Product A is 10 percent more expensive than Product B. Product B is 10 percent cheaper than Product C. If the price of the Product C is $300, how much cheaper is Product B than Product A?

A. $2.00

B. $3.00

C. $9.00

D. $15.00

E. $27.00

Question Discussion & Explanation

**Correct Answer** - E - (click and drag your mouse to see the answer)

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**Verbal**

Psychologist: Though popularized by the media and even some ill-informed scientists, photographic memory is a myth. Most of the top competitors at this summer’s national memory championship have brains that are neurologically indistinguishable from those of the general population and cognitive abilities that are seldom more than a standard deviation above average.

**Which of the following would LEAST strengthen the argument above?**

The competitors at this summer’s championship are typical of the segment of the population thought to have photographic memory.

People with photographic memories would have brains that are neurologically distinguishable from those of the general population.

Photographic memory requires cognitive abilities more than a standard deviation above average.

Photographic memory was initially proposed by a scientist whose theories of mind have since been widely discredited.

Photographic memory, if it exists, would be a requirement even to qualify for a national memory competition.

Question Discussion & Explanation

**Correct Answer** - D - (click and drag your mouse to see the answer)

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*Ross names one of its most popular professors as their new dean.*

There’s a new dean at one top business schools, the deans of two other top-ranked MBA programs talk about what it takes to get in and succeed, and NFL players are given an opportunity to learn business skills for their post-football lives. Catch all the trending stories for future business leaders.

It turns out that top-ranked University of Michigan’s Ross School of Business didn’t have to look very hard to find a replacement for its retiring dean, Alison Davis-Blake. Scott DeRue, one of the most popular Ross professors—also recently recognized as one of America’s outstanding professors under 40—will helm the business school this summer. “At the end of the day, I think it’s the experiences we are creating for students. And what we can promise them in terms of these transformational student experiences, really drawing on our commitment to the platform on action-based learning, world-class faculty, recruiters that are highly committed to the school and love our students,” said DeRue. (Poets & Quants)

Columbia Business School and Stanford Graduate School of Business are two of the world’s greatest MBA programs, with highly competitive admissions processes, top GMAT scores, globally recognized professors, and graduates who secure lucrative jobs and exciting careers. As the dean of CBS, Glenn Hubbard says, “We look for people who have their own way of being successful. They have a very entrepreneurial way of thinking. If you look at our most successful graduates, they are never cookie-cutter. They never fit a pattern. They are always on their own career trajectories and have their own personal ambitions.” Garth Saloner, Stanford GSB’s dean, notes that he’s been seeing similarly unpredictable outcomes, “What we’re finding in the data is that more of our students are going into technology companies than are going into consulting, which is a shift in the last 10 to 20 years.” (Nikei Asian Review)

A successful alumnus has just donated $25M to Cornell University’s Johnson School of Business, making it one of the school’s largest gifts the school has ever received. High-yield bond investor David Breazzano’s donation will help the school of business construct a new state of the art classroom center and establish the Breazzano Family Faculty Excellence Fund. Breazzano commented: “Johnson helped me discover my passion and aptitude, then helped me get my first job, and I did well at it because of my education. That solid foundation has helped me throughout my career. So I have a sense of gratitude. And I always knew I wanted to give back when I was in a position to do so.”

(Syrcause.com)

The entrepreneurial spirit is alive and well in today’s MBA students. In fact, many are so excited about starting their own businesses that they are launching startups while still in school. To grow their businesses, many of these busy students are taking to social media. Facebook reported 40 million business pages in April 2015; by December, that number had grown to 50 million. Meet Zarana Shah, a student at the George Washington University School of Business who launched two businesses—Get Junked, which sells home-made jewelry, and Goodness Gracious, which sells desserts. (BusinessBecause)

How long does the career of a successful football player (“football” to Americans; “American football” to everyone else) typically last? The answer is usually only a few years, or in the case of a select few, as many as 15 years. But the reality is that most players spend their adults lives doing something other than tackling, blocking, throwing the pigskin, and thrilling fans of all ages. One startling statistic shows that more than one in ten former football players file for bankruptcy after being in retirement for a decade. Knowing this to be the case, the National Football League has partnered with Ross Business School to give current player the skills they need to succeed after the stadium lights go out. “We’re trying to figure out how to take the bit of wealth we have now and make it last, grow long term and try not to get stuck in working 8-10 hours a day for the rest of our young lives,” says Eric Kush, who currently plays for the Los Angeles Rams. (*The Financial Times*)

*Ready to kick your GMAT score into top shape? Challenge yourself to our **free 20-minute workout**.*

The post New Dean Appointed at Ross School of Business appeared first on Business School Insider.

