119 INTRODUCTION TO QUANTITATIVE ABILITY Prior to starting the exam, you will be carefully walked through a very basic tutor- ial program explaining how to use the computer for this exam. The computer- adaptive GRE gives you 45 minutes to answer 28 quantitative questions. These questions are composed of Quantitative Comparisons and Math Ability (Multiple- Choice) Questions, and the question types are intermingled. You will be given a medium difficulty question to start with, and then the computer will adapt the level of questions you receive based on your responses to all the previous ques- tions. All of your work will be done on the scratch paper provided, and all of your answers will be recorded on the computer screen by using a mouse to fill in the appropriate ovals. You will not be allowed to go back to a previous question, so be sure to answer each question before you attempt to move to the next question. The Quantitative Section will generate a score from 200 to 800. Your score will be based on how well you do on questions presented and also on the number of ques- tions you answer. You should try to pace yourself so that you have sufficient time to consider every question. If possible, answer all 28 questions in this section. Guess if you need to. In this book — to assist you in understanding explanations and to direct your at- tention to different questions and answer choices — each question is given a num- ber, and letters have been placed inside the ovals of the answer choices. Note that on the actual exam, questions will not have numbers next to them and there will be no letters in the ovals. Introduction to Quantitative Comparison Quantitative Comparison questions require you to make a comparison between quantities in two columns. You are to decide if one column is greater, if the columns are equal, or if no comparison can be determined from the information given. Ability Tested Quantitative Comparison tests your ability to use mathematical insight, approxima- tion, simple calculation, or common sense to quickly compare two given quantities. Team-LRN 120 Part I: Analysis of Exam Areas Basic Skills Necessary This question type requires twelfth-grade competence in school arithmetic, alge- bra, and intuitive geometry. Skills in approximating, comparing, and evaluating are also necessary. No advanced mathematics is necessary. Directions You are given two quantities, one in column A and one in column B. You are to compare the two quantities and choose oval: A. if the quantity in Column A is greater; B. if the quantity in Column B is greater; C. if the two quantities are equal; D. if the comparison cannot be determined from the information given. Common Information: Information centered above columns refers to one or both columns. A symbol that appears in both columns represents the same thing in each column. Analysis ■ The purpose here is to make a comparison; therefore, exact answers are not always necessary. (Remember that you can tell whether you are taller than someone in many cases without knowing that person’s height. Comparisons such as this can be made with only limited or partial infor- mation—just enough to compare.) ■ Choice D—the comparison cannot be determined from the information given—is not a possible answer if there are values in each column, be- cause you can always compare values. ■ If you get different relationships, depending on the values you choose for variables, then the answer is always D. Notice that there are only four pos- sible choices here. ■ Note that you can add, subtract, multiply, and divide both columns by the same value, and the relationship between the columns will not change. Exception: You should not multiply or divide each column by negative numbers, because the relationship reverses. Squaring both columns is per- missible, as long as each side is positive. Team-LRN Suggested Approach with Sample Problems This section emphasizes shortcuts, insight, and quick techniques. Long and/or involved mathematical computation is unnecessary and is contrary to the pur- pose of this section. Samples Column A Column B 1. 