I replied to the student by discussing the *Process Pyramid for Sentence Correction.* Here is what the pyramid looks like.

Brevity

Clarity Specificity

Logic Grammar

**Logic and Grammar Come First**

You can see that the bottom level – the foundation of sentence correction – is logic and grammar, (including proper comparisons and parallelism). This is where your analysis should begin. If the answer choice has a flaw in grammar, such as subject-verb agreement or an error in logic, such as an illogical modifier then that answer choice should be eliminated.

This type of error is less subjective than something like an ambiguous modifier. That is why you should begin with logic and grammar, these errors are not a matter of judgment and the rules are easier to master. In particular students get a tremendous return on investment from mastering the rules of the common modifiers, including participial phrases, prepositions, appositives, and relative clauses.

**Next Clarity and Specificity**

The initial level of analysis should eliminate most answer choices based on flaws in grammar and logic. However, sometimes there will be more than one answer choice that has (or seems to have) no errors in grammar or logic. At this point you can move to clarity and specificity as a way to distinguish between answers. This is when it is appropriate to eliminate answer choices that have pronouns that are not clearly matched to antecedents.

The Official Guide for GMAT Review, 13^{th} Edition (written by the people who make the GMAT exam) states that a correct answer should avoid being “awkward, wordy, redundant, imprecise, or unclear” and that an answer that is any of these things can **be eliminated** **even if it is “free of grammatical errors.”** This group of secondary errors is referred to as problems with “rhetorical construction.”

The following answer choice is from question #44 of the sentence correction portion of the Official Guide 13^{th} Edition:

“The plot of the Bostonians centers on the active feminist, Olive Chancellor, and the rivalry with the charming and cynical cousin Basil Ransom, when they find themselves drawn to the same radiant young woman whose talent for public speaking has won her an ardent following.”

This answer choice is eliminated not for a grammatical flaw, but because it is lacks clarity and specificity. It is unclear in this particular answer choice that “Olive Chancellor is a party to the rivalry” with Basil Ransom.

**Finally, Brevity**

At the top of the Process Pyramid is Brevity. Most sentence correction questions do not require you to climb so high on the pyramid. It is only when two or more answers are logically and grammatically acceptable AND are each clear and specific that you need to bring brevity into the equation. However, the Official Guide describes many answer choices as “unnecessarily wordy.” So if you do find that you have two or more answer choices that satisfy the first two levels of the process pyramid only then do you eliminate the one that is “wordy.”

Looking for an error such as an ambiguous pronoun is fine; just make sure that you do so at the proper time. Use the process pyramid to organize errors and address those errors in the proper order: Grammar and logic, clarity and specificity, and finally, brevity.

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!

*David Newland* has been teaching for Veritas Prep since 2006, and he won the Veritas Prep Instructor of the Year award in 2008. Students’ friends often call in asking when he will be teaching next because he really is a Veritas Prep and a GMAT rock star! Read more of his articles here.

Last week, I discussed timing issues on a quantitative question, and many of the concepts covered are applicable to the verbal section as well. Maintaining a good pace and avoiding spending undue time on perplexing questions are fundamental elements of a good GMAT score. However, I wanted to delve further into a particular type of question that often causes timing issues on the exam. Particularly when exhausted near the end of the test, students often dread coming across protracted Reading Comprehension passages.

Reading Comprehension (or RC for friends and family) poses a unique challenge on the GMAT. Every quantitative question and every other type of verbal question is entirely self-contained. A question will ask you about something, and then the following problem will be a completely different question about a completely different topic. Reading Comprehension questions ask you three, four and even five questions about the same prompt, and the prompts can be dozens of lines. Indeed, the first question on Reading Comprehension expects you to read through the entire passage, creating an inherent timing concern. Surely you can’t be expected to read through the entire passage in 2 minutes? (You are expected to do so, and don’t call me Shirley.)

Indeed, you can read through the passage in about two minutes, but you’re unlikely to be able to both read the passage and answer the (first) question posed during that span. For RC questions, I often find the best strategy is to separate the passage from the questions. If you read the question first, you risk skewing the analysis of the passage towards the question you have in mind, so it’s best to read the passage first without reading the question on the opposite side of the screen. The goal of this initial reading is to be able to identify the main idea of each paragraph and the primary purpose of the passage as a whole. You can read the passage in about 2 minutes and then spend about 1.5 minutes on each question, yielding a total of 8 minutes for 4 questions, roughly what you’d expect to spend holistically.

Let’s try this approach on a GMAT Reading Comprehension passage. At the end of each paragraph, try to summarize the main idea in about 3-5 words. You can even write these words down if you want, but it should be sufficient to think about the ideas.

