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25. X and Y are both double digit numbers. If in each, the tens digit is greater by 3 than the tens digit, what is the greatest possible difference between X and Y? (A) 33 (B) 55 (C) 66 (D) 77 (E) 82 The best answer is C. Let’s pick the largest and the smallest numbers: 96 and 30. The difference between them is therefore (96 – 30 = 66) and so C is the right answer. 1. If 2A + 4B = 15 and A = -B, then 3B = (A) 15 (B) 17.5 (C) 22.5 (D) 25 (E) 27.5 The best answer is C. Replace A with –B: -2B + 4B = 15 2B = 15 B = 7.5. And therefore 3B = 7.5 x 3 = 22.5. 2. If (X + Y) 2 = Z + 2, X 2 + Y 2 = 10 and XY = 3, what is the value of Z? (A) 10 (B) 12 (C) 14 (D) 16 (E) 18 The best answer is C. (X + Y) 2 = X 2 + 2XY + Y 2 = 10 + 2 x 3 = Z + 2 16 = Z + 2 Z = 14. 3. If 22 2 a Z Y X and X 2 =2Y 2 , which of the following can be a value of a? (A) 2 Z (B) 2 Z (C) Z 2 (D) Z 2 (E) 2Z The best answer is A. X 2 =2Y 2 X 2 /Y 2 = 2. X 2 /Y 2 = 2 = Z/a 2 Z = 2a 2 2 Z a and so A is the right answer. 4. If every digit of a whole number is either a 4 or a 6, the number must be (A) divisible by 2 (B) divisible by 4 (C) divisible by 6 (D) divisible by 24 (E) odd The best answer is A. An even number is a number where the units’ digit is even. Since both 4 and 6 are even, the units’ digit must be even and the number is divisible by 2 and so A is the answer. 5. If 3X – 7 = 68, then 86 – 2X = (A) 24 (B) 36 (C) 48 (D) 52 (E) 65 The best answer is B. Solve: 3X – 7 = 68 3X = 75 X = 25. 86 – 2X = 86 – 50 = 36 and so B is the right answer. 6. Which of the following two lines are parallel? a b c d e (A) e and a (B) c and d (C) b and e (D) d and b (E) a and c The best answer is D. Two parallel lines have the same slope. The only two lines with the same slope among all the lines are b and d and therefore D is the right answer. 7. In which of the following answers, the fraction is equal to the number? (A) 4/5, 0.45 (B) 5/6, 0.825 (C) 1/5, 0.1 (D) 7/8, 0.875 (E) 8/9, 0.9 The best answer is D. The only answer where the fraction is equal to the number on the right is D. 1/8 is 0.125 and so 7/8 is equal to 0.875. 8. Steve buys a jeans and a t-shirt for $64.25. If the jeans cost $15.75 more than the t-shirt, how much does the t-shirt cost? (A) $14.5 (B) $17 (C) $21 (D) $23 (E) $24.5 The best answer is E. Let X be the cost of the t-shirt. The jeans cost X + 15.75. The jeans and the t-shirt both cost 2X + 15.75 = 64.25 2X = 64.25 – 15.75 = 48.5 X= $24.5. 9. Daniel bought a camera with 3 films for $104. If the camera costs 10 times more then a film, how much will 2 cameras and 2 films cost? (A) $56 (B) $112 (C) $158 (D) $176 (E) $182 The best answer is E. Let X be the price of a film and so the price of a camera is 10X. 10X + 3X = 96 13X = 104 X=$8 and a camera costs $80. 2 cameras with 2 films will cost 2(80) + 2(8) = $176. 10. When an integer X is divided by a prime number Y, the result is always a fraction. Which of the following proves that the statement above is not true? (A) X=23, Y=3 (B) X=14, Y=5 (C) X=19, Y=2 (D) X=21, Y=7 (E) X=29, Y=11 The best answer is D. X=21, which is an integer and Y=7, which is a prime number. X/Y = 21/7 = 3, which is not a fraction and therefore it proves that the statement is not definitely true. 11. In the figure above, there are three shaded squares blocked inside a larger square. What is the ratio between the shaded area to the area of the square ABCD? (A) 9:25 (B) 1:2 (C) 3:5 (D) 3:8 (E) 4:9 The best answer is A. The area of the shaded region is the sum of the areas of three squares: 2 x 2 + 1 x 1 + 2 x 2 = 9. The area of the rectangle ABCD is (5) x (5) = 25. The required ratio is therefore 9:25. 