The gre quatitative section 9 doc

d. 91 e. 440 79. Which of the following statements is (are) always true? (Assume a, b, and c are not equal to zero.) I. ᎏ 1 a ᎏ is less than a. II. ᎏ a 2 + a b ᎏ equals ᎏ b 2 + b a ᎏ when a equals b. III. ᎏ a b + + c c ᎏ is more than ᎏ a b ᎏ . a. II only b. I and II only c. I and III only d. II and III only e. I, II, and III 80. If bx Ϫ 2 ϭ k, then x equals a. ᎏ b k ᎏ ϩ 2. b. k Ϫ ᎏ 2 b ᎏ . c. 2 Ϫ ᎏ b k ᎏ . d. ᎏ k + b 2 ᎏ . e. k Ϫ 2  Answers 1. b. ᎏ n + 3 7 ᎏ + ᎏ n 4 –3 ᎏ ᎏ 4n +28 1 + 2 3n –9 ᎏ ᎏ 7n 1 + 2 19 ᎏ The numerators are the same, but the fraction in column B has a smaller denominator,denoting a larger quantity. 2. b. 1y + 0.01y = 2.2 10y + 1y = 220 Multiply each term by 100. 11y = 220 – THE GRE QUANTITATIVE SECTION– 230 0.1y = 2 Divide by 10 on each side. 3. c. The reciprocal of 4 is ᎏ 1 4 ᎏ ; Ί ᎏ 1 1 6 ᎏ = ᎏ 1 4 ᎏ ๶ . 4. b. 1 yard ϭ 3 feet and (0.5) or ᎏ 1 2 ᎏ yard ϭ 1 foot 6 inches. Therefore, (1.5) or 1 ᎏ 1 2 ᎏ yards ϭ 4 feet 6 inches. 5. c. Add: 5 ϩ 6 ϩ 7 ϩ 8 ϩ 9 ϭ 35; 6 ϩ 7 ϩ 8 ϩ 9 ϩ 10 ϭ 40; so x ϩ y ϭ 75; 5 ϫ 15 ϭ 75, so the two quantities are equal. 6. b. 8 ϫ 3 = 24 and 7 ϫ 3 = 21 + 2 – 23 Therefore, ▲ ϭ 3. Since 8 ϫ 7 ϭ 56, ᮀ = 6. 7. b. 4x = 4(14) – 4 4x = 56 – 4 4x = 52 x = 13 8. c. Rate = Distance Ϭ Time Rate = 36 miles Ϭ ᎏ 3 4 ᎏ hour (36) ᎏ 4 3 ᎏ = 48 miles/hour 9. d. ᎏ BC ϫ 2 AB ᎏ = 18, but any of the following may be true: BC Ͼ AB, BC Ͻ AB,or BC = AB. 10. a. ͙1,440 ෆ is a two-digit number, so you know that it is less than 120. 11. d. Since Gracie is older than Max, she may be older or younger than Page. 12. d. Since AD ϭ 5 and the area is 20 square inches, we can find the value of base BC but not the value of DC. BC equals 8 inches, but BD will be equal to DC only if AB ϭ AC. 13. c. Since y ϭ 50, the measure of angle DCB is 100 º and the measure of angle ABC is 80 º since ABCD is a parallelogram. Since x ϭ 40, z = 180 – 90 = 90 z – y = 90 – 50 = 40 14. a. In column A, d, the smallest integer, is subtracted from a, the integer with the largest value. 15. a. Since x ϭ 65 and AC ϭ BC, then the measure of angle ABC is 65 º , and the measure of angle ACB is 50 º . Since BCʈDE, then y ϭ 50 º and x Ͼ y. 16. c. From Ϫ5 to ϩ5, there are 11 integers. Also, from ϩ5 to ϩ15, there are 11 integers. 17. b. Since the area ϭ 25, each side ϭ 5. The sum of three sides of the square ϭ 15. – THE GRE QUANTITATIVE SECTION– 231 18. a. x ϭ 0.5 4x ϭ (0.5)(4) ϭ 2.0 x 4 ϭ (0.5)(0.5)(0.5)(0.5) ϭ 0.0625 19. b. The fraction in column A has a denominator with a negative value, which will make the entire frac- tion negative. 20. d. The area of a triangle is one-half the product of the lengths of the base and the altitude, and cannot be determined using only the values of the sides without more information. 