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Geometry Series Part 2: Inscribed Triangles

By posted October 27, 2010

To start off, let’s quickly review the essentials. These are formulas/concepts you must know:

  1. a² + b² = c², but only when a right triangle. If you don’t know it’s a right triangle, Pythagorean theorem does not apply!
  2. Common special right triangles include 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples.)
  3. 45-45-90 triangles are ALWAYS in the ratio 1:1:√2
  4. 30-60-90 triangle are ALWAYS in the ratio 1:√3:2
  5. Angles and opposite sides are in the same relative size order, but are NOT proportional.

Let’s continue with a standard diagram in which we have an equilateral triangle inscribed in a circle, which is inscribed in a square.

eq tri in circle in square 3

The center point of all three figures (triangle, circle, square) are all the same, but this is ONLY true if the triangle is equilateral. Therefore, if given ANY piece of information about the circle, square or triangle, we can derive the rest. We draw a perpendicular line from the center to the side of the triangle.

eq tri in circle in square 4

Note that the hypotenuses of the smaller triangles are equal to the radius of the circle. We also know that the smaller triangles are each 30-60-90 because you are taking the 120-degree internal angle from the circle’s center and cutting it in two. Here are your basic conversions:

r = ½d = ½s, where s is the side of the square.
The sides of the 30-60-90 triangles become ½r : (r√3)/2 : r respectively
The side of the equilateral triangle becomes 2*(r√3)/2 = r√3

If given the area of the square, we should be able to derive essentially any other information.

Area of an Equilateral Triangle

The area of an equilateral triangle equals (s²√3)/4. Memorize this. It will save you the time of drawing a 30-60-90 triangle, solving for the base, finding the height, multiplying and dividing by 2. That was long to write, imagine how long it takes to do!

If  the area of the square = 64 and we needed to find the area of the triangle, we just use the conversions above:

d = 8
r = 4
side of triangle = 4√3

Area of triangle = [(4√3)²√3]/4 = 16*3*√3 / 4 = 4*3*√3 = 12√3

Angle Relationships

eq tri in circle in square w angle 5

Another important rule is that the interior angle created from of two radii extending to the outside of the circle is exactly twice the measure of any angle on the circle extending to those same points.  In the image above, 2b = a. This information is never explicitly stated on tests, but will come up on quant questions over and over.

There are infinite variations of these concepts. Be flexible in your reasoning, and practice makes perfect!

Good luck!

Read other articles in this series:
Geometry Series pt 1, Circles inscribed in squares

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Argument Writing Task: Part 4

By posted October 14, 2010

After learning all the possible fallacies and how to spot them, it’s time to look at a real possible argument task. When I say “real,” I mean this could possibly be on your actual GRE, though the chances are very slim–approximately 1 out of 245. That statistic should not deter you, and you probably shouldn’t take it upon yourself to write practice essays for each and every prompt. For one thing, such a task would take a long time (don’t you have more important things to do?). Secondly, though these argument prompts are all ostensibly different, they repeat many of the same fallacies. To confirm this, just check out a hearty sample of prompts (perhaps twenty or thirty, or whatever you realistically have time for) and think about how your arguments against the prompts might overlap. The similarities will pleasantly surprise you. Here is one such prompt to get us started:

The Prompt

The following appeared in a memorandum written by the vice president of Nature’s Way, a chain of stores selling health food and other health-related products.

“Previous experience has shown that our stores are most profitable in areas where residents are highly concerned with leading healthy lives. We should therefore build our next new store in Plainsville, which has many such residents. Plainsville merchants report that sales of running shoes and exercise clothing are at all-time highs. The local health club, which nearly closed five years ago due to lack of business, has more members than ever, and the weight training and aerobics classes are always full. We can even anticipate a new generation of customers: Plainsville’s schoolchildren are required to participate in a ‘fitness for life’ program, which emphasizes the benefits of regular exercise at an early age.”

