August 20, 2012
Take a look at the picture with this blog. It’s an iconic optical illusion. Stare at it—what do you see? The picture is called the Great Wave off Kanagawa, painted by Katsushika Hokusai, a Japanese artist famed for his brilliant compositions. This drawing is of a wave, of course, but do you see the other wave, the reverse wave in the sky?
This image utilizes negative space. You take the whole frame, the great big rectangle, you block out that actual image—and what remains is, in its own right, an interesting picture.
To find the area of the shaded region, we need to subtract the area of the smaller inner circle from the large outer circle—the difference is the area of the ring.
But the concept extends beyond simple pictures and geometry questions. Probability problems sometimes operate on a similar principle, subtracting an easy-to-find probability of failure from 1, the total of all probabilities, the “whole frame,” as it were. Once you’ve subtracted all the failures, then whatever remains, the “negative space,” must be the chances of success!
The GMAT is a test of critical thinking. It tests your ability to find the most effective path to the solution. Sometimes, you’ll pick numbers, sometimes you’ll do the math directly, sometimes you’ll guess strategically. And sometimes, you’ll define the negative space around the answer, and solve that way. Today’s problem of the day is best solved by that principle. You can’t figure out how many arrangements follow the rule in the question stem. But you can find out how many arrangements don’t follow the rule, and subtract it from the total. Good luck!
Six children, Arya, Betsy, Chen, Daniel, Emily, and Franco, are to be seated in
a single row of six chairs. If Betsy cannot sit next to Emily, how many different
arrangements of the six children are possible?
Step 1: Analyze the Question
We have to arrange six children in six chairs, but two of
the children can’t sit together. We’re asked to calculate the
number of different arrangements of children.
Step 2: State the Task
We’ll calculate the number of possible arrangements of the
children. Then, we’ll subtract the number of ways Betsy can
sit next to Emily.
Step 3: Approach Strategically
The possible number of arrangements of six elements is
6! = 6 x 5 x 4 x 3 x 2 x 1= 720.
Now we’ll have to calculate the number of ways that would
violate the question stem by putting Betsy next to Emily.
If we number our seats from left to right, there are 5 ways
they can sit together if Betsy is on the left and Emily is on
Seats 1 & 2
Seats 2 & 3
Seats 3 & 4
Seats 4 & 5
Seats 5 & 6
And there are 5 more ways if Emily is on the left and Betsy
is on the right, for a total of 10. Now, for any one of those
10 ways, the four remaining children can be seated in 4!
ways: 4! = 4 x 3 x 2 x 1 = 24. So we need to subtract
24 x 10 = 240 ways that have Betsy and Emily sitting
together from our original total of 720: 720 – 240 = 480.
Answer choice (B).
July 8, 2012
“This is your last chance. After this, there is no turning back. You take the blue pill — the story ends, you wake up in your bed and believe whatever you want to believe. You take the red pill — you stay in Wonderland and I show you how deep the rabbit-hole goes.” – The Matrix
I guess that, since you are still reading this post, you are considering taking your prep to the next level and entering the GMAT matrix. Truly mastering the GMAT requires that you peel back the veneer of the test and understand the patterns behind the test or, as we commonly say here at Kaplan, learn the inner-calculus of the test. One of my favorite sayings about standardized tests like the GMAT is that “the GMAT never repeats, but it rhymes.” This is one of the great things about preparing for a standardized test. The test-maker must use a framework of question types, answer types, passages, etc. in order to truly standardize the test. There is a specific set of skills that the test-maker includes in the test each and every time. With that in mind, let’s peek inside the matrix of GMAT test prep.
In order to learn the patterns of the test and master the GMAT, there are a few key things to do and keep in mind as you prep.
1) The Problem Fades and the Concept Remains
As you prep, do so with an eye toward the bigger concept behind a particular question. Ask yourself what the test-makers are really testing with the question in front of you – what vital skill or skills are they including. Don’t fight with individual questions or let one miss throw you on that particular topic. The specifics of the questions (names, numbers, etc..) will be different on test day, but the concept being tested will absolutely be there. Identifying the concept being tested and/or the type of question focuses your approach and narrows your options. Get into the habit of identifying the question type and primary concept each time you approach a question in practice. The tougher questions are usually just a combination of more basic skills stacked on top of each other. By thinking strategically and identifying the concept(s) and question type, you take the mask off the question.
