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Geometry is an important part of any GMAT test-taker’s conceptual toolkit. On Data Sufficiency geometry questions, it’s especially key to have an intuitive feel for what is and is not solvable given certain bits of information. Consider the following difficult problem:

A circle having center O is inscribed in triangle ABC. What is the measure of angle BAC?

- The radius of the circle is 2.
- Segment OA has length 4.

(A) Statement (1)Â ALONE is sufficient, but statement (2)Â 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)Â TOGETHER are sufficient to answer the question asked, butÂ NEITHER statement ALONE is sufficient to answer the question asked.

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

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

There are two ways to attack a problem like this. At the end of this article is an extremely well-thought out, coldly-reasoned, logical,** academic** explanation. While interesting, in terms of the GMAT it is an **absolutely useless** way to think about the question. It just takes too much time! Instead, you should train yourself to approach problems like these intuitively. Here’s how:

First, check out this simple geometry tool. It’s a handy JAVA applet that lets you *see* how the problem works visually. If you take a few moments to play with the applet, you may be able to get a better *intuitive *feel for the restrictions created by statements (1) and (2).

Try keeping the radius the same while changing the size and shape of the triangle. Notice the angle BAC changing? Next try keeping the length of segment OA the same while moving the circle around to change its radius. See the angle changing again?

By thinking intuitively, you can tell that neither statement is sufficient on its own. When the radius and segment length are fixed, though, it’s another story. Once you know both these pieces of information, you can tell you’re dealing with a 30-60-90 right triangle (more on this below), so finding the measure of angle BAC is a matter working with triangle properties.

That’s why the answer here is C — statements (1) and (2) together are sufficient, but neither is sufficient alone.

The moral here is to avoid wordy reasoning in geometry whenever possible. Practice the art of visualization. You can’t use a nifty applet on test day, but you can draw pictures of extreme cases and move the segments around in your head. This kind of intuitive reasoning is essential on Data Sufficiency geometry questions —where time is short and diagrams are seldom drawn to scale.

Now that you’re thinking visually, take a look at the wordy explanation. Really — do not read the following explanation until you play with the applet! There are tons of sites out there like the one I mentioned above. Spending some time playing with the possible orientations of triangles and circles is going to build your geometric intuition, which will only help your GMAT score.

Here’s the **wordy explanation**. Look out.

Note that at the three points where the circle touches the triangle, the radius of the circle connecting these points to the center of the circle is going to be perpendicular to each of the triangle’s sides.

Statement 1 tells us that the radius of the circle is 2. Although this defines the circle entirely, there are many possible triangles in which a circle of radius 2 could be inscribed — imagine that the triangle that this circle sits inside of is NOT an equilateral triangle. It is certainly possible that triangle ABC is scalene, in which case each angle has a different measure. Since Statement 1 makes no mention of any of the triangle’s vertices, angles, or sides, note that by simply relabeling the vertices of a scalene version of triangle ABC, we could have different measures of angle BAC. Statement 1 is not sufficient.

Statement 2 tells us that segment OA has length 4. Although this gives us more specific information about the vertex named in angle BAC, note that, again, this statement makes no restrictions on the measure of the angle BAC. Imagine a really skinny acute triangle where angle BAC is very small, and the distance between point A and the segment BC is very close to the length of segment OA = 4. Or imagine a really fat obtuse triangle where angle BAC is very large, and the distance between point A and the segment BC is just slightly less than twice the length of segment OA = 4. In the first case, the measure of angle BAC is small, in the second case, the measure of angle BAC is close to 180 degrees. Statement 2 is not sufficient.

Taken together, the two statements tell us two important pieces of information about the smaller triangle AOD, where D is the point where the circle touches segment AC. Segment OD is a radius of the circle, so from statement 1 we know that it has length 2, and statement 2 tells us that OA has length 4. Knowing that the angle ODA is 90 degrees (remember that the radii of the circle are perpendicular to the triangle at each of the points of tangency) tells us that the triangle OAD is a right triangle. Furthermore, we know that the hypotenuse of the right triangle, OA, is twice the length of the leg, OD, the radius of the circle of length 2. This is beginning to look a lot like a 30-60-90 triangle. Noting that it would violate the triangle inequality if the leg DA were less than or equal to the length of leg OD, we can conclude that the leg OD = the radius of the circle is the shortest leg of the triangle, and we do indeed have a 30-60-90 triangle.

