![\"gm-tuaaof_img1\"](\"https:\/\/magoosh.com\/gmat\/files\/2014\/01\/gm-tuaaof_img1.png\")
<\/a><\/p>\n1) In the figure above, AC = BC = 8, angle C = 90\u00b0, and the circular arc has its center at point C.\u00a0\u00a0 Find the area of the shaded region.<\/p>\n
2)\u00a0 Employees at a company will vote for an executive team of five people from eight qualified candidates.\u00a0 The executive team consists of a president, a treasurer, and three warrant officers.\u00a0 If an executive team is considered different if any of the same people hold different offices, then how many possible executive teams could be selected from the eight candidates?<\/p>\n 3) In the above diagram, the 16 dots are in rows and columns, and are equally spaced in both the horizontal & vertical direction.\u00a0 How many triangles, of absolutely any shape, can be created from three dots in this diagram?\u00a0 Different orientations (reflections, rotations, etc.) and\/or positions count as different triangles. (Notice that three points all on the same line cannot form a triangle; in other words, a triangle must have some area.)<\/p>\n What all three of these problems have in common is: relying on formula alone will not help you.\u00a0 Complete explanations will come at the end of this article.<\/p>\n <\/p>\n Clearly, it\u2019s a good idea to know some GMAT math formulas<\/a> —– many of the best formulas to know are listed on that blog.\u00a0 Formulas can be wonderful shortcuts, very efficient time-savers.\u00a0\u00a0 Clearly it would be a mistake to walk into the GMAT Math section without knowing a single formula.<\/p>\n <\/p>\n Formulas are useful to a point, but one of the worst mistakes folks make when they study for GMAT Math is to think that all<\/span><\/i> they have to do to master the GMAT Math section is to memorize all the formulas.\u00a0 Similarly, the worst way to approach a GMAT math section is to approach each question asking only, “What’s the best formula to use for this question<\/i>?”\u00a0 That’s roughly the same as thinking that all I need to be the world’s best lion tamer<\/a> is a fancy hat that says “lion tamer”!\u00a0 In fact, the GMAT regularly designs math questions that predictably punish folks who blindly memorize formulas.\u00a0\u00a0 Most of the harder math problems on the GMAT have this property: if you approach the problem with the perspective of “what’s the best formula to use to solve this question?”, then you very likely will miss what the question is really about and get it wrong.\u00a0\u00a0 Here are some tips you can use to escape these traps.<\/p>\n <\/p>\n What do I mean by this?\u00a0 Yes, you have to remember some formulas for the GMAT.\u00a0 One option would be to memorize each one separately as an isolate mathematical factoid.\u00a0 Some people are exceptionally good memorizers, but for most people, this is not too helpful: in particular, if, in the stress of the real GMAT, you forget what you memorized, then you are out of luck.\u00a0\u00a0 Memorizing allows you no recourse if you forget, and it gives you no insight into mathematical logic.<\/p>\n Instead, I would recommend: understand the derivation of every formula: that is, understand the logical argument that underlies the formula and from which the formula arises.\u00a0 Think in terms of “why is it true?”: don’t stop at simply “what is true?”\u00a0 For example, both the permutation<\/a> and combination<\/a> formulas can be derived directly from the Fundamental Counting Principle<\/a>.\u00a0\u00a0 In another blog, I talked about the derivation of the formula for the area of an equilateral triangle<\/a>.\u00a0\u00a0 For most of the formulas you need to know for the GMAT, there is a context of logical argument that is also helpful to know.\u00a0 One exception is Archimedes<\/a>‘ formula for the area of a circle:<\/p>\n Yes, the great master Archimedes had a brilliant argument for why that formula is true, but that’s well beyond the kind of logic you need to employ on the GMAT.\u00a0 For this one, you can take a pass and simply memorize the formula, if you haven’t already.\u00a0 For most formulas, though, you should explore the logical of the context in depth.<\/p>\n <\/p>\n Problem-solving in mathematics is a subtle and sophisticated subject, and even in the limited context of the GMAT math section, the harder questions sometimes demand insightful approaches.\u00a0 Formulas are very cookie-cutter: if you have a formula to do something, it’s very good at doing that one thing, but that one thing is all it can do.\u00a0\u00a0 That’s quite different from the flexible analyses characteristic of mathematical thinking<\/a>.<\/p>\n Think about it this way.\u00a0 When you approach the GMAT math section, you should have a well-equipped tool box.