*Biologists have advanced two theories to explain why schooling of fish occurs in so many fish species. Because schooling is particularly widespread among species of small fish, both theories assume that schooling offers the advantage of some protection from predators. Proponents of theory A dispute the assumption that a school of thousands of fish is highly visible. Experiments have shown that any fish can be seen, even in very clear water, only within a sphere of 200 meters in diameter. When fish are in a compact group, the spheres of visibility overlap. Thus the chance of a predator finding the school is only slightly greater than the chance of the predator finding a single fish swimming alone. Schooling is advantageous to the individual fish because a predator’s chance of finding any particular fish swimming in the school is much smaller than its chance of finding at least one of the same group of fish if the fish were dispersed throughout an area.*

* However, critics of theory A point out that some fish form schools even in areas where predators are abundant and thus little possibility of escaping detection exists. They argue that the school continues to be of value to its members even after detection. They advocate theory B, the “confusion effect,” which can be explained in two different ways. Sometimes, proponents argue, predators simply cannot decide which fish to attack. This indecision supposedly results from a predator’s preference for striking prey that is distinct from the rest of the school in appearance. In many schools the fish are almost identical in appearance, making it difficult for a predator to select one. The second explanation for the “confusion effect” has to do with the sensory confusion caused by a large number of prey moving around the predator. Even if the predator makes the decision to attack a particular fish, the movement of other prey in the school can be distracting. The predator’s difficulty can be compared to that of a tennis player trying to hit a tennis ball when two are approaching simultaneously.*

*According to one explanation of the “confusion effect,” a fish that swims in a school will have greater advantages for survival if it *

*(A) **tends to be visible for no more than 200 meters.*

*(B) **stays near either the front or the rear of a school.*

*(C) **is part of a small school rather than a large school.*

*(D) **is very similar in appearance to the other fish in the school.*

*(E) **is medium-sized.*

This passage only has two main paragraphs, and really each one is mostly about a theory as to why fish form schools (theory C: to get business degrees). We can summarize the first paragraph as the evasion theory and the second paragraph as the confusion theory. Overall the passage is primarily concerned with differing theories as to why fish tend to regroup in many disparate situations.

Looking over the question, it is specifically concerned with the “confusion effect”, which was theory B in the second paragraph. We can now focus our attention on the second paragraph to answer the question about survival. Rereading the passage, nothing was mentioned about the front or back of a school, as well as the size of the school, which eliminates answer choices B and C. Answer choice E similarly makes decisions based on the size of the fish, which was only discussed in terms of small fish. We can fairly quickly eliminate this choice as being a medium sized fish was never even mentioned.

Only answer choices A and D remain. Answer choice A is mentioned in the general sense for all fish in schools, and so would be a dubious choice as a great advantage since it applies to all fish in a given school. This is equivalent to saying we should promote Bob because he breathes oxygen. Answer choice D offers a logical choice, which is almost verbatim in the middle of the second paragraph “*In many schools the fish are almost identical in appearance, making it difficult for a predator to select one.”* This answer lines up with the text and we’ve eliminated the other four choices, making D an easy selection (also possibly recalling memorable moments from Disney’s Finding Nemo).

The questions on Reading Comprehension tend to be somewhat less tricky than the other verbal sections (Sentence Correction and Critical Reasoning). This difference is somewhat due to the fact that reading through passages takes time and inherently contributes to the difficulty of the question. The trouble isn’t just finding the right answer, it’s reading through 300 words of drivel without falling asleep and then isolating the important aspect to answer the given question. Especially since the verbal section is the last section of this test, it’s important not to waste too much time and get mentally fatigued. A good timing strategy is crucial to getting the best possible result on your GMAT.

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

This problem speaks to the inherent time management skill required to succeed on the GMAT. Almost any question you will face on test day can be solved with a brute force approach. However, you won’t have a calculator and you will be under constant time pressure to complete each question fairly quickly, so simply running through every possible numerical combination seems like a fool’s errand. There may be a time when the brute force approach works, but it is like trying to break into someone’s e-mail by trying 00000001, 00000002, 00000003, etc until you find the correct password. You’d probably have more success with a logical approach (such as guessing birthdays or other important dates) than with trying every possible permutation until the lock opens.

Approaching the problem in a logical and methodical way should be your goal for both quant and verbal questions. The approach as such may vary a little, but pattern recognition and extrapolation are two skills that will come up over and over again. If you’ve ever asked a 5-year-old what 2 + 2 was, they generally answer 4. If you ask them what 1,002 + 1,002 was, you’d usually get a lot of blank stares and puzzled looks. (My attempts to explain that they are essentially the same question have led to more crying fits than I’d care to admit). The GMAT uses the same elements of misdirection to bait you into thinking this particular problem is one that you can’t solve.

Let’s look at a quant problem to get an idea of what we’re looking to do on these questions:

How many positive integers less than 250 are multiple of 4 but NOT multiples of 6?

(A) 20

(B) 31

(C) 42

(D) 53

(E) 64

This is the type of question that most people can get with unlimited time. You can simply go through every possible number from 1 to 249 and see if each number meets the criteria. Apart from going cross-eyed halfway through, you will also spend an atrocious amount of time on a question clearly designed to reward you for using logic. Let’s look at this question logically and see what we can determine.

