Tài liệu Master the Gre 2010 - Part 21 pdf

QUANTITATIVE COMPARISON STRATEGIES Just as with GRE Problem Solving, handling Quantitative Comparisons requires that you know the fundamentals of arithmetic, algebra, and geometry—no doubt about it—and that’s what the math review in the next part of this book is about. But the test designers craft Quantitative Comparisons to gauge not only your math knowledge, but also yourmental agility, flexibility, andcreativity inapplying it.Quantitative Compari- sons are designed, for example, to measure your ability to do the following: • See the dynamic relationships between numbers. • Recognize the easiest, quickest, or most reliable way to compare quantities. • Visualize geometric shapes and relationships between shapes. In this section of the chapter, you’ll learn some strategies and techniques that dem- onstrate these skills. These aren’t merely tricks and shortcuts. Your facility in applying these techniques, along with your knowledge of substantive rules of math, is exactly what the test designers are attempting to measure. Know What Concept Is Being Tested Each Quantitative Comparison question focuses on a particular math concept. The first thing to do is look at the centered information and the expressions in the columns to determine what’s being covered. This step can often help you get a head start in making the comparison. 4. 2x 2 1 9x 5 5 Column A Column B x 25 The correct answer is (D). Notice that the centered equation is quadratic, and that the two quantitative expressions essentially ask you to find the value of x. You know that quadratic equations have two roots, and this is the concept that’s probably being tested here. The two roots might be the same or they may differ. Now you know what you need to do and why. First, rewrite the centered equation in standard form: 2x 2 1 9x 2 5 5 0. Now, factor the trinomial expression into two binomials: (2x 2 1)(x 1 5) 5 0. It should now be clear that there are two different roots of the equation: 1 2 and 25. Since 1 2 .25, but 25= 25, the correct answer is choice (D). There May Be an Easier Way You shouldn’t have to perform involved calculations to make a comparison. A few simple calculations may be required, but if you’re doing a lot of number crunching or setting up and solving of equations, you’ve probably missed the mathematical prin- ciple that you’re being tested for. Put your pencil down and focus on the concept, not the process. Chapter 8: Quantitative Comparison 183 TIP The examples here involve a variety of math concepts and all are at least moderately difficult. If you have trouble with a concept, focus on it during the math review later in this part of the book. NOTE Not every Quantitative Comparison question you encounter on the GRE is suitable for one of these strategies, but some are—so it’s a good idea to become familiar with them. www.petersons.com 5. Column A Column B ~26! 11 ~26! 3 ~211! 8 ~211! 3 The correct answer is (A). To make the comparison, you could multiply each base number by itself multiple times, then figure out quotients. But that would be a horrendous waste of time, and most GRE test-takers would at least realize that they can simplify each fraction by canceling exponents: () () () − − =− 6 6 6 11 3 8 and () () () − − =− 11 11 11 8 3 5 Now, instead of multiplying 26 by itself 8 times, and then multiplying 211 by itself 5 times to compare the two products, look at the signs and the exponents for a moment. Quantity A is a negative number raised to an even power, so it must have a positive value. Quantity B is a negative number raised to an odd power, so it must have a negative value. There’s no need for all that calculating. Quantity A must be greater. Choose (D) Only If You Need More Information Remember: Pick choice (D) only if you can’t make a comparison without having more information. But if the comparison at hand involves numbers only, you’ll always be able to calculate specific numerical values for both expressions (assuming you have time to do the math). You certainly don’t need more information just to compare the size of two specific numbers. 6. x M y 5 x(x 2 y) Column A Column B (21 M 22) M (1 M 2) (1 M 22) M (21 M 2) The correct answer is (C). The centered information contains variables, but calculating the two quantities involves only numbers. Thus, the correct answer cannot be (D). Just by recognizing this fact, you’ve improved your odds of selecting the correct answer choice by 25 percent! Let’s go ahead and compare the two quantities. First, Quantity A: Apply the operation (defined by the symbol M) to each parenthesized pair, then apply it again to those results: (21 M 22) 521(21 2 [22]) 521(1) 521 (1 M 2) 5 1(1 2 2) 5 1(21) 521 Apply the defined operation again, substituting 21 for both x and y: (21 M 21) 521(21 2 [21]) 521(0) 5 0 PART IV: Quantitative Reasoning184 www.petersons.com Quantity A equals zero (0). Now let’s calculate Quantity B in the same manner: (1 M 22) 5 1(1 2 [22]) 5 1(3) 5 3 (21 M 2) 521(21 2 2) 521(23) 5 3 Apply the defined operation again, substituting 3 for both x and y: (3 M 3) 5 3(3 2 3) 5 3(0) 5 0 As you can see, Quantity (B) also equals zero (0). Consider All Possibilities for Variables When comparing expressions involving variables, pay careful attention to any cen- tered information that restricts their value.Aside from any such restriction, be sure to consider positive and negative values, as well as fractions and the numbers zero (0) and 1. Comparisons often depend on which sort of number is used. In these cases, the correct answer would be choice (D). 