When you’re asked a Yes/No Data Sufficiency question that asks whether an algebraic relationship is true, play The Imitation Game. Which means: if you can get one of the statements to directly imitate the question, you can definitively get the answer “yes” and prove that it’s sufficient.

Consider a few examples of questions that make for great Imitation Game candidates:

Is x – y > a – b?

(1) x + b > a + y

Here you can try to imitate the question with the statement. You want the statement to look more like the question, where x and y are paired together on the left and a and b are paired together on the right. so subtract y from both sides (to get it from the right to the left) and subtract b from both sides (to move it to the right), and the statement becomes:

x – y > a – b

Which directly answers the question “yes” – the question asks if the relationship is true, and by using the statement to imitate the question you can get the statement to directly answer it.

If the product abc does not equal 0, does a/b = c?

(1) bc = a

Here you can again use the statement to imitate the question, dividing both sides by b to get c on its own (which you’re allowed to do since no values are 0), and you have your answer:

c = a/b

Sometimes you’ll be able to imitate the question to get a definite “no” answer, which is still sufficient:

Is x – y > a – b?

(1) a > x and y > b

Here you can combine the inequalities to get them all in to one inequality. By adding the inequalities together (which you can do since the signs point in the same direction), you have:

a + y > x + b

And then you want to imitate the question, which has a and b on one side and x and y on the other. So subtract y and b from both sides to get:

a – b > x – y

Which is the opposite of the question, and therefore says “no, x – y is not greater than a – b” providing you with sufficient information.

The real lesson here? When you’re being asked a yes/no question with lots of algebra, it pays to play The Imitation Game. See if you can get the statement to imitate the question, and you’ll often find that it directly answers the question.

But be careful! As the second example showed you, you need to be careful when diving into algebra that you don’t:

*Divide by a variable that could be 0

*Multiply or divide by a variable in an inequality if you don’t know the sign

Keep those two caveats in mind and you can imitate math legend Alan Turing while you play the Data Sufficiency Imitation Game. And the winner is…you.

]]>**You’re being taken for a ride by a corrupt mechanic.**

Let’s explain. The GMAT testmakers are committed to testing the same concepts over and over again: Modifiers, Verbs, Pronouns, Parallel Structure, Logical Meaning… And at a certain point it’s difficult to make those concepts any harder; they are what they are. So the testmakers resort to a time-honored tradition among corrupt mechanics; when oil changes and tire rotations and front-end-alignments aren’t bringing in enough profit, what do corrupt mechanics do?

**They fix things that don’t need to be fixed. **

The corrupt mechanic never simply fixes, flushes, or replaces the part you came in asking about; he always “strongly recommends” that you add on another service. If you’re not careful, your $30 oil change becomes a several-hundred dollar outing and your car comes back with shiny new parts that replaced perfectly-functional components, all with a nice labor surcharge on top. As Seinfeld’s George Costanza put it:

*Well of course they’re trying to screw you! What do you think? That’s what they do. They can make up anything; nobody knows! “Why, well you need a new Johnson rod in here.” Oh, a Johnson rod. Yeah, well better put one of those on! *

Now, in defense of the GMAT testmakers, they’re not trying to steal your money for unnecessary services. But in their quest to reward the kinds of business skills that are associated with avoiding unnecessary expenses and wasted time on ineffective initiatives, the GMAT testmaker does act like a Corrupt Mechanic on Sentence Correction problems. By fixing problems that don’t need fixing, the testmaker steals your **attention**, not your money. And in doing so, the testmaker baits the unwitting into bad decisions, while also rewarding those who prioritize their Decision Points properly. Consider this example:

Immanuel Kant’s writings, while praised by many philosophers for their brilliance and consistency, are characterized by sentences so dense and convoluted as to pose a significant hurdle for many readers who study his works.