  1. 1. What’s the Argument?: The speaker claims that Nature’s Way, a health food store, should open in Plainsville, where “residents are highly concerned with leading healthy lives.” How did we gather this profile of Plainsville’s inhabitants? According to our speaker, three facts account for this description: 1. Increase in sales of exercise shoes and clothing; 2. The local health club is experiencing its highest rates of attendance, and 3. Plainsville’s schools are now mandating a fitness program.
  2. 2. The Problems with the Argument: I will enumerate potential problems I see with the speaker’s argument and reasoning, in no particular order, so to mimic your thought process and note-taking when you first come across an argument prompt:
    1. False correlation between exercise and health food: The speaker fallaciously correlates exercise with healthy eating habits. Nature’s Way is neither a health club nor a sporting goods store, but a health food store. While, ideally, a healthy lifestyle entails both exercise and healthy eating habits, the two are not mutually inclusive. With the convenience of fast food, our national eating habits, on average, are at their worst in history. Often, this guilt about eating habits encourages fast-food patrons to exercise, but not necessarily change their eating habits.

  1. Does buying exercise clothing necessarily cause exercise?: The speaker assumes that the increase in health-related items suggests that the residents of Plainsville are “highly concerned with leading healthy lives,” but there are other possible sources of these increases. The sale of running shoes and exercise clothing could be attributed to a fashion trend that prizes the aesthetic value–rather than the functional value–of such clothes; or, more simply, exercise clothes may be an inexpensive alternative to other clothing styles.
  1. An increase in health club attendance does not guarantee profits for Nature’s Way: Perhaps the local health club is full because of a lack of competition. The speaker refers to the club as “the local health club,” suggesting it’s the only one of its kind in Plainsville. If this is true, then high rates of attendance do not suggest an overwhelming increase in the citizens’ exercise.
  1. The compulsory exercise program is a poor indicator of future healthy lifestyles: The speaker mistakenly assumes that the compulsory “fitness for life” program enacted by schools will foster a new generation of health conscious individuals. Though we may applaud the efforts of schools to introduce such a program, we cannot assume that the program will have any lasting effect on the children’s lifestyles. In fact, mandating exercise in school, much like making beloved classics of literature “required texts,” may cause unintended opposition to exercise. Many children often willfully oppose orders given by parents and school teachers, not out of any sound reasoning, but because of sheer childhood obstinacy.

  1. Future interest in exercise?: Even if Plainsville residents are interested in health foods, how do we know the interest will continue in the future? After all, these changes in lifestyle habits are relatively recent; why shouldn’t we assume that they can easily revert back to unhealthy lifestyles?
  1. Competition?: The speaker fails to mention the possibility or lack of competing health food stores. How can we be sure that Nature’s Way will thrive despite its potential new competition?
  2. Suggestions for Improvement: To improve the argument, the speaker must show a correlation between exercise habits and healthy eating habits, perhaps through a survey or study. Also, the speaker should investigate the popularity of Plainsville’s health club and explain how Nature’s Way will plan to beat the competition.

What we have here is an abundance of information, not quite an essay. To write the essay, choose the best examples and develop them into coherent paragraphs. Don’t be afraid to integrate smaller fallacies into paragraphs: an abundance of information is not a bad thing, and, in fact, longer essays tend to receive higher scores. For practice, you may want to give yourself 30 minutes and write this essay, using your own words and, if you have them, your own arguments.

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Geometry Series Part 1: Circles Inscribed in Squares

By posted August 3, 2010

In this series, we will cover many types of geometric scenarios encountered on the GRE. A basic knowledge of simple formulas (area, perimeter, etc.) is essential, but there are numerous shortcuts to geometry questions that will save you time. Today, we’ll explore circles inscribed in squares.