7. p . 0 p Þ 1 Column A Column B p 23 p 4 The correct answer is (D). Here’s the general rule that applies to this problem: p p x x − = 1 . Hence, if p . 1, then p 23 must be a fraction less than 1 while p 4 is greater than 1, and Quantity B is greater. On the other hand, if p , 1, then the opposite is true. In short, which quantity is greater depends on whether 0 is greater than 1 or is a fraction between 0 and 1. Rewrite a Quantity to Resemble Another If you have no idea how to analyze a particular problem, try manipulating one or both of the expressions until they resemble each other more closely. You may be able to combine numbers or other terms, do some factoring, or restate an expression in a slightly different form. 8. Column A Column B 4 2 x 2 (2 1 x)(2 2 x) The correct answer is (C). Perhaps you recognized that (4 2 x 2 )isthe difference of two squares (2 2 and x 2 ) and that the following equation applies: a 2 2 b 2 5 (a 1 b)(a 2 b) If so, then you saw right away that the two quantities are the same. If you didn’t recognize this, you could restate Quantity B by multiplying (2 2 x)by (2 1 x) using the FOIL method: (2 1 x)(2 2 x) 5 4 1 2x 2 2x 2 x 2 5 4 2 x 2 Rewriting Quantity B, then simplifying it, reveals the comparison. The two quantities are equal. Chapter 8: Quantitative Comparison 185 ALERT! Remember: Just because a Quantitative Comparison question does involve one or more variables, it doesn’t necessarily mean that choice (D) is the correct answer. www.petersons.com Add or Subtract Across Columns to Simplify the Comparison If both expressions include the same term, you can safely “cancel” that term from each one either by adding or subtracting it. This technique may help simplify one or both of the expressions, thereby revealing the comparison to you. Remember: You don’t change the comparative value of two expressions merely by adding or subtracting the same terms from each one. 9. Column A Column B The sum of all integers from 19 to 50, including 19 and 50 The sum of all integers from 21 to 51, including 21 and 51 The correct answer is (B). The two number sequences have in common integers 21 through 50. So you can subtract (or cancel) all of these integers from both sides of the comparison. That leaves you to compare (19 1 20) in Column A to 51 in Column B. You can now see clearly that Quantity B is greater. Be Wary of Multiplying or Dividing Across Columns To help simplify the two expressions, you can also multiply or divide across columns, but only if you know for sure that the quantity you’re using is positive. Multiplying or dividing two unequal terms by a negative value reverses the inequality; the quantity that was the greater one becomes the smaller one. So think twice before performing either operation on both expressions. 10. x . y Column A Column B x 2 yxy 2 The correct answer is (D). To simplify this comparison, you may be tempted to divide both columns by x and by y. But that would be wrong! Although you know that x is greater than y, you don’t know whether x and y are positive or negative. You can’t multiply or divide by a quantity across columns unless you’re certain that the quantity is positive. If you do this, here is what could happen: Dividing both sides by xy would leave the comparison between x and y. Given x . y, you’d probably select (A). But if you let x 5 1 and y 5 0, on this assumption, both quantities 5 0 (they’re equal). Or let x 5 1 and y 521. On this assumption, Quantity A 521 and Quantity B 5 1. Since these results conflict with the previous ones, you’ve proven that the correct answer is choice (D). Solve Centered Equations for Values If the centered information includes one or more algebraic equations and you need to know the value of the variable(s) in those equations to make the comparison, then there’s probably no shortcut to avoid doing the algebra. PART IV: Quantitative Reasoning186 www.petersons.com 11. x 1 y 5 6 x 2 2y 5 3 Column A Column B xy The correct answer is (A). The centered information presents a system of simultaneous linear equations in two variables, x and y. The quickest way to solve for x is probably by subtracting the second equation from the first: Substitute this value for y in either equation. Using the first one: x x += = 16 5 Since both equations are linear, you know that each variable has one and only one value. Since x 5 5 and y 5 1, Quantity A is greater than Quantity B. Don’t Rely on Geometry Figure Proportions For Quantitative Comparison questions involving geometry figures, never try to make a comparison by visual estimation or by measuring a figure. Even if you see a note stating that a figure is drawn to scale, you should always make your comparison based on your knowledge of mathematics and whatever nongraphical data the question provides, instead of by “eyeballing” the figure. 