58. Is x + y Ͼ 2z ? (1) ᭝ABC is equilateral. (2) AD ⊥ BC 59. The circles in the diagram are concentric circles. What is the area of the shaded region? (1) The area of the inner circle is 25. (2) The diameter of the larger circle is 20. 60. Find the value of x. (1) The length of BC is 2͙ ෆ 3. (2) The length of AC is 4. A B C 30° x A B C D z x y – QUANTITATIVE PRACTICE TEST– 383 61. What is the value of a + b? (1) a 2 + b 2 = 13 (2) 62. Between what two numbers is the measure of the third side of the triangle? (1) The sum of the two known sides is 10. (2) The difference between the two known sides is 6. 63. What is the area of the circle? (1) The radius is 6. (2) The circumference is 12. 64. What is the positive value of z ? (1) 3y + z = 4 (2) z 2 – z = 12 65. Two cars leave the same city traveling on the same road in the same direction. The second car leaves one hour after the first. How long will it take the second car to catch up with the first? (1) The second car is traveling 10 miles per hour faster than the first car. (2) The second car averages 60 miles per hour. 66. In right triangle XYZ, the m∠y = 90 . What is the length of XZ? (1) The length of YZ = 6. (2) m ∠z = 45 67. Is ? (1) 3x = 6y (2) 68. What is the total cost of six pencils and four notebooks? (1) Ten pencils and nine notebooks cost $11.50. (2) Twelve pencils and eight notebooks cost $11.00. 69. What is the ratio of the corresponding sides of two similar triangles? (1) The ratio of the perimeters of the two triangles is 3:1. (2) The ratio of the areas of the two triangles is 9:1. x y 7 1 x y 7 y x 2b ϭ 12 a – QUANTITATIVE PRACTICE TEST– 384 70. What percent of the class period is over? (1) The time remaining is ᎏ 1 4 ᎏ of the time that has passed. (2) The class period is 42 minutes long. 71. Daniel rides to school each day on a path that takes him first to a point directly east of his house and then from there directly north to his school. How much shorter would his ride to school be if he could walk on a straight-line path directly to school from his home, instead of east and then north? (1) The direct straight-line distance from home to school is 17 miles. (2) The distance he rides to the east is 7 miles less than the distance he rides going north. 72. What is the slope of line m? (1) Line m intersects the x-axis at the point (4, 0). (2) The equation of line m is 3y = x – 4. 73. Jacob is a salesperson. He earns a monthly salary plus a commission on all sales over $4,000. How much did he earn this month? (1) His monthly salary is $855 and his total sales over $4,000 were $4,532.30. (2) His total sales for the month were $8,532.30. 74. Is ᭝ABC similar to ᭝ADE? (1) BC is parallel to DE (2) AD = AE 75. The formula for compounded interest can be defined as A = p (1 + r) n ,where A is the total value of the investment, p is the principle invested, r is the interest rate per period, and n is the number of periods. If a $1,000 principle is invested, which bank gives a better interest rate for a savings account, Bank A or Bank B? (1) The interest rate at Bank A is 4% compounded annually. (2) The total amount of interest earned at Bank B over a period of five years is $276.28. A D E BC – QUANTITATIVE PRACTICE TEST– 385 76. A fence has a square gate. What is the height of the gate? (1) The width of the gate is 30 inches. (2) The length of the diagonal brace of the gate is 30 ͙ ෆ 2 inches. 77. Find the area of the shaded region. (1) m ∠A = 43°. (2) AB = 10 cm. 78. A circle and a straight line are drawn on the same coordinate graph. In how many places do the two graphs intersect? (1) The equation of the circle is x 2 + y 2 = 25. (2) The y-intercept of the straight line is 6. 79. Michael left a city in a car traveling directly west. Katie left the same city two hours later going directly east traveling at the same rate as Michael. How long after Katie left will they be 350 miles apart? (1) An hour and a half after Katie left they are 250 miles apart. (2) Michael’s destination is 150 miles farther than Katie’s. 80. What is the area of the shaded region? (1) ᭝ABC is equilateral. (2) The length of is 16 inches.BC A O B C D A B C – QUANTITATIVE PRACTICE TEST– 386 Answer Explanations 1. d. The only prime numbers that satisfy this condition are 2 and 5. Since x Ͼ y, x = 5 and y = 2. There- fore, by substitution, 2 (5) + 2 = 10 + 2 = 12. 2. b. Convert 6% to its decimal equivalent of 0.06 and 14% to 0.14. The key word “product” tells you to multiply, so 0.06 × 0.14 = 0.0084, which is choice b. 3. b. 2 ᎏ 1 2 ᎏ miles divided by ᎏ 1 4 ᎏ is ten quarter miles. Since the first quarter mile costs x amount, the other nine quarter miles cost ᎏ 1 4 ᎏ x, so 9 × ᎏ 1 4 ᎏ x = ᎏ 9 4 ᎏ x. x + ᎏ 9 4 ᎏ x = ᎏ 4 4 ᎏ x + ᎏ 9 4 ᎏ x = ᎏ 13 4 ᎏ x. 4. a.The sum of the measures of the two shorter sides of a triangle must be greater than the longest side. Since 3 + 3 Ͼ 5, statement I works. Since 6 + 6 = 12 and 1 + 2 = 3, they do not form the sides of the triangle. The answer is statement I only. 5. a. If the average of four rounds is 78, then the total points scored is 78 × 4 = 312. If his score were to drop 2 points, that means his new average would be 76. A 76 average for five rounds is a total of 380 points. The difference between these two point totals is 380 – 312 = 68. He needs a score of 68 on the fifth round. 