Tài liệu Master the Gre 2010 - Part 19 pptx

Know When—and When Not—to Work Backward If a multiple-choice question asks for a number value, and if you draw a blank as far as how to set up and solve the problem, don’t panic. You might be able to work backward by testing the answer choices in turn. 8. A ball is dropped from 192 inches above level ground, and after the third bounce, it rises to a height of 24 inches. If the height to which the ball rises after each bounce is always the same fraction of the height reached on its previous bounce, what is this fraction? (A) 1 8 (B) 1 4 (C) 1 3 (D) 1 2 (E) 2 3 The correct answer is (D). The fastest route to a solution is to plug in an answer. Try choice (C) and see what happens. If the ball bounces up 1 3 as high as it started, then after the first bounce it will rise up 1 3 as high as 192 inches, or 64 inches. After a second bounce, it will rise 1 3 as high, or about 21 inches. But the problem states that the ball rises to 24 inches after the third bounce. Obviously, if the ball rises less than that after two bounces, it will be too low after three. So choice (C) cannot be the correct answer. We can see that the ball must be bouncing higher than one third of the way, so the correct answer must be a larger fraction, meaning either choice (D) or choice (E). You’ve already narrowed your odds to 50%. Try plugging in choice (D), and you’ll see that it works: 1 2 of 192 is 96; 1 2 of 96 is 48; and 1 2 of 48 is 24. Although it would be possible to develop a formula to answer question 8, it’s not worthwhile, considering how quickly and easily you can work backward from the answer choices. In multiple-choice questions, working backward from numerical answer choices works well when the numbers are easy and when few calculations are required, as in question 8. In other cases, applying algebra might be a better way. Chapter 7: Problem Solving 163 TIP In multiple-choice Problem Solving questions, numerical answer choices are listed in order of value from least to greatest. To work backward from the answer choices, start with choice (C). If choice (C) is too great, then you can assume choices (D) and (E) are incorrect and try choices (A) or (B) instead. www.petersons.com 9. How many pounds of nuts selling for 70 cents per pound must be mixed with 30 pounds of nuts selling at 90 cents per pound to make a mixture that sells for 85 cents per pound? (A) 8.5 (B) 10 (C) 15 (D) 16.5 (E) 20 The correct answer is (B). Is the easiest route to the solution to test the answer choices? Let’s see. First of all, calculate the total cost of 30 pounds of nuts at 90 cents per pound: 30 3 .90 5 $27. Now, start with choice (C). 15 pounds of nuts at 70 cents per pound costs $10.50. The total cost of this mixture is $37.50, and the total weight is 45 pounds. Now you’ll need to perform some long division. The average price per pound of the mixture turns out to be between 83 and 84 cents—too low for the 85-cent average given in the question. So you can at least eliminate choice (C). You should realize by now that testing the answer choices might not be the most efficient way to tackle this question. Besides, there are ample opportunities for calculation errors. Instead, try solving this problem algebraically by writing and solving an equation. Here’s how to do it. The cost (in cents) of the nuts selling for 70 cents per pound can be expressed as 70x, letting x equal the number that you’re asked to determine. You then add this cost to the cost of the more expensive nuts (30 3 90 5 2,700) to obtain the total cost of the mixture, which you can express as 85(x 1 30). You can state this algebraically and solve for x as follows: 70 2 700 85 30 70 2 700 85 2 550 150 15 10 xx xx x x +=+ +=+ = = ,() ,, At 70 cents per pound, 10 pounds of nuts must be added in order to make a mixture that sells for 85 cents per pound. Look for the Simplest Route to the Answer In many Problem Solving questions, there’s a long way and a short way to arrive at the correct answer. When it looks like you’re facing a long series of calculations or a complex system of equations, always ask yourself whether you can take an easier, more intuitive route to solving the problem. PART IV: Quantitative Reasoning164 www.petersons.com 10. What is the difference between the sum of all positive odd integers less than 32 and the sum of all positive even integers less than 32? (A) 32 (B) 16 (C) 15 (D) 1 (E) 0 The correct answer is (B). To answer this question, should you add up two long series of numbers on your scratch paper? In this case, it’s a waste of time, and you risk committing calculation errors along the way. A smart test-taker will notice a pattern and use it as a shortcut. Compare the initial terms of each sequence: even integers: [2, 4, 6, ]. . ., 30 odd integers: [1, 3, 5, ]. . ., 29, 31 Notice that for each successive term, the odd integer is one less than the corre- sponding even integer. There are a total of 15 corresponding integers, so the difference between the sums of all these corresponding integers is 15. But the odd-integer sequence includes one additional integer: 31. So the difference is 16, which is 31 – 15. Start with What You Know It’s easy to get lost in Problem Solving questions that are complex and involve multiple steps to solve. First, take a deep breath and start with information you know. Then ask yourself what you can deduce from it, leading yourself step-by-step to the solution. 