]]>If and are prime numbers, what is ?

(1) is a prime number.

(2) Both and are less than 5.

Question Discussion & Explanation

**Correct Answer** - C - (click and drag your mouse to see the answer)

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**Verbal**

After a number of migrant fishermen at the Rotunda Beach pier got sick from consuming locally fished hermit crabs, the Rotunda Beach city authorities posted signs cautioning against the consumption of the crabs. Tests revealed that the crabs were high in prestic acid, a dangerous chemical. But a marked decline in the number of goldfish found by the beach has led to the removal of the warning signs.

**Which of the following, if true, would best explain the removal of the signs?**

Goldfish found off Rotunda Beach cause hermit crabs to excrete prestic acid.

The amount of prestic acid produced by an individual goldfish has a direct positive correlation with the density of the goldfish population in that goldfish’s immediate vicinity.

Goldfish consume various parasites that often contain prestic acid.

Rotunda Beach is notoriously polluted.

Goldfish would not be able to inhabit the waters off Rotunda Beach if those waters were not saturated with a precise mixture of acids, including prestic acid.

Question Discussion & Explanation

**Correct Answer** - B - (click and drag your mouse to see the answer)

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How well do you know GMAT facts?

It’s time to test your GMAT knowledge. How much do you know about this important b-school entrance exam?

**1. The GMAT is one of the first elements in your application the adcoms will look at.**

**True!**

While the GMAT is not always the determining factor for admission, it can be used as a tool to weed out applicants and it can influence a busy reader to read an application a little more quickly and critically. It is also sometimes used as a screening tool by elite investment banking and management consulting firms. For them, 700 is the magic number.

**2. There is no way to recover from a low GMAT score.**

**False! **

It’s not easy to grab the attention of the adcom once you’ve lost it with a low GMAT, but it is possible. To do so, you’re going to need to truly blow the adcom readers away with an otherwise impeccable application. This includes writing out-of-this-world essays, creating an impressive MBA resume, presenting a flawless transcript, providing outstanding LORs, and – like it or not – having a less-than-ordinary background.

**3. An above average GMAT score can win you an automatic acceptance at some business schools.**

**False!**

No business school, even those that are lower ranked programs, will look at your GMAT score at the exclusion of everything else. If you earn a perfect GMAT score, but have no work experience, were kicked out of college due to cheating, and have six DUIs, then your score alone won’t be enough to gain you a seat in any MBA program. An impressive GMAT score must always be accompanied by an impressive application. However, some schools do treat a high score more favorably than others. A school working to boost its MBA rankings may be more inclined to accept students based on their high stats than others and may also be quicker to offer merit-based financial aid.

**4. Schools care about both the verbal and quant scores.**

**True!**

People tend to think that top b-schools care only about the quant score of the GMAT, but this isn’t true at all. Yes, being able to compute is important, but so is being able to communicate. Your verbal score indicates your ability to read and write – you can’t really go to business school where the language of instruction is English (or at least not successfully) if you don’t have some mastery over the English language.

**5. You should retake the GMAT as many times as you need until you hit your target score.**

**False!**

You SHOULD retake the GMAT two or three times, but retaking it more than five times may begin to negatively impact your application. Think of it as the law of diminishing returns for the GMAT: Putting forth effort to reach excellence is positive, until you’ve put forth so much effort that it begins to make you look bad – like a serial test-taker…or a person who keeps trying but who still can’t cut it. If after three re-tests you still don’t have the score you desire and there are no extenuating circumstances that are likely to change on the next retake, then you need to cast your school net wider to include programs that better meet your qualifications.