12. In which of the following sets of numbers is the sum of the numbers greater than their product? A B C D 2 2 1 1 2 2 (A) {1, 2, 3, 4} (B) {2, -2, 3, -3} (C) {1/2, 2, 3, 1/3} (D) {0, 1, -4, 3} (E) {12, -6, -2, 13} The best answer is C. Go over the sets of numbers and look for one set in which the sum is greater than the product. In the third set, the sum is greater than 5 yet the product is 0.5 x 2 x 3 x 1/3 = 1 and therefore this is the right answer. 13. In a local pool there are 5000 liters. Water is being poured into the pool at a rate of X liters per second and also, because of a leak, water is leaking out at a rate of Y liters per minute. After Z seconds, how many liters are in the pool? (A) 5,000 + Z(X – 60Y) (B) 5,000 – ZX +Y/60 (C) Z/X – Y/60 + 5,000 (D) Z(X – Y) + 5,000 (E) Z(X – Y/60) + 5,000 The best answer is E. In the beginning there are 5,000 liters in the pool. Every second, X liters come in and Y/60 liters come out. And therefore after Z seconds there are 5,000 + ZX – ZY/60 liters in the pool, or Z(X – Y/60) + 5,000 and so E is the right answer. 14. 55 percent of 80 are the same as 25 percent of which of the following numbers? (A) 176 (B) 180 (C) 184 (D) 188 (E) 192 The best answer is A. 55 percent of 80 are 50% of 80 + 5% of 80, which is 40 + 4 = 44. In order to find the number, divide 44 by 0.25 to get 44 x 4 = 176 and therefore A is the right answer. 15. o 34 o 43 A C B In the triangle above, which of the following must be true? (A) AC > AB (B) AB < CB (C) AB > AC (D) AC > AB (E) AC = CB The best answer is C. The third angle of the triangle is therefore 180 – 34 – 43 = 103 o . A side in front of a larger angle is larger and therefore AB > AC > CB. The only answer that is true is C. 16. Let @X be defined as (2X – 8)/(3X). If @A = -2, which of the following values can replace A? (A) -2 (B) -1 (C) 1 (D) 2 (E) 4 The best answer is C. Try out the answers to figure this one out. Try A=1: @1 = (2 – 8)/3 = -2 and so C is the right answer. 17. A new action was defined for every X: [X] = the largest possible integer, that is small or equal to X. For example: [3.6] = 3. What is the value of [6.4] x [1/4] + [2] x [7.178] ? (A) 14 (B) 16 (C) 18 (D) 20 (E) 22 The best answer is A. The expression is also equal to 6 x 0 + 2 x 7 = 14 and so A is the answer. 18. Let ~(A, B)~ be defined as the remainder of the partition A/B. What is the value of ~(163, 7)~ ? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 The best answer is B. 163 divided by 7 is 23 with a remainder of 2. The answer to the question is the remainder and so B is the right answer. 19. If 2X = 3Y, 2Y = 3Z, 2Z = 3R and 2R = 3S, what is the value of X/S? (A) 16/81 (B) 3/13 (C) 13/3 (D) 6/15 (E) 81/16 The best answer is E. X = 3Y/2 = 1.5Y = 1.5(1.5Z) = 1.5(1.5(1.5R) = 1.5(1.5(1.5(1.5S))) = S(3/2) 4 = S(81/16). X/S = (81/16) and therefore E is the right answer. 20. Which of the following is equal to the sum of two consecutive odd integers? (A) 345 (B) 361 (C) 382 (D) 396 (E) 411 The best answer is D. Let X be one of the integers and X + 2 the other. We know that X + X + 2 = 2X + 2 is equal to one of the answers and so subtract 2 from the answer, divide by two and see if you get an odd integer. All the answers except for C and D are disqualified since they’re odd. Check C: 382 – 2 = 380 380/2 = 190, which is even Check D: 396 – 2 = 394 394/2 = 197, which is odd and this is the answer. 21. If X is even and Y is odd, which of the following must be an integer? (A) XY/2 (B) (X + Y)/2 (C) XY/3 (D) (X + Y)/3 (E) (X + Y)/XY The best answer is A. In these types of questions, the easiest way is to pick up numbers. Let X=2 and =1. Answer A is an integer since XY/2 = 2/2 = 1 and so this is the right answer. 