problems practice phần 8 docx

a) Less than 32 cents. b) Less than 48 cents. c) Less than 96 cents. d) More than 48 cents. e) More than 96 cents. 24. A, B and C are three sequential integers that follow the rules: ABC < 0 and A < B < C. Which of the following might be true? a) A > -3. b) B/C 2 < A – B. c) B 2 A > C 2 (B-A) . d) AB < 0. e) AB/2C > 1. 25. If X-Y is positive, X and Y have the same sign. Which of the following must be true? a) X 2 > Y 2 . b) Y 2 > XY. c) 1/Y > 1/X. d) 1/Y 2 > 1/X 2 . e) XY > 0. 26. Z is a negative, even number. Which of the following is the greatest? fix a) Z z-1 . b) Z 2|z-1| . c) Z 2(z-1) d) Z z . e) 3Z z . 27. If -1 < X < 0, which of the following is the smallest? a) X 3 . b) -X 3 . c) 1/X. d) (1/X -3 ) -3 . e) (1/X -4 ) -3 . 28. A > 0.5 and B < 0.25. Which of the following must be true? a) A 2 > B 2 . b) A + B 2 < 0.75. c) A/B > 1. d) A 2 B 2 < 1. e) 2B < A. 29. Which of the following is equivalent to |X| + |X+3| < 5? a) -5 < X < 3 b) -4 < X < 1 c) X < -3 d) X < 1 e) X > -1 30. If X 2 < 1/Y, which of the following might be true? a) If X>0 then 1/X>Y. b) If X<0 then 1/X>Y. c) If Y<0 then X>0. d) If Y<0 then X=0. e) If X=Y then Y>1. 31. 1/3 < A < 6/7 and 0.5 < B < 8/5. Which of the following might be AB? a) 1 ½. b) 1 2/5. c) ½. d) 1/6. e) 1/8. 32. If |X| > 1. Which of the following is the greatest? a) (1/X) 4 . b) (1/X) -2 . c) 1/X – X 2 . d) X -6 (X -3 /X 4 ) -0.5 –X. e) |X|. 33. A ship can carry a cargo of 25.5 tons, and have a room for no more than 17 containers. What is the weight of each container, if all the containers are equal? a) Less than 1.5 tons. b) Less than 2 tons. c) Exactly 2.5 tons. d) More than 1.5 tons. e) More than 2 tons. 34. The phrase 1-(A -1 B) is negative only if: a) A < 0 and B < 0. b) B > A and AB > 0. c) B -1 < A and AB > 0. d) A+B > 1. e) AB > 1. 35. Which of the following fit the rule: XX  209 ? a) X < 0. b) X > 0. c) X > 5. d) X = 0. e) X < 20/9. 36. If A 3 > B 3 , what must be true? a) A 2 > B 2 . b) AB > 0. c) A 2 > B 3 . d) B 3 > A 2 . e) A > B. 37. The phrase (A+B) 2 ≥ 2AB is true only for: a) AB > 0. b) AB < 0. c) A+B > 0. d) Any A and B. e) Never. 38. If 1/X < 1 and 1/Y > -1, which of the following must be true when Y is an integer? a) Y 2 – X -1 > 0. b) Y 2 < X 2 . c) Y 3 + X 3 > 0. d) 1/(XY) > 1/X + 1/Y. e) |Y| < |X|. 39. The lottery price is greater than 10 million dollars and smaller than 35 million dollars, and is divided to 2 people or more, but no more than 10 people. What is the sum of money that each one of them gets? a) More than 10 million, less than 35 million. b) More than 100,000, less than 3.5 million. c) More than 1 million, less than 17.5 million. d) More than 1 million, less than 3.5 million. e) More than 3.5 million, less than 5 million. 40. If AB < 0 and A + B = 4/6. Which of the following must be true? a) A 2 > B 2 . b) |6A| > 4 -|6B| c) |A| -1 > |B| -2/6 d) A > B e) B > A 2 Explanations: 1. The best answer is A. After opening the brackets we get: X 2 -8X+16X <X 2 -4X+4 2. The best answer is C. In each answer, put the end points (including infinities) instead of X. In A, B and D, on of the end points is infinity, and we can easily see that infinity is a wrong answer. 3. The best answer is B. A 3 is a negative number, but B 2 is a positive number. Both |B| and |A 3 | are positive numbers. 4. The best answer is D. If -X 2 < -1 then X 2 > 1, and therefore: X > 1 or X < -1. If other words: |X| > 1. 5. The best answer is E. Since A 2 is always positive, B must be positive too (A 2 B > 0). 6. The best answer is A. Answers B to D contain infinities. Putting infinity instead of X wont give us a right answer. E is wrong since we can easily see that putting 2 instead of X is a right answer. 7. The best answer is C. If B 3 A 2 is negative, B must be negative (A can be negative or positive), Therefore –B is greater than B. 8. The best answer is E. Y can be ANY number that is smaller than 11, to minus infinity, And X can be either negative of positive. The multiplication of them can be any number. 9. The best answer is E. Since each of the numbers can be either negative of positive, and still follow the rule A < B < C, it’s impossible to conclude anything about their relationship. 10. The best answer is C. If both of the numbers are negative and A < B, the absolute value of A is GREATER than the absolute value of B, therefore the division A/B is a positive value that is GREATER than 1. 