11. In a group of 20 singers and 40 dancers, 20% of the singers are less than 25 years old, and 40% of the entire group are less than 25 years old. What portion of the dancers are younger than 25 years old? (A) 20% (B) 24% (C) 40% (D) 50% (E) 60% The correct answer is (D). To answer this question, you need to know (1) the total number of dancers and (2) the number of dancers younger than 25 years old. The question provides the first number: 40. To find the second number, start with what the question provides and figure out what else you know. Keep going, and eventually you’ll arrive at your destination. Of the whole group of 60, 24 are younger than 25 years. (40% of 60 is 24.) 20% of the 20 singers, or 4 singers, are younger than 25 years. Hence, the remaining 20 people younger than 25 must be dancers. That’s the second number you needed to answer the question. 20 is 50% of 40. Chapter 7: Problem Solving 165 www.petersons.com Search Geometry Problem Figures for Clues Some GRE geometry problems will be accompanied by figures. They’re there for a reason: The pieces of information provided in a figure can lead you, step-by-step, to the answer. 12. If O is the center of the circle in the figure above, what is the area of the shaded region, expressed in square units? (A) 3 2 p (B) 2p (C) 5 2 p (D) 8 3 p (E) 3p The correct answer is (E). This question asks for the area of a portion of the circle defined by a central angle. To answer the question, you’ll need to determine the area of the entire circle as well as what percent (portion) of that area is shaded. Mine the figure for a piece of information that might provide a starting point. If you look at the 60° angle in the figure, you should recognize right away that both triangles are equilateral (all angles are 60°) and, extended out to their arcs, form two “pie slices,” each one 1 6 the size of the whole pie (the circle). What’s left are two big slices, each of which is twice the size of a small slice. So the shaded area must account for 1 3 the circle’s area. You’ve now reduced the problem to the simple mechanics of calculating the circle’s area and then dividing it by 3. In an equilateral triangle, all sides are congruent. Mining the figure once again, notice length 3, which is also the circle’s radius (the distance from its center to its PART IV: Quantitative Reasoning166 www.petersons.com circumference). The area of any circle is pr 2 , where r is the circle’s radius. Thus, the area of the circle is 9p. The shaded portion accounts for 1 3 the circle’s area, or 3p. Sketch Your Own Geometry Figure A geometry problem that doesn’t provide a diagram might be more easily solved if it had one. Use your scratch paper and draw one for yourself. It will be easier than trying to visualize it in your head. 13. On the xy-coordinate plane, points R(7,23) and S(7,7) are the endpoints of the longest possible chord of a certain circle. What is the area of the circle? (A) 7p (B) 16p (C) 20p (D) 25p (E) 49p The correct answer is (D). There are lots of sevens in this question, which might throw you off track without at least a rough picture. To keep your thinking straight, scratch out your own rough xy-grid and plot the two points. You’ll see that R is located directly below S, so chord RS is vertical. Accordingly, the length of RS is simply the vertical distance from 23to7, which is 10. By definition, the longest possible chord of a circle is equal in length to the circle’s diameter. The circle’s diameter is 10, and thus its radius is 5. The circle’s area is p(5) 2 5 25p. Plug in the Numbers for “Defined Operation” Questions One of your 14 Problem Solving questions might very well be a so-called defined operation question. These are odd-looking problems that might strike you as being difficult. But they’re really not. In fact, the math usually turns out to be very easy. You’re being tested on your ability to understand what the problem requires you to do, and then to cross your t’s and dot your i’s as you perform simple arithmetical calculations. Chapter 7: Problem Solving 167 ALERT! GRE geometry figures provide helpful information for solving the problem, but they’re not intended to provide the answer through visual measurement. Make sure you solve the problem by applying your knowledge, not by looking at the figure. www.petersons.com 14. Let be defined for all numbers a, b, c, and d by 5 ac 2 bd.If x 5 , what is the value of ? (A) 1 (B) 2 (C) 4 (D) 5 (E) 10 The correct answer is (B). In defining the diamond-shaped figure as “ac 2 bd,” the question is telling you that whenever you see four numbers in a diamond like this, you should plug them into the mathematical expression shown in the order given. The question itself then requires you to perform this simple task twice. First, let’s figure out the value of x.Ifx is the diamond labeled as x, then a 5 5, b 5 4, c 5 2, and d 5 1. Now, we plug those numbers into the equation given, then do the simple math: x x x =×−× =− = ()()52 41 10 4 6 Now we tackle the second step. Having figured out the value of x, we can plug it into our second diamond, where a 5 6, b 5 10, c 5 2, and d 5 1.Again, plug in the numbers and do the math: (6 3 2) 2 (10 3 1) 5 12 2 10 5 2 As you can see, the math is very easy; the trick is understanding what the question is asking: which is to have you “define” a new math operation and then carefully plug in the numbers and work out the solution. With a little practice, you’ll never get a defined operation question wrong. THE DATA INTERPRETATION FORMAT Data interpretation is a special