**Continue to boost your GMAT IQ when you visit GMAT & MBA Admissions 101.**

**Related Resources:**

• GMAT & MBA Admissions Resource Page

• 3 Tips for Handling a Low GMAT Verbal Score

• Low GMAT Score? Don't Panic...Yet.

This article *originally appeared on *blog.accepted.com*.*

Applying to a top b-school? The talented folks at Accepted have helped hundreds of applicants get accepted to their dream programs. Whether you are figuring out where apply, writing your application essays, or prepping for your interviews, we are just a call (or click) away.

Contact us, and get matched up with the consultant who will help *you *get accepted!

The radius of the front wheels of a cart is half that of the rear wheels. If the circumference of the front wheels is 1 meter and the cart traveled 1 kilometer, how many revolutions did the rear wheels make?

A.

B.

C. 250

D. 500

E. 750

Question Discussion & Explanation

**Correct Answer** - D - (click and drag your mouse to see the answer)

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**Verbal**

Tiger beetles are such fast runners that they can capture virtually any nonflying insect. However, when running toward an insect, the beetles intermittently stop, and then, a moment later, resume their attack. Perhaps they cannot maintain their pace and must pause for a moment's rest; but an alternative hypothesis is that while running tiger beetles are unable to process the resulting rapidly changing visual information, and so quickly go blind and stop.

**Which of the following, if discovered in experiments using artificially moved prey insects, would support one of the two hypotheses and undermine the other?**

A When a prey insect is moved directly toward a beetle that has been chasing it, the beetle immediately turns and runs away without its usual intermittent stopping.

B In pursuing a moving insect, the beetles usually respond immediately to changes in the insect's direction, and pause equally frequently whether the chase is up or down an incline.

C The beetles maintain a fixed time interval between pauses, although when an insect that had been stationary begins to flee, the beetle increases its speed after its next pause.

D If, when a beetle pauses, it has not gained on the insect it is pursuing, the beetle generally ends its pursuit.

E When an obstacle is suddenly introduced just in front of running beetles, the beetles sometimes stop immediately, but they never respond by running around the barrier.

Question Discussion & Explanation

**Correct Answer** - C - (click and drag your mouse to see the answer)

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For access to the argument that I mention in the video, and all the other possible arguments, click here to download the PDF from mba.com.

Check out this week’s board!

The post GMAT Tuesday: Common Flaws in AWA – Sampling Error appeared first on Magoosh GMAT Blog.

]]>We recently published advice on how to solve questions involving fractions. The truth is that sometimes you will try one method to solve a question and that method does not turn out to be the right one. Don’t give up! Try another method. Of course, the trick is to select the correct method from the beginning, but let’s be realistic: This won’t happen 37 times on test day for all 37 Quant questions. Be prepared to change tack!

*What’s the best way to say thank you to your interviewer?*

You left the interview feeling good and loved everyone you met in the office. Although you may feel confident, you aren’t done with the interview process just yet.

As a proactive job seeker, you want to guarantee that you’ve done everything you can to secure that offer. From the moment you step out of the office, your thoughts should be on how you will follow up on the interview.

Once you leave the interview, pull out a notepad or your phone to write a few quick notes about specific conversation points and current events relevant to the company. Take note of how your qualifications can meet the job expectations. They’ll be fresh in your mind at this point and will be a huge help when you go to write thank you notes later on. Remembering key points of the conversation will show you were actively listening and interested.

Writing an email thanking the interviewer on the same day that your interview took place shows you are making this position a top priority before you even have it. The interviewer will get a sense that you care about making a good impression. It’s also important to send multiple thank you emails if you interviewed with more than one person.

The email doesn’t have to be anything long, but it should thank them for their time, reiterate some of what you discussed (as noted above), and state that you look forward to hearing back from them. Proper email etiquette can make or break your follow up, so remember to keep it professional.

