Tài liệu Master the Gre 2010 - Part 18 ppt

• How complex is this question? (How many steps are involved in solving it? Does it require setting up equations, or does it require merely a few quick calculations?) • Do I have a clue, off the top of my head, how I would begin solving this problem? Determine how much time you’re willing to spend on the problem, if any. Recognizing a “toughie” when you see it may save you valuable time; if you don’t have a clue, take a guess and move on. Step 2: Size Up the Answer Choices Before youattempt tosolve theproblem athand, examinethe answerchoices. Theycan provide helpful clues about how to proceed in solving the problem and what sort of solution you should be aiming for. Pay particular attention to the following: • Form: Are the answer choices expressed as percentages, fractions, or decimals? Ounces or pounds? Minutes or hours? If the answer choices are expressed as equations, are all variables together on one side of the equation? As you work through the problem, convert numbers and expressions to the same form as the answer choices. • Size: Are the answer choices numbers with extremely small values? Greater numbers? Negative or positive numbers? Do the answer choices vary widely in value? If they’re tightly clustered in value, you can probably disregard decimal points and extraneous zeros when performing calculations, but be careful about rounding off your figures. Wide variation in value suggests that you can easily eliminate answer choices that don’t correspond to the general size of number suggested by the question. • Other distinctive properties and characteristics: Are the answer choices integers? Do they all include a variable? Do they contain radical signs (roots) or exponents? Is there a particular term, expression, or number that they have in common? Step 3: Look for a Shortcut to the Answer Before plunging headlong into a problem, ask yourself whether there’s a quick way to determine the correct answer. If the solution is a numerical value, perhaps only one answer choice is in the ballpark. Or you might be able to identify the correct answer intuitively, without resorting to equations and calculations. Step 4: Set Up the Problem and Solve It If your intuition fails you, grab your pencil and do whatever computations, algebra, or other procedures are needed to solve the problem. Simple problems may require just a few quick calculations. However, complex algebra and geometry questions may require setting up and solving one or more equations. Step 5: Verify Your Selection Before Moving On After solving the problem, if your solution does not appear among the answer choices, check your work—youobviouslymade atleastone mistake.Ifyour solutiondoes appear amongthechoices,don’tcelebratequiteyet.Althoughthere’sagoodchanceyouranswer iscorrect,it’spossibleyouransweriswrong and thatthetestdesignersanticipatedyour Chapter 7: Problem Solving 153 NOTE Remember: The computerized GRE testing system adjusts the difficulty level of your questions according to previous responses. If you respond incorrectly to tough questions, fewer of them will come up later in that section. www.petersons.com error by including that incorrect solution as an answer choice. So check the question to verifythatyourresponsecorresponds to what thequestioncallsforinvalue,expression, units of measure, and so forth. If it does, and if you’re confident that your work was careful and accurate, don’t spend any more time checking your work. Confirm your response and move on to the next question. Applying the 5-Step Plan Let’s apply thesefive stepsto twoGRE-style ProblemSolving questions.Question 1is a story problem involving changes in percent. (Story problems might account for as many as one half of your Problem Solving questions.) This question is relatively easy— approximately 80 percent of test takers respond correctly to questions like this one: 1. If Susan drinks 10% of the juice from a 16-ounce bottle immediately before lunch and 20% of the remaining amount with lunch, approximately how many ounces of juice are left to drink after lunch? (A) 4.8 (B) 5.5 (C) 11.2 (D) 11.5 (E) 13.0 Step 1: This problem involves the concept of percent—more specifically, percent decrease. The question is asking you to perform two computations in sequence. (You’ll use the result of the first computation to perform the second one.) Percent questions tend to be relatively simple. All that is involved here is a two-step computation. Step 2: The five answer choices in this question provide two useful clues: Notice that they range in value from 4.8 to 13.0—a broad spectrum. But what general size should we be looking for in a correct answer to this question? Without crunching any numbers, it’s clear that most of the juice will still remain in the bottle, even after lunch. So you’re looking for a value much closer to 13 than to 4. You can eliminate choices (A) and (B). Notice that each answer choice is carried to exactly one decimal place, and that the question asks for an approximate value. These two features are clues that you can probably round off your calculations to the nearest “tenth” as you go. Step 3: You already eliminated choices (A) and (B) in step 1. But if you’re on your toes, you can eliminate all but the correct answer without resorting to precise calculations. Look at the question from a broader perspective. If you subtract 10% from a number, then 20% from the result, that adds up to a bit less than a 30% decrease from the original number. 30% of 16 ounces is 4.8 ounces. So the solution must be a number that is a bit larger than 11.2 (16 2 4.8). Answer choice (D), 11.5, is the only one that works. PART IV: Quantitative Reasoning154 ALERT! Many Problem Solving questions are designed to reward you for recognizing easier, more intuitive ways to find the correct answer—so don’t skip step 3. It’s worth your time to look for a shortcut. www.petersons.com Step 4: If your intuition fails you, work out the problem. First, determine 10% of 16, then subtract that number from 16: 16 3 0.1=1.6 16 2 1.6 = 14.4 Susan now has 14.4 ounces of juice. Now perform the second step. Determine 20% of 14.4, then subtract that number from 14.4: 14.4 3 0.2=2.88 Round off 2.88 to the nearest tenth (2.9), then subtract: 14.4 2 2.9 = 11.5 Step 5: The decimal number 11.5 is indeed among the answer choices. Before moving on, however, ask yourself whether your solution makes sense—in this case, whether the size of our number (11.5) “fits” what the question asks for. If you performed step 2, you should already realize that 11.5 is in the ballpark. If you’re confident that your calculations were careful and accurate, confirm your response choice (D), and move on to the next question. The correct answer is (D). Question 2 involves the concept of arithmetic mean (simple average). This question is moderately difficult. Approximately 60 percent of test takers respond correctly to questions like it. 2. The average of 6 numbers is 19. When one of those numbers is taken away, the average of the remaining 5 numbers is 21. What number was taken away? (A) 2 (B) 8 (C) 9 (D) 11 (E) 20 Step 1: This problem involves the concept of arithmetic mean (simple average). To handle this question, you need to be familiar with the formula for calculating the average of a series of numbers. Notice, however, that the question does not ask for the average but for one of the numbers in the series. This curveball makes the question a bit tougher than most arithmetic-mean problems. Step 2: Scan the answer choices for clues. Notice that the middle three are clustered close together in value. So take a closer look at the two outliers: choices (A) and (E). Choice (A) would be the correct answer to the question: “What is the difference between 19 and 21?” But this question is asking something entirely different, so you can probably rule out choice (A) as a “red herring” choice. Choice (E) might also be a red herring, since 20 is simply 19 + 21 divided by 2. If this solution strikes you as too simple, you’ve got good instincts! The correct answer is probably choice (B), (C), or (D). If you’re pressed for time, guess one of these, and move on to the next question. Otherwise, go to step 3. Chapter 7: Problem Solving 155 ALERT! In complex questions, don’t look for easy solutions. Problems involving algebraic formulas generally aren’t solved simply by adding (or subtracting) a few numbers. Your instinct should tell you to reject easy answers to these kinds of problems. www.petersons.com Step 3: If you’re on your toes, you might recognize a shortcut here. You can solve this problem quickly by simply comparing the two sums. Before the sixth number is taken away, the sum of the numbers is 114 (6 3 19). After removing the sixth number, the sum of the remaining numbers is 105 (5 3 21). The difference between the two sums is 9, which must be the value of the number removed. Step 4: If you don’t see a shortcut, you can solve the problem conventionally. The formula for arithmetic mean (simple average) can be expressed this way: AM = sum of terms in the set number