21. c. Let x ϭ the first of the integers. Then: sum ϭ x ϩ x ϩ 1 ϩ x ϩ 2 ϩ x ϩ 3 ϩ x ϩ 4 ϭ 5x ϩ 10 5x ϩ 10 ϭ 35 (given), then 5x ϭ 25. x ϭ 5 and the largest integer, x ϩ 4 ϭ 9. 22. a. ͙160 ෆ = ͙16 ෆ ͙10 ෆ = 4͙10 ෆ 23. c. Since the triangle is equilateral, x ϭ 60 and exterior angle y ϭ 120. Therefore, 2x ϭ y. 24. b. If ᎏ 2 3 ᎏ corresponds to 12 gallons, then ᎏ 1 3 ᎏ corresponds to 6 gallons. Therefore, ᎏ 3 3 ᎏ corresponds to 18 gal- lons, which is the value of column A. 25. c. Since the triangle has three congruent angles, the triangle is equilateral and each side is also equal. 3a ϩ 15 ϭ 5a ϩ 1 ϭ 2a ϩ 22 3a ϩ 15 ϭ 5a ϩ 1 14 ϭ 2a 7 ϭ a 26. d. Since x Ϫ y ϭ 7, then x ϭ y ϩ 7; x and y have many possible values, and therefore, x ϩ y cannot be determined. 27. b. ᎏ x 2 2 ᎏ = 18 x 2 = 36 x = 6 Therefore, AC ϭ 6͙2 ෆ and 6͙2 ෆ Ͼ 6. In addition, the hypotenuse is always the longest side of a right triangle, so the length of AC would automatically be larger than a leg. 28. c. Since the diagonal of the square measures 6͙2 ෆ , the length of each side of the square is 6. Therefore, AB ϭ 6, and thus, the perimeter ϭ 24. 29. c. Area = ᎏ 1 2 ᎏ (6)(6) = 18 30. c. AB ϭ BC (given) Since the measure of angle B equals the measure of angle C, AB ϭ AC. Therefore, ABC is equilateral and mЄA ϭ mЄB ϭ mЄC ϭ mЄB ϩ mЄC ϭ mЄB ϭ ϩ mЄA. – THE GRE QUANTITATIVE SECTION– 232 31. d. There is no relationship between a and f given. 32. d. The variable x may have any value between 64 and 81. This value could be smaller, larger, or equal to 65. 33. a. KL ϭ 24 ϩ length of AB, so KL Ͼ 23. 34. b. ͙144 ෆ = 12 and ͙100 ෆ + ͙44 ෆ = 10 + Ϸ 6.6 Ͼ 12 35. c. Because y ϭ z and AB ϭ AC, then x ϩ y ϭ x ϩ z. (If equal values are added to equal values, the results are also equal.) 36. c. ϫ = = ᎏ (3)( 3 12) ᎏ = 12 37. a. ᎏ 4 x ᎏ + ᎏ 3 x ᎏ = ᎏ 1 7 2 ᎏ ᎏ 1 3 2 x ᎏ + ᎏ 1 4 2 x ᎏ = ᎏ 1 7 2 ᎏ 3x + 4x = 7 x = 1 1 Ͼ –1 38. b. 0.003% ϭ 0.00003 0.0003 Ͼ 0.00003 39. c. ᎏ 4 k ᎏ % = ᎏ 4 k ᎏ Ϭ 100 = ᎏ 4 k ᎏ ϫ ᎏ 1 1 00 ᎏ = ᎏ 40 k 0 ᎏ 40. c. AB ϭ 3 inches ϩ 5 inches ϭ 8 inches BC ϭ 5 inches ϩ 4 inches ϭ 9 inches AC ϭ 4 inches ϩ 3 inches ϭ 7 inches Total ϭ 24 inches ϭ 2 feet 41. a. ᎏ 0 8 .8 ᎏ = ᎏ 8 8 0 ᎏ = 10 ᎏ 0 8 .8 ᎏ = ᎏ 8 8 0 ᎏ = ᎏ 1 1 0 ᎏ (0.8) 2 = 0.64 ͙0.8 ෆ = 0.89 0.8 ␲ = (0.8)(3.14) = 2.5 42. e. 17xy + 7 = 19xy 7 = 2xy 14 = 4xy 43. d. Average ϭ xy Sum Ϭ 2 ϭ xy Sum ϭ 2xy 3͙144 ෆ ᎏ 3 ͙3 ෆ ᎏ ͙3 ෆ 3͙48 ෆ ᎏ ͙ 3 ෆ – THE GRE QUANTITATIVE SECTION– 233 2xy ϭ x ϩ ? ? ϭ 2xy Ϫ x 44. c. This is a direct proportion. Let x ϭ length of the shorter dimension of enlargement. = = 2 ᎏ 1 2 ᎏ x = (4)(1 ᎏ 7 8 ᎏ ) ᎏ 5 2 x ᎏ = ᎏ 6 8 0 ᎏ x = 3 45. d. AEB ϭ 12 AE ϭ 8 AGD ϭ 6 AG ϭ 4 Area AEFG ϭ 32 Area ABCD ϭ 72 Area of shaded part ϭ 72 – 32 ϭ 40 46. c. Be careful to read the proper line (regular depositors). The point is midway between 90 and 100. 47. a. Number of Holiday Club depositors ϭ 60,000 Number of regular depositors ϭ 90,000 The ratio 60,000:90,000 reduces to 2:3. 