(A) so dense and convoluted as to pose

(B) so dense and convoluted they posed

(C) so dense and convoluted that they posed

(D) dense and convoluted enough that they posed

(E) dense and convoluted enough as they pose

To those who know their role in the GMAT, the verb difference along the right hand side of the answer choices should loom large. “Pose” (present) vs. “Posed” (past) is a very actionable decision and a very common decision on the test. Like an oil change or the replacement of brake pads, verb tense decisions are something you should do regularly! So what does the Corrupt Mechanic do? He takes something uncomfortable – the structure “so dense and convoluted as to…” – but that doesn’t need fixing, and it fixes it. And since that choice comes along the left-hand side, many of us go right along with that and eliminate A with a preference for the more-familiar structures in B and C, without ever realizing that we’ve been “Johnson rodded” into ignoring the ever-important verb tense decision at the ends of the choices.

That’s how the testmaker’s Corrupt Mechanic works in Sentence Correction. He changes things that didn’t need changing and dares you to accept those “repairs” as necessary. So how can you avoid these traps? Be a savvy customer. Know what you want before you start listening to the Corrupt Mechanic’s menu of possible changes; you want to make Verb, Modifier, Pronoun, and Parallelism decisions before you even listen to anything else. Make the common repairs first, and then with the choices that are left you can start to get creative with add-ons.

The GMAT testmakers act like Corrupt Mechanics when they write Sentence Correction problems, so beware that not every change is actually a necessary repair. It’s on you to determine which fixes truly need to be made, so stick to the recommended SC maintenance schedule – the errors most commonly tested – and you’ll avoid falling victim to the Corrupt Mechanic.

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*

**Be All About That Base.**

What does that mean? Nearly every exponent rule you’ll learn requires common bases. For example:

So when you’re presented with an exponent problem, one question you should always ask yourself is “Can I get all these terms to have the same base?”. That step allows you to use the exponent rules you’ve memorized to solve complicated problems. Consider an example:

For integers a and b, 16^a = 32^b. Which of the following correctly expresses a in terms of b?

(A) = 2^b

(B) a = 4^b

(C) a = 2^5b-4

(D) a = 4^5b-4

(E) a = 2^5b

Here there’s only one exponential term, 32 to the b power. But if you recognize that both 32 and 16 are powers of 2, you can quickly transform the problem, coming up with:

2^4 * a = (2^5)^b

And that allows you to dive right into exponent rules. First deal with the parentheses on the right, using the third exponent rule in the list above so that that term becomes 2^5b. That means you have:

2^4 * a = 2^5b

Then to isolate a, you divide both side by 2^4, getting to:

a = 2^5b / 2^4

And now since your bases are the same, you can use the second exponent rule in the list above to subtract the exponents and get to:

a = 2^(5b – 4), matching answer choice C.

More important than this problem is the lesson: when problems deal with exponents, and particularly with non-prime bases (like 16 and 32), one of your first mantras should be “All About That Base (no treble).” See if you can get multiple terms to have the same base, and you can simplify the expression using common exponent rules. Then, with your monster GMAT quant score, Harvard can take its Blank Space and write your name…

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*

Well, if you’re a GMAT student, you can think about what the odds mean in terms of probability and you can watch the announcers miss Critical Reasoning lesson after Critical Reasoning lesson. For example:

**Probability**

Before the last piece of confetti hits the turf on Sunday, oddsmakers will have posted their odds on next year’s winner. For example, New England and Seattle might open at 4:1, Green Bay might come in at 7:1, etc. And while you might look at those odds and think “if I bet $100 on the Packers I’ll win $700!” you should also think about what those mean. 7:1 for Green Bay is really a ratio: 7 parts of the money says that Green Bay will not win, and 1 part says that it will. So that’s a good bet if you think that Green Bay has a better than 1 out of 8 chance (so better than 12.5%) to win next year’s Super Bowl. And if those are, indeed, the odds (4:1 for two teams and 7:1 for another), Vegas is essentially saying that there’s a less than likely chance (1/5 + 1/5 + 1/8 = 52.5% chance that one of those two teams wins) that someone other than Green Bay, New England, or Seattle will win next year.

So consider what the probability of those bets means before you make them. Individually odds might look tempting, but when you consider what that means on a fraction or percent basis you might have a different opinion.

**Probability #2**

As you watch the Super Bowl, there’s a high likelihood that at some point the screen will start showing a line indicating the season-long field goal for either Steven Hauschka or Stephen Gostkowski (the Seattle and New England kickers…there’s a huge probability that someone named Steve will be incredibly important in this game!). And the announcers will use that line to say that it’s likely field goal range for that team to win or tie the game.