Firstly, it only cares about positive integers, so we can disregard zero. This is helpful because a lot of questions hinge on whether or not zero is included, but that won’t matter in this instance. Furthermore, only integers matter, and we’re looking for multiples of 4 but not 6. Your initial pass on a question like this might look might concentrate on the multiples of 4 and you might write (part of) the following sequence down:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100…

After writing a couple of dozen numbers, you may try to figure out the pattern and extrapolate from there. Numbers divisible by 6 are to be eliminated, so you could rewrite this sequence:

4, 8, 16, 20, 28, 32, 40, 44, 52, 56, 64, 68, 76, 80, 88, 92, 100…

Even with this, we have a long sequence of numbers, some of which are crossed off, and less than halfway through the entire sequence. Perhaps approaching the question from a more strategic approach would yield dividends:

The number must be divisible by 4 but not by 6. Calculating the LCM gives us 12, which means that every 12th number will be divisible by both of these numbers. We want the integers to be divisible by four, but not by six, so 12 is out. Along the way, we stop by 4 and 8, both of which are divisible by four but not by six. So every 12 numbers, our count goes up by two, and we start the pattern again. 1-12 will give two numbers that work. 13-24 will give two more numbers that work. 25-36 gives two more, 37-48 gives two more and 49-60 gives two more as well. Thus, through 60 numbers, we have 10 elements that are divisible by 4 and not 6.

From here, it might be easier to go up in bounds of 60, so we know that 61-120 gives 10 more numbers. 121-180 and 181-240 as well. This brings us up to 240 with 40 numbers. A cursory glance at the answer choices should confirm that it must be 42, as all the other choices are very far away. The numbers 244 and 248 will come and complete the list that’s (naughty or nice) under 250. Answer choice C is correct here.

There are other ways to get the right answer, but the fastest ones all hinge on pattern recognition. Figuring out that every 12 numbers gives two more answers can take us from 1 to 240 in one shot (20 sequences x 2). Alternatively, once finding 4 elements at 24, you can probably easily envision multiplying the total by 10 and getting to 240 straight away (like warping over worlds in Super Mario Bros).

Timing is one of the key elements being tested on the GMAT, and one of the goals of the exam is to reward those who have good time management skills. Given 10 minutes, almost everyone would get the correct answer to this question, but the exam wants to determine who can get it right in a fraction of that time. On the GMAT, as in business, timing is everything.

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One important thing to remember is that you won’t have a calculator on the exam, so blindly executing mathematical equations will be an exercise in futility. If the numbers seem large, the first thing to do is to determine whether the large numbers are required or just there to intimidate you. The difference between 15^2 and 15^22 is staggering, and yet most GMAT questions could use these two numbers interchangeably (think unit digit or factors).

Once you determine whether the bloated numbers truly matter, you need to ascertain how much actual work is required. If the question is asking you for something fairly specific, then you might need to actually compute the math, but if it’s a general or approximate number, you can often eyeball it (like proofreading at Arthur Andersen). Even if you end up having to execute calculations, you can usually estimate the correct answer and then scan the answer choices. Even in data sufficiency, determining how precise the calculations need to be can save you a lot of time and aggravation.

Let’s take a look at a question that can be somewhat daunting because of the numbers involved, but is rather simple if we correctly determine what needs to be done:

*If 1,500 is the multiple of 100 that is closest to X and 2,500 is the multiple of 100 closest to Y, then which multiple of 100 is closest to X + Y?*

* (1) X < 1,500 *

*(2) Y < 2,500*

*(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.*

The first step here is to try and understand what the question is asking. It can be a little confusing so you might have to read it more than once to correctly paraphrase it. Essentially some number X exists and some number Y exists, and the question is asking us what X + Y would be. The only information we get about X is that 1,500 is the closest multiple of 100 to it, meaning that X essentially lies somewhere between 1,450 and 1,550. Any other number would lead to a different number being the closest multiple of 100 to it. Number Y is similar, but offset by 1,000. It must lie between 2,450 and 2,550. At this point we may note that the problem would be exactly the same with 100 and 200 instead of 1,500 and 2,500, so the magnitude of the numbers is simply meant to daunt the reader.

Without even looking at the two statements, let’s see what we can determine from this problem: Essentially if we add X and Y together, the smallest amount we could get is (1,450 + 2,450 =) 3,900. The largest number we could get is (1,550 + 2,550 =) 4,100. The sum can be anywhere from 3,900 to 4,100, and therefore the closest multiple of 100 could be 3,900, 4,000 or 4,100, depending on the exact values of X and Y. This tells us that we have insufficient information through zero statements, which isn’t particularly surprising, but it also sets the limits on what we need to know. There aren’t dozens of options; we’ve already narrowed the field down to three possibilities.

(1) X < 1,500

Looking at statement 1, we can narrow down the scope of value X. Instead of 1,450 ≤ X ≤ 1,550, we can now limit it to 1,450 ≤ X < 1,500. This reduces the maximum value of X + Y from 4,100 to under 4,050. This statement alone has eliminated 4,100 as an option for the closest multiple of 100, but it still leaves two possibilities: 3,900 and 4,000. Statement 1 is thus insufficient.

(2) Y < 2,500

Looking at statement 2 on its own, we now have an upper bound for Y, but not for X. This will end up exactly as the first statement did, as we can now limit the value of Y as 2,450 ≤ Y < 2,500. This is fairly clearly the same situation as statement 1, and we shouldn’t spend much time on it because we’ll clearly have to combine these statements next to see if that’s sufficient.