22. E is an even number. To which of the following E 3 doesn’t necessarily divide to with out a remainder? (A) 2 (B) E (C) E 3 (D) 8 (E) 16 The best answer is E. Every even number in the power of 3 is divisible by 1, itself, itself in the power of 3 and 8 and therefore E is the only answer left. Try any number in order to validate this principle. 23. If the first angle in a triangle is equal to two thirds of the second angle and the second angle is equal to one fifth of the third angle, what is the measurement of the median angle in the triangle? (A) 18 (B) 27 (C) 26 (D) 84 (E) 135 The best answer is B. Let the median angle (the one we are looking for) be X. We know that the first angle is 2X/3 and that the third angle is 5X. The sum of the angles in the triangle is equal to 180 o and therefore X + 2X/3 + 5X = 180 20X/3 = 180 X = 27 o and so the right answer is B. 24. It takes Gloria 7.5 hours to knit 6 sweaters. If Cynthia can knit three times faster, how many minutes would it take her to knit 3 sweaters? (A) 15 (B) 25 (C) 50 (D) 75 (E) 100 The best answer is D. Cynthia can knit three times faster and therefore it takes her 2.5 hours to knit 6 sweaters and half that time to knit 3 sweaters. Half of 2.5 hours is one hour and a quarter, which is 75 minutes. 25. Three carpenters, working at a constant rate, can build a cabin in 7 hours. How many hours would it take 7 carpenters, working at a constant rate, to built the same cabin? (A) 2 (B) 3 (C) 4.5 (D) 6 3 1 (E) 7 The best answer is B. Instead of three carpenters, now we have 7, which is 2.5 times the manpower. If there are 2 and a third times the workers it would cut the time by 2 and a third also and therefore it would take 7/ 3 1 2 = 3 hours to built the same cabin and B is the right answer. 1. (7 + 11) 2 = (A) (7 x 11)(7 + 11) (B) (11 – 7)(11 + 7) (C) 4 2 (D) 10 2 + 8 2 (E) 324 The best answer is E. First add the numbers in the parenthesis and only then relate to the power. (7 + 11) 2 = 18 2 = 324 and so E is the right answer. 2. (17 + 19) = (A) (19 – 17)(17 – 19) (B) 46 (C) 1917 1917 22 (D) 6 2 /2 (E) 79 – 11 The best answer is C. Answer C can also be written as )1917( )1917( )1917)(1917( 1917 1917 22 . 3. The average (arithmetic mean) of 4 numbers is 100. If three of the numbers are 20, 60 and 80, what is the fourth number? (A) 120 (B) 180 (C) 220 (D) 240 (E) 360 The best answer is D. Let the fourth number be X. Use the average formula: 100 4 608020 X X = 400 – 20 – 80 – 60 = 240 and so D is the right answer. 4. If the average (arithmetic mean) of 7 numbers is 343, what is sum? (A) 49 (B) 187 (C) 343 (D) 1520 (E) 2401 The best answer is E. The sum of the numbers is actually the average multiplied by the amount of numbers, thus 343 x 7 = 2401 and so the right answer is E. 5. If 6X + 9Y = 23, what is the value of (15X + 22.5Y)? (A) 57.5 (B) 54.5 (C) 49 (D) 37 (E) 31.5 The best answer is A. Take 6X + 9Y = 23 and multiply it by 2.5: (6 x 2.5)X + (9 x 2.5)Y = 23 x 2.5 15X + 22.5Y = 57.5 and so the right answer is A. [...]... dollars, of 42 0 markers? (A) 140 N (B) 14N (C) 1.4N (D) 140 /N (E) N/1 .4 The best answer is C The cost of 3 markers is N cents 42 0/3 = 140 and so it will cost 140 N cents to buy 42 0 markers 140 N = 1.4N dollars and so C is the right answer 14 If 7 6 13 , what is the value of 3Y? Y 7 7 (A) (B) (C) (D) (E) 35 28 21 14 6 The best answer is C Multiply the expression by 7Y: 49 + 6Y = 13Y 7Y = 49 Y=7 The... are produced in 7 hours and so one propeller is produced in 7 /4 hours, which is 7 x 60 /4 = 105 minutes and so C is the right answer 6 If A = 7, what is the value of A2(B – 3A) in terms of B? (A) 1029 – 49 B (B) 49 A – 1029 (C) 21B – 1029 (D) 1029B – 49 (E) 49 B – 1029 The best answer is E If A = 7, A2(B – 3A) = 49 (B – 21) = 49 B – 49 x 21 = 49 B – 1029 and so X 3 E is the right answer A B C 7 In the figure... times is 2.25 fit inside 