Where’s the flawed logic? If that’s the longest field goal he’s made all year, is it really likely that he’ll make another one from a similar spot with all that pressure? Or, in the case of a low-scoring game like many predict between these two elite defenses, how likely is either kicker to make two consecutive field goals from a relatively far distance?

Sports fans are pretty bad with that probability. Say that a kicker has been 70% accurate from over 50 yards. Is it likely that he’ll make two straight 50-yard field goals on Sunday (assuming he gets those attempts)? Check the math: that’s 7/10 * 7/10 or 49/100 – it’s less than likely that he makes both! Even a kicker with 80% accuracy is only 8/10 * 8/10 = 64% likely to make two in a row…meaning that fail to perform that feat 1 out of every 3 times he had the chance! Think of the probability while announcers talk about field goals as a near certainty on Sunday.

**Critical Reasoning**

The announcers on Sunday will try to use all kinds of data to predict the outcome, and in doing so they’ll give you plenty of opportunities to think critically in a Critical Reasoning fashion. For example:

“For the last 40 Super Bowls, the team with the most rushing yards has won (some massive percent) of them; it’s important for New England to get LeGarrette Blount rolling early.”

This is a classic causation/correlation argument. Do the rushing yards really win the game? It could very well be true (Weaken answer!) that teams that build a big lead and therefore want to run out the clock run the ball a lot in the second half (incomplete passes stop the clock; runs keep it going). Winning might cause the rushing yards, not the other way around.

Similarly, the announcers will almost certainly make mention at halftime of a stat like:

“Team X has won (some huge percentage) of games they were leading at halftime, so that field goal to put them up 13-10 looms large.”

Here the announcer isn’t factoring in a couple big factors in that stat:

-A 3-point lead isn’t the same as a 20-point lead; how many of those halftime leads were significantly bigger?

-You’d expect teams leading at halftime to win a lot more frequently; based on 30 minutes they may have shown to be a better team plus they now have a head start for the last 30 minutes. Over time those factors should bear out, but in this one game is a potentially-flukey 3-point lead significant enough?

Regardless of how you watch the game, it can provide you with plenty of opportunities to outsmart friends and announcers and sharpen your GMAT critical thinking skills. So while Tom Brady or Russell Wilson runs off the field yelling “I’m going to Disneyland!”, if you’ve paid attention to logical flaws and probability opportunities during the game, you can celebrate by yelling “I’m going to business school!”

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*

Some will say it’s a heinous act committed by serial cheaters. Others will say it’s a minor violation and that “everybody does it.” And still others will say it’s an inadvertent mistake that happened to run afoul of a technicality. What does it mean for you, a GMAT aspirant?

**Be careful about honest mistakes that could be construed as cheating!**

While the NFL isn’t going to kick the Patriots out of the Super Bowl, the Graduate Management Admission Council won’t hesitate to cancel your score if you’re found to be in violation of its test administration rules. So beware these rules that honest examinees have accidentally violated:

**1. You cannot bring “testing aids” into the test center.**

Don’t bring an Official Guide, a test prep book, or study notes into the test center with you. You may want to have notes while you’re waiting to check in, but if you’re caught with “study material” in your hands during one of your 8-minute breaks – which has happened to students who were rearranging items in their lockers to grab an apple or a granola bar – you’ll be in violation of the rule, and GMAC has cancelled scores for this in the past. Don’t take that risk! Leave watches, cell phones, and study aids in your car or at home so that there’s no chance you violate this rule simply by having a forbidden item in your hand during a break.

**2. You cannot talk to anyone about the test during your administration.**

You’ll be at the test center with other people, and someone’s break might coincide with yours. Holding a restroom door or crossing paths near a drinking fountain, you might be tempted to socially ask “how is your test going?” or sympathetically mention “man these tests are hard.” But since those innocent phrases could be seen as “talking about the test” you would technically be in violation of the rule, and GMAC has cancelled scores for this in the past. Your 8-minute break isn’t the time to make new friends – don’t take the risk of being caught talking about the test.