Now we can conclude that angle OAD is equal to 30 degrees, being opposite the shortest leg, but what about angle OAE? We need to know the measure of this angle in order to find the measure of angle BAC. Symmetry applies here. All statements applied to segments OD and DA also apply to OE and EA because triangles AOD and AOE are congruent — they share side OA, angles OEA and ODA are equal (right angles), and the triangles have their shortest sides equal to the radius of the circle. Hence the measure of angle BAC is twice the measure of angle OAD: 2 Ã— 30 = 60. The statements together are sufficient.

]]>Nothing, that is, except yourself. On test day, your pulse quickens, your hands sweat, and your mind races. You find yourself concentrating on everything that *isn’t* the test in front of you. You actually begin to worry about the fact that you are worrying, until you realize that you have just spent 5 minutes staring at the first question on the GMAT quantitative section without even *beginning *to find a solution.

Sound familiar?

If it does, you are not alone. Everyone, at one time or another, has felt the negative effects of test performance anxiety. It’s a horrible feeling — both during the test and afterward — knowing that you *and only you* were the direct cause of your subpar score.

Despite what you may think about your innate test ability (“I’m just bad at taking tests,” “I always choke,” etc. etc.), there are ways to deal with this. Check out this concrete plan, after the jump.

You may have heard about or even tried drugs called beta blockers in order to calm your nerves. You also might know some people that swear by powerful stimulants like caffeine and Adderall to boost mental performance and focus on the day of the test. Both of these solutions have drawbacks. For one, all drugs have side effects — beta blockers can make you feel slow and lethargic (not exactly ideal states for an intense test), while stimulants can cause your mind and body to race out of control. The main problem with using drugs to optimize performance is that you don’t really solve the problem — the next time you have to perform under pressure, you’ll need the drug again. Effectively, you’ll be writing yourself a lifelong prescription for a mental crutch every time you need to perform. This situation can only inhibit long-term improvement.

Drugs solve the physiological problems of performance anxiety by addressing the physiology of the phenomenon. While they may lower your heart rate or boost your dopamine levels, they cannot truly act to optimize your mental state. To do this you need to effectively hack your own mind.

What makes test-day different from taking practice tests in your living room? There are the visual stimuli: the unfamiliar appearance of the testing center, the slightly annoying monitor, the presence of other test-takers in the room. There is also the smell of the room, the sound of other keyboards a-clicking (or the feel of the uncomfortable headphones if you choose to use them), the unfamiliar chair, the locker room before the test, etc., etc. And then there is the unfamiliar feeling that *this one matters*. You are not in Kansas anymore.

The classic symptoms of performance anxiety are merely responses to these changes in stimuli. Understand this, and you can take a few simple steps to engineer these responses to work to your benefit.

1. Keep a record of your GMAT practice test scores with notes. What did you do the day you scored your best? Did you exercise? Did you listen to music? Or did you come home from work, eat a big bowl of mac and cheese and watch The Office? Write these things down and pay attention to them. It doesn’t matter what they are. You’re just looking for a set of 2-3 things that you *enjoy *doing — things that you can do consistently every single day to put you in a good test-taking state of mind. Let’s call these things your “zone-ins” — you will use them to zone in to your best state of mind before every test you take.

2. Suppose your two zone-ins are eating mac and cheese and watching The Office. These are particularly good zone-ins because they touch upon more than one of your five senses. Mac and cheese is a taste stimulus; The Office is both a visual and an aural stimulus. Suppose you are a month away from taking your test. (You will need at least a month to make this process work). Do these two or three activities in the same order before *every single* practice test that you take.

Say you come home from work at 6, eat your mac and cheese, watch an episode of The Office online, and then take your practice test. Take a break before you do any studying after you take the test. What you want to do is to isolate “taking the test well” into a multi-step process that involves both eating your mac and cheese and watching The Office as matter-of-fact precursors to the actual GMAT. Spend about two weeks doing this until it becomes *a habit*.