12 12/2.5 = 4 and a remainder and therefore D is the right answer 7 It takes Adam one hour and 40 minutes to put together one puzzle How many complete puzzles can Adam put together in 14 hours? (A) 9 (B) 8 (C) 7 (D) 6 (E) 5 The best answer is B 2 hours 3 14 14 14 3 42 2 2 Divide 14 by 1 to get: 8 2 5 5 5 5 3 1 3 3 One hour and 40 minutes is 1 And so Adam will... According to the question: A = 0.15D, B = 0.22D and C = 0 .41 D The value of 2A + B + C is 2(0.15D) + 0.22D + 0 .41 D = 0.93D and so B is the right answer 9 If X is 40 % of Y and Y is 20% of Z, what is Z in terms of X? (A) 0.08X (B) 1.25X (C) 3.5X (D) 12.5X (E) 15X The best answer is D According to the question: X = 0.4Y and Y = 0.2Z X = 0.4Y = 0 .4( 0.2Z) = 0.08Z Z = 1/0.08 x X = 12.5X and so D is the right... probability of 2/9 and so C is the right answer 14 If O is an odd number between 1 and 14 inclusive, what is the probability that O would be smaller than 8? (A) 2/9 (B) ½ (C) 1/3 (D) 4/ 7 (E) 3/7 The best answer is D Enumerate all the odd numbers between 1 and 14: 1, 3, 5, 7, 9, 11 and 13 Out of these 7 numbers, only 4 are smaller than 8 and therefore the probability is 4/ 7 15 A is divisible by 3 If B = 2A +... consists of 4 isosceles triangles? (A) One (B) Two (C) Three (D) Four (E) Five The best answer is B By simply adding two diagonals to the square, it will divide into 4 identical triangles, each is an isosceles and so the right answer is B 4 A C.N.C machine produces 120 parts in 4 hours At this rate, in how many minutes can the C.N.C machine produce 15 parts? (A) 0.25 (B) 0.5 (C) 15 (D) 30 (E) 45 The best... The value of 3Y is therefore 21 15 What percent of 150 are 12? (A) 4. 5% (B) 8% (C) 6% (D) 12% (E) 14% The best answer is B Use the answers Try answer B: 8 8 15 120 150 12 and so the right answer is B 100 10 10 16 If A (4, 3), B(X, 3) and C(5, X) are three vertices of a triangle, what is its area? (A) 24 (B) X 2 + 12 (C) 24 + X 2 (D) X 2 (E) 0.5X 2 – 3.5X + 6 The best answer is E Draw an axis... 6 The value of answer is B 1 3 Y 2 is 1 3 3 8 2 since 2 = 8 and therefore the right 2 If 4X + 5 = 3X + 17, what is the value of X/Y assuming that Y is a quarter of X? (A) 2 (B) (C) (D) (E) 3 4 5 6 The best answer is C Solve: 4X + 5 = 3X + 12 X = 12 Y is a quarter of X, which is ¼ x 12 = 3 X/Y = 12/3 = 4 and so the right answer is C A B D 3 C The figure above is of a square What is the least number... following is even? (A) A + 4 (B) A2 + 2 (C) (A – 1)2 (D) 2A + 1 (E) A – 2 The best answer is C Take A = 5 for example Only answer C is even, (5 – 1)2 = 42 = 16 and therefore the right answer is C 12 3X, 4X and 5X are three angles in a triangle, what is the value of X? (A) 8 (B) 12 (C) 15 (D) 18 (E) 22 The best answer is C The sum of all the angles in the triangle is 180o 3X + 4X + 5X = 180 o 12X =... produces 120 parts in 4 hours and so it produces 120 /4 = 30 parts in one hour 15 parts are half of 30 and therefore it will take the machine 60/2 = 30 minutes to produce this amount of parts 5 A propeller lathe factory produces 4 propellers in 7 hours At this rate, in how many minutes can the factory produce one part? (A) (B) (C) (D) (E) 15 75 105 120 150 The best answer is C 4 propellers are produced . of 80 + 5% of 80, which is 40 + 4 = 44 . In order to find the number, divide 44 by 0.25 to get 44 x 4 = 176 and therefore A is the right answer. 15. o 34 o 43 A C B In the triangle. (E) N/1 .4 The best answer is C. The cost of 3 markers is N cents. 42 0/3 = 140 and so it will cost 140 N cents to buy 42 0 markers. 140 N = 1.4N dollars and so C is the right answer. 14. If. 100 4 608020 X X = 40 0 – 20 – 80 – 60 = 240 and so D is the right answer. 4. If the average (arithmetic mean) of 7 numbers is 343 , what is sum? (A) 49 (B) 187 (C) 343 (D)