21 × 43 × 56 44 × 21 × 57 Canceling (or dividing) 21 from each side leaves 43 × 56 44 × 57 The rest of this problem should be done by inspection, because it is obvious that column B is greater than column A without doing any multiplication. You could have attained the correct answer by actually multiplying out each column, but you would then not have enough time to finish the section. The correct answer is B. Column A Column B 2. 7 3 5 2 8 5 ## 5 2 11 4 8 5 ## Because both sides have the factors 2 ⁄ 5 and 5 ⁄ 8 , you may eliminate them from each column. Now compare 3 ⁄ 7 and 4 ⁄ 11 by cross-multiplying upward, and you get Because 33 is greater than 28, 3 ⁄ 7 > 4 ⁄ 11. The correct answer is A. Always keep the columns in perspective before starting any calculations. Take a good look at the value in each column before starting to work on one column. 3 7 4 11 33 28 121 Introduction to Quantitative Ability Team-LRN Samples Column A Column B 3. 40% of 60 60% of 40 There is no need to do any calculations for this problem. Column A can be written ( 40 ⁄ 100 ) × 60. Column B can be written ( 60 ⁄ 100 ) × 40. You should note that both columns have (40 × 60)/100. The correct answer is C. Column A Column B 4. 7 6 3 2 8 After looking at each column (note that the answer could not be D because there are values in each column), compute the value on the left. Because you are taking a cube root, simply divide the power of 7 by 3 leaving 7 2 , or 49. There is no need to take 2 out to the 8th power; just do as little as necessary: 2 2 = 4 2 3 = 8 2 4 = 16 2 5 = 32 STOP It is evident that 2 8 is much greater than 49; the correct answer is B. Approximating can also be valuable while remembering to keep the columns in perspective. As you keep the columns in perspective, check to see if the value in each col- umn increases or decreases from the starting point. Sample Column A Column B 5. (.9) 8 (1.01) 4 In Column A, a fractional value (a value less than 1) is multiplied by itself many times. So its value becomes increasingly smaller. (For example, 1 ⁄ 2 × 1 ⁄ 2 = 1 ⁄ 4 ; 122 Part I: Analysis of Exam Areas Team-LRN 1 ⁄ 4 × 1 ⁄ 2 = 1 ⁄ 8 , and so forth). In Column B, a number greater than 1 is multiplied by itself; its value grows larger. So Column B is greater. The correct answer is B. As you keep the columns in perspective, notice if the signs (+, −) in each col- umn are different. If they are, you don’t need to work out the problem. Samples Column A Column B 6. (−10) 100 (−10) 101 A negative number multiplied an even number of times will yield a positive prod- uct. A negative number multiplied an odd number of times will yield a negative product. Since Column A will be positive and Column B will be negative, A is greater. The correct answer is A. Column A Column B 7. .05 − .125 .1 Subtracting in Column A, you get .05 − .125 =−.075. Our difference is a negative number. Thus, the positive value in Column B must be greater. The correct answer is B. Column A Column B 8. x + ya+ b Because coordinates (x,y) are in quadrant III, they are both negative, so their sum is negative. Because coordinates (a, b) are in quadrant I, they are both positive, so their sum is positive. Therefore, Column B is greater than Column A. The correct answer is B. • (a,b) • (x,y) (0,0) 123 Introduction to Quantitative Ability Team-LRN The use of partial comparisons can be valuable in giving you insight into find- ing a comparison. If you cannot simply make a complete comparison, look at each column part by part. Sample Column A Column B 9. 57 1 65 1 - 58 1 63 1 - Because finding a common denominator would be too time consuming, you should first compare the first fraction in each column (partial comparison). Notice that 1 ⁄ 57 is greater than 1 ⁄ 58 . Now compare the second fractions and notice that 1 ⁄ 65 is less than 1 ⁄ 63 . Using some common sense and insight, if you start with a larger number and subtract a smaller number, it must be greater than starting with a smaller number and subtracting a larger number, as pointed out below. The correct answer is A. Often, simplifying one or both columns can make an answer evident. Samples Column A Column B 10 . a, b, c, all greater than 0 a(b + c) ab + ac Using the distributive property on Column A to simplify gives ab and ac; there- fore, the columns are equal. The correct answer is C. 1 57 