Most of the time, you can eventually figure out what’s happening, but sometimes you missed an important point near the beginning and just can’t understand the situation. As frustrating as this situation may seem, imagine if, at the end of the conversation, everyone turned to you and asked you to give your detailed opinion on the debate!

On the GMAT, you will frequently be parachuted into a situation that is already in progress. This type of scenario discombobulates most people, because we’re used to a gradual progression starting from the beginning. Since you won’t be at the beginning, you will need to figure out the beginning and the end given what you know from your position in the middle. (In essence, you’re Malcolm). You may not immediately know how to solve the issue, but you can deduce the beginning by seeing where you are in the middle and attempting to reverse engineer the process.

In many ways, this is similar to the dichotomy between multiplication and division. They are, in effect, the exact same operation (multiplying by 2 is dividing by ½ and vice versa). However, people tend to find multiplication easier because you’re going forward. Going backwards is typically harder, in no small part because your brain is not used to going in that (one) direction. When you do something a hundred times a day, it becomes second nature. If you start something for the first time on the GMAT, it may seem almost impossible to solve.

Let’s look at an example of a problem that starts you off in the middle of the action:

*A term a _{n} is called a cusp of a sequence if a_{n} is an integer but _{an+1} is not an integer. If a_{5} is a cusp of the sequence a_{1}, a_{2},…,a_{n},… in which a_{1} = k and a_{n} = -2(a_{n-1 }/ 3) for all n >1, then k could be equal to:*

*3**16**108**162**243*

Sequences are excellent examples of this parachuting phenomenon because you typically need to have the previous entry in order to find the next element (like a scavenger hunt!). If you find a_{3}, you should be able to find a_{4}. But if you have a_{4}, it’s a lot harder to identify a_{3}. Since you tend to have the pattern, you have to start at the beginning to uncover the progression.

This particular sequence is made easier if you manipulate the algebra a little to get a more manageable form. Instead of the way the sequence is defined, change the pattern to a_{n} = -2/3 a_{n-1}. This small change highlights the fact that the new element is just the old element multiplied by -2/3. And since the question hinges on when the sequence changes from integers to non-integers, it’s really the denominator that will be of interest to us.

Since this is fairly abstract, let’s go through plugging in answer choice A to see what happens to the series. If k = 3, then the second element of the series would be -2/3 (3). This gives us just -2, and is still an integer. However, the next iteration, a_{3}, would call for -2/3 (-2), which is 4/3, and not an integer. Indeed, this sequence is just calling for us to continually divide by 3, and then determine when the result will no longer be an integer. Clearly, answer choice A won’t be the right choice, as we just found that a_{3} was not an integer, and thus a_{2} would be the “cusp” as defined in the question.

Now, using the brute force approach of plugging in each answer choice will eventually yield the correct answer, but it can be tedious and time-consuming. A more logical approach would involve determining that we need a number that has many 3’s in its prime factors. Every time we divide by 3, we will get another integer, provided that we still have 3’s in the numerator. Once we’re left with a number that is not a multiple of 3, the sequence will spit out a non-integer, and the previous number will be the cusp. Using the prime factorization of the four remaining answer choices, we get:

16 = 2^4

108 = 2 * 54 –) 2 * 2 * 27 –) 2^2 * 3^3

162 = 2 * 81 –) 2 * 3 * 27 –) 2 * 3^4

243 = 3 * 81 –) 3^5

So as we can see, one answer choice has three 3’s, the other has four and the final one has five (the seventh would be Furious). How many 3s do we actually need? Well if the fifth one must be the cusp, then we need to divide by 3 four separate times to get rid of all the 3s. After that, the fifth element will be an integer (also, an action movie), and the sixth element will be a non-integer. Since answer choice D is our educated guess, let’s double check our answer by executing the sequence on 162.

A_{1} = 162

A_{2} = -2/3 (162) = -108

A_{3} = -2/3 (-108) = 72

A_{4} = -2/3 (72) = -48

A_{5} = -2/3 (-48) = 32

A_{6} = -2/3 (32) = -64/3.

This is exactly what we wanted. We can see that each time we are multiplying the previous item by 2/3 and changing the sign. Once we get to 32, that is just 2^5 and dividing it by 3 will no longer yield an integer.

If you’d gone through the complete trial and error process, you’d quickly see that answer choices A and B are incorrect. Answer choice C, 108, comes pretty close, but cusps at A_{4}, not A_{5}. If you then pick answer choice D, 162, you find that you get to 108 on the second iteration, and you can skip the next four steps because you just did them. Finally, answer choice E is a tempting number to start testing with, because it is a perfect exponential of 3. However, you will get to an integer at A_{6}, and thus you need a number with fewer 3s in the numerator.