Although not a requirement, a blank thank you card in which you can write a handwritten note goes a long way. As in your email, thank them for their time and bring up conversation points that will help them recall your interview. Whereas emails get read and then buried in the interviewer’s inbox, a physical card sits on their desk. It’s a gesture that shows you’re a job seeker who takes initiative.

Andrew LaCivita, chief executive of milewalk Inc., believes two of the most important aspects of the thank you notes are “speed and thoughtfulness.” An email can be sent quickly, but a physical card shows that you took the time to think about the interviewer and care about the position.

It’s important to be patient in waiting to hear back from the company. If they tell you that you can expect to hear back in a week and you still haven’t heard from them after that, however, it’s okay to send another email.

There’s a fine line between politely reminding them that you look forward to hearing back and annoying them because you’re still waiting for their response. Sending a short follow up from your last email is an acceptable way to see where they are in the selection process.

If you receive news that you were not offered the job after your interview, remain courteous and thank them again for their consideration. It’s also beneficial to politely ask what you could have done better in the interview or if they have any recommendations on areas you can improve for future interviews. This way, you’re making the best of the situation and turning it into a learning experience.

*Thinking about business school? Challenge yourself to our **free 20-minute GMAT workout** to start shaping up your application.*

The post Job-Seeking Journal Part 4: How to Follow Up appeared first on Business School Insider.

]]>Is point closer to point than to point ?

(1) Point lies on the line

(2) Point lies on the line

Question Discussion & Explanation

**Correct Answer** - A - (click and drag your mouse to see the answer)

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**Verbal**

Advocates argue that five-cent bottle deposits charged on beverage containers are necessary for environmental protection because they help to ensure that plastic and glass bottles as well as aluminum cans are recycled. This is, the advocates say, because the five-cent redemption programs provide a strong incentive to return the used containers to recycling facilities. However, a recent study found that states without a bottle deposit had more success in implementing comprehensive recycling programs, which include paper, plastics, and steel, in addition to the beverage containers, than did states with a bottle deposit law.

**The answer to which of the following questions would be most useful in analyzing the significance of the study referenced above?**

A) Did any of the states surveyed lose revenue on the bottle deposit program?

B) Do the citizens of the states that were studied prefer five-cent redemption programs on beverage containers?

C) When the five-cent deposit programs were implemented, were the citizens of the states that began programs as enthusiastic about recycling as the citizens of the other states?

D) Did citizens of the states with and without bottle deposit programs purchase comparable numbers of beverages in plastic, glass and aluminum containers?

E) Where the bottle deposit and comprehensive recycling programs given equal funding?

Question Discussion & Explanation

**Correct Answer** - C - (click and drag your mouse to see the answer)

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1) In quadrilateral ABCD, is angle D ≤ 100 degrees?

__Statement #1__: AB = BC

__Statement #2__: angle A = angle B = angle C

2) Point P is a point inside triangle ABC. Is triangle ABC equilateral?

Statement #1: Point P is equidistant from the three vertices A, B, and C.

__Statement #2__: Triangle ABC has two different lines of symmetry that pass through point P.

3) ABC is an equilateral triangle, and point D is the midpoint of side BC. A is also a point on circle with radius r = 3. What is the area of the triangle?

__Statement #1__: The line that passes through A and D also passes through the center of the circle.

__Statement #2__: Including point A, the triangle intersects the circle at exactly four points.

4) ABCD is a square, and EFGH is a square, each vertex of which is on a side of ABCD. What is the ratio of the area of square EFGH to the area of square ABCD?

__Statement #1__: AE:AB = 4:7

__Statement #2__: The ratio of the area of triangle AHE to the area of square EFGH is 0.24

5) In the diagram above, the four triangles ABE, CBE, ADE, and CDE are all equal, and CD = 5. What is the area between the two circles?

__Statement #1__: AE = 3

__Statement #2__: angle BEC = 90 degrees

6) In trapezoid JKLM, KL//JM, and JK = LM = 5. What is the area of this trapezoid?

__Statement #1__: KL = 10 and JM = 15

__Statement #2__: angle J = 60 degrees

7) In the diagram above, ADF is a right triangle. BCED is a square with an area of 12. What is the area of triangle ADF?