of terms in the set In the question, you started with six terms. Let a through f equal those six terms: 19 6 = +++++abcdef 114 =+++++abcdef fabcde=−++++ ( ) 114 Letting f = the number that is removed, here’s the arithmetic mean formula, applied to the remaining five numbers: 21 5 = ++++abcde 105 = a + b + c + d + e Substitute 105 for (a + b + c + d + e) in the first equation: f =1142 105 f =9 Step 5: If you have time, check to make sure you got the formula right, and check your calculations. Also make sure you didn’t inadvertently switch the numbers 19 and 21 in your equations. (It’s remarkably easy to commit this careless error under time pressure!) If you’re satisfied that your analysis is accurate, confirm your answer and move on to the next question. The correct answer is (C). Question 3 involves the concept of proportion. This question is moderately difficult. Approximately 50 percent of test takers respond correctly to questions like it. PART IV: Quantitative Reasoning156 ALERT! On the GRE, committing a careless error, such as switching two numbers in a problem, is by far the leading cause of incorrect responses. www.petersons.com 3. If p pencils cost 2q dollars, how many pencils can you buy for c cents? [Note: 1 dollar = 100 cents] (A) pc q2 (B) pc q200 (C) 50pc q (D) 2pq c (E) 200pcq Step 1: The first step is to recognize that instead of performing a numerical compu- tation, you’re task in Question 3 is to express a computational process in terms of letters. Expressions such as these are known as literal expressions, and they can be perplexing if you’re not ready for them. Although it probably won’t be too time- consuming, it may be a bit confusing. You should also recognize that the key to this question is the concept of proportion. It might be appropriate to set up an equation to solve for c. Along the way, expect to convert dollars into cents. Step 2: The five answer choices provide two useful clues: Notice that each answer choice includes all three letters (p, q, and c)—therefore so should your solution to the problem. Notice that every answer but choice (E) is a fraction. So anticipate building a fraction to solve the problem algebraically. Step 3: Is there any way to answer this question besides setting up an algebraic equation? Yes. In fact, there are two ways. One is to use easy numbers for the three variables; for example, p =2,q =1,andc = 100. These simple numbers make the question easy to work with: “If 2 pencils cost 2 dollars, how many pencils can you buy for 100 cents?” Obviously, the answer to this question is 1. So plug in the numbers into each answer choice to see which choice provides an expression that equals 1. Only choice (B) works: 2 100 200 1 1 ( ) ( ) ( ) ( ) = Another way to shortcut the algebra is to apply some intuition to this question. If you strip away the pencils, p’s, q’s and c’s, in a very general sense the question is asking: “If you can by an item for a dollar, how many can you buy for one cent?” Since one cent (a penny) is 1 100 of a dollar, you can buy 1 100 of one item for a cent. So you’re probably looking for a fractional answer with a large number such as 100 in the denominator (as opposed to a number such as 2, 3, or 6). Choice (B) is the only that appears to be in the right ballpark. And choice (B) is indeed the correct answer. Chapter 7: Problem Solving 157 NOTE On the GRE, expect to encounter two or three “story” problems involving literal expressions (where the solution includes not just numbers but variables as well). www.petersons.com Step 4: You can also answer the question in a conventional manner, using algebra. (This is easier said than done.) Here’s how to approach it: • Express 2q dollars as 200q cents (1 dollar = 100 cents). • Let x equal the number of pencils you can buy for c cents. • Think about the problem “verbally,” then set up an equation and solve for x: “p pencils is to 200q cents as x pencils is to c cents.” “The ratio of p to 200q is the same as the ratio of x to c” (in other words, the two ratios are proportionate). Therefore: p q x c200 = pc q x 200 = Step 5: Our solution, pc q200 , is indeed among the answer choices. If you arrived at this solution using the conventional algebraic approach (step 4), you can verify your solution by substituting simple numbers for the three variables (as we did in step 3). Or if you arrived at your solution by plugging in numbers, you can check you work by plugging in a different set of numbers, or by thinking about the problem conceptually (as in step 3). Once you’re confident you’ve