(1) X < 1,500

(2) Y < 2,500

Combining the two statements, we can see that the value of X is: 1,450 ≤ X < 1,500 and the value of Y is 2,450 ≤ Y < 2,500. If we tried to solve for X + Y, the value could be anywhere between 3,900 and 4,000 (exclusively), so 3,900 ≤ X+Y < 4,000. This still leaves us in limbo between two possible values. To illustrate, let’s pick X to be 1,460 and Y to be 2,460. Both satisfy all the given conditions and give a sum of 3,920, which is closest to 3,900. If we then picked X to be 1,490 and Y to be 2,490, we’d get a sum of 3,980. The second situation clearly gives 4,000 as the closest multiple. If we can solve the equation using valid arguments and yield two separate answers, we have to pick answer choice E.

These types of questions can be daunting because of the big numbers and the ambiguous wording, but the underlying material on these questions will never be something that can’t be solved in a matter of minutes. The difficulty often lies in determining how much work we really need to do to solve the question at hand. The old adage is that you get A for effort, but that’s applicable when you tried earnestly and failed. On the GMAT, you want to put in as much effort as is needed, but the only A you want to get is for Awesome GMAT Score (admittedly an AGMATS acronym).

You may wonder why such a simple rule is often overlooked. The problem is often one of perspective. When evaluating five different choices, it is easy to concentrate on the differences among the options given and ignore the rest of the world (like watching Game of Thrones). Whichever choice you select must merge effortlessly with the rest of the sentence. If it doesn’t, the answer choice selected cannot possibly be the correct answer.

It’s surprisingly easy to overlook this aspect of sentence correction. However, there’s a simple strategy to combat this inertia: (i.e. There’s an app for that) we must ensure to pay special attention to the first and last words of the underlined portion. These are the connector words that link the sentence fragment back to the rest of the sentence. It’s possible that there is only one such word if the underlined portion is at the beginning or at the end. As long as the whole sentence isn’t underlined (which brings a whole different set of problems to the table), pay attention to the connector word(s) and any syntax that must be respected.

Let’s look at a typical Sentence Correction question to illustrate the point:

*To Josephine Baker, Paris was her home long before it was fashionable to be an expatriate, and she remained in France during the Second World War as a performer and an intelligence agent for the Resistance*

*(A) **To Josephine Baker, Paris was her home long before it was fashionable to be an expatriate*

*(B) **For Josephine Baker, long before it was fashionable to be an expatriate, Paris was her home*

*(C) **Josephine Baker made Paris her home long before to be an expatriate was fashionable *

*(D) **Long before it was fashionable to be an expatriate, Josephine Baker made Paris her home*

*(E) **Long before it was fashionable being an expatriate, Paris was home to Josephine Baker*

Since I’ve spent three paragraphs discussing the perils of ensuring that the underlined portion flows flawlessly with the rest of the sentence, let’s start the discussion there. The underlined portion ends with a comma, and then there’s immediately an “*and she*” that we cannot modify. This means the subject of the underlined portion must unequivocally be “Josephine Baker”, lest we not have a clear antecedent for the pronoun. Let’s look at the answer choices one by one and eliminate them if they do not make logical and grammatical sense until only one remains.

The original answer choice “*To Josephine Baker, Paris was her home long before it was fashionable to be an expatriate*“ doesn’t work because the sentence contains a modifier error. The sentence is also set up so that Paris seems to be the subject, making the “she” pronoun unclear (is this referring to Paris Hilton, perhaps?) This sentence is grammatically incorrect, and the transition into the rest of the sentence highlights this discrepancy.

Moving on, answer choice B “*For Josephine Baker, long before it was fashionable to be an expatriate, Paris was her home” *suffers from the same ambiguity. We can mentally strike out the modifier “long before it was fashionable to be an expatriate” as it adds nothing to the grammatical structure of the sentence. This leaves us with “For Josephine Baker,…, Paris was her home, and she…”. This time the pronoun should refer back to Paris, clearly incorrect. In the best case this sentence is hopelessly unclear, and in the worst case it’s inadequate and unnecessary (Some would argue that’s another Paris Hilton reference).

Answer choice C “*Josephine Baker made Paris her home long before to be an expatriate was fashionable” *actually works fairly well with the rest of the sentence. However it’s often the first answer choice to be eliminated because of the phrasin*g “long before to be an expatriate”, *which is clearly wrong. The underlined portion must gel with the rest of the sentence, but that is not the only criterion that matters.

*Answer choice D “Long before it was fashionable to be an expatriate, Josephine Baker made Paris her home”, *seems to work*. *It puts the modifier at the beginning of the sentence and clearly identifies Josephine Baker as the subject. The rest of the sentence flows naturally from this sentence. D should be the correct answer, but we should still eliminate E for completion’s sake.

Answer choice E “*Long before it was fashionable being an expatriate, Paris was home to Josephine Baker” *recreates the same problem that’s pervaded this sentence since answer choice A. This sentence clearly has Paris as a subject, and everything after the comma naturally refers to Paris. Answer choice E is incorrect, cementing our decision that answer D is correct (Final answer, Regis).

On sentence correction problems, it’s very easy to get so enthralled by the underlined text that you ignore the rest of the sentence. While the underlined portion is the most important part, focusing exclusively on those words makes you lose perspective and gives you a fishbowl mentality (Orange Is the New Black style). The words that aren’t underlined may be indispensable to selecting the correct answer, especially the connector words that link the underlined text back to the rest of the sentence. To see the big picture, sometimes you have to make sure to connect the dots.