__Statement #1__: angle DCF = 75 degrees

__Statement #2__: AB:EF = 3

8) FGHJ is a rectangle, such that FJ = 40 and FJ > FG. Point M is the midpoint of FJ, and a Circle C is constructed such that M is the center and FJ is the diameter. Circle C intersects the top side of the rectangle, GH, at two separate points. Point P is located on side GH. What is the area of triangle FJP?

__Statement #1__: One of the two intersections of Circle C with side GH is point P, one vertex of the triangle FJP.

__Statement #2__: One of the two intersections of Circle C with side GH is point R, such that RH = 7

9) Points A, B, and C are points on a circle with a radius of 6. Point D is the midpoint of side AC. What is the area of triangle ABC?

__Statement #1__: Segment BD passes through the center of the circle.

__Statement #2__: Arc AB has a length of 4(pi)

10) In the figure, ABCD is a trapezoid with BC//AD, AB = CE, BE//CG, and angle AEB = 90°. Point M is the midpoint of side BD. Point F, not shown, is a vertex on triangle EFG such that EF = FG. Is point F inside the trapezoid?

__Statement #1__: BE = EG

__Statement #2__: FG//CD

Full solutions will come at the end of this article.

Here are two previous blogs on GMAT DS questions about Geometry.

GMAT Data Sufficiency: Congruence Rules

GMAT Geometry: Is It a Square?

One big difference between Geometry on the PS questions and Geometry on the DS questions is that for all the PS questions, unless otherwise noted, you know that all diagrams are drawn as accurately as possible. That is the written guarantee of the test writers. By contrast, no guarantee at all accompanies the diagrams on the DS questions. Consider the following diagram.

This triangle appears equilateral. There is no guarantee that it is exactly equilateral, with three exactly equal sides and angles exactly equal to 60 degrees. If this were diagram given on a PS questions, we would know that the triangle is at least *close* to equilateral: all the side lengths are close to one another, and the angles are close to 60 degrees. We would know that much on a PS question. If this diagram were given on a DS question, then triangle ABC could be *absolutely any triangle on the face of the Earth*. It could be a right triangle, or a triangle with a big obtuse angle, or a tall & thin triangle, or a short & wide triangle, or etc. It could be any triangle at all. Aside from the bare fact that ABC is some kind of triangle, we can deduce nothing from the diagram on a DS question. Other than the bare facts of what’s connected to what, you can deduce nothing about lengths, angles, and shapes of figures given on DS questions. They may be 100% accurate or they may look nothing like the the shape described by the two statements.

Because of this, some DS questions are a real test of your capacity for spatial reasoning and geometric imagination. Many DS Geometry questions, including ones here, test your capacity to imagine how different the spatial scenario might be.

If this is not a natural gift for you, I strong recommend drawing out shapes on paper. Even get a ruler, compass, and protractor, and practice constructing specific shapes. Use straws or some other straight items to construct triangles in which you can adjust the sides and the angles. Strive to visualize and picture physically every rule of geometry you learn. By working with shapes you can see, and working with your hands, you will be engaging multiple parts of your brain that will give you a much deeper understanding of geometry.

If the above discussion gave you some insights, you may want to look back at those practice problems before jumping into the explanations below. If you don’t understand something said in an explanation here, draw it yourself, and explore the different possibilities within the constraints. The point of geometry is to see.

1) The figure is drawn as a square, but on GMAT DS, there’s no reason to assume the figure is drawn anywhere to scale.

If both statements are true, then the figure could be a square, in which the answer to the prompt question would be “yes,” or it could be this figure:

For this figure, all the conditions are met, and angle D is considerably larger than 100°; thus, the answer to the prompt question is “no.”

We could get either a “yes” or a “no” to the prompt consistent with these conditions, even with both statements put together.

Answer = **(E)**

2) __Statement #1__: As it turns out, for any triangle of any shape, there is *some* point that is equidistant from all three vertices: this is center of the circle that passes through all three vertices.

