statement II and III first since they look relatively easy.

II. ms/pv

Think of the case when m/p and s/v are both less than 1. When you multiply them, they will become even smaller. Say .2*.3 = .06. So the product ms/pv may not lie between m/p and s/v.

Tip: When working with number properties, you should imagine the number line split into four parts:

  • less than -1
  • between -1 and 0
  • between 0 and 1
  • greater than 1

Numbers lying in these different parts behave differently. You should have a good idea about how they behave.

III. s/v – m/p

Think of a case such as this:

…………. 0 ………………………… m/p … s/v ……….

s/v – m/p will be much smaller than both m/p and s/v and will lie somewhere “here”:

…………. 0 ……… here ………………… m/p … s/v ……….

So the difference between them needn’t actually lie between them on the number line.

Hence s/v – m/p may not be between m/p and s/v.

I. (m+s)/(p+v)

This is a little tricky. Think of the four numbers as N1, N2, D1, D2 for ease and given fractions as N1/D1 and N2/D2.

(m+s)/(p+v)

= [(m+s)/2]/[(p+v)/2]

= (Average of N1 and N2)/(Average of D1 and D2)

Now average of the numerators will lie between N1 and N2 and average of the denominators will lie between D1 and D2. So (Average of N1 and N2)/(Average of D1 and D2) will lie between N1/D1 and N2/D2. Try to think this through.

We will try to explain this but you must take some examples to ensure that you understand it fully. When is one fraction smaller than another fraction?

When N1/D1 < N2/D2, one of these five cases will hold:

  • N1 < N2 and D1 = D2 . For example: 2/9 and 4/9

Average of numerators/Average of denominators = 3/9 (between N1/D1 and N2/D2)

  • N1 < N2 and D1 > D2. For example: 2/11 and 4/9

Average of numerators/Average of denominators = 3/10 (between N1/D1 and N2/D2)

  • N1 << N2 and D1 < D2. For example: 2/9 and 20/19 i.e. N1 is much smaller than N2 as compared with D1 to D2.

Average of numerators/Average of denominators = 11/14 (between N1/D1 and N2/D2)

  • N1 = N2 but D1 > D2. For example: 2/9 and 2/7

Average of numerators/Average of denominators = 2/8 (between N1/D1 and N2/D2)

  • N1 > N2 but D1 >> D2. For example: 4/9 and 2/1

Average of numerators/Average of denominators = 3/5 (between N1/D1 and N2/D2)

In each of these cases, (average of N1 and N2)/(average of D1 and D2) will be greater than N1/D1 but smaller than N2/D2. Take some more numbers to understand why this makes sense. Note that you are not expected to conduct this analysis during the test. The following should be your takeaway from this question:

Takeaway: (Average of N1 and N2)/(Average of D1 and D2) will lie somewhere in between N1/D1 and N2/D2 (provided N1. N2, D1 and D2 are positive)

(m+s)/(p+v) must lie between m/p and s/v.

Answer (B)

Are you studying for the GMAT? We have free online GMAT seminars running all the time. And, be sure to find us on Facebook and Google+, and follow us on 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!

GMAT Tip of the Week: Oh Thank Heaven For Seven-Eleven

GMAT Tip of the WeekA week after the Fourth of July, a lesser-known but certainly-important holiday occurs each year. Tomorrow, friends, is 7-11, a day to enjoy free Slurpees at 7-11 stores, to roll some dice at the craps table, and to honor your favorite prime numbers. So to celebrate 7-11, let’s talk about these two important prime numbers.

Checking Whether A Number Is Prime

Consider a number like 133. Is that number prime? The first three prime numbers (2, 3, and 5) are easy to check to see whether they are factors (if any are, then the number is clearly not prime):

2 – this number is not even, so it’s not divisible by 2.
3 – the sum of the digits (an important rule for divisibility by three!) is 7, which is not a multiple of 3 so this number is not divisible by 3.
5 – the number doesn’t end in 5 or 0, so it’s not divisible by 5

But now things get a bit trickier. There are, in fact, (separate) divisibility “tricks” for 7 and 11. But they’re relatively inefficient compared with a universal strategy. Find a nearby multiple of the target number, then add or subtract multiples of that target number. If we want to test 133 to see whether it’s divisible by 7, we can quickly go to 140 (you know this is a multiple of 7) and then subtract by 7. That’s 133, so you know that 133 is a multiple of 7. (Doing the same for 11, you know that 121 is a multiple of 11 – it’s 11-squared – so add 11 more and you’re at 132, so 133 is not a multiple of 11)

This is even more helpful when, for example, a question asks something like “how many prime numbers exist between 202 and 218?”. By finding nearby multiples of 7 and 11:

210 is a multiple of 7, so 203 and 217 are also multiples of 7
220 is a multiple of 11, so 209 is a multiple of 11

You can very quickly eliminate numbers in that range that are not prime. And since none of the even numbers are prime and neither are 205 and 215 (the ends-in-5 rule), you’re left only having to check 207 (which has digits that sum to a multiple of 3, so that’s not prime), 211 (more on that in a second), and 213 (which has digits that sum to 6, so it’s out).

So that leaves the process of testing 211 to see if it has any other prime factors than 2, 3, 5, 7, and 11. Which may seem like a pretty tall order. But here’s an important concept to keep in mind: you only have to test prime factors up to the square root of the number in question. So for 211, that means that because you should know that 15 is the square root of 225, you only have to test primes up to 15.

Why is that? Remember that factors come in pairs. For 217, for example, you know it’s divisible by 7, but 7 has to have a pair to multiply it by to get to 217. That number is 31 (31 * 7 = 217). So whatever factor you find for a number, it has to multiply with another number to get there.

Well, consider again the number 211. Since 15 * 15 is already bigger than 211, you should see that for any number bigger than 15 to be a factor of 211, it has to pair with a number smaller than 15. And as you consider the primes up to 15, you’re already checking all those smaller possibilities. That allows you to quickly test 211 for divisibility by 13 and then you’re done. And since 211 is not divisible by 13 (you could do the long division or you could test 260 – a relatively clear multiple of 13 – and subtract 13s until you get to or past 211: 247, 234, 221, and 208, so 211 is not a multiple of 13. Therefore 211 is prime.

Are you studying for the GMAT? We have free online GMAT seminars running all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!

By Brian Galvin

Interpreting the Language of the GMAT

Ron Point_GMAT TipsEveryone who writes the GMAT must speak English to some degree. Since English is the default language of business, the GMAT is administered exclusively in that language. Some people feel that this is unfair. If you take an exam in your mother tongue, you tend to do better than if you took the exam in your second, third or even fourth language (I consider Klingon as my fourth language). However, even if you’re a native English speaker, the GMAT offers many linguistic challenges that make many people feel that they don’t actually speak the language. (¿Habla GMAT?)

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
B) 616
C) 672
D) 900
E) 1024

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!”

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.