On test day, you might be able to recognize patterns or you might have to bite the bullet and try each answer choice one by one. However, if you recognize that you need to determine what happens at the beginning before moving on to the middle and the end, you’ll have more success. You always need to understand the pattern, and that starts at the beginning. If you keep this strategy in mind, you won’t find yourself stuck in the middle (with you).

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.*

For the purposes of this exercise it is useful to divide the complete application into the following streams of work:

- Tests: GMAT or GRE, and TOEFL (for most international applicants);
- School selection: school visits, desktop research, primary research or informational interviews with alums;
- Online applications: essays, resume and general information;
- Recommendations: selecting recommenders and preparing them to write amazing recommendations.

Roughly speaking, the above work streams are listed in the order they should be approached. There is certainly some overlap between the different streams, and you should build in some flexibility in your timeline. The best way to develop a timeline is to work backwards from admissions deadlines. Starting with the online application, it generally takes about 3 months to complete the essays, resume (which might have to tweaked for each school) and gather all information you need to complete the online application. For a September deadline, it means you should start brainstorming and drafting essays in early June. It is generally a good idea to complete the resume first as it serves to create a summary of who you are and what you have achieved.

Regarding your recommenders, you should prepare them to write those amazing letters of recommendations. Don’t just tell them which schools you applying to and send them the email with instructions. Instead, provide them with an updated resume, relevant examples of leadership and an overview of what themes you are trying to convey in your application. (Having started the essays and resume already you will be well prepared for this.) This should happen about 6-8 weeks before the first deadline, or by early July for a September deadline.

School selection starts with desktop research, includes class visits (international applicants should try to attend local information sessions), as well informational interview with alumni and current students. Be sure to check the visiting schedule well in advance, as most schools do not offer class visits around final examinations. Try to complete these by April (after that things you run out of options).

Standardized test results serve as an important indicator of academic abilities in the Admissions Committee’s eyes. If you are striving for admissions to a top tier b school, you should be aiming to get it around average for that school. This means you might have to retake it more than once. (Non-traditional applicants, including military, get a break typically.) Your GMAT and GRE also serve to inject some realistic expectations into your short list of schools. Hence, getting tests out of the way early is really ideal. For a typical applicant the tests should be completed about 4 months prior to the first deadline. For September deadlines, it means the official GMAT should be taken in April, allowing sufficient time to retake the test if necessary.

In summary, for a September deadline, here are some milestones you should try to hit in order put yourself in the best possible situation when applying to b school :

- GMAT: complete by April for a September deadline (4 months prior);
- School selection: finalize by June, after taking the GMAT, and extensive primary and secondary research (3-4 months before deadline);
- Recommendations: provide recommenders with all necessary information to write amazing letters of recommendations in early July (2 months before deadline);
- Online applications: begin essay drafts in early June (3 months before deadline), and iterate many, many times!

Want to craft a strong application? Call us at 1-800-925-7737 and speak with an MBA admissions expert today. Click here to take our Free MBA Admissions Profile Evaluation! As always, be sure to find us on Facebook and Google+, and follow us on Twitter!

*By Marcus D.* Learn more about him here, or find the expert who’s right for you here! Visit our Team page today.

Let’s apply this logic to an extremely challenging 700+ level Data Sufficiency question*:

We’re given the following:

*In the figure shown, point O is the center of the circle and points B, C, and D lie on the circle. If the segment AB is equal to the length of line segment OC, what is the degree measure of angle BAO?*

*The degree measure of angle COD is 60**The degree measure of angle BCO is 40*

That is a complicated-looking figure. Your instinct might be that you don’t have time to draw it, but these kinds of questions will be designed specifically to thwart our intuition if we attempt to do too much work in our heads. So the first thing to do is draw the figure on our scratch pad, and mark the relationships we’re given. We’re told that segment CO is equal to AB, so we’ll designate that relationship. We’ll also call angle BAO, which we’re asked about, ‘x.’ Now we have the following:

Fight the impulse to jump to the statements now. In a harder question like this, we’ll benefit from taking more time to derive additional relationships from the question stem. Psychologically, this is often a struggle for test-takers. You’re conscious of your time constraint. You want to work quickly. The trick is to trust that this pre-statement investment of time will allow you to evaluate the information provided in the statements more efficiently, ultimately *saving* time.

Now the name of the game is to try to label as much of this figure as we can without introducing a new variable. Notice that segments CO and BO are both radii of the circle, so we know those are equal. Our diagram now looks like this:

Next, look at triangle ABO. Notice that segments AB and BO are equal. If angles opposite equal angles are equal to each other, we can then designate angle AOB as ‘x’ because it must be equal to angle BAO, as those two angles are opposite sides that are of equal length. Moreover, if the three interior angles of a triangle will sum to 180, the remaining angle, ABO, can be designated 180-2x. This gives us the following.