You know that you’re not a cheater, but as most New Englanders feel today it’s very possible to be considered a cheater if you end up on the wrong side of a rule, however accidentally. Learn from the lessons of test-takers before you: avoid these common mistakes and ensure that the score you earn is the score you’ll keep.

*By Brian Galvin*

Thou doth protest too much. Meaning:

*We all think we can write verbal questions better than the authors of the test.*

When it comes to GMAT verbal questions, we critique but don’t solve Critical Reasoning problems, we correct rather than solve Sentence Correction problems, and we try to write but don’t thoroughly read Reading Comprehension questions. And this hubris can be the death of your GMAT verbal score, even if it comes from a good place and a good knowledge base.

Wander into a GMAT class or scan a GMAT forum and you’ll see and hear tons of comments like:

“I feel like the question should say *people* and not *individuals*.”

“I would never use the word imply like that.”

“I don’t think that’s the right idiom.”

“I would have gotten it right if it said X…I think it should have said X.”

Or you’ll hear questions like:

“But what if answer choice D said *and* and not *or*?”

“If that word were different would my answer choice be right? And if so would it be more right than B?”

And while these questions often come from a genuine desire to learn, they more often come from a place of frustration, and they’re the type of hypothetical thinking that doesn’t lend itself to progress on this test. Even if it’s not always perfect, **the GMAT chooses its words very carefully**. When the word in the Reading Comprehension correct answer choice isn’t the word you were hoping it would be (but it’s close), they picked that word for a reason – it makes the problem more difficult. When none of the Sentence correction answer choices match the way you or your classmates would have phrased it, that’s not a mistake – that’s an intentional device to make you eliminate four flawed answers and keep the strange-but-correct one. The GMAT can’t always match your expectations, not just because doing so would make it too easy but also because it’s trying to test other critical-thinking skills. It has to test your ability to see less-clear relationships, to make logical decisions amidst uncertainty, to find the least of five evils, and it has to punish you for jumping to unwarranted conclusions.

GMAT verbal is constructed carefully, and as you study it you have to learn how to answer questions more effectively, not to write better questions. The only thing you get to write on test day is the AWA essay; everything else you must answer on the GMAT’s terms, not on your own, so as you study you have to resist the urge to protest the problem and instead learn to see the value in it.

So as you study, remember your mission. Your job isn’t to find a flaw with the logic of the question, but rather with the logic of the four incorrect answers. When you get mad at a wrong answer, use that energy to attack the next problem with the lessons you learned from that frustrating mistake. Take the GMAT as it is and don’t try to justify your mistakes or fight the test.

Save your writing energy for the AWA essay; on the verbal section, you only get to answer the problem in front of you. When you accept that the test is what it is and commit yourself to learning how to attack it through critical thinking and not just general angst, you’ll have a competitive advantage over most frustrated examinees. Think like the testmaker, but don’t try to be the testmaker.

*By Brian Galvin*

As a GMAT student who wants to make 2015 the year of the elite MBA acceptance letter, how can you be among the disappointingly-few who keep up this week’s excellence exuberance?

Keep it simple.

The problem with most New Year’s Resolutions and GMAT study plan’s is that they’re far too ambitious. Hatched over eggnog and 7-10 days of paid vacation, these plans are destined to failure because they’re way too much for anyone to adhere to in the long term. They often read like:

“I’ll get up 90 minutes before I normally do and study over a healthy breakfast, then after work three days a week I’ll go the library, and every Saturday I’ll take a practice test and spend Sunday mornings with a tutor reviewing it all.”

“I’ll take a leave of absence from work so that I can study 40-50 hours a week for three months, then I’ll take the GMAT in the spring and get a high score, then volunteer all summer to demonstrate my community service, then apply round 1 to Harvard/Stanford/Wharton, and maybe throw Yale or London Business School in the mix as a safety school.”

“I’ll turn off my smartphone and give up social media for the next few months, study at least 90 minutes a day, and….”

And the problem with those study plans? You’ll resent them within a week, just like most New Year’s Resolvers resent their no-carb / all-lettuce diets and overpriced gym memberships. You have to come up with a study plan that:

1) You can fit in to your lifestyle so that you can keep to it.