3. Beginning roughly two weeks before the test, slowly taper down your zone-ins to processes that take progressively less time, but still retain the essence of the experiences. For example, instead of cooking and eating an entire box of mac and cheese, try making it the night before and reheating a few spoonfuls in the microwave. And after you’ve done that, maybe cut it down to a quick cold spoonful out of the fridge. And after that, maybe you just need to smell it. Or if you’re watching The Office, maybe instead of watching a full episode, you just watch the opening 6 minutes of an episode. And then you cut it down to maybe a 2-minute webisode. And then maybe you only need to hear the opening music or the sound of Michael Scott’s voice to make you think, “I’m watching The Office.”

4. The final step is to take these boiled-down essences of your zone-ins and find a way to use them on the day of the test. Maybe before you leave the locker room to enter the testing room, you eat a spoonful of mac and cheese out of a Ziploc bag and watch a webisode of The Office on your iPhone. Remember, these activities could be anything, as long as they reflect the feel and essence of your original zone-ins and are done in the identical order that you practiced and tapered. If you have done the process slowly, over at least a month, you will find yourself walking into the test room with confidence and scoring as if you were taking the test in your own living room.

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*Nate Burke is a Content Developer at Knewton, specializing in GMAT prep.*

In Part I of this series, I talked about approaching wordy GMAT questions as a businessperson would–by carefully reading these questions the first time around in order to absorb all information. The following GMAT problem has inspired me to expand this approach questions to include the actual process of decision-making:

A square countertop has a square tile inlay in the center, leaving an untiled strip of uniform width around the tile. If the ratio of the tiled area to the untiled area is 25 to 39, which of the following could be the width, in inches, of the strip?

I. 1

II. 3

III. 4a. I only

b. II only

c. I and II only

d. I and III only

e. I, II, and III

In case you haven’t figured out the answer, the strip could be any width. Answer choice **E is correct**. How do we arrive at this answer?

The problem-solving version of this question is taken from p. 85 of the GMAT Quantitative Review, 1st Edition. The solution given is computational; the basic steps are:

Let [pmath]t^2[/pmath] = area of the square tile inlay

Let [pmath]s^2[/pmath] = area of the entire countertop

Then [pmath](t^2)/(s^2 – t^2) = 25/39[/pmath], so that [pmath](t^2)/(s^2) = 25/(25 + 39) = 25/64[/pmath].

Taking the square root of both sides, we see that t/s = 5/8, which means that the length of the side of the square tile inlay is 5/8 the length of the side of the entire square countertop. At this point, the solution in the QR proceeds to write an expression for the width of the strip, w, in terms of these two variables: w = (s – t)/2, and then substitute an expression for t in terms of s to obtain an expression for w in terms of one variable, s:

[pmath]t/s = 5/8[/pmath] –> [pmath]t = (5/8)s[/pmath]. Then [pmath]w = (s – (5/8)s)/2 = (3/16)s[/pmath].

Presumably, you need to be able to write w = ks for some constant k, in order to see that w can take on any positive value. This is a lot of computation, and it will surely put you beyond the scant average of 2 minutes per question that you should allow. Is there a better way?

Imagine that this problem was posed as a data sufficiency question. It could look something like this:

A square countertop has a square tile inlay in the center, leaving an untiled strip of uniform width around the tile. What is the width, in inches, of the strip?

1. The ratio of the tiled area to the untiled area is 25 to 39.

2. The ratio of the tiled area to the total area of the countertop is 25 to 64.

Granted, this version of the question is slightly easier, because statement II gives you a clue to what your first step should be. But the same insight is necessary to solve this, and I maintain that it does not require computation. From either statement, you can reach the point reached in the solution given above: t = (5/8)s. Instead of plowing ahead with more computation, take a step back and use your imagination. The only constraint given by the problem is a proportion relating the length of a side of the tiled area to the length of a side of the table. FURTHERMORE, the question asks for width of the strip IN INCHES. This is slightly peculiar, given that the only other information about the table (a ratio of areas) does not mention ANY UNITS OF MEASUREMENT. What if the question had said “in feet.” Or “in millimeters.” Or “in miles.” Any one of these is possible given the information in the statements. The fact is that there is no way to determine absolute length measurements in terms of a specific unit if only proportions are known. A simpler question might look like this:

If a red spherical balloon contains twice the volume of a blue spherical balloon, which of the following can be the surface area of the red balloon, in square meters?

