24HourAnswers.com. Online Homework. Focused Exercises for Math SAT. Skill Set 30: Miscellaneous Problems

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1 24HourAnswers.com Online Homework Focused Exercises for Math SAT Skill Set 30: Miscellaneous Problems Many of the problems in this exercise set came from The College Board, writers of the SAT exam.

2 1. If each of a man's three sons works 1 8f as fast as he does, and the man does a job in 3 hours, how many hours does it take his sons working together to do the job? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 2. If k is a positive even integer, then (k + 1)(k + 2) could equal which of the following? (A) 10 (B) 20 (C) 30 (D) 40 (E) If a and x represent real numbers for which x a, which of the following statements could be true? I. a > 0 II. a = 0 III. a < 0 (A) None (B) II only (C) III only (D) I and II only (E) II and III only

3 4. If x is an integer greater than 1 and if y x 1 xf, which of the following must be true? I. y x II. y is an integer. III. xy > x 2 (A) I only (B) III only (C) I and II only (D) I and III only (E) I, II, and III 5. If a and b are positive integers and a b 2 7, what is the value of a? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7

4 6. If the rectangular box with no lid shown above is cut along the vertical edges and flattened, which of the following figures best represents the result?

5 7. In the figure above, a path from point A to point D is determined by moving upward or to the right along the grid lines. How many different paths can be drawn from A to D that do not include either B or C? (A) Two (B) Four (C) Six (D) Eight (E) Sixteen 8. A certain triangle has two angles that have the same measure. If the lengths of two of the sides of the triangle are 50 and 30, what is the least possible value for the perimeter of the triangle? 9. There is the same number of boys and girls on a school bus when it departs from school. At the first stop, 4 boys get off the bus and nobody gets on. After the first stop, there are twice as many girls as boys on the bus. How many girls are on the bus? (A) 4 (B) 6 (C) 8 (D) 12 (E) Stacey noted that she is both the 12th tallest and the 12th shortest student in her class. If everyone in the class is of a different height, how many students are in the class? (A) 22 (B) 23 (C) 24 (D) 25 (E) 34

6 11. Edna and Nancy leave the house of a common friend at the same time and walk for 4 hours. Edna walks due east at the average rate of 4 kilometers per hour and Nancy walks due north at the average rate of 3 kilometers per hour. What is the straight-line distance between them, in kilometers, at the end of 4 hours? (A) 4 (B) 5 (C) 12 (D) 16 (E) Z W Y X The sequence above may be changed in either of two ways. Either two adjacent letters may be interchanged or the entire sequence may be reversed. What is the least number of such changes needed to put the letters into alphabetical order from left to right? (A) 2 (B) 3 (C) 4 (D) 5 (E) Let S be the set of all integers that can be written as n 2 1, where n is a nonzero integer. Which of the following integers is in S? (A) 16 (B) 28 (C) 35 (D) 39 (E) A "simple square" is any integer greater than 1 that has only three positive integer factors - itself, its square root, and 1. Which of the following is a simple square? (A) 121 (B) 100 (C) 81 (D) 64 (E) 33

7 15. What is the greatest three-digit integer that has a factor of 10? 16. A recipe for chili for 20 people requires 4 pounds of beans. At this rate, how many pounds of beans are required to make chili for 150 people? 17. NOTE: Figure not drawn to scale. In the figure above, ABCD is a rectangle. Points A and C lie on the graph 3 of y px, where p is a constant. If the area of ABCD is 4, what is the value of p? 18. The number that results when an integer is multiplied by itself CANNOT end in which of the following digits? (A) 1 (B) 4 (C) 5 (D) 6 (E) 8

8 19. According to a certain recipe, 25 pounds of flour are needed to make 300 rolls. At this rate, how many pounds of flour are needed to make 12 rolls? (A) 1 (B) 2 (C) 3 (D) 4 (E) P (3, 2) Q (7, 2) R (7, 4) The coordinates of points P, Q, and R in the xy-plane are given above. What is the perimeter of PQR? (A) 12 (B) 14 (C) 6 p20 w (approximately 10.47) w (D) 6 p32 (approximately 11.66) w (E) p 164 (approximately 12.81)

9 21. The letter H is symmetric with respect to two different lines, as shown by the dotted lines in the figure above. Which of the following letters is symmetric with respect to at least two different lines? 22. How many integers in the set of all integers from 1 to 100, inclusive, are not the square of an integer? (A) 19 (B) 50 (C) 81 (D) 89 (E) 90

10 23. The figure above shows the route of Diane's trip from her house to her job. Diane travels 16 miles from A to B, 15 miles from B to C, and 4 miles from C to D. If she were able to travel from A to D directly, how much shorter, in miles, would the trip be? (A) 5 (B) 8 (C) 10 (D) 11 (E) If n is any negative number, which of the following must be positive? (A) n f 2 (B) 2n (C) n + 2 (D) 2 (E) n

11 25. Which of the following numbers is between 1 f 1 and 5 4 (A) 0.14 (B) 0.15 (C) 0.19 (D) 0.21 (E) The following are coordinates of points in the xy-plane. Which of these points is nearest the origin? (A) (0,@ 1) h j (B) 0, 1 2 i k f f? (C) (D) h j h j 1f 2 2 1f 1, 2 2 i k f i k f b c In a certain game, points are assigned to every word. Each q, x, and z in a word is worth 5 points, and all other letters are worth 1 point each. What is the sum of the points assigned to the word "exquisite"? (A) 21 (B) 17 (C) 16 (D) 13 (E) If there is no waste, how many square yards of carpeting is needed to cover a rectangular floor that is 12 feet by 18 feet? (1 yard = 3 feet) (A) 8 (B) 16 (C) 24 (D) 30 (E) 216