Some Things to Remember

circle in square dr tri 1

  • The center of the square is the same point as the center of the circle
  • Draw lines! Depending on what the stimulus asks for, draw in lines that create simple shapes. (Squares can be turned into triangles, for example.)
  • Shared angles will normally not be explicitly stated, unless necessary.
  • Trust the pictures, but not too much. Inferences must be drawn from fact. Just because it looks like 90-degrees doesn’t mean it is! (Many of these common inferences will be detailed in this series.)
  • Lengths cannot be negative. Be careful in DS questions that pose equations in the context of quadratic equations with two solutions. If one solution is negative and the other is positive, only the positive solution remains and the information is sufficient.

For circles:

  • d=2r and all lines from the center to the exterior equal r.
  • C = 2πr = πd
  • A = πr²
  • NEVER use 2πr² unless you are adding the areas of identical circles!
  • Tangent lines create right angles with the radius that meets that tangent.
  • If you know r, you know everything about the circle!
  • Use π = 22/7 with caution. Remember 22/7 > π.

For squares:

  • The diagonal equals s√2, since it creates 45-degree angles.
  • The intersection of the diagonals creates a right angle.
  • When a circle is inscribed inside a square, the side equals the diameter.

Usually, you will be provided with one bit of information that tells you a whole lot, if not everything. If given the length of the side of the square in the above image, we can actually find the length of the hypotenuse of the internal triangle (s = d = 2r, so the hypotenuse = (s√2)/2).

Shaded Areas

Find the large area and subtract the small area from it. When dealing with circles along with other figures, eliminate answer choices that ONLY have π’s in them or don’t have any at all. Typically, your answer will look like x + yπ.

Two important takeaways:

  1. Never assume without proof.
  2. Follow the trail.

Post below with other helpful tips for your fellow GREers.

Next Lesson: Inscribed Triangles.

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Simple Quantitative Strategies, Part 2

By posted July 26, 2010

This is the second installment in the simple quantitative strategies series. These strategies are here to remind you that there is just a bit more to studying than simply cramming math material. Yes, learning the processes is more than half the battle, but outsmarting the test-writers is very important too. Use the test format to your advantage, don’t overwork, and anticipate the clever gambits so often deployed by the ETS. Beating the GRE quantitative requires a balanced combination of math skills and test-taking skills–these strategies will help improve the latter.

  1. BallparkingTo ‘ballpark’ is to roughly approximate. In terms of GRE quantitative strategy, ballparking essentially means thinking about mathematical figures in a vague, imprecise, but nonetheless common sense manner. When we are overwhelmed by figures and calculations, it’s easy to make mistakes. Moving a decimal one unit could transform a correct answer into a wrong answer, no matter how many correct steps you painstakingly went through. This is where ballparking plays a significant role.Let’s look at a crude example:What’s 32.33 % of 50?A. 5.125
    B.16.165
    C. 35.685
    D.50.350
    E. 70.195

    Any relaxed, common sense thinker will probably be able to answer this question without doing a calculation. But, when you’re in the middle of a timed test, things change. Anxiety sets in, and you go into human calculator mode. You see a question like this and immediately start calculating the product.

    Step back, though, and look at the simplicity of the question. 32.33 percent is awfully close to one third. A third of 50 is a little more than 15 (15*3= 45). The only thing close to that is B; it can be no other answer.

  1. Avoid TrapsNearly every multiple choice math problem has trap answers, or attractors. These types of answers catch your eye for one reason or another, often making the problem appear a little bit easier than it actually is. You may notice the anticipated answer in the choices and think that you won’t have to finish the problem, but remember that such a choice is probably a trap.Looks look at an example of what this might look like:1. The price of a T-shirt was reduced by 20%. Then, during a special sale, the price was reduced another 20%. What was the total percentage discount from the original price?a. 25%
    b. 36%
    c. 40%
    d. 42%
    e. 50%

    You may read this question and think that a 20 percent discount plus another 20% discount equals a 40% discount. Seeing 40% as an answer choice, you may be inclined to choose it and move on. Unfortunately, you’ve just missed a pretty easy question. Did you really think that the test would give you a question that required such minimal effort as adding 10 and 10? It’s nice to dream, isn’t it? But, let’s get real. Just perform the calculations as necessary.

    First, why not imagine the shirt is 100 bucks to start.