48. b. I is not true; although the number of depositors remained the same, one may not assume that inter- est rates were the cause. II is true; in 1984, there were 110,000 depositors. Observe the largest angle of inclination for this period. III is not true; the circle graph indicates that more than half of the bank’s assets went into mortgages. 49. c. (58.6%) of 360 º ϭ (0.586)(360 º ) ϭ 210.9 º 50. e. (Amount Invested) ϫ (Rate of Interest) = Interest or Amount Invested = ᎏ Rate In o t f e I r n es t t erest ᎏ Amount invested in bonds = ᎏ x d b o % llars ᎏ or x Ϭ ᎏ 10 b 0 ᎏ or x( ᎏ 10 b 0 ᎏ ) or (x)( ᎏ 10 b 0 ᎏ ) or ᎏ 10 b 0x ᎏ Since the amount invested in bonds = ᎏ 10 b 0x ᎏ , the amount invested in mortgages must be 2( ᎏ 10 b 0x ᎏ ) dollars, or ᎏ 20 b 0x ᎏ , since the chart indicates that twice as much (58.6%) is invested in mortgages as is invested in bonds (28.3%). 4 ᎏ x 2 ᎏ 1 2 ᎏ ᎏ 1 ᎏ 7 8 ᎏ longer dimension ᎏᎏ shorter distance – THE GRE QUANTITATIVE SECTION– 234 51. d. Draw altitudes of AE and BF. ᎏ 1 2 ᎏ (b 1 + b 2 )h = ᎏ 1 2 ᎏ (10 + 2)6 = = 36 square units 52. d. Factor x 2 ϩ 2x Ϫ 8 into (x ϩ 4)(x Ϫ 2). If x is either Ϫ4 or 2, then x 2 ϩ 2x Ϫ 8 ϭ 0. 53. a. Set up a proportion. Let x ϭ the total body weight in terms of g. ᎏ w to e t i a g l h b t o o d f y sk w e e le ig to h n t ᎏ = ᎏ 1 7 0 0 , , 0 0 0 0 0 0 g g r r a a m m s s ᎏ = ᎏ x g ᎏ ᎏ 1 7 ᎏ = ᎏ x g ᎏ x = 7g 54. b. Between 1 P.M. and 3:52 P.M., there are 172 minutes. There are three intervals between the classes. Therefore, 3 ϫ 4 minutes, or 12 minutes, is the time spent in passing to classes. That leaves a total of 172 Ϫ 12, or 160, minutes for instruction, or 40 minutes for each class period. 55. e. (Average)(Number of items) ϭ Sum (x)(P) ϭ Px (y)(N) ϭ Ny ᎏ Numb S e u r m of items ᎏ = Average ᎏ P P x + + N Ny ᎏ = Average 56. b. Select the choice in which the value of n is greater than the value of d in order to yield a value of ᎏ n d ᎏ greater than 1. 57. a.mЄc ϩ mЄd ϭ 180 ° ,but mЄc  mЄd. mЄa ϭ mЄd (vertical angles) mЄa ϭ mЄe (corresponding angles) mЄf ϭ mЄb (corresponding angles) mЄf ϭ mЄc (alternate interior angles) 58. b. Sum ϭ (0.6)(4) or 2.4 0.2 ϩ 0.8 ϩ 1 ϭ 2 x ϭ 2.4 Ϫ 2 or 0.4 0 2 4 6 8 10 2 4 6810 12 14 A B C D E F 2 10 6 – THE GRE QUANTITATIVE SECTION– 235 . to ϩ5, there are 11 integers. Also, from ϩ5 to ϩ15, there are 11 integers. 17. b. Since the area ϭ 25, each side ϭ 5. The sum of three sides of the square ϭ 15. – THE GRE QUANTITATIVE SECTION 231 18 of the lengths of the base and the altitude, and cannot be determined using only the values of the sides without more information. 21. c. Let x ϭ the first of the integers. Then: sum ϭ x ϩ x ϩ. 0.0625 19. b. The fraction in column A has a denominator with a negative value, which will make the entire frac- tion negative. 20. d. The area of a triangle is one-half the product of the lengths