chosen the correct expression among the five choices, confirm your choice, and then move on to the next question. The correct answer is (B). PROBLEM-SOLVING STRATEGIES To handle GRE Quantitative questions (Problem Solving and Quantitative Compari- sonsalike),you’llneed to be well-versedin thefundamental rulesof arithmetic, algebra, and geometry:Your knowledge of these basics is, to a large extent, what’s being tested. (That’s what the math review later in this part of the book is all about.) But when it comes to Problem Solving questions, the GRE test designers are also interested in gauging your mental agility, flexibility, creativity, and efficiency in problem solving. Morespecifically,theydesignthesequestionstodiscoveryourabilitytodothefollowing: • Manipulate numbers with a certain end result already in mind. • See the dynamic relationships between numbers as you apply operations to them. • Visualize geometric shapes and relationships between shapes. • Devise unconventional solutions to conventional quantitative problems. • Solve problems efficiently by recognizing the easiest, quickest, or most reliable route to a solution. This section of the chapter will help you develop these skills. The techniques you’ll learn here are intrinsic to the GRE. Along with your knowledge of substantive rules of math, they’re precisely what GRE Problem Solving questions are designed to measure. PART IV: Quantitative Reasoning158 NOTE Don’t worry if you didn’t fully understand the way we set up and solved this problem. You’ll learn more about how to handle GRE proportion questions in this book’s math review. 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. www.petersons.com Read the Question Stem Very Carefully Careless reading isby far theleading cause of wrong answers in GREProblem Solving, so bedoubly sureyou answerthe precise question that’s being asked, andconsider your responses carefully. Ask yourself, for example, whether the correct answer to the question at hand is one of the following: • an arithmetic mean or median • a circumference or an area • a sum or a difference • a perimeter or a length of one side only • an aggregate rate or a single rate • a total time or average time Also check to make sure of the following: • you used the same numbers provided in the question. • you didn’t inadvertently switch any numbers or other expressions. • you didn’t use raw numbers where percentages were provided, or vice versa. Always Check Your Work Here are three suggestions for checking your work on a Problem Solving question: Do a reality check. Ask yourself whether your solution makes sense for what the question asks. (This check is especially appropriate for story problems.) For questions in which you solve algebraic equations, plug your solution into the equation(s) to make sure it works. Confirm all of your calculations. It’s amazingly easy to commit errors in even the simplest calculations, especially under GRE exam pressure. Scan Answer Choices for Clues For multiple-choice questions, scan the answer choices to see what all or most of them have in common—such as radical signs, exponents, factorable expressions, or frac- tions. Then try to formulate a solution that looks like the answer choices. Chapter 7: Problem Solving 159 ALERT! No calculators are provided or allowed during the test, which makes calculating on your scratch paper and checking those calculations that much more crucial. www.petersons.com 4. If a Þ 0or1,then 1 2 2 a a − = (A) 1 22a − (B) 2 2a − (C) 1 2a − (D) 1 a (E) 2 21a − The correct answer is (A). Notice that all of the answer choices here are fractions in which the denominator contains the variable a. Also, none have fractions in either the numerator or the denominator. That’s a clue that your job is to manipulate the expression given in the question so that the result includes these features. First, place the denominator’s two terms over the common denominator a. Then cancel a from the denominators of both the numerator fraction and the denominator fraction (this is a shortcut to multiplying the numerator fraction by the reciprocal of the denominator fraction): Don’t Be Lured by Obvious Answer Choices When attempting multiple-choice Problem Solving questions, expect to be tempted by wrong-answer choices that are the result of common errors in reasoning, in calcula- tions, and in setting up and solving equations. Never assume that your solution is correct just because you see it among the answer choices. The following example is a variation of the problem on page 155. 