Problem Solving format designed to gauge your ability to read and analyze data presented in statistical charts, graphs, and tables, and to calculate figures such as percentages, ratios, fractions, and averages based on the numbers you glean from graphical data. Expect to find three to five data interpretation questions (possibly in sets of two or three) mixed in with your other Quantitative Reasoning questions. Each question in a data interpretation set pertains to the same graphical data. Each question (and each set) involves either one or two distinct graphical displays. Four types appear on the GRE most frequently: tables, pie charts, bar graphs, and line charts. Note the following key features of GRE data interpretation questions: PART IV: Quantitative Reasoning168 NOTE Expect 3 to 5 data interpretation questions on the exam. Some may appear in sets pertaining to the same graphical data. www.petersons.com • Important assumptions will be provided. Any additional information that you might need to know to interpret the figures will be indicated above and below the figures. Be sure to read this information. • Some questions might ask for an approximation. This is because the test makers are trying to gauge your ability to interpret graphical data, not your ability to crunch numbers to the nth decimal place. • Many of the numbers used are almost round. This feature relates to the previous one. The GRE rewards test takers who recognize that rounding off numbers (to an appropriate extent) will suffice to reach the correct answer. • Some questions may be long and wordy. Solving a data interpretation problem may call for multiple steps involving various graphical data, so the questions can be lengthy. In fact, you may have more trouble interpreting the questions than the graphical data. • Bar graphs and line charts are drawn to scale. That’s because visual esti- mation is part of what’s required to analyze a bar graph or line chart’s graphical data. However, pie charts will not necessarily be drawn to scale. (You’ll interpret them strictly by the numbers provided), and visual scale is irrelevant when it comes to analyzing tables. • Figures are not drawn to deceive you or to test your eyesight. In bar graphs and line charts, you won’t be asked to split hairs to determine precise values. These graphs and charts are designed for a comfortable margin for error in visual acuity. Just don’t round up or down too far. • You may need to scroll vertically to see the entire display. Some vertical scrolling might be required to view the entire display, especially the information above and below the chart, graph, or table. If so, don’t forget to scroll up and down as you analyze each question. The 5-Step Plan Here’s the 5-step approach that will help you handle any data interpretation question (or set of questions). Just ahead, we’ll apply this approach to a two-question set. STEP 1: LOOK AT THE “BIG PICTURE” FIRST Before plunging into the question(s), read all the information above and below the figure(s). Look particularly for the following: • Totals (dollar figures or other numbers) • Whether the numbers are expressed in terms of hundreds, thousands, or millions • How two or more figures are labeled • Whether graphical data is expressed in numbers or percentages STEP 2: READ THE ENTIRE QUESTION CAREFULLY Readtheentire question verycarefully.As you doso,dividethequestionintoparts,each of which involves a distinct step in getting to the answer. Pay particular attention to what the question asks for. For example: Chapter 7: Problem Solving 169 NOTE Data interpretation questions are most often based on tables, pie charts, bar graphs, and line charts. However, you might encounter some other type of graphic, so don’t be surprised to see something else. www.petersons.com • Does the question ask for an approximation or a precise value? • Does the question ask for a percentage or a raw number? • Does the question ask for a comparison? • Does the question ask for an increase or a decrease? In breaking down the question into tasks, look for a shortcut to save yourself pencil work. STEP 3: PERFORM THE STEPS NEEDED TO REACH THE ANSWER Look for a shortcut to the answer. For questions calling for approximations, round numbers up or down (but not too far) as you go. STEP 4: CHECK THE CHOICES FOR YOUR ANSWER Ifan “approximation”questionasksfora number,find the choiceclosesttoyour answer. Look for other answer choices that are too close for comfort. If you see any, or if your solution is nowhere near any of the choices, go to step 5. STEP 5: CHECK YOUR CALCULATIONS Check all of your calculations and make sure the size and form (number, percentage, total, etc.) of your solutionconforms with what the question asks.Check your rounding technique. Did you round off in the wrong direction? Did you round off too far? APPLYING THE 5-STEP PLAN Let’s applythe5-step approachtoa pairofquestions, based onthefollowing tworelated pie charts. We’ll perform step 1 before we even look at the first question based on this data. Step 1: Size up the two charts and read the information above and below them. Notice that we’re dealing only with one company during one year. Notice also that dollar PART IV: Quantitative Reasoning170 TIP Use rounding and estimation to answer a data interpretation question—but only if the question calls for an approximation. www.petersons.com totals are provided, but that the pie segments are all expressed only as percentages. That’s a clue that your tasks for this set of data might be to calculate dollar amounts for various pie segments. Now that you’ve familiarized yourself with the graphical display of data, proceed to step 2 for the first question. Step 2: Carefully read the question, but not the five lettered answer choices yet. (This question is near the lowest level of difficulty; about 85% of test takers would answer it correctly.) 