2) Mastery First, Timing Later
When you first start to prep, it should not be about timing. Your GMAT timing will not improve in the right way until you are better able to read the patterns in the test, so don’t fret about timing until you have built the content mastery to back it up. When you get to the point that you can spot the concept being tested and the strategy you should use to tackle the question when you glance at the question, then you are ready to turn your attention to timing practice.
3) Review every explanation
In order to build the mastery noted above, it’s important to review every explanation, even when you get a question correct. Too many people neglect this step, but I can’t emphasize enough how important it is to truly being able to see the matrix of the test! Often, people prep by just doing question after question without fully reviewing their approach. While it’s important to check the answer, it is equally important to review why you chose the answer that you did. Your goal on the GMAT is to be correct AND efficient. The hardest thing to do is to read all of the explanations for the questions you got right. You want to make sure that you 1) got the right answer for the right reason and 2) are approaching the question in the most efficient way.
4) Make mistakes with a growth mindset
In order to really master the GMAT, your response to mistakes needs to be appropriate. Too many test-takers let mistakes define their performance and dampen their eventual improvement. Mistakes are good if viewed in the right light. We learn more from our mistakes than our successes. They are the ideal opportunities for improvement. Mistakes represent the glitches in your ability to read the matrix that must be identified in order to be fixed. It’s much better to make these mistakes before test day so that you have the ability to fix, reinforce, and strengthen your approach. Therefore, don’t be afraid to make mistakes in your practice. Make them boldly so that you can sharpen your mastery as you fix them!
5) Use full-length practice tests to target areas of study
In your prep and mastery building, full-length tests are important, but it’s not the score that is the vital component. While a score is important as you track performance, it is more important to track the trends in your performance. The first thing is to make sure that you understand the basics about how the test is scored. For instance, long strings of missed questions are typically worse than missing the same number of questions spread out. Also, there is a steep penalty for questions left unanswered at the end of a section. Your review of the test should take these things into account. It’s also important to understand your own strengths and weaknesses. As you review, put your questions into categories: 1) answered correctly and with the best strategic approach, 2) answered correctly but could tweak the approach to make it more efficient, 3) answered incorrectly because of a misread or silly error (face-palm moment), 4) answered incorrectly but the content is somewhat familiar, and 5) answered incorrectly and the content needs a lot of review. These categories are placed in the order of approach as well. Tackle the issues that are easiest to fix first to quickly boost and tighten your approach, and formulate your study plan on the basis of this analysis. Taking test after test does not do as much for you as using your test performance as a tool to inform the practice that you will do. Take another practice test after 15-20 hours or so of solid practice and repeat the analysis.
6) The How Matters (Just not in the way that you think…)
How you answer the questions on the test definitely matters; however, it’s with an eye towards efficiency rather than the official, linear quantitative or verbal route. The test-makers do not require you to show your work or demonstrate the “proper” steps. As a matter of fact, stop thinking about the textbook route as the “proper” route! Too many test-takers feel like they are somehow copping out by using a strategy or shortcut to get to the answer, but that is actually preferred. You are not only being tested on your content knowledge, but more importantly, you are being tested on your ability to recognize the best and most efficient approach to a question. Take the red pill, learn to read the test matrix, and use that to master the GMAT!
How you prep matters! Prep with an eye towards learning the patterns behind the test and improvement will happen. Combine your quest for mastery with a solid study plan to maximize your improvement. Experts see and use patterns! While you contemplate what your approach will be – red pill or blue pill – I hope to see you soon in the GMAT matrix!
June 28, 2012
In poetry, a rose is a rose is a rose. On GMAT problems with central angle “slices” in circles, a fraction is a fraction is a fraction.
This may seem like common sense. Cut a pizza into six slices. If you cut it evenly, each slice now has one-sixth the cheese, one sixth the crust, and an angle of one sixth the way around a circle—that is, 60 degrees. However, though this may seem obvious, it’s actually a very useful technique for resolving certain geometry problems.
Consider the following Data Sufficiency question:
In a radius 6 circle, two points A and B are connected to the center, point O. What is angle AOB?
1) The length of the minor arc defined by sector O is 1.5π
2) The area of the sector defined by angle AOB is 4.5π
It looks all GMAT-like and formal, but if you actually think about it…, it’s just the pizza I described above. We’re taking a slice, and want to know what’s the angle of that slice. The “cheese” of the pizza is the area of a radius 6 circle, which is the radius squared times pi, or 36π. The “crust” is it’s circumference, which in this case would be the diameter times pi, 12π.