    Take 20% off of 100, and you get 80.

    Take 20% off of 80 (80 / 5 = 16) and you get 64.

    We went from a 100 dollar shirt to a 64 dollar shirt. That’s a difference of 100-64=36.

    Thus, the total discount is $36, B.

These two strategies may appear simple, but they can mean big points in a pressured testing environment. So, when you practice, think about these techniques! They will keep you from making careless mistakes on easy to intermediate problems, which can end up making a huge difference on a computer adaptive test.

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Issue Writing Task: Part 4

By posted July 2, 2010

In our last post, we learned about the intricacies of GRE Issue pre-writing and the introduction. Both are tremendously important to write a quality essay, but, let’s face it, they are really just leading up to the body of your essay. The body of an essay contains the evidence and reasons for an argument. Each paragraph should adopt one unified reason in support of an overall argument. How many body paragraphs should there be? You could probably get away with as few as two, but I’d say three body paragraphs is a good minimum to shoot for. Any more paragraphs than that is just good insurance.

1. Starting the Body: If you have written a solid outline of your essay, then, by all means, follow it. If you haven’t yet figured out the order of your ideas, but you have grouped them into coherent sections, then begin with what you feel is easiest to write (a paragraph that is easy to write is often convincing and logical). Remember, if you realize that a different order of body paragraphs would make more sense, then you can arrange them later. Taking a computer test has its perks—take advantage of them.

Each one of your body paragraphs should begin with a topic sentence that tells the reader what the paragraph is about. Since the essay graders are not spending much time on each essay, make their lives easier by providing a roadmap to follow. A clear essay makes a happy grader; a happy grader makes a happy test-taker.

If you decide to qualify the prompt statement and choose to present opposing sides of an argument, then paragraph order is very important. For example, if you’re evaluating the argument that “political leaders should withhold information from the public,” you might want to argue that while political leaders cannot be expected to divulge the embarrassing minutia of their personal lives, they must remain honest in order to avoid falling into demagoguery and to uphold the values of democracy. What I’ve done with this thesis is concede a point to my opposition, which I will acknowledge in the beginning of my essay, but then I will end strong with my “honest is the best policy” argument. It would be nonsense to end my evidence with a concession. The last body paragraph is the one that will most strongly resonate with your reader, so if you opt for a thesis like this one, start with your concession and move into your strong argument.

2. Conclusion: No matter how tired you are after writing the body paragraphs, you must write a conclusion. Some may argue that the conclusion paragraph is often superfluous or redundant, but it is still a convention that you adhere to—at least for the GRE. The conclusion is meant for you to remind the reader of the main thrust of your essay. In your conclusion, restate your thesis, preferably in different words. If you can, try to think of a larger implication of your argument. Ask yourself “so what” after you’ve written these 500 words, and maybe a broader implication will come to you. If it doesn’t, don’t worry: an insightful flourish at the end of an essay may help put you into 6 territory, but don’t stress about forcing brilliance if it’s not coming naturally.

3. Revise: Allow yourself around 8-10 minutes to revise your essay. Watch out for awkward phrasing, inappropriate diction, and poor grammar. While you are reading through for these mechanical errors, think about the logical flow of your essay. You do have that copy/paste function to rearrange sentences and paragraphs, so use it to your advantage. When rereading your essay, keep these things in mind:

-Don’t be too one-sided. While it’s fine to adopt a strong position, don’t be afraid to acknowledge other viewpoints or anticipate objections.

-Pay attention to flow. Each paragraph should flow naturally to the next. This is easier said than done, but, sometimes, all you’ll need is a transitional phrase or sentence to do the trick.

-Avoid unnecessary repetition. Under the time constraint, you may notice yourself repeating key phrases over and over. If it seems tiresome as you read, cut it down. Redundancy is a sign of immature writing, and while essay graders may acknowledge it as a common side effect of timed writing, it’s best to cut it out if you can.