If all three angles of the triangle are acute, then the point is inside the triangle. If the triangle is a right triangle, then this center is always the midpoint of the hypotenuse. If the triangle has an obtuse angle, then the center is outside the triangle. All Statement #1 tells us is that triangle ABD has three acute angles. Beyond that, we know nothing. Statement #1, alone and by itself, is **not sufficient**.

__Statement #2__: A triangle that has a line of symmetry is isosceles. Let’s say that one line of symmetry goes through vertex A and point P. This would mean that AB = AC and that angle B = angle C. Now, let’s say that another line of symmetry goes through vertex B and point B. This would mean that AB = BC and angle A = angle C. Putting those together, we get three equal angles and three equal sides: an equilateral triangle. If a triangle has two separate lines of symmetry, it must be an equilateral triangle. We can give a definitive “yes” to the prompt question on the basis of this statement. Statement #2, alone and by itself, is **sufficient**.

Answer = **(B)**

3) Statement #1 tells us that the center of the circle is on the line of symmetry of the triangle through point A, but the triangle could be any size. In the diagram below, this line of symmetry is blue, and triangles of four different sizes are shown.

There actually would be an infinite number of possible triangle sizes on the basis of this statement alone. This statement is wildly **insufficient**.

Forget about Statement #1. With Statement #2 alone, a variety of off-center triangles with four intersection points are possible:

Notice AB is a chord of the circle as well as a side of the triangle. This chord could be a medium length chord or anything up to the full diameter, and different sides of the triangle would result in different areas. We still cannot give a definitive answer to the prompt question. This statement, alone and by itself, is **insufficient**.

Combined statements. If the center of the circle is on the line of symmetry of the triangle, then this places significant constraints on the number of intersections. For tiny triangles, they would simply intersect at point A and not reach the circle on the other side: one point of intersection, so this doesn’t work.

Larger triangles would touch the circle in three places, at the three vertices: this also doesn’t work.

Slightly larger, and those two vertices at B and C would “poke out” of the triangle, producing five points of intersection: Point A plus four other points.

The only way we will get exactly four points is when the sides get long enough and the side BC drops low enough that it is tangent to the circle.

The altitude of this triangle, AD is exactly equal to the diameter. We could use the ratios of the 30-60-90 triangle to figure out the sides, and thus figure out the area. If the sides get any longer, then side BC would break contact with the circle, and there would be only three points of intersection. This triangle, with the point of tangency at D, is the only triangle on this line of symmetry that has exactly four intersection points, and we can compute its area.

The combined statements allow us to give a numerical answer to the prompt question, so together, the statements are sufficient.

Answer = **(C)**

4) Statement #1: since we care only about ratios, we can set any lengths that are convenient. Let AE = 3 and AB = 7: then BE = 3. The figure is symmetrical on all four sides, so, for example, AH = 3. This means AEH is a right triangle with legs of 3 and 4—that is, a 3-4-5 triangle! The hypotenuse HE = 5. That’s the side of the smaller square, and 7 is the side of the larger square. The ratio of areas is 25/49. This statement leads directly to a numerical answer to the prompt question. This statement, alone and by itself, is **sufficient**.

Now, forget all about statement #1.

Statement #2: triangle to small square = 0.24 = 24/100 = 6/25. Let’s say that the central square has an area of 25 and one triangle has an area of 6. This means that four triangles together would have an area of 24. The big square equals the central square plus four triangles: 24 + 25 = 49. The ratio of the two squares = 25/49. This statement also leads directly to a numerical answer to the prompt question. This statement, alone and by itself, is **sufficient**.

Each statement sufficient on its own. Answer =** (D)**

5) __Statement #1__: If AE = 3, then it must be true that EC = 3, because the triangles are all equal. Also, AB = BC = CD = AD = 5. Because the four angles meeting at point E are all equal, it must be true that each one equals 90 degrees. Thus, we have four right triangles, and each one has a leg of 3 and an hypotenuse of 5. Thus, each must be a 3-4-5 triangle. This allows us to see that the radius of the smaller circle is EC = 3 and the radius of the larger circle is BE = 4. From these, we could figure out the areas and then subtract these areas to find the area between them. This statement allows us to arrive at a numerical answer to the prompt question. Statement #1, alone and by itself, is **sufficient**.