No reason to stop here. Notice that angles ABO and CBO lie on a line. Angles that lie on a line must sum to 180. If angle ABO is 180-2x, then angle CBO must be 2x. Now we have this:

Analyzing triangle CBO, we see that sides BO and CO are equal, meaning that the angles opposite those sides must be equal. So now we can label angle BCO as ‘2x.’ If angles CBO and BOC sum to 4x, the remaining angle, BOC, must then be 180-4x, so that the interior angles of the triangle will sum to 180.

We’ve got enough at this point that we can very quickly evaluate our statements, However, there is one last interesting relationship. Notice that angle COD is an exterior angle of triangle CAO. An exterior angle, by definition, must be equal to the sum of the two remote interior angles. So, in this case, Angle COD is equal to the sum of angles BCO and BAO. Therefore COD = 2x + x = 3x, which I’ve circled in the figure. (Triangle CAO is outlined in blue in the figure below to more clearly demarcate the exterior angle.)

That’s a lot of work. Determining all of these relationships will likely take close to two minutes. But watch how quickly we can evaluate our statements if we’ve done all of this preemptive groundwork:

Statement 1: Angle COD = 60. We’ve designated angle COD as 3x, so 3x = 60. Clearly we can solve for x. Sufficient. Eliminate BCE.

Statement 2: Angle BCO = 40. We’ve designated angle BCO as 2x, so 2x = 40. Clearly we can solve for x. Sufficient. Answer is D.

Notice, all of the heavy lifting for this question came before we even so much as glanced at our statements.

**Takeaway**: For a challenging Data Sufficiency question in which you’re given a lot of information in the question stem, the best approach is to spend some time taming the complexity of the problem before examining the statements. When you work out these relationships, try to minimize the number of variables you use when doing so, as this will simplify your calculations once you’re ready to go to the statements. Most importantly, don’t do too much work in your head. There’s no need to rely on the limited bandwidth of your working memory if you have the option of putting everything into a concrete form on your scratch pad.

*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 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. *

We saw that simple and compound interest (compounded annually) in the first year is the same. In the second year, the only difference is that in compound interest, you earn interest on previous year’s interest too. Hence, the total two year interest in compound interest exceeds the two year interest in case of simple interest by an amount which is interest on year 1 interest.

So a question such as this one is very simple to solve:

Question 1: Bob invested one half of his savings in a bond that paid simple interest for 2 years and received $550 as interest. He invested the remaining in a bond that paid compound interest (compounded annually) for the same 2 years at the same rate of interest and received $605 as interest. What was the annual rate of interest?

(A) 5%

(B) 10%

(C) 12%

(D) 15%

(E) 20%

Solution:

Simple Interest for two years = $550

So simple interest per year = 550/2 = 275

But in case of compound interest, you earn an extra 605 – 550 = $55

This $55 is interest earned on year 1 interest i.e. if rate of interest is R, it is

55 = R% of 275

R = 20

Answer (E)

The question is – what happens in case you have 3 years here, instead of 2? How do you solve it then? Here is a small table of the difference between simple and compound interest to help you.

Say the Principal is P and the rate of interest if R

It gets a bit more complicated though not very hard to solve. All you need to do is solve a quadratic, which, if the values are well thought out, is fairly simple to solve. Let’s look at the same question adjusted for three years.

Question 2: Bob invested one half of his savings in a bond that paid simple interest for 3 years and received $825 as interest. He invested the remaining in a bond that paid compound interest (compounded annually) for the same 3 years at the same rate of interest and received $1001 as interest. What was the annual rate of interest?

(A) 5%

(B) 10%

(C) 12%

(D) 15%

(E) 20%

Simple Interest for three years = $825

So simple interest per year = 825/3 = $275

But in case of compound interest, you earn an extra $1001 – $825 = $176

What all is included in this extra $176? This is the extra interest earned by compounding.

This is **R% of interest of Year1 + R% of total interest accumulated in Year2**

This is **R% of 275** + **R% of (275 + 275 + R% of 275)** = 176

(R/100) *[825 + (R/100)*275] = 176

Assuming R/100 = x to make the equation easier,

275x^2 + 825x – 176 = 0

25x^2 + 75x – 16 = 0

25x^2 + 80x – 5x – 16 = 0

5x(5x + 16) – 1(5x + 16) = 0

x = 1/5 or -16/5

Ignore the negative value to get R/100 = 1/5 or R = 20

*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!*

*30 minutes is not a lot of time, many say, and because an effective essay needs to be well-organized and well-written it is therefore impossible to write a 30-minute essay.*

Let’s discuss the extent to which we disagree with that conclusion, in classic AWA style.

In the first line of a recent blog post, the author claimed that writing an effective AWA essay in 30 minutes was impossible. That argument certainly has at least some merit; after all, an effective essay needs to show the reader that it’s well-written and well-organized. But this argument is fundamentally flawed, most notably because the essay doesn’t need to “be” well-written as much as it needs to “appear” well-written. In the paragraphs that follow, I will demonstrate that the conclusion is flawed, and that it’s perfectly possible to write an effective AWA essay in 30 minutes or less.