-Check for consistency. Does your intro address the topic? Does your body address your intro? Does your conclusion address your body? All parts of your essay should work together as a whole, and they should directly address the prompt

Just follow these steps and you’ll have a quality GRE essay. The only way to test your skills, however, is to practice. Luckily, the GRE website gives you all the possible topics, so you have no excuses—start writing!
Read other articles in this series:

Issue Writing Task pt. 1
Issue Writing Task pt. 2
Issue Writing Task pt. 3

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Issue Writing Task: Part 3

By posted July 1, 2010

Last time, we looked at the single most important step in writing your essay: brainstorming. After all, you can’t write a solid essay without solid ideas. Unfortunately, great ideas alone will not get you the grade. That’s where organization comes in. Once you have your brilliant ideas down on paper, follow these steps:

1. Adopt a position / Articulate your thesis: When you look down at the overflowing mass of ideas you have written, your first step is to identify each idea as “agreeing with” or “disagreeing with” the prompt statement. In our previous example statement, “Over the past century, the most significant contribution of technology has been to make people’s lives more comfortable,” you might have jotted down “advances in medicine,” “automotive safety,” “machines relieve factory worker of monotonous work,” and “internet allows for ease of communication.” To quickly identify the stance of these ideas, write down “pro” or “con” next to each; “pro” indicates that the idea supports the statement, and “con” indicates that it opposes the statement. “Advances in medicine,” for example, deserves a “con” since it argues that there are more prodigious technological achievements than those that make us comfortable. “Machines relieve factory worker of monotonous work,” however, is an example in favor of the statement, so it deserves a “pro.” After you have labeled your pieces of evidence, organize these ideas into body paragraphs. Try to see where ideas cohere; if some ideas are weak, don’t use them. Fewer, finely tuned arguments are better than a bulk of crude ones.

After you organize your ideas, you should start to see a coherent argument forming. Remember, your argument can be one-sided, or it can qualify the conditions of the prompt statement.

2. Introduction: Your introduction should first clearly articulate the argument in the given statement. Show the reader that you understand the implications of the issue at hand. Then, articulate your stance on the issue, indicating your agreement, disagreement, or qualification of the statement’s argument. Don’t get too specific with your evidence here, but do give an informative outline of your main arguments.

Your thesis, which is essentially a sentence or two that outlines your argument, should go at the end of the introduction. Ideally, your thesis statement should be organically integrated into your introduction. The purpose of your introduction is to build up to your argument, so we don’t want the thesis seem forced or out of place.

Don’t worry too much about refining your thesis statement at this stage. In fact, you may choose to write the thesis after you’ve written your body because your argument may slightly change during the writing process. If you’re taking a computer-based exam, this is no big deal. No need to leave a chunk of blank space—the magic of word processing takes care of this.

These two stages of the writing process should take about 6 minutes; when combined with the initial brainstorming stage, it should take about 9 minutes tops. That doesn’t seem like a lot of time, but you really want to devote the bulk of your time to the body paragraphs. Not only are the body paragraphs the most important and most heavily weighted part of the essay, but the process of writing them will help you refine your own ideas. Very often, writing the body paragraphs leads to a more fine-tuned thesis, so do not strictly limit your argument before you begin writing.

Next time, we’ll look at how to construct the body paragraphs. Stay tuned, and in the meantime, practice with Grockit or write some sample outlines and introductions.

Read other articles in this series:

Issue Writing Task pt. 1
Issue Writing Task pt. 2

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GRE Issue Writing Task: Part 2

By posted June 28, 2010

Last time, we looked at what the Issue Writing task is and what exactly it asks of us. Your job is to present your perspective on an issue. You may agree with, disagree with, or qualify the given statement, but you must defend your perspective with evidence and a convincing argument. The range of topics is very broad, but the specific content of each question does not really allow you to tailor one essay–say, on Abraham Lincoln and Martin Luther King–to all possible essay prompts.