__Statement #2__: This statement tells us something we already could figure out from the prompt information. Technically, this statement is tautological. A tautological statement is one that contains no new information, nothing new that we couldn’t figure out on our own; examples of tautologies are “My favorite flavor of ice cream is the flavor I like most” or “Today is the day after yesterday.” Like those statements, Statement #2 adds nothing to our understanding. Statement #2, alone and by itself, is **not sufficient**.

Answer = **(A)**

6) This is question that demands visual insight.

__Statement #1__: Think about these lengths. The top, KL is twice the length of the slanted sides, and the bottom, JM, is three times the length. This means that we could build this trapezoid from five equilateral triangles.

With other combinations of four lengths, we would be able to get different quadrilaterals resulting (e.g. changing the tilt of a rhombus). With these lengths (5, 10, 5, 15), there is no other quadrilateral possible. (Try this with physical items with lengths in the ratio 1:2:1:3 to see for yourself.) Thus, we know all the angles. We know that each equilateral has side of 5, so we could figure out the area of each equilateral, then multiply by five. Thus, we can find the area on the bases of this statement alone. Statement #1, alone and by itself, is **sufficient**.

__Statement #2__: If we know just this, then the shape could have any width. It could be relative narrow or a mile-wide. We cannot determine a unique area on the basis of this statement alone. Statement #2, alone and by itself, is **not sufficient**.

Answer = **(A)**

7) We know the area of the square, so the side of the square is

Thus, we know the length of the vertical leg, CE, in right triangle CEF, and we know the horizontal leg, BC, in right triangle ABC. Furthermore, these two triangles must be similar to teach other and similar to the larger triangle, ADF, because all the angles are the same.

__Statement #1__: We know triangle CDE is a half a square, so it’s a 45-45-90 triangle. Angle DCE = 45 degrees. Well,

(Angle ECF) = (Angle DCF) – (Angle DCE) = 75 – 45 = 30 degrees

This means that CEF is a 30-60-90 triangles, and so is triangle ABC because they are similar. In each, we know the length of one side, so we could find the other sides and solve for the areas. Thus, we could find the area of the entire triangle ADF. This statement leads directly to a numerical answer to the prompt question. Statement #1, alone and by itself, is **sufficient**.

__Statement #1__: This is interesting. We know that triangles ABC and CEF are similar, so they are proportional. Let AB:BC = r. Then CE:EF = r as well.

Now, notice that both BC and CE are sides of the square. Let BC = CE = s.

Now, multiply those two fractions together, and the s terms will cancel.

This the ratio of the longer leg to the shorter leg in the 30:60:90 triangle. We know the sides of the square, so we can find all the lengths in triangles ABC and CEF, which would allow us to find all the areas. Thus, we could find the area of the entire triangle ADF. This statement leads directly to a numerical answer to the prompt question. Statement #2, alone and by itself, is **sufficient**.

Each statement is sufficient on its own. Answer = **(D)**

8) We know the diameter of the circle is FJ = 40, so its radius is r = 20. FJ = 40 is also the base of the triangle in question. We need the height of the triangle in order to find its area.

__Statement #1__: We know point P is one of the two points where the circle intersects side GH, the top of the rectangle. We still don’t know how tall the rectangle is. We know the height must be less than 20, so that the circle can intersect it, but we certainly don’t know the exact height.

Without an exact height, we cannot compute an exact area. Statement #1, alone and by itself, is **not sufficient**.

__Statement #2__: Construct Point Q, the midpoint of GH, and draw in segments MQ and MR. MQ joins midpoints of opposite sides of a rectangle, so this would be perpendicular to both FJ and GH.

We know that MR is a radius, so it has a length of 20. We know that QH is half the length of GH, so QH = 20. We know that RH = 7. Notice

QR + RH = QH

QR = QH – RH = 20 – 7 = 13

Now, look at right triangle MQR. We know the hypotenuse MR = 20. We know the horizontal leg QR = 13. We could use that most extraordinary mathematical theorem, the Pythagorean Theorem, to find the length of QM. On GMAT Data Sufficiency, we don’t have to carry out the actual calculation: it results in an ugly radical expression anyway. It’s enough to know that we could find the numerical value of QM, the height of the rectangle.