Most conspicuously, the author leans on the 30-minute limit for writing the AWA essay, when in fact the 30 minutes only applies to the amount of time that the examinee spends actually typing at the test center. In fact, much of the writing can be accomplished well beforehand if the examinee chooses paragraph and sentence structures ahead of time. For this paragraph, as an example, the transition “most conspicuously” and the decision to refute that claim with “in fact” were made long before I ever stopped to type. So while the argument has merit that you only have 30 minutes to TYPE the essay, you actually have weeks and months to have the general outline written in your mind so that you don’t have to write it all from scratch.

Furthermore, the author claims that the essay has to be well-written. While that’s an ideal, it’s not a necessity; if you’ve followed this post thus far you’ve undoubtedly seen a number of organizational cues beginning and then transitioning within each paragraph. However, once a paragraph’s point has been established the reader is likely to follow the point even if it’s a hair out of scope. Does this sentence add value? Maybe not, but since the essay is so well-organized the reader will give you the benefit of the doubt.

Moreover, while the author is correct that 30 minutes isn’t a lot of time, he assumes that it’s not sufficient time to write something actually well-written. Since the AWA is a formulaic essay – like this one, you’ll be criticizing an argument that simply isn’t sound – you can be well-prepared for the format even if you don’t see the prompt ahead of time. Knowing that you’ll spend 2-3 minutes finding three flaws in the argument, then plug those flaws into a template like this, you have the blueprint already in place for how to spend that time effectively. Therefore, it really is possible to write a well-written AWA in under 30 minutes.

As discussed above, the author’s insistence that 30 minutes is not enough time to write an effective AWA essay lacks the proper logical structure to be true. The AWA isn’t limited to 30 minutes overall, and if you’ve prepared ahead of time the 30 minutes you do have can go to very, very good use. How do I know? This blog post here took just under 17 minutes…

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*

Let’s focus on this disconnect first. If the GMAT provided you airtight arguments that were absolutely perfect, there would be no simple way to strengthen or weaken them. As such, the arguments provided inevitably have some kind of gap in logic contained between the conclusion and the evidence that theoretically supports that conclusion. Your goal is to identify that gap and either attempt to seal it up (strengthen) or rip it apart (weaken).

Of course, a dozen different answers could all weaken the same conclusion, so it’s not always possible to predict the exact answer ahead of time. However, all the answers that weaken the conclusion stem from the same gap (not banana republic) in logic, whereby the evidence provided does not quite support the conclusion stated. If you can identify the conclusion and the gap in logic, you tend to do quite well on these types of questions.

Let’s look at an example to illustrate this point:

*Researchers have recently discovered that approximately 70% of restaurant lemon wedges they studied were contaminated with harmful microorganisms such as bacteria and fungal pathogens. The researchers looked at numerous different restaurants in different regions of the country. Most of the organisms had the potential to cause infectious disease. For that reason, people should not order lemon wedges with their drinks.*

*Which of the following, if true, would most weaken the conclusion above?*

*A. The researchers could not determine why or how the microbial contamination occurred on the lemon wedges.*

*B. The researchers failed to investigate contamination of restaurant lime wedges by harmful microorganisms.*

*C. The researchers found that people who ordered the lemon wedges at restaurants were equally likely to contact the diseases caused by the discovered bacteria as were people who did not order lemon wedges.*

*D. Health laws require lemons to be handled with gloves or tongs, but the common practice for waiters and waitresses is to handle them with their bare hands.*

*E. Many factors affect the chance of an individual contracting a disease by coming into contact with bacteria that have nothing to do with lemons. These factors include things such as health and age of the individual, as well as the status of their immune system.*

There is a lot of text to review for this question, so let’s begin by identifying the conclusion. (Pauses an appropriate amount of time for review). The final sentence “For that reason, people should not order lemon wedges with their drinks” is the conclusion. In fact, the first three words can be removed, as they simply point to the fact that everything previous to that sentence is evidence to back up the ultimate conclusion. The passage concludes that we should not order lemon wedges (Antilles).

Let’s examine the evidence provided to back this up: 70% of the wedges observed are contaminated, and this contamination can lead to infectious diseases. Furthermore, the study was conducted in various locations across the country. This means we can’t weaken the conclusion by simply going two towns over. Apart from that, the sky’s the limit.

At first blush, this passage seems like a classic causation/correlation problem. The majority of lemon wedges are contaminated, so we shouldn’t order the lemon wedges in order to avoid falling ill. Well what if something else (say the water) was contaminated, leading to tainted lemon wedges. Then we’d avoid the wedges without avoiding the underlying cause of the diseases. In the general sense, avoiding the lemon wedges may not have the desired effect because there is nothing guaranteeing that it is solely the wedges that cause infectious diseases.