The essay prompts run the gamut of intellectual disciplines, including anthropology, sociology, history, law and government, political science, philosophy, the fine arts, the performing arts, literature, physical science, and economics. Fortunately, you’ll have your choice of two prompts to choose from. You do not need to have a specialized knowledge in any one of these disciplines, but if you do, it will undoubtedly facilitate your writing and ideation. Fortunately, you’ll have your choice of two statements to choose from. So, if there is one or two topics you want to avoid, chances are you’ll have your chance to avoid them. ETS has been kind enough to actually show us all the possible topics beforehand here: http://www.ets.org/gre/general/prepare/sample_questions/analytical/issues/index.html. Yes, there are hundreds, but it helps to look at a bunch of these, and feel free to practice with them.

For many students, the first looming question about an essay assignment is length. Most authorities suggest that the issue writing task be at least 400 words, which, in 45 minutes, is rather brief. That being said, length is never really the main goal. A concise, well-articulated essay of 400 words will be better than a wordy, redundant, and trite essay of 700 words. At the same time, a 700 word essay with many convincing examples and articulate prose will be better than a vague 400 word essay without concrete examples. In the end, length should not be on your mind: clear writing, convincing examples, and a solid argument should be your focus.

Now, let’s look at the writing process. Beginning a timed essay will probably be the most intimidating part, so make sure you develop a system for writing them. Here’s the first, and in my opinion, most important, step for writing the essay:

  1. Brainstorm: As soon as you decide which of the two choices you’d like to write on, begin the brainstorming process. Jot down some reasons for and against the issue. You may already have a personal opinion about the issue, but set that aside. Let your ability to reason an argument do the choosing for you. Based on the reasons you brainstorm, you may want to argue for the issue, against it, or qualify it.

When brainstorming, it is important to stay on track. Always keep that quoted statement in mind, and reread it to come up with new ideas. Before you jump into pro and con arguments, briefly sum up the statement’s argument on paper. For example, let’s look at an actual prompt from the GRE website: “”Over the past century, the most significant contribution of technology has been to make people’s lives more comfortable.”

What’s the author trying to say? Simply put, the argument is that in the 20th century, the most important accomplishment of technology has been to make people more comfortable. Immediately, you should think of important technological accomplishments that don’t fit into this narrow category. Advances in medicine, for example, allow people to live longer. Yes, you may concede that certain drugs have been engineered to reduce human suffering and thus make people more comfortable, but on a grand scale, medicine has accomplished much more than comfort. Don’t forget, you’ll want to acknowledge what statement’s argument. Technology has indeed made people comfortable: automated machines have reduced the monotony of factory labor, computer engineering has allowed for the construction of safer and more efficient vehicles, roads, etc, and the internet has allowed us to more easily keep in touch with distant friends and families. Certainly these facts fall into the author’s argument, but it’s your job to assess their “significance” in the face of other technological achievements.

Brainstorming is all about parsing the author’s statement into manageable parts that inspire ideas. The statement tells you what to include and exclude in your essay. Never assume that you can just write an essay about ‘technology’ and avoid the statement’s argument. Each statement is specific, and your response should be the same.

Read other articles in this series:

Issue Writing Task pt. 1

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GRE Issue Writing Task: Part 1

By posted June 23, 2010

45 minutes of your entire exam will be devoted to the Issue Writing task, so even though it may not be the most famous section of the test, do not take it lightly. Do not assume that, because this is simply a timed essay, you do not have to study for it. Though practicing writing may be an even bigger pain than practicing multiple choice questions, you still have to do it to increase your chances of a high score.

Your job in Issue Writing is to present your perspective on an issue. The Issue will consist of two elements: a statement of your task and a 1-2 sentence topic which is a statement of opinion on an issue. Your statement of task will always be the same: “present your perspective on the following issue; use relevant reasons and/or examples to support your viewpoint.” The topic might look like this: “The objective of science is largely opposed to that of art: while science seeks to discover truths, art seeks to obscure them.”

Before you see your topic,  the testing system will present you with more directions specific to the task;

1. Writing on any topic other than the one presented is unacceptable.

2.The topic will appear as a brief statement on an issue of general interest.

