MULTIPLE PROBABILITIES To find the probability that two or more events will occur, add the probabilities of each. For example, in the problem above, if we wanted to find the probability of drawing either a red or blue marble, we would add the probabilities together. The probability of drawing a red marble = ᎏ 1 3 4 ᎏ . And the probability of drawing a blue marble = ᎏ 1 5 4 ᎏ . Add the two together: ᎏ 1 3 4 ᎏ + ᎏ 1 5 4 ᎏ = ᎏ 1 8 4 ᎏ = ᎏ 4 7 ᎏ . So, the probability for selecting either a blue or a red would be 8 in 14, or 4 in 7. Helpful Hints about Probability ■ If an event is certain to occur, the probability is 1. ■ If an event is certain not to occur, the probability is 0. ■ If you know the probability an event will occur, you can find the probability of the event not occurring by subtracting the probability that the event will occur from 1. Special Symbols Problems The last topic to be covered is the concept of special symbol problems. The GRE will sometimes invent a new arithmetic operation symbol. Don’t let this confuse you. These problems are generally very easy. Just pay atten- tion to the placement of the variables and operations being performed. Example: Given a ⌬ b ϭ (a ϫ b ϩ 3) 2 , find the value of 1 ⌬ 2. Solution: Fill in the formula with 1 being equal to a and 2 being equal to b. (1 ϫ 2 ϩ 3) 2 ϭ (2 ϩ 3) 2 ϭ (5) 2 ϭ 25. So, 1 ⌬ 2 ϭ 25. Example: Solution: Fill in variables according to the placement of number in the triangular figure: a ϭ 1, b ϭ 2, and c ϭ 3. ᎏ 1– 3 2 ᎏ + ᎏ 1– 2 3 ᎏ + ᎏ 2– 1 3 ᎏ = ᎏ – 3 1 ᎏ + –1 + –1 = –2 ᎏ 1 3 ᎏ b c a 2 31 If = _____ + _____ + ____ _ a − b a − c b − c c b a Then what is the value of . . . – THE GRE QUANTITATIVE SECTION– 210 Tips and Strategies for the Official Test You are almost ready to begin practicing. But before you begin the practice problems, read through this sec- tion to learn some tips and strategies for working with each problem type. Quantitative Comparison Questions ■ It is not necessary to find the exact value of the two variables, and often, it is important not to waste time doing so. It is important to use estimating, rounding, and the eliminating unnecessary informa- tion to determine the relationship. ■ Attempt to make the two columns look as similar as possible. For example, make sure all units are equal. This is similar to a strategy given in the problem solving section, and it is even more applicable here. This is also true if one of the answer choices is a fraction or a decimal. If this is the case, make the other answer into an improper fraction or a decimal, which ever is going to make the choices the most similar. ■ Eliminate any information the two columns share. This will leave you with an easier comparison. For exam- ple, if you are given the two quantities: 5(x ϩ 1) and 3 (x ϩ 1), and told that x is positive, you would select the first quantity because you can eliminate the (x ϩ 1) from both. That leaves you to decide which is greater, 5 or 3. This has become a very easy problem resulting from eliminating information the two quantities shared. ■ Substitute real values for unknowns or variables. If you can do so quickly, many of the comparisons will be straightforward and clear. The process of substituting numbers should be used in most QC questions when given a variable. However, be sure to simplify the equation or expression as much as possible before plugging in. ■ The QC section tests how quick, creative, and accurate you can be. Do not get stuck doing complex computations. If you feel yourself doing a lot of computations, stop and try another method. There is often more than one way to solve a problem. Try to pick the easiest way. ■ Make no assumptions about the information listed in the columns. If the question requires you to make assumptions, then choose answer d. For example, if one