15. Which of the following pairs of divisions, each pair considered separately, accounted for more than 50 percent of Company XYZ’s income during the year? I. Division A and Division B II. Division C and Division D III. Division A and Division C (A) I only (B) III only (C) I and II only (D) I and III only (E) I, II, and III Just by reading the question, you know that it involves only the chart on the left (“Income”) and that you won’t need to convert percentages into dollar figures. So it’s a straightforward question that you should not consider skipping. Nevertheless, be sure to read it very carefully: You’re looking for any pair that accounted for “more than 50 percent” of XYZ’s income, not 50 percent or more. (This distinction makes the dif- ference between the correct answer choice and an incorrect one, as you’ll soon see.) Step 3: You need to combine three different percentage pairs (indicated by Roman numerals I, II, and III). Be careful to combine the correct percentages from the chart on the left. Here are the results: I. Division A and Division B (38% + 12% = 50%) II. Division C and Division D (20% + 30% = 50%) III. Division A and Division C (38% + 20% = 58%) Step 4: The only pair among the three that accounted for more than 50% of XYZ’s income is pair III. Because only pair III meets that criterion, the correct answer must be (B): “III only.” If you have extra time, go to step 5. Step 5: Check the pie chart on the left and the three division pairs (I, II, and III) again to make sure the percentages you used and your addition are correct. Once you’ve confirmed they are, go on to the next question. The correct answer is (B). Let’s apply the same five-step approach to a moderately difficult question. (About 50% of test takers would respond correctly to questions like it.) Step 1: You’ve already performed this step, so move to step 2. Chapter 7: Problem Solving 171 NOTE The Roman numeral format illustrated in Question 15 is not used very frequently on the GRE. Expect no more than one or two Problem Solving questions in this format. www.petersons.com Step 2: Carefully read the question, but not the five answer choices yet: 16. During year X, by approximately what amount did Division C’s income exceed Division B’s expenses? (A) $125,000 (B) $127,000 (C) $140,000 (D) $180,000 (E) $312,000 This question involves three tasks: (1) calculate Division C’s income; (2) calculate Division B’s expenses; and (3) compute their difference. There’s no shortcut to these three tasks, although you know from the question stem that calculating approximate values will probably suffice. Go on to step 3. Step 3: Division B’s expenses accounted for 26% of XYZ’s total expenses, given as $495,000. Rounding off these figures to 25% and $500,000, Division B’s expenses totaled approximately $125,000. Income from Division C sales was 20% of total XYZ income, given as $1,560,000. Rounding this total down to $1,500,000, income from Division C sales was approximately $300,000. Income from Division C sales exceeded Division B’s expenses by approximately $175,000. Step 4: Scan the answer choices. The only one that’s close to this solution is choice (D). If you have extra time, go to step 5. Step 5: Make sure that you started with the right numbers. Did you compare C’s income with B’s expense (and not some other combination)? If you’re satisfied that the numbers you used were the right ones and that your calculations are okay, confirm your response and move on to the next question. The correct answer is (D). DATA INTERPRETATION STRATEGIES Applying the five-step approach to the two sample questions in the previous section highlighted certaintips, techniques, andstrategies fordata interpretation.Here’s alist of them. (The last one applies to certain other types of graphical data.) Don’t Confuse Percentages with Raw Numbers Most data interpretation questions involveraw data as well as proportion—in terms of either percent, fraction, or ratio.Always ask yourself: “Is the solution to this problem a rawnumberoraproportionalnumber?”(You can besurethatthetestdesignerswillbait you with appropriate incorrect answer choices.) Be Sure to Go to the Appropriate Chart (or Part of a Chart) for Your Numbers When itcomes to GREdata interpretation,carelessly referring to the wrongdata isthe leading cause ofincorrectanswers. To makesure you don’t committhiserror,point your finger to the proper line, column, or bar on the screen; put your finger right on it, and don’t move it until you’re sure you’ve got the right data. PART IV: Quantitative Reasoning172 ALERT! Be sure to consult the appropriate portion of the chart, graph, or table for the information you need to answer a data interpretation question. Your ability to find the right data is a big part o f what’s being tested. www.petersons.com . figures. They’re there for a reason: The pieces of information provided in a figure can lead you, step-by-step, to the answer. 12. If O is the center of the. term, the odd integer is one less than the corre- sponding even integer. There are a total of 15 corresponding integers, so the difference between the sums