This means that you factor in your hobbies and, yes, limitations. If you’re not a morning person, you won’t keep to a schedule of studying every morning before work. If you thrive on a good workout, giving up your soccer league or gym regimen completely won’t work either. And friends, family, work functions, etc. are always important.

2) You can build on.

The best study plans are those that start a bit smaller and build into something more robust, like a “Couch to 5k (or marathon)” training program. If you want to run a marathon, you start with a couple miles and build up to 18-20 milers as your body is ready for it. If you want a 700 on the GMAT, you start with a handful of study sessions per week and build into longer sessions when they’re more purposeful and you know what you’re using the time to work on.

3) Focus on achievement, not activity.

Veritas Prep emphasizes the famous John Wooden quote “never mistake activity for achievement”, meaning that simply spending 4 hours studying Sentence Correction, for example, isn’t going to get the job done; it’s the quality of study that helps. So hold yourself accountable for goals, not time spent. Think in terms of “I want to do 25 SC problems focusing on major error categories first, then thinking of logical meaning second”

or “I’m going to practice applying right triangle principles to geometry problems” or “I’m going to do a timed drill to force myself to think more quickly.” Give your study sessions themes and achievement goals, and they’ll not only be more productive but they’ll also be more fun.

So what does a productive, sustainable study schedule look like?

*It’s firm but flexible. Plan to study at least 3 times per week, but let yourself move Tuesday’s session to Wednesday if you get tickets to a Tuesday concert or you work late and just need to blow off steam with a run. You have to get those sessions in, but you don’t have to resent them or go through the motions just to stick to your (probably arbitrary) schedule.

*It’s achievement-driven. Your study sessions have themes and goals, not just durations.

*It’s reasonable. Know yourself and your preferences and limitations.

Very few people can study for hours every day, so schedule something you can commit to – a few sessions per week, maybe two weeknights and one weekend morning, or something that you know you can hold yourself accountable to.

*It’s custom-built. Think about when you’ve been most successful in other academic pursuits and try to replicate that. Do you study better in the morning? In the evening? With friends or music? Alone? After a good workout? With a snack? Build your plan around your own successes.

*It’s built to expand. 2-3 study sessions a week may very well not be enough for you, so be honest with yourself once you’ve up and running. Do you need more time to master algebra? Do you need to build in a class or On Demand program to supplement your practice? Do you have enough time for practice tests? Once you’re committed to a bsseline study regimen, you need to be honest with yourself about what you need, and at that point it’s often easier to bite the bullet and dive into something more intense.

But in the beginning, make sure you have a schedule/plan that you won’t quit before your neighbors even take their Christmas lights down.

January is a great time to make plans for self-improvement, but most of those plans never live to see February. To ensure that your New YEAR’s Resolution to succeed on the GMAT isn’t limited to one month or less, resolve to plan on something that will last. If you can do that, we’ll see you back in this GMAT Tip of the Week every Friday until you have that score you’re looking for.

*By Brian Galvin*

If your New Year’s Resolution is to make 2015 the year that you ace the GMAT, you can take a lesson from this time of year. The darkest points always give way to enlightenment, and that secret will get you through some very difficult GMAT problems. There are two very common structures for challenging GMAT quant problems:

1) It looks easy, but the last step or two are tricky.

2) It looks impossible, but once you’ve found the right foothold it gets easy quickly.

This post is all about #2, those problems where it looks incredibly dark right up until that moment that you reach enlightenment. Veritas Prep’s own Jason Sun recounts the first quant question en route to his official 780 score: “I stared at a nasty sequence problem for probably 45 seconds with my jaw open thinking ‘there’s no way to solve this’. Then I remembered the strategy of starting with small numbers and finding a pattern, and 10 seconds later the answer was obvious.”

That’s common on the GMAT, and step one for you is to realize that problems are designed to look like that. When things look darkest, have faith that they’ll clear up. Here are a few ways that that occurs on the GMAT.

**Calculations look awful, but work themselves out before you get to the answer.**

Consider this problem:

If the product of the integers a, b, c, and d is 1,155 and if a > b > c > d > 1, then

what is the value of a – d?