of the questions asks for the root of x 2 , you cannot assume that the answer is a positive root. Remember that x 2 will have two roots, one posi- tive and one negative. Do not let the test fool you. Be aware of the possibility of multiple answers. ■ If one or both of the expressions being compared have parentheses, be sure to evaluate the expression(s) to remove the parentheses before proceeding. This is a simple technique that can make a large difference in the similarity of the two comparisons. For example, if you are comparing the binomial (x Ϫ 2)(x Ϫ 2) with the trinomial x 2 Ϫ 4x ϩ 4, first remove the parentheses from the product of (x Ϫ 2)(x Ϫ 2) by multiplying the two binomials. The product will be the trinomial x 2 Ϫ 4x ϩ 4. You can clearly see that they are equal. ■ Perform the same operation to both columns. This is especially useful when working with fractions. Often, finding an LCD and multiplying both columns by that number helps to make the comparison easier. Just keep in mind that, like working in an equation, the operation must be performed exactly the same in each column. – THE GRE QUANTITATIVE SECTION– 211 Problem-Solving Questions Problem-solving questions test your mathematical reasoning skills. This means that you will be required to apply several basic math techniques for each problem. Here are some helpful strategies to help you improve your math score on the problem-solving questions: ■ Read questions carefully and know the answer being sought. In many problems you will be asked to solve an equation and then perform an operation with the resulting variable to get an answer. In this situation, it is easy to solve the equation and feel like you have the answer. Paying special attention to what each question is asking, and then double-checking that your answer satisfies this, is an important technique for performing well on the GRE. ■ Sometimes it may be best to try one of the answers. Many times it is quicker to pick an answer and check to see if it is a solution. When you do this, use response c. It will be the middle number and you can adjust the outcome to the problem as needed by choosing b or d next, depending on whether you – THE GRE QUANTITATIVE SECTION– 212 ANSWER SHEET 1. abcde 2. abcde 3. abcde 4. abcde 5. abcde 6. abcde 7. abcde 8. abcde 9. abcde 10. abcde 11. abcde 12. abcde 13. abcde 14. abcde 15. abcde 16. abcde 17. abcde 18. abcde 19. abcde 20. abcde 21. abcde 22. abcde 23. abcde 24. abcde 25. abcde 26. abcde 27. abcde 28. abcde 29. abcde 30. abcde 31. abcde 32. abcde 33. abcde 34. abcde 35. abcde 36. abcde 37. abcde 38. abcde 39. abcde 40. abcde 41. abcde 42. abcde 43. abcde 44. abcde 45. abcde 46. abcde 47. abcde 48. abcde 49. abcde 50. abcde 51. abcde 52. abcde 53. abcde 54. abcde 55. abcde 56 abcde 57. abcde 58. abcde 59. abcde 60. abcde 61 abcde 62. abcde 63. abcde 64. abcde 65. abcde 66. abcde 67. abcde 68. abcde 69. abcde 70. abcde 71. abcde 72. abcde 73. abcde 74. abcde 75. abcde 76. abcde 77. abcde 78. abcde 79. abcde 80. abcde need a larger or smaller answer. This is also a good strategy when you are unfamiliar with the informa- tion the problem is asking. ■ When solving word problems, look at each phrase individually, then rewrite each in math language. This is very similar to creating and assigning variables, as addressed earlier in the word-problem sec- tion. In addition to identifying what is “known” and “unknown,” also take time to translate operation words into actual symbols. It is best when working with a word problem to represent every part of it, phrase by phrase, in mathematical language. ■ Make sure all the units are equal before you begin. This will save a great deal of time doing conversions. This is a very effective way to save time. Almost all conversions are easier to make at the beginning of a problem