11. The best answer is E. It’s easy to see that 5 2 – 20 = 5, and that doesn’t follow the rule. 12. The best answer is D. The value of Y 2 – X is negative, therefore cannot be under a square root. There’s no need to check anything else. 13. The best answer is B. The given phrase is equivalent to: -(3 B /3 A ). The phrase inside of the brackets is greater than 1, and therefore the phrase is smaller than –1. 14. The best answer is C. Conclude that B is greater than A, and their both positive. The terms in A and B are always positive. A/B is smaller than 1, therefore cannot be greater than (B/A) 2 which is greater than 1. 15. The best answer is D. D is the only negative number among the answers. Y 12 is positive, –(X-Y) is negative and X 5 is positive. 16. The best answer is E. E is the only answer that gives us the possibility of a positive value. Almost any positive number instead of X we’ll give us a negative phrase under the square root. 17. The best answer is C. A 4 /B is always a negative number, and so is A/2. The sum of them is a negative number too. 18. The best answer is B. B 2 is a positive number between 0 and 1. A is a negative number which is smaller than –1. The sum of these two is always negative. 19. The best answer is B. There is no specific relationship between A and B, therefore you cannot determine which of them is greater. 20. The best answer is A. The given phrase is equivalent to 4A 2 + 0.6 > B, and if B > 0 we can power the phrase by 2, and get answer A. 21. The best answer is C. The left side is always a negative number, and cannot be greater than a positive number. 22. The best answer is B. B is the only answer that includes 3, which cannot be put instead of X (division by zero). 23. The best answer is C. 2 times 32 are 64. The total price is 32+64 cents or less, which is less than 96 cents. 24. The best answer is B. From the three rules we know that A, B and C are negative numbers, and A must be – 3 or less (otherwise the multiplication will be 0). B/C 2 is a negative phrase that must be smaller than A – B, which is a positive phrase. 25. The best answer is C. If X is greater than Y, The relationship between the opposite numbers turns upside down, weather they are either both negative or positive. 26. The best answer is B. The phrase 2|Z-1| is a positive, even number, therefore the value of the whole phrase must be a positive number, larger than 1. 27. The best answer is D. The phrase is equivalent to (1/X) 9 , which is smaller than –1, and is certainly smaller than 1/X. 28. The best answer is E. No matter if B is negative of positive, 2B is always smaller than 0.5 and therefore – smaller than A. 29. The best answer is B. We can eliminate answers C, D and E, that include infinities. Putting the endpoints of A won’t give us the right answer. 30. The best answer is A. C and D are impossible answers since X 2 (positive) cannot be greater than 1/Y (negative). E is an impossible since a number greater than 1 cannot be smaller than it’s opposite. B cannot be true since a negative number (X) cannot be greater than a positive one (Y). 31. The best answer is C. A can be ½ and B can be 1. 32. The best answer is B. (1/X) -2 is equivalent to X 2 which is a positive number, greater than 1. 33. The best answer is B. 17 times 1.5 is exactly 25.5, and 17 is the MAXIMUM number of containers. 34. The best answer is B. If B>A then the phrase B/A is greater than 1. A and B must have the same sign. 35. The best answer is C. The phrase under the square root mustn’t be negative, therefore X must be greater than 20/9. 36. The best answer is E. There is no info about the signs, the best thing we can do is put a triple root on both sides. 37. The best answer is D. A 2 + 2AB + B 2 is always greater or equal to 2AB. 38. The best answer is A. The absolute value of Y and X is greater than 1,therefore a positive integer (Y 2 ) is greater than a fraction (X -1 ). 39. The best answer is C. Take the greatest sum of money (35) and divide it by the smaller number of people (2) – you get the maximum amount (17.5). Take the smallest sum of money (10) and divide it by the smaller number of people (10) – you get the minimum amount (1). 40. The best answer is B. Changing the structure of the phrase gives us: |A| + |Y| > 4/6. That must always be true if A and B have different signs. 