Statement 1) tells us that the length of the arc/crust is 1.5π, which is 1.5/12 = 1/8 of the circumference. And an eighth is an eighth is an eighth. Our slice, which goes one eighth of the way around the outside of the circle, and it goes an eighth of the way around the inside of our circle as well—its central angle is 1/8 * 360 = 45 degrees. Sufficient!
And statement 2) says that the area of our slice/sector is 4.5π, or 4.5/36 = 1/8 the total area of the whole pie. Once again, an eighth is an eighth is an eighth, so by the exact same reasoning we get 45 degrees. Sufficient!
This is a pretty basic rule, but it’s widely applicable to many circle problems. It can be especially useful with central triangles, which have central angles by definition. Take a look at today’s question of the day to test your skills, and when you get it right, treat yourself to a slice of pepperoni as a reward!
What is the area of the circle above with center O?
(1) The area of D AOC is 18.
(2) The length of arc ABC is 3π.
Step 1: Analyze the Question Stem
This is a Value question. For any Circle question, we
only need one defining parameter of a circle (area,
circumference, diameter, or radius) in order to calculate
any of the other parameters. Also, all radii of the same
circle will have the same length. So AO = CO. That makes
the triangle an isosceles right triangle (or 45-45-90) for
which we know the ratio of the sides. Not only would the
circle’s circumference, diameter, or radius be sufficient,
but information that gave us any side length of triangle
AOC would also be sufficient, as it would give us the length
of the radius.
Step 2: Evaluate the Statements
Statement (1): We are given the area, and we already know
that the base and the height are equal. So if we call the
radius of the circle r, then the area of the triangle is equal to
1/2 (Base)(Height) = 1/2 (r)(r) = 1/2 r ^2 = 18
Remember that we are not asked to calculate the actual
value of r. Because we have set up an equation with
one variable, and we know in this case that r can only
be positive since it is part of a geometry figure, we have
enough information to determine the area of the circle (the
solution would have been r ^2 = 36, r = 6).
Therefore, Statement (1) is sufficient. We can eliminate
choices (B), (C), and (E).
Statement (2): Because O is the center of the circle and
angle AOC measures 90 degrees, we know that the length
of arc ABC is one-fourth of the circumference. Because this
statement allows us to solve for the circumference, it is
(D) is correct.
January 31, 2012
I watched the Oscar-nominated film Master and Commander when it came out in theaters, and to this day a particularly ghastly scene lingers in my mind. While fighting through a brutal, sudden storm, the captain of the ship (a very macho Russell Crowe) is forced to make a horrible decision. Two of his men have gone overboard, clinging to a snapped mast floating in the ocean. The mast is still connected to the boat by its rigging, and as the squall blows the boat onward, the deck slowly lists to the side and takes on water, dragged down by the sodden wood and cloth. The men on the mast beg for help, knowing the freezing water is not survivable. But the captain knows that he has seconds before the mast will capsize the boat. Without hesitation, he takes an axe to the ropes, dooming the men in the water but saving the lives of every crew member still aboard.
If you’ve spent any time taking CATs in preparation for your test, you probably already know where I’m going with this. But if you’re new to GMAT prep, here is what you need to know: not only is the GMAT is strictly timed, but is also harshly penalizes unanswered questions at the end of a section. That means that you may have to make the same decision that the Captain did: if a single problem is dragging you down, you have to cut your losses to prevent your ship from sinking.
The signs that it’s time to guess on a problem are clear, though they may require several practice CATs before they become obvious. As a general rule, it’s time to guess if you’ve spend more than a minute looking at any problem without figuring out a how to start solving. Also, you should be aware of the recommended time to answer each question type (2 minutes for Quantitative problems and for Critical Reasoning, 1.5 minutes for Reading Comp questions, 1 minute for Sentence Correction). If you find you’ve doubled the allotted time on a single problem, it’s time to guess even if you feel like you’re ‘close’ to the answer. You’re already behind; you can’t afford to fall behind further.
But unlike the captain, you shouldn’t lose sleep at night for sending problems to an early grave. Remember, the test-makers design the test with guessing in mind. Guessing strategically is the sign of a skilled test-taker, not a struggling one. For one, the GMAT runs on an adaptive algorithm and adjusts to your performance. A super-hard problem that you need to guess on may be a sign that you’re getting your best score ever! Moreover, the test gives you easier questions as a chance to ‘prove yourself’ when you get other questions wrong. You can make for an incorrect guess by nailing those easier questions. But you can’t make up lost time without guessing on another problem—especially if you end up getting the correct solution, making the next adaptive problem even harder. No single problem will get you a good score by itself. Good scores come from getting the most questions right in the most effective manner.