6. e. Suppose Celeste worked for 8 hours each day for 5 consecutive days. Her total pay would be found by finding her total hours (8 × 5 = 40) and then multiplying 40 by her pay per hour ($9.50). Since you are only multiplying to solve the problem, the expression is 9.50 × d × h or 9.50dh. 7. e. To make this problem easier, assume the initial cost of the jacket was $100. The first markdown of 20% would save you $20, bringing the cost of the jacket to $80. For the second markdown, you should be finding 20% of $80, the new cost of the jacket. 20% of 80 = 0.20 × 80 = 16. If you save $16 the sec- ond time, the final cost of the jacket is 80 – 16 = $64. Since the initial cost was $100, $64 is 64% of this price. 8. d. First calculate the number of letters completed by 30 typists in 20 minutes. Let x = the number of letters typed by 30 typists and set up the proportion . Cross-multiply to get 20x = 1,440. Divide both sides by 20 and get x = 72. Since 20 minutes is one-third of an hour, multiply 72 × 3 = 216 to get the total letters for one hour. 9. d. This problem can be solved by using the simple interest formula: interest = principal × rate × time. Remember to change the interest rate to a decimal before using it in the formula. I = (1,000)(0.05)(2) = $100. Since $100 was made in interest, the total in the bank account is $1,000 + $100 = $1,100. 10. a. Using the rules for exponents, choice a simplifies to 2 5 and choices b, c, and d simplify to 2 6 = 64. Choice e becomes 27 × 81, which is obviously much larger than 64. typists letters ϭ 20 48 ϭ 30 x – QUANTITATIVE PRACTICE TEST– 387 11. d. Let x = the number of liters of the 40% solution. Use the equation 0.40x + 0.20(35) = 0.35 (x + 35) to show the two amounts mixed equal the 35% solution. Solve the equation: 0.40x + 0.20(35) = 0.35(x + 35) Multiply both sides by 100 in order to work with more compatible numbers: 40x + 20(35) = 35(x + 35) 40x + 700 = 35x + 1,225 Subtract 700 on both sides: 40x + 700 – 700 = 35x + 1,225 – 700 Subtract 35x from both sides 40x – 35x = 35x – 35x + 525 Divide both sides by 5: x = 105 liters of 35% iodine solution 12. b. Let x = the part of the floor that can be tiled in 1 hour. Since Steve can tile a floor in 6 hours, he can tile of the floor in 1 hour. Since Cheryl can tile the same floor in 4 hours, she can tile of the floor in 1 hour. Use the equation , where represents the part of the floor they can tile in an hour together. Multiply each term by the LCD = 12x. . The equation simplifies to 2x + 3x = 12. 5x = 12. Divide each side by 5 to get hours. Since 0.4 times 60 minutes equals 24 minutes, the final answer is 2 hours 24 minutes. 13. a. The length of one side of a square is equal to the square root of the area of the square. Since the area of the squares is 9, 16, and 25, the lengths of the sides of the squares are 3, 4, and 5, respectively. The triangle is formed by the sides of the three squares; therefore, the perimeter, or distance around the tri- angle, is 3 + 4 + 5 = 12. 14. c. Suppose that the shoes cost $10. $10 – 10% = 10 – 1 = $9. When the shoes are marked back up, 10% of $9 is only 90 cents. Therefore, the markup must be greater than 10%. = , or about 11%. 15. b. Note that the figure is not drawn to scale, so do not rely on the diagram to calculate the answer. Since the angles are adjacent and formed by two intersecting lines, they are also supplementary. Com- bine the two angles and set the sum equal to 180. 2x + 3x – 40 = 180. Combine like terms and add 40 to both sides. 5x – 40 + 40 = 180 + 40. 5x = 220. Divide both sides by 5 to get x = 44. Then 2x = 88 and 3x – 40 = 92. The smaller angle is 88. 16. b. x, x + 2, and x + 4 are each two numbers apart. This would make x + 2 the average of the three numbers. If x + 2 = 33, then x = 31. 17. d. It costs d for the first 100 posters plus the cost of 25 additional posters. This translates to d + 25e, since e is the cost of each poster over the initial 100. 18. d. If the volume of the cube is x 3 , then one edge of the cube is x. The surface area of a cube is six times the area of one face, which is x times x. The total surface area is 6x 2 . 19. c. The next larger multiple of two would be x – 3 + 2, which is x – 1. In this case, remember that any even number is a multiple of two and all evens are two numbers apart. If x – 3 is a multiple of two, you can assume that it is also an even number. This number plus two would also produce an even number. 11 1 9 % $1 $9 x ϭ 12 5 ϭ 2.4 12x × 1 6 ϩ 12x × 1 4 ϭ 12x × 1 x 1 x 1 6 ϩ 1 4 ϭ 1 x 1 4 1 6 5x 5 ϭ 525 5 – QUANTITATIVE PRACTICE TEST– 388 . Suppose that the shoes cost $10. $10 – 10% = 10 – 1 = $9. When the shoes are marked back up, 10% of $9 is only 90 cents. Therefore, the markup must be greater than 10% . = , or about 11%. 15. b using it in the formula. I = (1,000)(0.05)(2) = $100 . Since $100 was made in interest, the total in the bank account is $1,000 + $100 = $1 ,100 . 10. a. Using the rules for exponents, choice a simplifies. 31. 17. d. It costs d for the first 100 posters plus the cost of 25 additional posters. This translates to d + 25e, since e is the cost of each poster over the initial 100 . 18. d. If the volume of the