Now let’s look at the answer choices, keeping in mind that the correct answer choice should weaken the conclusion that the wedges are somehow responsible for any potential illness.

Answer choice A, “the *researchers could not determine why or how the microbial contamination occurred on the lemon wedges”, *doesn’t help in any real way. Just because you don’t understand how a virus works doesn’t make it any less dangerous to you (e.g. the Walking Dead). The problem is still the lemon wedges, even if no one is sure why. This answer choice can be eliminated.

Answer choice B, “the *researchers failed to investigate contamination of restaurant lime wedges by harmful microorganisms”* is quite obviously out of scope. Lime wedges have very little to do with lemon wedges (despite what Sprite says), so the cleanliness of the lime wedges is irrelevant to avoiding the lemon wedges. It is possible to be tempted by this answer choice if you conflate lemon with lime, especially if you’re tired, but a thorough analysis convincingly knocks this choice out.

Answer choice C, “the *researchers found that people who ordered the lemon wedges at restaurants were equally likely to contact the diseases caused by the discovered bacteria as were people who did not order lemon wedges” *is spot on. We had predicted that the problem was about lemon wedges being correlated to infectious disease without necessarily causing them. This answer choice tells us that people who didn’t order the lemon wedges were exactly as likely to fall sick as those who did. Therefore, avoiding the lemon wedges (the conclusion) will have no effect on your likelihood of feeling sick. This will be the correct answer, but we should look through the remaining two choices nonetheless.

Answer choice D, “health l*aws require lemons to be handled with gloves or tongs, but the common practice for waiters and waitresses is to handle them with their bare hands.”* is almost certainly true, but does not weaken the conclusion. Newsflash: Not everyone follows health code guidelines. (I’ve seen Ratatouille). If anything, knowing such an uncouth practice is commonplace would strengthen the idea of not ordering lemon wedges. Answer choice D is incorrect, as our goal is to weaken the conclusion.

Finally, answer choice E, “Man*y factors affect the chance of an individual contracting a disease by coming into contact with bacteria that have nothing to do with lemons. These factors include things such as health and age of the individual, as well as the status of their immune system” *is also true, but orthogonal to the issue of lemon wedges. Perhaps you could claim that healthy people have fewer risks in ordering lemon wedges, but still it would be a health risk. This answer does not weaken the conclusion in any way, and must therefore be discarded as well.

As indicated before, your prediction might not match exactly the correct answer choice, but it will exploit the gap in logic between the conclusion and the evidence. There will inevitably be (at least) one disconnect between the conclusion and the supporting evidence presented, your goal is to identify and elaborate upon that gap. If you successfully do that on test day, you can go toast your score with a celebratory drink, lemon wedges and all.

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!

The problem with this line of thinking is that our goal on the test isn’t simply to answer the questions correctly, but to do them within the confines of a challenging time constraint. So while it might feel more satisfying for the quantitatively-inclined to solve a complicated system of equations than it would feel to use a strategy, a strictly algebraic approach can be counterproductive, even if done correctly.

Take this Official Guide* question, for example:

*During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine’s average speed for the entire trip? *

A. (1800 – x) /2

B. (x + 60) /2

C. (300 – x ) / 5

D. 600 / (115 – x )

E. 12,000 / ( x + 200)

Here’s what happens if we do this algebraically: let’s say that the total distance traveled is ‘D.’ If x% of the trip is spent traveling 40mph, then this distance can be represented as (x/100)*D. This means that the remaining distance, during which Francine will be traveling at 60mph, will be [1 – (x/100)]*D.

Here’s what this will look like in a standard rate table:

R | T | D | |

Part 1 | 40 | [(x/100) * D]/40 | (x/100) * D |

Part 2 | 60 | [[1 – (x/100)]*D]/60 | [1 – (x/100)]*D |

Total | Ugh | D |

Well, good luck. Incidentally, this is how the Official Guide solves this question in their explanations. This approach will get you to the answer. But it will likely be difficult and time-consuming.

So rather than suffer through the brutal algebra required above, we can pick numbers. I always appreciate symmetry in my math problems, so let’s say that Francine went the same distance at 40mph as she did at 60mph. If this is the case, then she went 50% of the distance at 40mph, and x = 50.

Next, we can pick any distance we like for both parts of the trip. To make the arithmetic as simple as possible, let’s pick a number that’s a multiple of both 40 and 60. 120 will work nicely. Now our table will look like this:

R | T | D | |

Part 1 | 40 | 120 | |

Part 2 | 60 | 120 | |

Total |

Life is much improved. We can see that Francine spent 3 hours going 40mph and 2 hours going 60mph, so now we can fill in the rest of the table:

R | T | D | |

Part 1 | 40 | 2 | 120 |

Part 2 | 60 | 3 | 120 |

Total | R | 5 | 240 |

Solving for R, we get R*5 = 240. R = 48.