12. Column A Column B yz The correct answer is (D). If you were to measure the two angles (y° and z°) by eye, you might conclude that they are the same or that one is slightly greater than the other, and then select choice (A), (B), or (C) accordingly. But that would be a mistake. Your task is to compare y and z without resorting to visual measurement. Given that any circle contains a total of 360°, you know Chapter 8: Quantitative Comparison 187 www.petersons.com that y+z=360 2 220 = 140. Also, since x, y, and z are supplementary (their sum is 180 because the three angles combine to form a straight line), you know that x = 40. But the problem does not supply enough information for you to determine the degree measure of either y or z. (All you know is that their sum is 140.) PART IV: Quantitative Reasoning188 www.petersons.com SUMMING IT UP • The 45-minute Quantitative Reasoning section tests your proficiency in per- forming arithmetical operations and solving algebraic equations and inequalities. It also tests your ability to convert verbal information into mathematical terms; visualize geometric shapes and numerical relationships; interpret data presented in charts, graphs, tables, and other graphical displays; and devise intuitive and unconventional solutions to conventional mathematical problems. • Quantitative Comparison is one of two basic formats for questions in the Quan- titative Reasoning section of the GRE (the other is Problem Solving). • All Quantitative Comparison questions require you to compare two quantities, one in Column A and one in Column B. You choose whether the quantity in Column A is greater, the quantity in Column B is greater, the quantities are equal, or the relationship cannot be determined from the information given. These four are the only answer choices, and they’re the same for all Quantitative Comparison questions. Information concerning one or both of the quantities to be compared is centered above the two columns. • You’ll make fewer calculations and solve fewer equations for Quantitative Com- parison questions than for Problem Solving questions. You’re being tested mainly on your ability to recognize and understand principles, not your ability to work step-by-step toward a solution. • Follow and review the six basic steps for handling GRE Quantitative Comparison questions outlined in this chapter and apply them to this book’s Practice Tests. Then review them again just before exam day. Chapter 8: Quantitative Comparison 189 www.petersons.com Math Review: Number Forms, Relationships, and Sets OVERVIEW • Percents, fractions, and decimals • Simplifying and combining fractions • Decimal place values and operations • Simple percent problems • Percent increase and decrease • Ratios and proportion • Arithmetic mean, median, mode, and range • Standard deviation • Arithmetic sequences • Geometric sequences • Permutations • Combinations • Probability • Summing it up In this chapter, you’ll focus first on various forms of numbers and relation- ships between numbers. Specifically, you’ll review the following basic tasks involving percents, decimal numbers, and fractions (including ratios): • combining fractions using the four basic operations • combining decimal numbers by multiplication and division • comparing numbers in percentage terms • comparing percent changes with number changes • rewriting percents, fractions, and decimal numbers from one form to another • determining ratios between quantities and determining quantities from ratios • setting up equivalent ratios (proportions) • handling fractions and ratios • handling ratios involving more than two quantities • solving proportion problems with variables chapter 9 191 Later in this chapter, you’ll explore the following topics, all of which involve descriptive statistics and sets (defined groups) of numbers or other objects: • Arithmetic mean—of simple average—and median (two ways that a set of numbers can be described as a whole) • Standard deviation (a quantitative expression of the dispersion of a set of mea- surements) • Arithmetic sequences (where there is a constant difference between successive numbers) • Geometric sequences (where each term is a constant multiple of the preceeding one) • Permutations (the possibilities for arranging a set of objects) • Combinations (the possibilities for selecting groups of objects from a set) • Probability (the statistical chances of a certain event, permutation, or combi- nation occurring) PERCENTS, FRACTIONS, AND DECIMALS Any real number can be expressed as a fraction, a percent, or a decimal number. For instance, 2 10 , 20%, and 0.2 are all different forms of the same quantity or value. GRE math questions often require you to rewrite one form as another as part of solving the problem at hand. You should know how to write any equivalent quickly and confi- dently. To rewrite a percent as a decimal, move the decimal point two places to the left (and drop the percent sign). To rewrite a decimal as a percent, move the decimal point two places to the right (and add the percent sign). 95% 5 0.95 0.004 5 0.4% To rewrite a percent as a fraction, divide by 100 (and drop the percent sign). To rewrite a fraction as a percent, multiply by 100 (and add the percent sign). Percents greater than 100 are equivalent to numbers greater than 1. 810% 5 810 100 5 81 10 5 8 1 10 3 8 5 300 8 % 5 75 2 % 5 37 1 2 % Percents greater than 100 or less than 1 (such as 457% and .067%) can be confusing, because it’s a bit harder to grasp their magnitude. Here’s an illustrative example in the Quantitative Comparison format. (If you need to review this format, see Chapter 8.) PART IV: Quantitative Reasoning192 ALERT! Although the initial sections of this chapter cover the most basic of all the math review topics in this book, don’t skip them. The skills covered here are basic building blocks for other, more difficult types of questions covered in later sections of this math review. www.petersons.com . to each parenthesized pair, then apply it again to those results: (21 M 22) 521( 21 2 [22]) 521( 1) 521 (1 M 2) 5 1(1 2 2) 5 1 (21) 521 Apply the defined operation. 521 Apply the defined operation again, substituting 21 for both x and y: (21 M 21) 521( 21 2 [21] ) 521( 0) 5 0 PART IV: Quantitative Reasoning184 www.petersons.com Quantity