5. The average of 6 numbers is 19. When one of those numbers is taken away, the average of the remaining 5 numbers is 21. What number was taken away? (A) 2 (B) 6.5 (C) 9 (D) 11.5 (E) 20 PART IV: Quantitative Reasoning160 www.petersons.com The correct answer is (C). This question contains two seemingly correct answer choices that are actually wrong. Choice (A) would be the correct answer to the question: “What is the difference between 19 and 21?” But this question asks something entirely different. Choice (E) is the other too-obvious choice. 20 is simply 19 1 21 divided by 2. If this solution strikes you as too simple, you have good instincts. You can solve this problem quickly by simply comparing the two sums. Before the sixth number is taken away, the sum of the numbers is 114 (6 3 19). After taking away the sixth number, the sum of the remaining numbers is 105 (5 3 21). The difference between the two sums is 9, which must be the value of the number taken away. Size Up the Question to Narrow Your Choices If a multiple-choice question asks for a number value, you can probably narrow down the answer choices by estimating the size and type of number you’re looking for. Use your common sense and real-world experience to formulate a “ballpark” estimate for word problems. 6. A container holds 10 liters of a solution that is 20% acid. If 6 liters of pure acid are added to the container, what percent of the resulting mixture is acid? (A) 8 (B) 20 (C) 33 1 3 (D) 40 (E) 50 The correct answer is (E). Common sense should tell you that when you add more acid to the solution, the percent of the solution that is acid will increase. So, you’re looking for an answer that’s greater than 20—either choice (C), (D), or (E). (By the way, notice the too-obvious answer, choice (A); 8 liters is the amount of acid in the resulting mixture.) If you need to guess at this point, your odds are one in three. Here’s how to solve the problem: The original amount of acid is (10)(20%) 5 2 liters. After adding 6 liters of pure acid, the amount of acid increases to 8 liters, while the amount of total solution increases from 10 to 16 liters. The new solution is 8 16 , or 50%, acid. Know When to Plug in Numbers for Variables In multiple-choice questions, if the answer choices contain variables (such as x and y), the question might be a good candidate for the “plug-in” strategy. Pick simple numbers (so the math is easy) and substitute them for the variables. You’ll need your pencil and scratch paper for this strategy. Chapter 7: Problem Solving 161 TIP Remember: Number choices are listed in order of value— either ascending or descending. This feature can help you zero in on the most viable among the five choices. www.petersons.com 7. If a train travels r 1 2milesinh hours, which of the following represents the number of miles the train travels in 1 hour and 30 minutes? (A) 36 2 r h + (B) 3 2 r h + (C) r h + + 2 3 (D) r h + 6 (E) 3 2 ~r +2! The correct answer is (A). This is an algebraic word problem involving rate of motion (speed). You can solve this problem either conventionally or by using the plug-in strategy. The conventional way: Notice that all of the answer choices contain fractions. This is a tip that you should try to create a fraction as you solve the problem. Here’s how to do it: Given that the train travels r 1 2milesinh hours, you can express its rate in miles per hour as r h + 2 .In 3 2 hours, the train would travel this distance: 3 2 236 2 ⎛ ⎝ ⎞ ⎠ + ⎛ ⎝ ⎞ ⎠ = +r h r h The plug-in strategy: Pick easy-to-use values for r and h. Let’s try r 5 8 and h 5 1. Given these values, the train travels 10 miles (8 1 2) in 1 hour. So obviously, in 1 1 2 hours the train will travel 15 miles. Start plugging these r and h values into the answer choices. You won’t need to go any further than choice (A): 36 2 38 6 21 30 2 15 r h + = + = () () , or Choice (E) also equals 15, 3 2 (8 + 2) = 15. However, you can eliminate choice (E) out of hand because it omits h. Common sense should tell you that the correct answer must include both r and h. PART IV: Quantitative Reasoning162 ALERT! The plug-in method can be time-consuming, so use it only if you don’t know how to solve the problem in a conventional manner. www.petersons.com . comparing the two sums. Before the sixth number is taken away, the sum of the numbers is 114 (6 3 19). After removing the sixth number, the sum of the remaining. either the numerator or the denominator. That’s a clue that your job is to manipulate the expression given in the question so that the result includes these