Not so bad. So we know that if x = 50, the average rate should be 48. Now all we have to do is plug 50 in place of ‘x’ in all the answer choices, and once we get to 48, we’ll have our answer.

Before we proceed, let’s think about this from the perspective of the question-writer for a moment. If we were trying to make this question more challenging, where would we put the correct answer? Considering that the average test-taker will start with A and work her way down, it makes sense to put the correct answer towards the bottom of our options, as this will require more work for the test-taker. Let’s get around this by starting with E and working our way up.

E. 12,000 / (x + 200)

Substituting 50 in place of ‘x’ we get:

12,000 / 250

Rather than doing long division, I’ll rewrite 12,000 as 12*1000 to get

12*1000/250

That becomes 12 * 4 = 48. That’s what we want. We’re done. The answer is E.

Takeaway: There are no style points on the GMAT. We don’t want the approach that would most impress our fellow test-takers, we want the approach that gets us the right answer in the shortest amount of time. Percent questions that involve variables are excellent opportunities for simplifying matters by picking numbers.

Moreover, when we find ourselves in a situation that requires testing the answer choices, we want to remember that the problem will be more challenging if the correct answer is D or E, so while this won’t always be true, it is the case often enough that it’s beneficial to start by testing E and systematically working our way up. As soon as we have our answer, we’re finished. We can save the impressive mathematical flourishes for our finance classes.

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

Today we have such a question for you – you could get really lost in it or could solve it in a few seconds if you take the right track. The trick is starting on the right track and that is why you have 2 mins per question available to you else 40 secs per question would have been sufficient!

**Question**: This game season, five divisions are going to play. Out of all the teams in each division, 6, 9, 12, 13 and 14 teams have qualified from the respective divisions. Each division will hold its own tournament – where a team is eliminated from the tournament upon losing two games – in order to determine its champion. The five division champions will then play in a knock-off tournament – a team is eliminated as soon as it loses a game – in order to determine the overall champion. Assuming that there were no ties and no forfeits, what is the maximum number of games that could have been played in order to determine the overall league champion?

(A) 89

(B) 100

(C) 102

(D) 107

(E) 112

**Solution**: Is it a max-min problem? Perhaps, but which guiding principle of max-min will we use to solve this problem? First think on your own how you will solve this problem.

Will you focus on the method or the result i.e. will you worry about who plays against whom or just focus on each result which gives one loss and one win? If you don’t worry about the method and just focus on the result, you can use a concept of mixtures here. In mixture questions, we focus on one component and how it changes. Here, we need to keep track of losses. Let’s focus on those and forget about the wins. As given, there were no ties so every loss has a win on the other side.

Every time a game is played, someone loses. We can give at most 2 losses to a team since after that it is out of the tournament. Don’t worry about against who it plays those two games. Whenever a team loses 2 games, it is out. The team could have won many games but we are not counting the wins and hence, are not concerned about its wins. As discussed, we are counting the losses so each win of that team will be counted on the other side i.e. as a loss for the other team.

Consider the division which has 6 teams – what happens when 12 games are played? There are 12 losses and each team gets 2 losses (we can’t give more than 2 to a team since the team gets kicked out after 2 losses), so all are out of the tournament. But we need a winner so we play only 11 games so that the winning team gets only 1 loss. We want to maximize the losses (and hence the number of games), therefore the winning team must be given a loss too.

So maximum number of games that can be played by the district in its own tournament = 2*6 – 1 = 11

Similarly, the division with 9 teams can play at most 2*9 – 1 = 17 games.

The division with 12 teams can play at most 2*12 – 1 = 23 games.

The division with 13 teams can play at most 2*13 – 1 = 25 games.

The division with 14 teams can play at most 2*14 – 1 = 27 games.

This totals up to 11 + 17 + 23 + 25 + 27 = 103 games

Now we come to the games between the district champions.

We have 5 teams. 1 loss gets a team kicked out. If the teams play 4 games, there are 4 losses and 4 teams get kicked out. We have a final winner!

Hence the total number of games = 103 + 4 = 107

There are a lot of variations you can consider for this question.

Say, if we need to minimize the number of games, how many total games would have been played?

Notice that the only games you can avoid are the ones in which the 5 district champions lost. You do still need 2 losses for each team to get the district champion and one loss each for four district champions to get the winner. Hence, at least 107 – 5 = 102 games need to be played.

Look at it in another way:

To kick out a team, it needs to have 2 losses so if the district had 6 teams, there would be 5*2 = 10 games played.

Similarly, the division with 9 teams will play at least 2*8 = 16 games.

The division with 12 teams will play at least 2*11 = 22 games.

The division with 13 teams will play at least 2*12 = 24 games.

The division with 14 teams will play at least 2*13 = 26 games.