(A) 2

(B) 8

(C) 10

(D) 11

(E) 14

Upon first glance, 1155 and four variables might look really messy. But take the first step – you know it’s divisible b y 11 and that you have to factor it. 1100 is 11*100 and 55 is 11*5, so you have 11*105. And 105 is much easier to divide out since it ends in a 5. That’s 21*5, which is 7*3*5. Once you’ve factored it down, it’s 11*7*5*3, which are all prime, so when 1 has to be less than any of these, that’s exactly a, b, c, and d. You need the biggest minus the smallest, and 11-3 is 8. What may have looked like a big, intimidating number was actually not so bad once you took the first step. It’s always darkest before the light goes on.

**The problem is abstract, but comes into focus when you test small numbers.**

What is the units digit of 2^40?

(A) 2

(B) 4

(C) 6

(D) 8

(E) 0

2^40 is an insanely large number. You’ll never be able to calculate it. But if you take the first few steps with small numbers, you’ll see a pattern:

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

2^5 = 32

2^6 = 64

2^7 = 128

2^8 = 256

And since you only care about the units digits, you should see a pretty firm pattern emerging. 2, 4, 8, 6, 2, 4, 8, 6. If you repeat through this pattern, you’ll see that every 4th number is a 6, and since 2^40 will be the finish of the tenth run of that every-fourth-number cycle, the answer has to be 6. The GMAT loves to give you problems with big or abstract numbers that seem unfathomable, but if you test properties with small numbers you can often find a pattern or some other way to determine what you have.

**It’s always the last place you look.**

Another common theme is specific to geometry problems – the GMAT often constructs them so that a seemingly irrelevant piece of information (like the measure of a far, far away angle, or the area of a figure when you’re only solving for the length of one line) is crucial to the answer…it’s just that you don’t even consider filling in that piece of information that seems so far away from what you’re really trying to solve for. So FILL IN EVERYTHING! Even if it seems irrelevant, fill in every piece of information you can solve for and you’ll give yourself a better shot of finding that unlikely relationship that cracks the code.

**You’re not supposed to be able to solve for it, but you can estimate or use answer choices.**

Plenty of GMAT questions beg you to do some horrifying math, but if you look at the answer choices ahead of time you can see that they’re either spread incredibly far apart and ready to be estimated or they have easy-to-plug-in properties that allow you to just test them. It’s crucial to remember that the GMAT isn’t a test of pure math, but of problem solving using math. Heed this advice: if you think the calculations are too detailed to do in two minutes, you’re probably right. That’s when you should look to estimate or backsolve.

So if your GMAT study sessions are growing longer as the daylight does, keep this wisdom in mind. It always looks darkest before sunrise, and the same is true of many tough GMAT quant problems. As you struggle through practice problems, pay attention to all those times that the solution wasn’t nearly as bad as it seemed it would have to be upon first glance.

*By Brian Galvin*

The other answer we can give you? Sarah Koenig would do pretty darned well on Data Sufficiency questions, where often it’s just as important to determine what you don’t know as it is to determine what you do. While the internet buzzed with theories certain that Adnan did it, that Jay did it, that a recently-released serial killer did it, Koenig was often ridiculed for being so noncommittal in her assessment of whether Adnan is guilty or not. But that’s an important mentality on Data Sufficiency questions, as one of the common ways that the GMAT will bait you is giving you information that seems overwhelmingly sufficient (The Nisha call! The phone was in Leakin Park!) but that leaves just enough doubt (Why did Jay’s story change so much?) that you can’t prove a definitive answer. And like the jury in the Serial case, we all have that tendency to jump to conclusions (“well if he didn’t kill her, who did?”) and filter out information that we don’t like (Christina Gutierrez’s performance…). This Serial-themed Data Sufficiency problem should exemplify (forgive the lack of subscript formatting, but a sequence problem in a Serial blog post seemed fitting):

The infinite (serial) sequence a1, a2, …, an, … is such that a1 = x, a2 = y, a3 = z,a4 = 3 and an = a(n-4) for n > 4. What is the sum of the first 98 terms of the sequence?