rather than at the end. Sometimes a person can get so excited about getting an answer that he or she forgets to make the conversion at all, resulting in an incorrect answer. Making the conversions at the start of the problem is definitely more advantageous for this reason. ■ Draw pictures when solving word problems if needed. Pictures are always helpful when a word problem doesn’t have one already, especially when the problem is dealing with a geometrical figure or location. Many students are also better at solving problems when they see a visual representation. Do not make the drawings too elaborate; unfortunately, the GRE does not give points for artistic flair. A simple draw- ing, labeled correctly, is usually all it takes. ■ Avoid lengthy calculations. It is seldom, if ever, necessary to spend a great deal of time doing calculations. This is a test of mathematical concepts, not calculations. If you find yourself doing a very complex, lengthy calculation — stop! Either you are not doing the problem correctly or you are missing a much easier solution. ■ Be careful when solving Roman numeral problems. Roman numeral problems will give you several answer possibilities that list a few different combinations of solutions. You will have five options: a, b, c, d, and e.To solve a Roman numeral problem, treat each Roman numeral as a true or false statement. Mark each Roman numeral with a “T” or “F” on scrap paper, then select the answer that matches your “T’s” and “F’s.” These strategies will help you to do well on the GRE, but simply reading them will not.You must practice, prac- tice, and practice. That is why there are 80 problems in the following section for you to solve. Keep in mind that on the actual GRE, you will only have 28 problems in the Quantitative section. By doing 80 problems now, it will seem easy to do only 28 questions on the test. Keep this in mind as you work through the practice problems. Now the time has come for all of your studying to be applied; the practice problems are next. Good luck! – THE GRE QUANTITATIVE SECTION– 213 Practice Directions: In each of the questions 1–40, compare the two quantities given. Select the appropriate choice for each one according to the following: a. The quantity in column A is greater. b. The quantity in column B is greater. c. The two main quantities are equal. d. There is not enough information given to determine the relationship of the two quantities. Column A Column B 1. n Ͼ 1 ᎏ n + 3 7 ᎏ + ᎏ n 4 –3 ᎏ ᎏ 7n + 7 19 ᎏ 2. 0.1y + 0.01y = 2.2 0.1y 20 3. the reciprocal of 4 Ί 4. 3 feet, 5 inches 1.5 yards 5. x = 6 + 7 + 8 + 9 + 10 y = 5 + 6 + 7+ 8+ 9 5(15) x ϩ y 6. 5678 ϫ 73 170▲4 3974ٗ0 414,494 value of ▲ value of ٗ 7. 4x = 4(14) – 4 x 14 1 ᎏ 16 – THE GRE QUANTITATIVE SECTION– 214 8. Cindy covered 36 miles in 45 minutes. Cindy’s average speed 48 miles/hour (in miles/hour) 9. length of AB length of BC 10. 120 ͙1,440 ෆ 11. Page is older than Max and Max is younger than Gracie. Page’s age Gracie’s age 12. 4 inches length of DC 13. z – y 40 14. a Ͼ b Ͼ c Ͼ d Ͼ 0 a Ϫ dbϪ c A B D C x˚ x˚ y˚ y˚ z˚ y = 50 ABCD is a parallelogram A B D C AREA OF = 20 in. 2 ABC AD = 5 INCHES AND AD BC A B C AB BC ABC AREA OF = 18 – THE GRE QUANTITATIVE SECTION– 215 . add the probabilities together. The probability of drawing a red marble = ᎏ 1 3 4 ᎏ . And the probability of drawing a blue marble = ᎏ 1 5 4 ᎏ . Add the two together: ᎏ 1 3 4 ᎏ + ᎏ 1 5 4 ᎏ =. + 7 19 ᎏ 2. 0.1y + 0.01y = 2.2 0.1y 20 3. the reciprocal of 4 Ί 4. 3 feet, 5 inches 1 .5 yards 5. x = 6 + 7 + 8 + 9 + 10 y = 5 + 6 + 7+ 8+ 9 5( 15) x ϩ y 6. 56 78 ϫ 73 170▲4 3974ٗ0 414,494 value. binomial (x Ϫ 2)(x Ϫ 2) with the trinomial x 2 Ϫ 4x ϩ 4, first remove the parentheses from the product of (x Ϫ 2)(x Ϫ 2) by multiplying the two binomials. The product will be the trinomial x 2 Ϫ 4x