3. You are free to accept, reject, or qualify the statement.

4.You should support your perspective with reasons and/or examples from such

sources as your experience, observation, reading, and academic studies.

5. You should take a few minutes to plan your response before you begin typing.

6. You should leave time to reread your response and make any revisions you think

are needed.

Contact one of Grockit’s expert GRE tutors and find out how you can get personalized feedback on your practice essays.

Read more »

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Quantitative Comparison Strategies: Part 2

By posted June 18, 2010

This is the second installment of the quantitative comparison strategies; below, you’ll find traditional and alternative strategies to solving quantitative comparisons.

Factoring:
Factoring is another popular way to simplify both expressions in order to make a comparison easier. Factoring doesn’t just mean pulling an x out of an expression. You can, and should, factor with constants (real, known numbers) in order to simplify. Let’s check out a few examples:

Example 1. 9^99 – 9^98          OR                   9 ^98

Hmm. There’s no way I’m calculating this one. Let’s try to factor out a 9^98. It helps to remember your basic multiplication and exponent rules, e.g. (x^5)(x)= x^6

9^98 (9 – 1)                  OR                            9^98

= 9^98 (8)                     OR                           9^98

What’s bigger: a huge number times 8 or that huge number by itself? Clearly, A is bigger.

Example 2. 5x + 5y / x + y         OR                   5

Here’s a chance for some good ol’ fashioned factoring.

5( x +y ) / x +y                           OR                     5

(X+Y) on top and bottom cancel, giving you: 5 OR 5

Easy choice. They’re equal.                  

Simplify by multiplying or dividing both sides by the same value:
Let’s look at the problem we did above for the factoring method. We can actually use another method to figure out the answer. This just goes to show you that there is usually more than one way to solve a math problem. Though there is usually one “fastest” method, there are some rare cases when there are multiple quick methods to solving a problem.

1. 9^99 – 9^98             OR                     9 ^98

Why don’t we divide both sides by 9^98?

1. 9^99 – 9^98 / 9^98        OR           9 ^98 / 9^98

9^1 – 9^0                         OR                     9^0

9-1 OR 1 → 8 OR 1→A

Warning: Do not simplify by multiplication or division unless you know the quantity you are using is positive.

Let’s look at an example where you would not want to use this method:

3x                  OR               4x

If we tried to simplify by dividing by x, we’d think that we arrived at this:

3                   OR                  4

We might choose B as a result, but we’d be wrong. When in doubt, use your common sense. It’s easy to see that, when comparing 3x and 4x, the answer must be D because when x= 0, the values are equal, or when we use a negative number, 3x is larger.

The Fallback Strategy: Use 1, 0, a fraction, and a negative number as testers.
When you are testing variable expressions, the most reliable method to test them is to be thorough with the types of numbers you use. You must use a negative, a positive, a fraction, and a zero; we suggest that you use simple numbers in order to save yourself time and avoid any calculation errors.

Let’s check out some examples that show us why it’s necessary to be thorough.

Example 1: If x>0, y> 0, z= 0…

3z (2x +5y)                 OR                  3x (2z+5y)

When a zero is on the outside, as in our A value, the whole value is zero. Since all the other values are positive, we can be confident that our second value is larger.

Example 2: If x<0, y >0, z=0

3z (2x +5y)             OR                       3x (2z+5y)

We still have zero for our first value, but we’re not in the clear quite yet. When negative numbers are involved, always test them. Testing our second value gives us 3(-1) (2*0 + 5*2). What we have is a negative multiplied by a positive, so we know the answer is negative. Our first value is larger.

Example 3: If 3x= 4y

x                               OR                         y

We know that if x and y are positive, then x is greater than y, e.g. if x is 4 then y is 3. But what if they are negative? If x is -4, then y is -3. So in that case, y is greater. And, what if x is zero? Well in that case, y is zero also, so both values are equal.