(1) x = 5

(2) y + z = 2

As people unpack the mystery in this problem, they start to see what’s going on. If an = a(n-4), then each term equals the term that came four prior. So the sequence really goes:

x, y, z, 3, x, y, z, 3, x, y, z, 3…

So although it looks like a pretty massive mystery, really you’re trying to figure out x, y, and z because 3 is just 3. And here’s a common way of thinking:

Statement 1 is not sufficient, but it gets you one of the terms. And Statement 2 is not sufficient but it gets you two more. So when you put them together, you know that the sum of one trip through the 4-term sequence is 5 + 2 + 3 = 10, so you should be able to extrapolate that to the whole thing, right? Just figure out how many trips through will get you to term 98 and you have it; like the Syed jury, you have the motive and the timeline and the cell phone records and Jay’s testimony, so the answer has to be C. Right?

But let’s interview Sarah Koenig here:

Sarah: The pieces all seem to fit but I’m just not so sure. Statement 2 looks really bad for him. If we can connect those dots for y and z, and we already have x, we should have all variables converted to numbers. Literally it all adds up. But I feel like I’m missing something. I can definitely get the sum of the first 4 terms and of the first 8 terms and of the first 12 terms; those are 10, and 20, and 30. But what about the number 98?

And that’s where Sarah Koenig’s trademark thoughtfulness-over-opinionatedry comes in. There is a giant hole in “Answer choice C’s case” against this problem. You can get the sequence in blocks of 4, but 98 is two past the last multiple of 4 (which is 96). The 97th term is easy: that’s x = 5. But the 98th term is tricky: it’s y, and we don’t know y unless we have z with it ( we just have the sum of the two). So we can’t solve for the 98th term. The answer has to be E – we just don’t know.

Now if you’ve heard yesterday’s episode, think about Dana’s “think of all the things that would have to have gone wrong, all the bad luck” rundown. “He lent his car and his phone to the guy who pointed the finger at him. That sucks for him. On the day that his girlfriend went missing. That’s awful luck…” And in real life she may be right – that’s a lot of probability to overcome. But on the GMAT they hand pick the questions. On this problem you can solve for the 97th term (up to 96 there are just blocks of 4 terms, and you know that each block sums to 10, and the 97th term is known as 5) or the 99th term (same thing, but add the sum of the 98th and 99th terms which you know is 2). But the GMAT hand-selected the tricky question just like Koenig hand-selected the Adnan Syed case for its mystery. GMAT Data Sufficiency questions are like Serial…it pays to be skeptical as you examine the evidence. It pays to think like Sarah Koenig. Unlike Jay, the statements will always be true and they’ll always be consistent, but like Serial in general you’ll sometimes find that you just don’t have enough information to definitively answer the question on everyone’s lips. So do your journalistic due diligence and look for alternative explanations (Don did it!). Next thing you know you’ll be “Stepping Out!!!” of the test center with a high GMAT score.

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*By Brian Galvin*

However…

There are three knee-jerk questions that you should plug (if not chug) into your brain to ask yourself every time you see a Min/Max problem before you ask that fourth question “What’s my strategy?”:

- Do the numbers have to be integers?
- Is zero a possible value?
- Are repeat numbers possible?

In the Veritas Prep Word Problems lesson we refer to these problems as “scenario-driven” Min/Max problems precisely because of the above questions. The scenario created by the problem drives the whole thing, related mainly to those three above questions. Consider these four prompts and ask yourself which ones can definitively be answered:

*#1: “Four friends go fishing and catch a total of 10 fish. How many fish did the friend with the highest total catch?”*

*#2: “Four friends go fishing and catch a total of 10 fish. If no two friends caught the same number of fish, how many fish did the friend with the highest total catch?”*

*#3: “Four friends go fishing and catch a total of 10 fish. If each friend caught at least one fish but no two friends caught the same number, how many fish did the friend with the highest total catch?”*

*#4: “Four friends go fishing and catch a total of 10 pounds of fish. If each friend caught at least one fish but no two friends caught the same number, how many pounds of fish did the friend with the highest total catch?”*

Hopefully you can see the progression as this set builds. In the first problem, there’s clearly no way to tell. Did one friend catch all ten? Did everyone catch at least two and two friends tied with 3? You just don’t know. But then it gets interesting, based on the questions you need to ask yourself on all of these.