This totals up to 10 + 16 + 22 + 24 + 26 = 98 games

Now we have 5 champions and they will need to play at least 4 games to pick a winner.

Therefore, at least 98 + 4 = 102 games need to be played.

You can try other similar variations – what happens when a team is kicked out after it loses 3 games instead of 2? What happens if you don’t have the knock-off tournament and instead need each district champion to lose 2 games to get knocked out?

*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!*

We’re still firmly entrenched in the first third of the year, and if 2015 is the year that you plan to conquer the GMAT you’re in luck. Why?

Because your GMAT study plan should include three phases:

**1) Learn**

One of the most common mistakes that GMAT studiers make is that they forget that they need to learn before they can execute. Are you keeping an eye on the stopwatch on every question you complete? Are you taking multiple practice tests in your first month of GMAT prep? Have you uttered the phrase “how could I ever do this in two minutes???”? If so, you’re probably not paying nearly enough attention to the learning phase. In the learning phase you should:

- Review core skills related to the GMAT by DOING them and not just by trying to memorize them. You were once a master of (or maybe a B-student at) factoring quadratics and identifying misplaced modifiers and completing long division. Retrain your mind to do those things well again by practicing those skills.
- Learn about the GMAT question types and the strategies that will help you attack them efficiently. For this you might consider a prep course or self-study program, or you can always start by reviewing prep books and free online resources.
- Take as much time as you need to complete and learn from problems. You’ll learn a lot more from struggling through a problem in six minutes than you will from taking two minutes, giving up, and then reading the typewritten solution in the back of the book. Let yourself learn! Again, it’s critical to learn by doing – by actively engaging with problems and talking yourself into understanding – than it is to try to memorize your way to success. The stopwatch is not your friend in the first third of your preparation!
- Embrace mistakes and keep a positive attitude. The GMAT is a hard test; most people struggle with unfamiliar question formats (Data Sufficiency, anyone?) and challenging concepts (without a calculator, too). Recognize that it will take some time to learn/re-learn these skills, and that making mistakes and thinking about them is one of the best ways to learn.

**2) Practice**

Regardless of how you’ve studied, you’ll need to complete plenty of practice to make sure you’re comfortable implementing those strategies and using those skills on test day. Once you’ve developed a good sense of what the GMAT is testing and how you need to approach it, it’s time to spend a few weeks devouring practice problems. Among the best sources include:

- The Official Guide for GMAT Review series
- The GMATPrep Question Pack
- The Veritas Prep Question Bank

In this phase, you can start concerning yourself with the stopwatch a little and you’ll want to identify weaknesses and common mistakes so that you can emphasize those. Particularly with GMAT verbal, the more official problems you see the more you develop a feel for the style of them, so it’s important to emphasize practice not just for the conscious skills but also for those subconscious feelings you’ll get on test day from having seen so many ways they’ll ask you a question.

**3) Execute**

Before you take the GMAT you should have taken several practice tests. Practice tests will help you:

- Work on pacing and develop a sense for how much time you’ll need to complete each section. From there you can develop a pacing plan.
- Determine which “silly” mistakes you tend to make under timed pressure and exam conditions, and be hyperaware of them on test day.
- Develop the kind of mental stamina you’ll need to hold up under a 4-hour test day. Verbal strategies can be much easier to employ in a 60-minute study session than at the end of a several-hour test! Make sure that at least a few times you take the entire test including AWA and IR for the first hour.
- Continue to see new problems and hone your skills.

While it’s not a terrible idea to take a practice test early in your study regimen and another partway through the Practice phase, most of your tests should come toward the end of your study process. Why? Because the learning and practice phases are so important. You can’t execute until you’ve developed the skills and strategies necessary to do so, and you won’t do nearly as effective a job of gaining and practicing those if you’re not allowing yourself the time and subject-by-subject focus to learn with an open mind.

So be certain to let yourself learn with a natural progression via the GMAT Study Rule of Thirds. Learn first; then focus on practice; then emphasize execution via practice tests. Studying in thirds is the best way to ensure that you get into a school that’s your first choice.

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*

In many ways, data sufficiency questions are like being in a foreign land. Even if you understand the rules, you’re often not as comfortable as in your native environment that you’ve acclimated to over many years (e.g. an Englishman in New York). It is normal to feel a little discombobulated, especially at first. However, once you’ve done a few (hundred) data sufficiency questions, you tend to get a feel for the question type. One issue still eludes a lot of test takers: When is it enough?

Data sufficiency is asking about (drum roll, please) when the data is sufficient. It’s pretty easy to disprove something if you can find a counter-example right away, but if you struggle with finding definitive proof, how long should you try to work at it.