So much is at stake, actually, that some of the greatest minds in the world have dedicated time to breaking down all the possibilities; Nate Silver’s website gives the US a slightly better than 75% chance of moving through, with those possibilities including:

-An outright win against Germany

-A draw with Germany (around which a popular conspiracy theory is growing, given that a draw puts both teams through)

-A close loss to Germany with a Portugal win (but not blowout) over Ghana

-A close loss to Germany with more overall goals scored in the tournament than a victorious Ghana

Given all the situations – all requiring math, encompassing all the permutations available and including probabilities…all GMAT-relevant terms – some of these great minds have put together helpful infographics that can shed light on the scenarios…and help you study for the GMAT’s Integrated Reasoning / Graphics Interpretation section. How? Consider this infographic (click to enlarge):

This graphic has a lot of similarities to some you may see on the Integrated Reasoning section of the GMAT. It’s a “unique graphic” – not a standard pie chart, bar graph, line graph, etc. – so it includes that “use reasoning and logic to figure out what’s happening” style of thinking that you’ll almost certainly find on at least one Graphics Interpretation problem. And like many GI problems on the GMAT – even those classic bar graphs, etc. – this one has a potentially-misleading scale or display if you’re not reading carefully and thinking critically. Most notably:

*If Nate Silver is right (as he usually is) and the US is better than a 3-1 favorite to advance, why is there so much red on this graph?!*

And here’s where critical thinking comes into play:

1) What’s more likely – that both Germany and Ghana win 4-0, or that they each win 1-0? Soccer history tells us that 4-0 wins are quite rare, but 1-0 wins are fairly common. The blue Germany 1-0 / Ghana 1-0 box, though, is the same size as the red 4-0/4-0 box, making the scale here a little misleading. This graph does not incorporate probability into its cell size, so it treats all outcomes as equally likely, therefore skewing the red-vs-blue dynamic. On Integrated Reasoning, you may well have to consider a chart’s scale and determine whether it can accurately be extrapolated into something like probability!

2) This graph only expands “__________ side wins” into scores for three teams: Germany, Ghana, and Portugal. Why doesn’t it do so for the USA, or include the goals scored in a US-Germany tie? Likely because this graph is designed for an American audience, and the American side’s “what if?” scenarios are the same for *all* wins – if the US wins, it finishes #1 in the group and moves on – and for draws, in which the US would finish second. It’s only if the US loses that any other situations matter – by how much did the US lose? what was the score of the other match? – so in order to save space and draw attention to the meaningful “what ifs” this graph treats all US > Germany scenarios with one column. Which works for the purpose of this graph, but leads to another really misleading takeaway if all you’re looking at is blue vs. red – the blue columns for the US are wildly consolidated (and it’s all noted correctly so it’s not “wrong”), so you have to read carefully and think critically in order to understand what the graph truly displays.

Note that this is in no way a “misleading graphic” – it’s a well-constructed infographic to talk about all the possibilities that could happen and change US fortunes tomorrow. It’s just that the maker of the graphic chose to display the valid information in a certain way, one that may mislead the eye if the user is not being careful and thinking critically. That’s also very true of GMAT Integrated Reasoning – the graphics you see will be valid and meaningful, but you’ll need to read them carefully and think logically to avoid making assumptions or drawing flawed conclusions. And as this graphic shows, sometimes your mind’s initial reaction needs to be checked by some critical thinking.

So when you see Graphics Interpretation problems on the GMAT Integrated Reasoning section, be careful. What may seem obvious or too-good-to-be-true (like, it hurts to say, a 2-1 lead into the 95th minute) may require that little extra attention to detail to gain the result that you’re looking for, the one that gets you through to the next stage where you want to be.

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*

As can be expected, different quantitative questions will pertain to different mathematical notions. However some more advanced questions will begin to blur the lines (#BlurredLines) between multiple concepts. A question can ask you to solve an equation using variables from a given shape, incorporating geometry, algebra and even arithmetic concepts in one fell swoop. It’s important to note that all these seemingly disparate topics you’re studying while preparing for the GMAT can be combined into one question. These questions tend to be more difficult, but mostly because they require more steps, and therefore more opportunities to make mistakes.

The mathematical concepts don’t have to be any harder on these questions; the simple fact of merging them into a Frankenstein’s monster question can make the problem harder than the sum of its parts. (The question wants you to use your BRAINS). Add to this the time pressure of having to solve such questions in roughly two minutes, and you can imagine how longer questions combining various elements can frustrate even the most experienced student.

Let’s review a question and examine the various pitfalls we can fall into:

If you select two cards from a pile of cards numbered 1 to 10, what is the probability that the sum of the numbers is less than the average of the pile?

(A) 1/100

(B) 2/45

(C) 2/25

(D) 4/45

(E) 1/10

The first hurdle here is interpreting the question. To paraphrase, if I were to choose two random cards, would their sum be less than a certain other number. This is essentially a probability question, as evidenced by the answer choices as fractions. However there are a couple of elements to keep in mind. The first task is to determine the average of the pile.