And finally, here is a trick from one of my students to make it easier when it’s time to guess: remind yourself that there are experimental questions. Of course, it’s a sucker’s game to ‘outsmart’ the test by guessing which questions are experimental. But when that nagging feeling of “I can get this one with a few more minutes” is making you hesitate to forge ahead, remind yourself that even if you do spend a few more minutes, and even if you do end up getting the correct answer as a result, there is a small but real chance that the five minutes you spent were for nothing—you could have been clinging to an experimental question all along.
For today’s question of the day, pretend you’re behind schedule. There is a way to narrow the answer choices down two only a few options very quickly. See if you can make an educated guess in under 45 seconds, before you go back to solve it for real. Good luck!
In the figure above, point O is the center of the semicircle, and PQ is parallel to OS.
What is the measure of ∠ROS?
Analyze: What shapes do we see in this mess? A semicircle and three isosceles triangles (we know they’re isosceles because each of them has two sides that are radii of the semicircle).
Because they’re isosceles, what else can we label? We know that for all three triangles, the two “top” angles are equal. So, angle OPQ is 70°, angle OQR is x°, and ORS is x + 1°.
What else does the Q-stem tell us? That PQ is parallel to OS. That means we can find corresponding angles.
Where is a corresponding angle to angle OQP found? Angle QOS. It must also be 70°.
Why might that be of interest? Because PO and QO are both transversals.
Task: Good analysis. What are we being asked for? The measure of angle ROS.
And what do we know about triangles and their internal angles? They sum to 180°. so, < ROS + 2(x +1) = 180.
Approach strategically: All right. We know a lot about triangles QOR and ROS now. What will the sum of their combined angles be? 360. Each triangle has 180°.
Give us an equation that contains all the info we have. 360 = 70 + 4x + 2.
Solve that for x. 360 = 4x + 72 → 288 = 4x → 72 = x.
And how can we use that to calculate angle ROS? 180 = 2(72 + 1) + < ROS → 180 = 146 + < ROS → 34 = < ROS.
Answer (a) is correct.
Confirm: That wasn’t so bad actually. Notice how we used triangles, circles, and even good ol’ lines and angles to get
the solution. Now, what if you didn’t have time for those steps. Is there a guessing strategy available? The triangles
are pretty close and the lines are parallel, so x° must be close to 70°. That would get us down to (a) or (B). Given that
<QOR has angles of x° and < ROS has angles of (x + 1)°, it makes sense to go with (a).
By the way, what’s up with answers (D) and (E)? They’re traps in case we solved for x or (x + 1) instead of what we’re supposed to.
April 20, 2011
Word problem, geometry, and algebra combined….check out this sample GMAT problem, where the biggest challenge is proper set up.
The area of circle O is added to its diameter. If the circumference of circle O is then subtracted from this total, the result is 4. What is the radius of circle O?
The key to solving this problem within the two-minute time frame on the GMAT is realizing what it is really testing. As is the case with many GMAT problems, this is not the type of question it seems to be at first. Many students, seeing information about a circle, start drawing a picture. If this were really a geometry problem, that would be the correct first step. However, this is actually an algebra problem in disguise.
The correct first step is to translate the information in the problem into an equation. ‘The area of a circle is added to its diameter’ becomes pr2 + d. As diameter is twice the radius, we can rewrite d as 2r, making the expression pr2 + 2r. Next, we are told that the circumference of the circle is subtracted from this total, making the expression pr2 + 2r – 2pr. Finally, we know that the result is 4. So, the entire equation is pr2 + 2r – 2pr = 4. As the problem asks for the radius of the circle, all we need to do now is solve for r, which can be done in the following manner:
pr2 + 2r – 2pr = 4
pr2 + 2r – 2pr – 4 = 0
At this point, be sure to note that you have a quadratic, which can be factored to:
(pr + 2)(r – 2) = 0
As is the case with most quadratics, this equation has two solutions. Either pr + 2 = 0, in which case r = -2/p, or r-2 = 0, in which case r = 2.
Looking at the answer choices, you will notice that both of these are listed as options. But, because this problem is referring to a circle’s radius, which can only have a positive value, r must be positive and thus must equal 2.