1. X is an even number, which of the following is odd? a) X 2 . b) (X +1) 2 c) (X+2) 2 d) X 3 + X e) 2X 2 2. X is an integer, which of the following must be even? a) X (X +2) + 2 b) X (X +1) +1 c) X (X + 1) d) X 2 e) X 3 +1 3. X, Y, Z, and w are integers. The expression X-Y-Z is even and the Expression Y-Z-W is odd. If X is even what must be true? a) Y-Z must be odd. b) W must be even. c) W must be odd. d) Z must be even. e) Z must be odd. 4. X is an even number and Y is a positive odd number. Which of the following expressions cannot be even? a) (XY) Y b) X 3 Y 3 c) X 3 d) XY e) Y 2 5. X is a prime number bigger than 10. Also, Y = X+X 3 +X 5 +X 7. What is definitely true about Y? a) Y is a prime number. b) Y is odd. c) Y is even. d) Y is divisible by 3. e) Y is divisible by 7. 6. P is divisible by 4; Q is divisible by 3. Which of the following is definitely odd? a) Q (P+1) b) 2P+3Q c) PQ 2 d) P 2 Q 3 e) None of the answers. 7. A and B are integers. If 2A-B = B-A, than which of the following is true? a) B<A b) A<B c) A and B are even. d) A is even and B is either even or odd. e) B is even. 8. A is even and B is odd. Which of the following expressions cannot be an integer? a) 1 1   B A b) B A 1  c) 2 1 A B  d) 64 A e) B A A  4 9. T and S are prime numbers larger than 2. Which of the following expressions cannot an integer? a) 3 TS  b) 2 TS  c) 2 TS  d) 11 TS  e) 4 2 TS  10. A and B are integers. The expression (A+1)(B+1) is even. What can be said about A and B? a) They are both even numbers. b) At least one of them is even. c) At least one of them is odd. d) They are both odd. e) Nothing can be said surly on A and B. 11. In orange county one fifth of the people are gathering mushrooms and one seventh of the people are collecting apples. What can be the number of people in Orange County? a) 60. b) 42. c) 85. d) 140. e) 252. [...]... digit numbers) a) b) c) d) e) 35 If 170 180 173 1 98 146 2  32  a  a , what is a? a) b) c) d) e) 64 16 8 4 2 36 If 1/3 < A < 2, which of the following expressions can have the largest value? a) b) c) d) e) A2/3 A A/2+1/3 A + 1/3 A3-4 37 What is the smallest possible common multiple of three integers, each larger than 26? a) b) c) d) e) 27 54 84 6 19,656 21,924 38 What is the smallest possible common... village? a) b) c) d) e) 792 540 1 98 132 346 13 Danny can divide his herd into 5 equal parts and also to 6 equal parts, but not to 9 equal parts How many cows can Danny have in his herd? a) b) c) d) e) 155 336 180 120 456 14 The number of bunnies in Peter’s yard increases by 4 times every week How many weeks will it take for the number of bunnies to be equally divided by 8 assuming that he started out... and 6 but not by nine, the answer is D 14 The best answer is B In the beginning he has 3, after one week he has12 and after the second week he has 48 and that can be equally divided by 8 15 The best answer is B From the given expression we learn that X /8 = 25 x 7, and so it cannot be divided equally by 2 16 The best answer is B The best way is to place numbers (X+Y) = 19 and so 19/5 gives a remainder... and Q The smallest number therefore is 3 18 The best answer is D Factorize C: C = A x A x A Therefore C can be equally divided into 1, A, A2, and A3 =C 4 numbers all together 19 The best answer is D Let’s try A = 22 In expression D: 22-10=12 12/11 = 1 with a remainder of 1 20 The best answer is B If she bought 2 boxes (22 eggs) than she has 48 eggs left 48 eggs are divided equally into eight boxes... 26? a) b) c) d) e) 27 54 84 6 19,656 21,924 38 What is the smallest possible common multiple of two integers, both bigger than 260? a) b) c) d) e) 261 262 524 12,542 18, 244 39 What is the units’ digit of (9)7(17)3(3)3 ? a) b) c) d) e) 3 5 7 8 9 40 If a, b, c and d are consecutive integers (a . (CBA and BCA are 3 digit numbers). a) 170. b) 180 . c) 173. d) 1 98. e) 146. 35. If aa  322 , what is a? a) 64. b) 16. c) 8. d) 4. e) 2. 36. If 1/3 < A < 2, which. b) 54. c) 84 6. d) 19,656. e) 21,924. 38. What is the smallest possible common multiple of two integers, both bigger than 260? a) 261. b) 262. c) 524. d) 12,542. e) 18, 244. . has12 and after the second week he has 48 and that can be equally divided by 8. 15. The best answer is B. From the given expression we learn that X /8 = 25 x 7, and so it cannot be divided