With #2, two big restrictions are in play. Fish must be integers, so you’re only dealing with the 11 integers 0 through 10. And if no two friends caught the same number there’s a limited number of unique values that can add up to 10. But the catch on this one should be evident after you’ve read #3. Zero *IS* possible in this case, so while the totals could be 1, 2, 3, and 4 (guaranteeing the answer of 4), if the lowest person could have caught 0 (that’s where “min/max” comes in – to maximize the top value you want to minimize the other values) there’s also the possibility for 0, 1, 2, and 7. Because the zero possibility was still lurking out there, there’s not quite enough information to solve this one. And that’s why you always have to ask yourself “is 0 possible?”.

#3 should showcase that. If 0 is no longer a possibility *AND* the numbers have to be integers *AND* the numbers can’t repeat, then the only option is 1 (the new min value since 0 is gone), 2 (because you can’t match 1), 3, and 4. The highest total is 4.

And #4 shows why the seemingly-irrelevant backstory of “friends going fishing” is so important. Pounds of fish can be nonintegers, but fish themselves have to be integers. So even though this prompt looks very similar to #3, because we’re no longer limited to integers it’s very easy for the values to not repeat and still give wildly different max values (1, 2, 3, and 4 or 1.5, 2, 3, and 3.5 for example).

As you can see, the scenario really drives the answer, although the fourth question “What is my strategy?” will almost always require some real work. Let’s take a look at a couple questions from the Veritas Prep Question Bank to illustrate.

**Question 1:**

Four workers from an international charity were selling shirts at a local event yesterday. Did one of the workers sell at least three shirts yesterday at the event?

(1) Together they sold 8 shirts yesterday at the event.

(2) No two workers sold the same number of shirts.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

(C) Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient

(D) EACH statement ALONE is sufficient to answer the question asked

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed

Before you begin strategizing, ask yourself the three major questions:

1) Do the values have to be integers? YES – that’s why the problem chose shirts.

2) Is zero possible? YES – it’s not prohibited, so that means you have to consider zero as a min value.

3) Can the numbers repeat? That’s why statement 2 is there. With the given information and with statement 1, numbers can repeat. That allows you to come up with the setup 2, 2, 2, and 2 for statement 1 (giving the answer “NO”) or 1, 2, 2, and 3 (giving the answer “YES” and proving this insufficient).

But when statement 2 says on its own that, NO, the numbers cannot repeat, that’s a much more impactful statement than most test-takers realize. Taking statement 2 alone, you have four integers that cannot repeat (and cannot be negative), so the smallest setup you can find is 0, 1, 2, and 3 – and with that someone definitely sold at least three shirts. Statement 2 is sufficient with really no calculations whatsoever, but with careful attention to the ever-important questions.

**Question 2:**

Last year, Company X paid out a total of $1,050,000 in salaries to its 21 employees. If no employee earned a salary that is more than 20% greater than any other employee, what is the lowest possible salary that any one employee earned?

(A) $40,000

(B) $41,667

(C) $42,000

(D) $50,000

(E) $60,000

Here ask yourself the same questions:

1) The numbers do not have to be integers.

2) Zero is theoretically possible (but probably constrained by the 20% difference restriction)

3) Numbers absolutely can repeat (which will be very important)

4) What’s your strategy? If you want the LOWEST possible single salary, then use your answer to #3 (they can repeat) and give the other 20 salaries the maximum. That way your calculation looks like:

x + 20(1.2x) = 1,050,000

Which breaks out to 25x = 1,050,000, and x = 42000. And notice how important the answer to #3 was – by knowing that numbers could repeat, you were able to quickly put together a smart strategy to minimize one single value.

The larger lesson is crucial here, though – these problems are often (but not always) fairly basic mathematically, but derive their difficulty from a situation that limits some options or allows for more than you’d think via integer restrictions, the possibility of zero, and the possibility of repeat values. Ask yourself these four questions, and your answer to the first three especially will maximize your efficiency on the strategic portion of the problem.

*By Brian Galvin*