1. 1

2. 2

3. 3ANY of these numbers could be the surface area.

GMAT questions force the test-taker to make decisions about quantitative matters as quickly and as accurately as possible–very often without having to make calculations. The key behind these questions is to realize that computation is not necessary–a sense of what CAN be computed, however is indispensable.It makes sense that questions like these are on the GMAT. Very often in world of business, folks with an MBA are faced with the following decision: Should we spend all this time to do this computational task?

In the world of management, understanding when to ask this question can often be the difference between money wasted and money well-spent. On the GMAT, understanding this question will raise your score.

]]>Are you stubborn? There are many situations in which stubbornness would help a human being. A stubborn nature can be extremely useful when trying to accomplish a complicated goal over an extended period of time. Building a house, traveling a long distance, hunting for food, and courting someone are all activities central to human history that require, at least to some degree, a knock-down, drag-out, brick-headed resolve to get the damn thing done NO MATTER WHAT.

It is natural, therefore, that you would want to spend 5 minutes on the first question in the GMAT quant section. It’s okay. Natural and cultural forces have optimized our problem-solving heuristics in a certain way; recognizing which ones actually are optimal in certain situations is the key to good performance. In any situation in life OTHER than the GMAT quant section, thinking really hard and creatively about a problem until a solution is found (even if it is for an extended period of time) will usually be of value. Not on the GMAT.

We have a saying here at Knewton–answer when you are 90% sure, and move on. This is easy to say and extraordinarily difficult to do–precisely for the reasons outlined above. The reason why it is absolutely necessary, though, has to do with optimization. When you take a computer-adaptive test, you are given a powerful tool–the test itself. The test can work in your favor if you allow it to. The idea is that, over time, the test will be able to calculate your ability and give you questions that reflect it. If you find that you are given a particularly tricky question, DO NOT TAKE IT PERSONALLY. Do not assume that you are stupid for not having any idea how to approach it. Do not interpret it as a challenge that must be overcome by sheer intellectual willpower. The question was generated by the computer as the next step in its program to determine where you fall on the curve. The computer generated the question with the expectation that you’d spend roughly 2 minutes on it and move onto the next question. If your approach creates a situation that in any way deviates from these basic assumptions, the algorithm is designed to output a score that is a poor approximation of your abilities–and it will always err on the side of portraying you as stupid.

The long and short of it then is to focus your stubborn impulses in useful ways. Be stubborn about maintaining a rigorous study schedule. Be stubborn about attending your live Knewton GMAT classes. Be stubborn about learning and memorizing exponent rules and common powers. Be stubborn about studying the both the explanations of practice questions you answered incorrectly AS WELL AS practice questions that, while answered correctly, still gave you a tough time. Be stubborn about timing yourself on the easiest questions and work to improve the time it takes to answer them. Be stubborn about spending three weeks waking up at the time you will be waking up on test day to avoid being overly tired. Be stubborn about taking the test again if you happen to have a bad day. In short, be stubborn about improving your score on every single day leading up to the test.

But on test day, let it go. Just answer the question and move on.

]]>The GMAT quantitative section is different from most math tests. You don’t usually see Data Sufficiency questions outside the GMAT, for one thing. They’re tricky, and mastering them requires a high level of familiarity. The good news is that the answer choices are the same for every question, and precise calculations are often unnecessary.

Then there are the word problems. All that text takes a long time to read. With 37 questions to do within a scant 75-minute period, you have an average of about two minutes to answer each question. It can be nerve-racking to spend almost half of this precious time just parsing out questions that are essentially prose versions of a company’s balance sheet.

*photo by stuartpilbrow*

Maybe it seems silly to you to have to read through a lengthy explanation of two trains traveling on parallel tracks at different rates, when it would be a lot simpler to just look at a well-labeled diagram. After all, there is a reason why balance sheets, graphs, and diagrams exist, right?

There is a reason behind the test-maker’s strategy, however. These questions are testing how well you can take information that’s disorganized, messy, and portrayed in a slightly illogical way, and turn it into a *correct decision.* As a banker, manager, CEO, COO, CFO or any other leader in the world of business, you’ll have to make decisions based on information from people that “report” to you. It’s likely that these people won’t have the type of intense training you’ll have had at business school. Chances are, they’ll communicate with you in a somewhat disorganized, messy, and slightly illogical way.