We don’t know the exact position of point P, but it’s somewhere on GH, and every point on GH has the same height above FJ, so this height would be equal to the height of the triangle. Thus, we could find the height of the triangle, and therefore the area. On the basis of this statement, we could give a numerical response to the prompt question. Statement #2, alone and by itself, is **sufficient**.

Answer = **(B)**

9) __Statement #1__: This one guarantees that BD is a line of symmetry in the diagram, so triangle ABC would have to be isosceles, but it could be any one of a number of a different sizes & shapes.

In all these examples, AB = BC and (angle A) = (angle C). The triangle could be equilateral, but it doesn’t have to be. These three examples have different areas, so this statement, by itself does not guarantee that we could calculate an exact area. Statement #1, alone and by itself, is **not sufficient**.

Now, forget all about statement #1.

__Statement #2__: We know that the radius is r = 6, so

Thus, we know that arc AB is 1/3 of the entire circumference. Therefore, it must occupy an angle of 1/3 of 360 degrees: arc AB must occupy 120 degrees.

In an equilateral triangle, all three angles would be 60 degrees and all three arcs would be 120 degrees. Here, all we know is that one arc, AB, is 120 degrees, and other two arcs could be other values. Thus, angle C must be 60 degrees, but other other angles can be other values.

In all three of those diagrams, AB is a 120 degree arc and angle C is 60 degrees. The triangle could be equilateral, but it doesn’t have to be. Statement #2, alone and by itself, is **not sufficient**.

Now combine the statements. From the first statement, we know that AB = BC and (angle A) = (angle C). From the second statement, we know that (angle C) = 60 degrees. Well, that would mean that (angle A) = 60 degrees as well, and that leaves exactly 60 degrees for angle B. If we have three 60 degree angles, we know that ABC is equilateral. If we know the radius of a circle, then we can calculate the area of an equilateral triangle with its three vertices on the circle (this would involve subdividing the equilateral into six 30-60-90 triangles).

With the combined information of both statements, we can find a definitive answer for the prompt question. Together, the statements are **sufficient**.

Answer = **(C) **

10) Start with what we know from the prompt. We know BCGE is a rectangle with two parallel vertical sides that are perpendicular to two parallel horizontal sides.

We know that ABE and CGD are right triangles with the same length vertical legs and the same length hypotenuses, so by the Pythagorean theorem, the third sides must be equal, AE = DG, and the two triangles are equal in every respect.

We know that entirely figure is symmetrical around a vertical line down the middle. The trapezoid is entirely symmetrical, and isosceles triangle EFG is also symmetrical. Suppose we constructed the midpoint of EG and called it Q. Then, line MQ would be the symmetry line of both the trapezoid and the isosceles triangle. This line MQ would be parallel to BE and CG, and it would be perpendicular to BC and EG. If we extended MQ above and below the trapezoid, we would be guaranteed that point F would lie somewhere on this line.

For this problem, I am going to jump ahead to the combined statements. Statement #1 tells us that BCGE is a square. Statement #2 tells that the sides of the trapezoid are parallel to the sides of the isosceles triangle (by symmetry, the parallelism must be true on both the right and the left side). Even with all this information, we cannot give a definitive answer to the prompt question.

You see, the missing piece are the lengths of AE and DG. By the symmetry of the diagram, we know AE = DG, but we don’t know how this size compares to BM = MC. In the diagram, it appears that DG < MC, but because this is a GMAT DS diagram, we can’t believe sizes on the diagram.

If DG < MC, then point F will be above M, outside of the trapezoid, as seen in the diagram on the left. If DG = MC, then point P will coincide with point M. If DG > MC, then point F will be below point M, inside the trapezoid.

Because we don’t know how the AE = DG length compares to the BM = MC length, we don’t know where point F falls, and we can’t give a definitive answer to the prompt question. Even combined, the statements are **insufficient**.

Answer = **(E)**

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