\u00a0 The individual formulas you know are tools, but each one is a highly specific tool, useful for only one thing.\u00a0\u00a0 Other more widely applicable tools are skills such as estimation<\/a> or backsolving<\/a> or picking numbers<\/a>.\u00a0\u00a0 It’s important to have in your “tool box” a mix of tools, some more general and some more specific; what’s more important, though, it knowing when and how to use them.<\/p>\n <\/p>\n In the post How to do GMAT Math Faster<\/a>, I talk in depth about left-brain vs. right-brain thinkers.\u00a0 Left-brain dominant thinkers want to know “what to do<\/i>,” and they love formulas, because it’s always very clear what to do with a formula.\u00a0\u00a0 A more right-brain perspective focuses on “how to look at the problem<\/i>“, on the question of the best way to frame a problem, the best problem-solving perspective to adopt.\u00a0 Often, on a challenging GMAT problem, when one adopts the best perspective, what to do becomes obvious.\u00a0\u00a0 Given the correct perspective, many test-takers could solve the problem, but most folks get that problem wrong because it’s hard to come up with that right perspective on one’s own.\u00a0\u00a0 Coming up with the best perspective for a problem, the best way to frame a problem, takes time.\u00a0 It’s a right-brain pattern-matching skill: these always take time & experience to master.\u00a0 It’s very important to read solutions carefully: left-brain thinkers will want to jump ahead to the formulas, to “what to do<\/i>“, and may be frustrated because they knew all those parts already.\u00a0 To get the most from solutions, it’s very important to study the very beginning, any reasons given for making one perceptual choice rather than another.\u00a0 Sometimes, it may be helpful to solve the problem in more than one way, to experience the difference in different solution routes.\u00a0 Sometimes, the perceptual choices are hard to get from the solution, and the only way to find an answer is to post a question to an expert in the forums<\/a>, asking them: “why do we get the solution this way but not in that way<\/i>?”\u00a0 Through asking questions and investigating, over time one develops more of a sense for problem-solving perspective.<\/p>\n <\/p>\n Yes, know the GMAT formulas, but don’t think they are a magic bullet.\u00a0\u00a0 You will get considerably more from understanding the logic behind each formula, than you will from blind memorization of only the formula.\u00a0 Finally, it’s most important to develop the problem-solving perspective, which will help you to see which formulas to use when.\u00a0 Here’s another practice question from inside the Magoosh product:<\/p>\n 4) http:\/\/gmat.magoosh.com\/questions\/124<\/a><\/p>\n If you would like to share your own experience with formulas, or would like to clarify something I have said, let us know in the comments section.<\/p>\n <\/p>\n 1) There’s not a single formula we can use to get the answer, but by combining a few formulas, we can calculate this.<\/p>\n First, think about the circle.\u00a0 The circle has radius r = 8, so its total area would be<\/p>\n This entire figure is a quarter of the circle, so that area would be<\/p>\n Now, the shaded area (technically known as a circular segment<\/b>), would have an area of<\/p>\n (circular segment) = (quarter circle) \u2013 (triangle ABC)<\/p>\n Well, we already have the area of the quarter circle.\u00a0 The triangle would have an area of (1\/2)bh = 32.\u00a0 Therefore, the area of the segment is Answer = (B)<\/b><\/p>\n 2) We can use a quick formula for this.\u00a0 We have to go back to the Fundamental Counting Principle<\/a> and think this through.<\/p>\n Choice #1: for the president, we have eight choices<\/p>\n Choice #2: for the treasurer, we have seven remaining choices<\/p>\n Choice #3: we have to pick 3 warrant officers from the remaining 6.\u00a0 This would be<\/p>\n The FCP tells us to multiply the number of choices in each selection:<\/p>\n 8*7*20 = 56*20 = 1120<\/b> choices\u00a0 — Answer = (D)<\/b><\/p>\n 3) This is a very subtle counting problem.\u00a0\u00a0 This post on difficult counting problems<\/a> may provide some insights.<\/p>\n The basic strategy will be to calculate the total number of sets of three points we can select in this diagram, and then subtract the small number of sets of three collinear points.<\/p>\n The first step: in how many ways can we select three points from a set of 16 points?<\/p>\n<\/a><\/p>\n\n\t(A) 56
\n\t(B) 120
\n\t(C) 210
\n\t(D) 1120
\n\t(E) 6720\n<\/ol>\n<\/a><\/p>\n\n\t(A) 516
\n\t(B) 528
\n\t(C) 1632
\n\t(D) 3316
\n\t(E) 3344\n<\/ol>\nFormulas are useful<\/h2>\n
But BEWARE!<\/h2>\n
Remember, but don’t memorize<\/h2>\n
<\/a><\/p>\nMathematical thinking<\/h2>\n
Perspective<\/h2>\n
Summary<\/h2>\n
Practice problem solutions<\/h2>\n
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