Given 10 numbers, we could simply sum them up and divide by 10, but it’s probably much faster to recognize that the mean of an evenly spaced set is equal to the median of the set. A set with 10 numbers has a median that’s the average of the 5th and 6th elements (Not the Bruce Willis movie). Conveniently, the 5th element is 5 and the 6th element is 6, yielding an average of 5.5. Since we’re dealing with integers, we must now determine the number of possibilities that give a sum of 5 or less.

The options are limited enough that we can just reason out the choices. A good strategy is just to assume that the first card is a 1, and figure out what numbers work for the second number. If we pick 1, the next smallest card is 2. Thus the possibility (1,2) works. Similarly, we can see that (1,3) and (1,4) will work. (1,5) is too big, so we can stop there as any other option would only be bigger than this benchmark. It’s worth noting that the question is set up so that there’s no repetition, thus the option (1,1) cannot be considered. If the first card picked is a 1, there are three options that will keep the average below 5.5 (like a Russian judge at the Winter Olympics).

Next, supposing that the first card were a 2, there would be the separate option of (2,1). Since the order matters, (2,1) is not the same as the aforementioned (1,2). This is another valid choice. (2,2) is eliminated because of duplication, leaving us only with (2,3) that will also work if the first card is a 2. Since (2,4) is too big, we don’t need to examine any further. That’s two more options to add to our running tally.

Continuing, if the first card were a 3, then (3,1) and (3,2) would work. (3,3) is above the average, and it is a duplicate, so it can be eliminated for either reason. That gives us two more options for our running tally. The final option is to start with a 4, giving (4,1). Anything bigger is above the average. Similarly, anything starting with 5, 6, 7, 8, 9 or 10 will be above the average. Only eight options work out of all the possibilities.

The question is almost over, but there is one final trap we need to avoid before locking in our answer. The stimulus purported 10 different cards to select. If we were to compile all the possibilities, a natural total to think of would be 100 (10×10). However, since there is no replacement, we’re first selecting from 10 choices, and then from 9 choices. Exactly as a permutation of two selections out of 10, this gives us a total of 90 possible choices. If there are eight options that satisfy the conditions out of 90 choices, then the correct answer must be 8/90, which simplifies to 4/45. Answer choice D.

Examining the answer choices, we can see some of the more obvious traps. Compiling eight options out of 100 choices would give us the erroneous 2/25 fraction in answer choice C. Overlooking the lack of replacement would give us 10 total choices (the same eight plus (1,1) and (2,2) out of 100 possibilities, or answer choice E. The exam is designed to ask tricky questions, which means that the answer choices will often be answers you can get if you make a single calculation error or unfounded assumption. Be vigilant until the end of the question, as you don’t want to spend a full two minutes on a complicated question just to falter at the finish line. Questions can have many aspects to consider and many steps to execute, but by continuously thinking in a logical manner, you can solve any GMAT question. Remember that even the longest journey begins with a single step.

As anyone who’s actively studying for the GMAT knows, you must determine whether you have sufficient data with each statement separately, and then possibly combine them if you still have not determined sufficiency. This leads most assiduous students to spend most of their time determining the relationship between the statements and the question stem. If the question were true (which it always must be), would that guarantee one specific answer? Would such a definitive answer be guaranteed if I used the other statement instead? What if I used both statements?

Allow me to pose one more rhetorical question: what happens when the exam throws a spanner in the works? The exam is designed to zigzag to avoid always asking questions in the same way. Sometimes these winding paths lead to counter-intuitive questions, which can confound unprepared test takers. One such tactic is to provide too much information (#TMI) so that test takers get perplexed as to what they’re supposed to solve.

Let’s look at an example that isn’t particularly difficult, but can cause students to feel stress and spend undue time on a question they inherently know how to solve:

*If the average (arithmetic mean) of the five numbers x, 7, 2, 16 and 11 is equal to the median of the five numbers, what is the value of x?*

* (1) 7 < x < 11*

*(2) x is the median of the five numbers*

*(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.*

Looking at the question, we are being asked to solve for x. One specific value is needed here, as a range of values would be useless. Ignoring the statements, a lot of information is provided in the question stem. The average of the five numbers is also the median of the same numbers, so it behooves us to put them in order to give loose boundaries on x. The question specifically doesn’t put them in order for us to not necessarily see the limits as easily. In order, the set would be {x, 2, 7, 11, 16}.

Once we have an ordered set, we can easily solve for x. The first hint is that the mean and the median are the same, which we know to be true for sets that are equally spaced. That isn’t very helpful here as the spacing is not even between the four elements we already have, much less when we introduce x, but it’s a natural place for our thinking to initially go. The next step might be to use the logic that x is also the mean of the set, which can be solved algebraically or logically within a couple of steps.

Using algebra, we know that the sum of the five terms is equal to the average times the number of terms. We can then set up the equation: (x+2+7+11+16)/5=x

Which can then be mathematically combined: (36+x)/5=x

Multiplying both sides by 5 to eliminate the denominator: (36+x)=5*x

Moving x to the same side: 36=4*x

Thus: 9=x

We can also get the answer using logic, especially since the GMAT usually gives integers in this situation, so you only have a couple of values of x to plug in to find that it must be 9.