1 65 − 1 58 1 63 − Larger Smaller Larger Smaller 124 Part I: Analysis of Exam Areas Team-LRN Column A Column B 11. a > 0 b > 0 c > 0 (3a)(3b)(3c) 3abc Multiplying column A gives (3a)(3b)(3c) = 27abc. Because a, b, and c are all posi- tive values, 9abc will always be greater than 3abc. The correct answer is A. Column A Column B 12 . Number of prime numbers 5 between 3 and 19 The prime numbers between 3 and 19 are 5, 7, 11, 13, and 17. The correct answer is C, since there are 5 primes. If a problem involves variables (without an equation), substitute in the num- bers 0, 1, and -1. Then try 1⁄2, and 2 if necessary. Using 0, 1, and -1 will often tip off the answer. Samples Column A Column B 13 . a + bab Substituting 0 for a and 0 for b gives the following: 0 + 0 (0) Therefore, 0 = 0. Using these values for a and b gives the answer C. But when you multiply two numbers, you don’t always get the same result as when you add them, so try some other values. Substituting 1 for a and −1 for b gives the following: 1 + (−1) 1(−1) Therefore, 0 >−1 and the answer is now A. 125 Introduction to Quantitative Ability Team-LRN Anytime you get more than one comparison (different relationships), depend- ing on the values you choose, the correct answer must be D, the relationship cannot be determined. Notice that if you had substituted the values a = 4, b = 5; or a = 6, b = 7; or a = 7, b = 9; and so on, you would repeatedly get the answer B and may have chosen the incorrect answer. The correct answer is D. Column A Column B 14 . x < y < z x + y + z xyz Substituting 0 for x, 1 for y, and 2 for z, gives (0) + (1) + (2) (0)(1)(2) Therefore, 3 > 0. Now substituting −1 for x, 0 for y, and 1 for z gives (−1) + (0) + (1) (−1)(0)(1) Therefore, 0 = 0. Because different values give different comparisons, the correct answer is D. Column A Column B 15 . x > y > 0 x and y are integers x xy x + _i y xy y + _i Plug in values for x and y such that x > y > 0, and x and y are integers. For exam- ple, let y = 1 and x = 2. This gives 2 21 2 + ^h 1 12 1 + ^h 2 3 2 ^h 1 3 1 ^h 2 9 > 1 3 126 Part I: Analysis of Exam Areas Team-LRN Using these values, 9 ⁄ 2 , or 4 1 ⁄ 2 , is greater than 3, so Column A is greater. Using other values such that x > y > 0 will always give the same relationship. Column A is greater. The correct answer is A. Sometimes you can solve for a column directly, in one step, without solving and substituting. If you have to solve an equation or equations to give the columns values, take a second and see if there is a very simple way to get an answer before going through all of the steps. Sample Column A Column B 16 . 4x + 2 = 10 2x + 14 Hopefully, you would spot that the easiest way to solve for 2x + 1 is directly by dividing 4x + 2 = 10 by 2, leaving 2x + 1 = 5. Therefore, 5 > 4 Solving for x first in the equation and then substituting would also have worked but would have been more time consuming. The correct answer is A. Redrawing and marking diagrams and figures can be very helpful for giving insight into a problem. If you are given a diagram or figure on the screen, quickly redraw it on your scratch paper. Remember that diagrams and figures are meant for positional information only. Just because something “looks” a certain way is not enough reason to choose an answer. 127 Introduction to Quantitative Ability Team-LRN Sample Column A Column B XZ = YZ 17. xy Even though x appears larger, this is not enough. Mark in the diagram as shown. Notice that you should mark things of equal measure with the same markings, and since angles opposite equal sides in a triangle are equal, x = y. The correct answer is C. If you are given a description of a diagram or a geometry problem without a diagram, you should make a sketch. When in doubt, “draw.” This may tip off a simple solution. Sample Column A Column B 18 . Perimeter of an equilateral Perimeter of a square with side triangle with side length 5x length of 4x Simply sketch and label each geometric figure as follows: 5x 4x 4x 5x5x 4x4x XY Z X X° y° Y Z 128 Part I: Analysis of Exam Areas Team-LRN [...]