March 30, 2011
Some of the most difficult GMAT geometry problems are those that feature multiple figures (for example, a half-circle inscribed within a rectangle, or other combination of geometric shapes). Multiple figure problems also tend to be somewhat time consuming, so do not expect to see more than one or two on test day. Additionally, these problems are usually considered advanced, so you are more likely to see them when you are doing well on the GMAT. The key to answering multiple figure problems correctly is to follow two specific steps.
Step 1 of Multiple Figure Problems
The first step is to determine all of the information you KNOW about the figure. Use the lines and angles you are given to fill in any lengths or degrees that are not written on the initial figure. When doing this look for parallel lines, perpendicular lines and triangles primarily, but you can use any rules of geometry within this step.
Step 2 of Multiple Figure Problems
Once you have all of the information written on your figure (that you, of course, have redrawn in your scratch work), start to look for familiar shapes hidden in the multiple figures. These shapes can be triangles, quadrilaterals or any other shape that will all you to solve. If the problem features variables, the shape will usually involve the parts of the figure in which they show up.
By following these two steps, you will be able to solve complex geometry problems in the fastest manner possible, which leads, ultimately, to a higher score. For examples of Multiple Figures problems, check out this Kaplan GMAT video.
February 14, 2011
A Checklist for Untangling Multiple Figures in Geometry
On the most advanced GMAT content, you will see complicated geometric figures. While these questions are often the most daunting questions, you (as the prepared and confident test taker) will realize that the GMAT is only testing a couple concepts.
The geometric concepts you need to know on test day include the equations for finding the area, circumference, and volume of triangles, circles, and quadrilaterals. That is it. Nothing else. While those are straight forward topics, Multiple Figures on the GMAT push your understanding of these concepts to the extreme.
When presented with a complex image that doesn’t easily fit into the category of circle, triangle, or quadrilateral, you want to proceed by following the check list below to identify the relevant component of the image that will unlock the simplicity mentioned above:
- Are there parallel lines?
This may be one of the most helpful questions you can ask. If there are parallel lines, you can deduce a significant amount of information from the properties of parallel lines and the subsequent angles of the intersecting lines.
- Are there recognizable angles?
As you are studying for the GMAT, you will come across several common triangles that conform to common, simplistic structure, including 30-60-90 degree triangles and 45- 45- 90 triangles. These angles have important properties that can unlock the right answer.
- If it is circle, can I figure out the radius or diameter?
Of course, if you can calculate either the radius or the diameter, you can calculate the other. However, for any image containing a circle, the radius is the key to solving the problem. Identify creative ways of locating it.
- Do I know the internal angles?
The internal angles of triangles sum to 180. The internal angles of quadrilaterals sum to 360. Can you backsolve with these totals into the correct answer?
Consider the above four points a simple checklist to approach the most difficult GMAT geometry content. Many times, you’ll be required to employ several of the points above to uncover the right answer. These represent just the points you want to start evaluating.
Remember, you have to memorize certain formulas and be able to apply them to complex situations. The combination of memorization and strategy together is the key to a fantastic score. Good luck on geometry!
December 27, 2010
Circle problems are among the most common types of geometry questions that appear on the GMAT. As such, you must make sure that you are fully prepared for these problems on test day.
The first key to circle questions is understanding what a circle really is. A circle is defined as a collection of all of the points that are equidistant from a center point. This distance is defined as the radius of the circle and the diameter is defined as twice the radius. For this reason, the radius of a circle is the key measurement when working with circles. On circle problems, knowing or solving for the radius will almost always be essential.
After the radius, the most important number to understand is π. π is defined as the ratio between the circumference of a circle and its diameter. Thus, the formula for finding the circumference of a circle is 2πr. You should also know the formula for the area of a circle, which is πr2.
Once you understand the fundamentals and formulas of a circle, you must also be prepared to calculate sector areas and arc lengths. Sector area is the area of a slice of the circle, and arc length is the distance between two points along the circle. These are both calculated by setting up a ratio of the angle measure the arc or sector creates in the center of the circle to 360. This ratio is equal to both the sector area to the total area of the circle and the arc length to the total circumference of the circle.
Finally, when thinking in terms of Data Sufficiency and having ENOUGH information to solve a given problem, keep in mind that if you have any ONE of the following: Area, Circumference, Radius, or Diameter, you can solve for all of the others! By keeping these rules in mind you will be able to solve the vast majority of circle problems quickly, saving time for more advanced problems.
For examples in action and more tips on circles, see the Kaplan GMAT Video on Circles.