At this point, after a four step algebraic problem or a couple of educated guesses, we have done everything necessary to correctly answer this problem. (Gasp!) We have, in fact, solved the value of x without using either statement! I know the answer must be 9 from the information given uniquely in the question stem (is that answer choice F?) After solving the question, let’s look at the two statements and see which of the five answer choices we should select.

Statement 1 tells us that x is between 7 and 11. This was given in the question stem because the x was the median. In other words, statement 1 doesn’t give any new information, so it seems that it’s somewhat superfluous (TMI?). However, the question format specifically asks: “If statement 1 were true, could we solve for x”? And the answer is that, yes, absolutely we can solve that x is 9 if statement 1 were true. The fact that we can solve it without statement 1 doesn’t invalidate that we can solve it with statement 1. Specifically, statement 1 alone is sufficient to answer the question, which narrows the possible correct answers to A and D.

Statement 2 tells us that x is the median of the five numbers, which is the same information as statement 1. Statement 2 thus implies statement 1, and whatever the answer to statement 1, the same will hold for statement 2. The answer on such questions can thus only be D or E, since both statements give redundant information. Since statement 1 was true, statement 2 must also be true. Thus, each statement alone is sufficient, which is a verbatim transcript of answer choice D.

In actuality, you can solve this question without using either statement, but that option is not valid in Data Sufficiency. It’s not so much do I need the statement, but rather if the statement were true, would that guarantee the uniqueness of the answer. Since either statement alone guarantees one definitive answer, the answer must be D. On test day, you don’t want to waste undue time or second guess yourself if the question pattern isn’t exactly what you expect. Understand the rules of the game and approach each question logically. Those two tenets should be sufficient to get the right answer, even if you feel that the question has given you TMI.

Students who were never very fond of math in high school (and even kindergarten) often struggle with the math on the GMAT, but this is somewhat expected. If you never liked a topic, you probably never spent hours thinking about it or doing exercises in your leisure time (think of people who dislike cardio). However, many students who traditionally excel at math struggle just as much as the students who never cared for the subject. This frustration can be even more pronounced when it’s about a topic you’ve traditionally excelled at over your life.

Delving into the topic a little, the GMAT does not allow you to have a calculator with you during the exam because the calculator is a crutch that will end up doing the work for you. Naturally, in every conceivable real world situation, you will have a calculator with you, but finding ways to get the correct answer is an important aspect of business. When a decision needs to be made in a split second, you cannot always reach for your calculator. Worse than that, a calculator is clearly faster and more accurate than you, but we cannot (yet) be replaced by computers because computers cannot think as humans do (#Skynet). If the goal of the GMAT was to ensure that all students could perform complex mathematical calculations, you’d have a TI graphing calculator attached to your arm. The goal of the exam is to make you think, and nothing mitigates independent thinking like a calculator.

So how does the exam test math if it won’t give you complex math? Basically by giving you simple math and expecting you to solve it quickly. Simple math does not necessarily mean small numbers. In fact, large, unwieldy numbers are a great way to validate that you understand the underlying concept rather than utilize a brute force approach to solve the problem.

Let’s look at a very simple math question that helps to underline the kind of math problems you should be able to execute quickly:

What is 1,800 / 2.25?

(A) 400

(B) 500

(C) 650

(D) 800

(E) 850

On the actual GMAT, you might only see this question if you’re scoring in the bottom quintile of the test. However, you can easily have a calculation such as this to execute as part of a larger problem. Either way, getting the correct answer on a question such as this should ideally take you 30 seconds or less.

There are many ways to get the correct answer here, and the method chosen has a lot to do with personal preference. As someone who is comfortable with mental math, I would immediately attempt to approximate this equation. If it were simply 1,800 / 2, the answer would be 900. Since 2.25 is bigger than 2, the answer must be a little smaller. This narrows the choice down to likely either D or E. Rounding 2.25 to 3 would yield a division with a quotient of 3, further cementing the elimination of answer choices A and B. However between 800 and 850, the choice is pretty close, so we might need a more precise approach.

One common strategy is to convert the decimal into a fraction. Using algebraic rules, this might simplify our math quite a bit. 1,800 / 2.25 is the same as 1,800 / (9/4). This equation might seem equally daunting, but remember that division is the same thing as multiplication, and dividing by 9/4 is the same as multiplying by 4/9 (this property holds for all numerators and denominators). If I turned this into 1,800 * 4/9, I can think of it as two separate steps: (1,800 * 4) / 9, or (1,800 / 9) * 4 (commutative property). The second is clearly much easier to process, and you end up with 200 * 4, or 800. The answer must thus be D and can be seen fairly cleanly using fractions.

You can also get the answer by using reverse-engineering. Simply put, an equation of 2.25 * x = 1,800 would yield the same x, so you can think of this equation as backwards. If x were 1, the product would be 2.25, which is clearly not the right answer. How can I get closer to the actual product? Well if I set x to be 4, then the product would be 9. From 9, I might be able to see that I could set x to be 40 and then 400, giving 90 and 900 respectively. Once I’m at 900, I simply double x (from 400 to 800) and get the correct answer. This strategy can be helpful for those who dislike division and prefer to work with multiplication.