... DE is parallel to AB 40 C y Team-LRN C Area of DEBC 2 Area of rectangle ABCD 3 135 Part I: Analysis of Exam Areas y z A 10 B x C y > 2x 40° 5 47 E z 48 % AB D % AE Answers and Explanations for Practice Quantitative Comparison Questions Easy to Moderate 1 A Because 43⁄ 4 is equivalent to 4.75, Column A is greater 2 B Converting each fraction to a decimal (dividing numerator by denominator) gives 82 for... equals 3 tells nothing about the value of x x B C xy 3 A D ABCD is a rhombus 37 A First factor: m2 − 5m − 24 = 0 (m − 8)(m + 3) = 0 Now set each equal to 0: m−8=0 m=8 m+3=0 m = −3 Since both 8 and −3 are less than 10, Column A is greater 140 Team-LRN Introduction to Quantatitive Ability Above Average to Difficult 38 A To find the number of degrees in the interior angles of a pentagon, use the formula 180... to you and difficult to work with, change the number slightly (but remember what you’ve changed) to something easier to work with Sample Column A Column B C 3 88° 4 19 5 c Becasue the 88° shown in the figure is unfamiliar to work with, change it to 90° for now so that you may use the Pythagorean theorem to solve for c a2 + b2 = c2 Solve for c as follows: (3)2 + (4)2 = c2 9 + 16 = c2 25 = c2 Therefore,... numbers 132 Team-LRN Introduction to Quantitative Ability Practice Quantitative Comparison Questions Easy to Moderate Column A B n° Column B A m° C 3 1 4.78498 4 ⁄4 AB = BC 11 13 2 9⁄ 11 11 n m 12 53 27 x>0 3 x x+ 2 x x- 2 6x + 18y = 12 Questions 4–6 refer to the diagram 13 x + 3y 2 C x+y=4 xy = 0 A ∆ABC is equilateral CD is a median 4 EABC + EBAC y Questions 15–18 refer to the diagram B ECDB 5 AD 6 AB... because if the smallest angle were equal to 60°, the three angles would sum to greater than 180°, which isn’t possible So Column A is greater B A 142 C Team-LRN Introduction to Quantatitive Ability 46 A The area of DEBC is 3⁄ 4 the area of ABCD, or: area DEBC 3 units 3 area ABCD = 4 units = 4 Column A is greater x A E x B x D 47 x C x A If angle y were equal to 2x, then in the triangle, y would be 60°... angle (CBD) in ∆CDB 5 C The definition of median is that it divides the side it intersects into two equal parts 6 A Since ∆ ABC is equilateral, AB = BC Thus AB + BD must be more than BC alone 136 Team-LRN Introduction to Quantatitive Ability 7 D As the only condition for plugging in values for x and y is that together they must equal 0, the values for x and y may vary For instance, both x and y may equal... Since the radius is 4, and π is about 3.14 π(4)2 Area of square with side 7 is 49 3.14 × 16 50.24 Answers 24–25 refer to the diagram A D C O O is the center 138 Team-LRN B Introduction to Quantatitive Ability % C AC = 2 ^EBh, since an inscribed angle is half of the arc is subtends (connects to) % 25 A Since EAOB is a central angle, it equals the measure of AB, and since EADC is outside % % the circle but... definite relationship can be determined 34 B Because a, b, c, and d are each greater than 0, they are therefore positive In Column A, the denominator is greater than the numerator, so the fraction equals less than 1 In Column B, the numerator is greater than the denominator, so the fraction equals more than 1 Therefore, Column B is greater 35 D If y is 0, columns A and B each equal 25, and so the columns could...Introduction to Quantitative Ability Now it is evident that the perimeter of an equilateral triangle with side 5x is 3(5x) = 15x The perimeter of a square with side 4x is 4(4x) = 16x Since 4x and 5x represent lengths of sides, x must be a positive number Therefore, 15x < 16x The correct answer is B If you are given information that is unfamiliar to you and difficult to work with, change the... The correct answer is D Because the height of each right circular cylinder is necessary to obtain the volume, no comparison can be made On occasion, you will actually have to solve information centered between the columns or information in the columns You should be able to work these quickly Remember, if it takes too long, you’re probably doing it wrong Samples Column A 22 Column B The value of 5x − . 119 INTRODUCTION TO QUANTITATIVE ABILITY Prior to starting the exam, you will be carefully walked through a very basic tutor- ial program explaining how to. mouse to fill in the appropriate ovals. You will not be allowed to go back to a previous question, so be sure to answer each question before you attempt to