Suppose a question asks whether X^2 = Y^3, that is asking whether any perfect square is also a perfect cube, you could spend a lot of time meandering towards a solution. What if we try 2^3, which gives 8? Well 8 isn’t a perfect square of any number, so we keep going. 3^3 is 27, which isn’t a perfect square of any number either. How far should we go? The next number, 4^3, gives 64, which is a perfect square, so we found an example relatively quickly, but we could conceivably spend several minutes calculating various permutations. Imagine a question asking if X^2 = Z^5 and see how long it would take to find an example.

The good news is that the question is almost always solvable using logic, algebra and mathematical properties. The bad news is it’s not always obvious how to proceed with these definitive approaches, and the brute force strategy is often employed. We can try various options and see if any of them work, while at the same time looking for patterns that tend to repeat or signal the underlying logic of the situation. While this strategy certainly has its place, it can sometimes be very wearisome.

Let’s look at a data sufficiency question that highlights this issue:

*W, X, Y and Z represent distinct integers such that WX * YZ = 1,995. What is the value of W?*

* WX*

** YZ*

*_____*

*1,995*

*X is a prime number**Z is not a prime number*

*Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.**Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.**Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.**Each statement alone is sufficient to answer the question.**Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.*

This question can be very tempting to start off with brute force. We can limit our choices by looking at the unit digits. If the unit digit of the product is 5, then there are only a few digits that are possible for X and Z. They all have to be odd, and, more than that, one of them must be exactly 5, as no other digits combine to give a 5. If one of them is 5, the other one is some odd number, 1, 3, 5, 7 or 9. Unfortunately, multiple options exist at both prime (3, 5 and 7) and non-prime (1, 9) for these digits, so it will be hard to narrow down the choices (where’s a dart board when you need one?)

Let’s look at this problem another way, which is: these two numbers must multiply to 1,995. We know one number ends with a 5, so we arbitrarily set it to be 25 and see what that gives if we set the other number to be 91. That comes to 2,275, which is way above what we need. How about 25 * 81, that yields 2,025. That’s too big, but just barely. How about 25 * 79? That will give us 1,975, which is slightly too small. We can’t get 1,995 with 25, but that’s all we’ve demonstrated so far. We can eliminate some choices as number like 15 can never be multiplied by a 2-digit number and yield 1,995, but there are still numerous choices to test.

It’s pretty easy to see how the brute force approach when you have dozens of possibilities will be very tedious. There’s another element that’s even worse, which is let’s say you manage to find a combination that works (such as 21 * 95), how can you be sure that this is the only way to get this product? Short of trying every single possibility (or calling the Psychic Friends hotline), you can’t be sure of your answer.

This problem thus requires a more structured approach, based on mathematical properties and not dumb luck. If two numbers multiply to a specific product, then we can limit the possibilities by using factors. We thus need to factor out 1,995 and we’ll have a much better idea of the limitations of the problem.

1,995 is clearly divisible by 5, but the other number might be hard to produce. The easiest trick here is to think of it as 2,000, and then drop one multiple of 5. Since 2,000 is 5 x 400, this is 5 x 399. Now, 399 is a lot easier than it looks, because it’s clearly divisible by 3 (since the digits add up to 21, which is a multiple of 3). Afterwards, we have 133, which is another tough one, but you might be able to see that it’s divisible by 7, and actually comes to 7 x 19. Finally, since 19 is prime, we have the prime factors of 1,995: 3 x 5 x 7 x 19.

How does this help? Well there may be 16 factors of 1,995, but the limitations of the problem tell us that we only have two two-digit numbers. Thus something like 15 * 133 breaks the rules of the problem. Our only options to avoid 3-digits are 19*3 and 5*7 or 19*5 and 3*7. This gives us either 57 * 35 or 95 * 21. At least at this point we’re 100% sure that these are the only two-digit permutations that combine to give 1,995.

Let’s get back to the problem. Statement 1 tells us that X (the unit digit of the first number) is prime, which knocks out 21 from the running. However the three other options all end with a prime unit digit, meaning that any of them are still possible. At this point it’s very important to note that the problem specified that W, X, Y and Z were all distinct integers. Since they must all be different, the option of 57 * 35 is not valid because the 5 is duplicated. As such, the only option is 95*21, and the prime number restriction confirms that it’s really 95 * 21 (and not 21 * 95). Variable W must be 9, and thus this statement ends up being sufficient.

Statement 2 essentially provides the same information, as Z is not a prime number and thus necessarily 1 given our choices. This confirms that the multiplication is 95 * 21 and W is still 9. Either statement alone is sufficient, so answer choice D is the correct option here. It’s important to note how close this question was to being answer choice B, as the non-prime limitation ensured we knew where the 1 was. But the fact that these digits had to be distinct changed the answer from B to D, reinforcing the adage that you should read the questions carefully.

This question can be solved without factors, but it is very hard to confidently answer it using only a brute-force approach. Solving through mathematics and number properties is not always the easiest route to success on data sufficiency. Sometimes you can write down a few options and see exactly how the problem will unfold, but if you use concrete concepts, you’ll know when it’s been enough.