How do you answer more GMAT quantitative questions correctly? Take the same steps that a good businessperson would take in order to make a decision:

- Calmly and carefully obtain information.
- Think analytically.
- Decide without second-guessing yourself.

Though all three steps are required for all GMAT math questions, the bulk of the work required for overly-wordy math problems comes in the first step: calmly and carefully obtaining information. It doesn’t pay to read the prompt and answer choices as quickly as possible. Getting the question correct requires a thorough knowledge of all information in the prompt. Skimming will only increase your risk of misreading or omitting important facts. Consider the following question, taken from the 2nd edition of the GMAT “Quantitative Review”:

One week a certain truck rental lot had a total of 20 trucks, all of which were on the lot Monday morning. If 50 percent of the trucks that were rented out during the week were returned to the lot on or before Saturday morning of that week, and if there were at least 12 trucks on the lot that Saturday morning, what is the greatest number of different trucks that could have been rented out during the week?

A: 18

B: 16

C: 12

D: 8

E: 4

This long-winded question boils down to:

Find the maximum value for t if (1/2)t + (20 – t) â‰¥ 12

This inequality is easily reduced using algebra (pop quiz: solve this.) The tasks of reading the question and interpreting it into a suitable equation are a lot more time-consuming. If you felt rushed and skipped the simple phrase “of that week,” then the incorrect answer choices C and D would become far more compelling than if you had read the question accurately.

The key here is to read the question as if you had all the time in the world—*the first time through.* Read it carefully. Absorb every word. Write down expressions and equations. Misreading a question will either lead to an incorrect response or a reread. Having to reread a question means that your first reading was a waste of time, and wasted time, much like an incorrect response, will always add up to a lower score.

**“The best way to win at Russian Roulette is to not play at all.Â Or you could just have the other guy go first and then run away quickly.” **–Unknown

In the heyday of paper-based tests, the cure-all for the common ill of “getting stuck” on a question was simple: Skip the question and simply move on.Â The rationale behind this strategy was that work done for other questions on the test might illuminate simple key concepts that were overshadowed by things like early-morning drowsiness, test anxiety, tip-of-the-tongue syndrome, etc, etc,.

Things have changed. The GMAT, like many other standardized tests, is administered on a computer. Though the question formats have remained roughly the same, the switch to computer-based-testing has rendered the “skip the question” strategy obsolete.Â On a computer-based-test, it is impossible to skip a question. The best that a time-constrained student with a total conceptual block can do is to guess and hope for the best outcome of what is essentially a game of Russian roulette with a 5-chambered revolver.

An ideal strategy for “getting stuck” within the context of a computer-based test is thus constrained to the narrow confines of the maximum time allotted per question. Consider that at the beginning of this time interval, WITHOUT EVEN HAVING READ THE QUESTION, a test-taker has already been granted a 20% chance of answering correctly by guessing randomly. This fact alone has consequences. If the student is still at a random-guess level of confidence at the end of the 2-3 minute maximum-time-per-question-interval, then all that the student has accomplished on this question was to lower his or her score. Every second that passes without progress lowers the score incrementally (and that’s not even accounting for the added adverse effects of things like stress-induced second-guessing, etc. etc.).

What is a student to do then, should he “get stuck?” If the path to the correct answer is obscured and overgrown, how does one trim the hedges?

Here is a list of quick things that can help:

Write down ANYTHING from the question on a sheet of scrap paper.

Draw a picture.

Mouth words from tricky passages slowly as you read them.

Cover the answer choices and look only at the question prompt.

Cover the question and look only at the answer choices.

In quantitative sections, re-write messy expressions in as many ways as you can.

In data-sufficiency questions, move on to the second statement if the first statement makes no sense.

If you are stuck, your only goal is to change the situation!

Here are some general things that may help also:

Take a deep breath.

Lower your shoulders.

Monitor how quickly your eyes are moving–chances are that if you’re stressed and stuck, they’re moving and reading at a rate that is faster than your brain can deal with.

Slow down.

And finally, if it’s been a few minutes to no avail:

GUESS.

The nice thing about this guessing game is that you hear a “click” every time…

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