It turns out that the answer must be D.
Example 4: If x > 0 and x does NOT equal 1…

x ²                        OR           x

So x must be positive and cannot be one. So in that case, it must be A right? Common Not so fast. Remember, our special numbers to test are negatives, positives, zeroes, and of course, fractions. Fractions have some very special properties. If we multiply a fraction by itself, it happens to become smaller, not bigger (.5 * .5 = .25).

So in that case, our answer is D.

There you have it. When you practice on Grockit, try using different strategies to figure out which work best for you; remember, speed and accuracy are crucial!

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Quantitative Comparison Strategies Pt 1

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Quantitative Comparison Strategies: Part 1

By posted June 14, 2010

Quantitative Comparison problems are not like standard math problems you’ll find on the SAT or a common standardized test. Your job is not to find the correct answer amongst a group of answers (i.e. multiple choice) or to find the correct answer and write it in. You merely have to find out if one of the two expressions is larger, smaller, or equal to the other, or if such information is impossible to calculate. This may sound like a pain, but notice that you can get away with much less.

The art of quantitative comparison problems is getting away with the bare minimum to save time. Many quantitative comparisons are designed to look time consuming, which is a good indicator that there is a much faster way to solve the problem. Let’s look at some strategies that pinpoint those faster ways to solve quant comps: when to calculate, when not to calculate, and how to quickly compare variable expressions.

Avoid Unnecessary Calculation

If, in your practice, you notice yourself doing endless calculations, you are doing unnecessary work. The GRE will not make you do endless calculations on paper, even if such a strategy appears to be the most obvious way to reach an answer.

Before we examine certain question types, let’s look at a couple simple examples to show how immediate calculation can be an inefficient strategy:

1. 3569                               OR                           3(10) + 5 (10²) + 6 (10¹) + 9 (10^0)

If I saw this problem without thinking, I might multiply out the second column (3 times 10³ is 3000, etc). Such an approach is self-defeating. There is a simple trick here. Notice that 3569 is the same thing as saying 3000 + 500 + 60 + 9, which is what the expression on the right is really saying. Calculation is not necessary, and I know that both expressions are equal.

2. 31 x 32 x 33 x 34 x 35                               OR                       32 x 33 x 34 x 35 x 36

Again, if you don’t quickly examine the two expressions intelligently, you might jump into calculation, which would take you quite a while (not to mention leave you vulnerable to errors). Since we are just comparing the two expressions, we can cross out the numbers that appear in both expressions, that is, 32, 33, 34, and 35. Thus, we are left with a simple comparison: 31 OR 36; now it’s quite clear that B is greater.

Simplify

When presented with two baffling expressions, always think of ways to simplify before you calculate. Let’s check out this example:

1. 2,000,000 / 200, 000                        OR                             1,000 / 100

When you see many zeros in fractions like this, your first instinct should be to cross out matching zeros. If I have 2,000,000 in the numerator and 200,000 in the denominator, I should just eliminate five zeros from the top and 5 zeros from the bottom; now, my expression is simply 20/2 = 10. Same idea for column b: 1,000/100 = 10/1 = 10.

Simplify by Adding/Subtracting Same Value

1. 4x +5                  OR                                3x +6

I could approach this problem a few ways. First, I could use the tried and true plug-in method, where I would test a few simple numbers (preferably something like -2, 0, 2, and .5–you want to use a positive, a negative, 0, and a fraction). Don’t forget, though, that you can manipulate both expressions to make the comparison simpler. As long as I add or subtract the same number or variable from these expressions, I’ll have the same relationship between the two expressions. Remember, when choosing numbers to add or subtract, our goal is to make the relationship simpler, so:

Subtracting 5 from both sides gives us:

4x OR 3x+1

Subtracting 3x from both sides gives us:

x OR 1

Now, look how simple it is? The comparison is any number (x) OR 1, which is clearly indeterminate. Our answer is D.

Stay tuned for some more strategies for quantitative comparisons that will save you time and increase accuracy. In the meantime, try some of these strategies in Grockit!

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