Overall, it doesn’t matter which strategy you use (in fact you may use an entirely different approach and still get the correct answer. There is no “correct” strategy on the GMAT, only the Machiavellian notion that you must get the correct answer, by algebra, deduction, induction, strategic guessing or even dumb luck. Being able to solve math questions in roughly as long as it would take to solve if you had a calculator will help you realize why the tool is not allowed on the exam. In the best case, you can turn math on its ear and appreciate the nuanced way the GMAT tests your understanding of these fundamental concepts.

The downside of Data Sufficiency questions is that, by not necessarily going through the calculations, it’s very possible to misinterpret the question or reach a premature conclusion without considering every option. While there is no formal requirement of actually calculating anything, I do recommend trying to cement your answer by plugging in a few numbers to confirm your theory. In the worst case, your hunch is validated and you feel confident. In the best case, you recognize a simple mistake or assumption you took for granted and you avoid a glaringly incorrect choice (like Decca records passing on The Beatles in 1962).

Another common trap students fall into on data sufficiency is misunderstanding the information given. If the question is asking you for x, and you think it’s asking you for y, your chances of getting the right answer are reduced to lucky guesses and finger slips of the mouse (much like Australia’s chance of winning the 2014 World Cup). Avoiding doing the math also makes it harder to see if you go down the wrong path. In some instances, it’s worth writing down some numbers just to see what happens. Sometimes just seeing what doesn’t work will lead you down the path of the correct answer.

Let’s highlight this principle with a Data Sufficiency question that a lot of people can narrow down to two choices, but then pick incorrectly:

*A certain car rental agency rented 25 vehicles yesterday, each of which was either a compact car or a luxury car. How many compact cars did the agency rent yesterday?*

* (1) The daily rental rate for a luxury car was $15 higher than the rate for a compact car.*

*(2) The total rental rates for luxury cars was $105 higher than the total rental rates for compact cars yesterday*

*(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.*

Looking at what’s provided in the question stem, there are two types of cars being rented. The total number of cars rented is 25, and every car is either compact or luxury. We only have to determine how many compact cars were rented, so something as small as the number of luxury cars rented would solve our problem very quickly. Looking at the statements, we only have information about prices. The daily rate for the compact car is 15$ less than the luxury vehicle. That’s great (and a little unrealistic), but it doesn’t help us answer the question about the number of vehicles. Statement 2 also talks about money, this time talking about the total revenue instead of a per-car basis. This doesn’t help either, so answer choices A, B and D are all out.

This type of question visibly needs you to combine statements in order to get anywhere. There is a danger in combining statements without thinking, because there is often a relationship that’s just hard enough to detect linking the two statements that gets test-takers thinking they’re on the right track. In this question, the fact that 105$ is 7 times the luxury car premium of 15$ makes it feel like 7 more luxury cars were rented than compact cars. This type of connector is hard enough to see that people feel encouraged that they’ve stumbled upon something useful. Unfortunately, when you’re feeling clever is when you’re most vulnerable to fall into a GMAT trap (Something about pride going before a fall).

Let’s delve into these numbers a little. If 7 more luxury cars got rented than compact cars, and the numbers add up to 25, then that means the company rented 16 luxury cars and 9 compacts. If we stop here, we might think that the answer is C. However, applying arbitrary numbers might make us realize the error of our ways. Let’s say a compact car is 100$ an hour (easy number to work with). This makes the luxury cars 115$. We can quickly calculate that the compact cars will bring in exactly (9×100) 900$. The luxury cars will bring in well over 1600$. These two numbers don’t respect the 105$ difference mentioned in statement 2. Why is that? Maybe I picked the wrong prices? Let’s go smaller: 20$ compacts and 35$ luxury cars. That’s 180$ for the compacts and 525$ for the luxury cars. We’re getting closer, but this still doesn’t work. What’s happening?

The number of cars we chose (16 luxury cars and 9 compacts) has a solution, but it’s not one that makes any real world sense. Solving for the two equations and two unknowns with our chosen number of cars:

L+C = 25

L(x+15) = C*x+105

Replacing L by 16 and C by 9

16(x+15) = 9*x+105

16x+240 = 9x+105

7x = -135

x = -19.286

That’s right, this solution works if we give people 19$ to rent compact cars and only 4$ to rent out luxury cars. Clearly this solution does not work in the real world because it does not mean what we expected. On test day, you don’t have to go through the actual math to solve for x, but being able to recognize that renting out 16 cars at a 15$ premium will yield at least (16*15) = 240$ more dollars for the luxury line than the compact line. The relationship of 7 additional cars only works if we rent a total of 7 cars, all luxury liners. Any other rental will throw off this delicate balance, highlighting that it was nothing but a mathematical mirage.

So what’s the answer to this question? As many of you probably figured out, it’s just going to be answer choice E. There are multiple values that will work (and even be positive) for the two constraints given. Many test takers can solve these questions without having to write a single digit down. However, if you’re ever unsure, write down a few numbers and see what they tell you. The reason some people dislike math is the same reason some people love math: it tells the truth. If your understanding of the question is shoddy, a couple of concrete numbers will tell you more than all the x’s and y’s in an alphabet soup (or a Jerry Springer show).