12 29. How many positive three-digit integers have the hundreds digit equal to 3 and the units digit (ones digit) equal to 4? (A) 10 (B) 19 (C) 20 (D) 190 (E) If the volume of a cube is 8, what is the shortest distance from the center of the cube to the base of the cube? 31. (A) 1 (B) 2 (C) 4 (D) 2 (E) 2 2 p w w In the figure above, three wires are braided. That is, by starting in the order A, B, C, the outer left wire A is brought over wire B to the middle position, forming the order shown in step 1, then the outer right wire C is brought to the new middle position shown in step 2, and so on, alternately bringing each new left and each new right wire to the middle. At what numbered step does the braid first repeat the original order A, B, C? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7

13 32. In rectangle ABCD, point E is the midpoint of BC. If the area of quadrilateral ABED is 2 3f, what is the area of rectangle ABCD? (A) 1 f 2 (B) 3 f 4 (C) 8 f 9 (D) 1 (E) 8 f An hour-long television program included 20 minutes of commercials. What fraction of the hour-long program was not commercials? 34. NOTE: Figure not drawn to scale. In the figure above, PQST is a rectangle and URST is a square. PU = 5 and UT is a positive integer. If the area of PQST must be more than 10 but less than 30, what is one possible value of UT? 35. How many different positive three-digit integers can be formed if the three digits 4, 5, and 6 must be used in each of the integers? (A) Three (B) Four (C) Six (D) Eight (E) Nine

14 36. If 1 and k > 0, which of the following has the greatest value? (A) 2kx (B) 4kx 2 (C) 6kx 3 (D) 8kx 4 (E) 10kx If x 3 f is an integer, then x must be 2 (A) a negative integer (B) a positive integer (C) a multiple of 3 (D) an even integer (E) an odd integer 38. If n and p are integers greater than 1 and if p is a factor of both n + 3 and n + 10, what is the value of p? (A) 3 (B) 7 (C) 10 (D) 13 (E) If k is a positive integer, what is the least value of k for which integer? v w u5k t f is an 3 (A) 3 (B) 5 (C) 15 (D) 25 (E) 60

15 40. The figures above represent three pieces of cardboard. All angles of the cardboard pieces are right angles, all short sides have length 1, and all long sides have length 2. Which of the following figures could be made from only the three pieces of cardboard without overlapping or cutting them? (A) None (B) I only (C) II only (D) III only (E) I and II

16 41. If the degree measures of the angles of a triangle are in the ratio 2:3:4, by how many degrees does the measure of the largest angle exceed the measure of the smallest angle? (A) 20 (B) 30 (C) 40 (D) 50 (E) 60

17 42.,43.,44. Alyssa makes a number wheel to represent the integers from 0 through 99, inclusive. The short hand points to the tens digit, and the long hand points to the units digit. For example, the number wheel above shows 07, which we would we would write as 7. Which of the following represents the sum of the two integers represented on the two number wheels above?

18 Which of the following is the next greater prime number after the prime number represented above? (A) 15 (B) 17 (C) 33 (D) 37 (E) 41 Exactly how many integers can be represented on this number wheel? (A) 91 (B) 98 (C) 99 (D) 100 (E) 101

19 45. Exactly 1 2f yard of ribbon is needed to make a certain bow. Which of the following lengths of ribbon could be used to make the bow with the least amount remaining? (A) 2 5f yd (B) 3 5f yd (C) 3 4f yd (D) 1 3f yd (E) 2f yd If x is an integer and x 7 f is an integer, which of the following must be 2 true? (A) I only (B) II only (C) III only (D) I and II (E) I and III I. x is odd. II. x is a multiple of 7. III. x 5 f is an integer. 2 1f 47. A recipe calls for 7 tablespoons of milk. This amount is equivalent to 3 how many teaspoons of milk? (3 teaspoons = 1 tablespoon)

20 48. {2, 5, 6, 7, 10} How many different pairs of unequal numbers can be chosen from the set above so that their sum is greater than 10? (Do not consider a pair such as 5, 2 to be different from 2, 5.) (A) 7 (B) 10 (C) 14 (D) 32 (E) 60

21 49. In the figure above, the three diameters divide the circle into six equal regions. If the circle is rotated 120 in a clockwise direction in its plane, which of the following represents the resulting circle?

22 50. Three business partners are to share profits of $24,000 in the ratio 5:4:3. What is the amount of the least share? (A) $1,200 (B) $3,000 (C) $6,000 (D) $8,000 (E) $10, Add 3 to x. Divide this sum by 4. Subtract 2 from this quotient. Which of the following is the result obtained by performing the operations described above? (A) 5 f 4 (B) x 1 f 4 (C) x 3 f 2 (D) 8 f 4 (E) x 1 f If j and k are integers and j k 2j 4, which of the following must be true? I. j is even. II. k is even. III. j is even (A) None (B) I only (C) II only (D) III only (E) I, II, and III

23 53. AB +BA CD4 In the correctly worked addition problem above, each letter represents a different nonzero digit. What is one possible value of the two-digit number represented above as AB? 54. The product of two integers is between 102 and 115. Which of the following CANNOT be one of the integers? (A) 5 (B) 10 (C) 12 (D) 15 (E) If a and b are different positive integers and 5a b 32, what is the sum of all possible values of a? (A) 6 (B) 11 (C) 15 (D) 18 (E) 21

24 56. The positive difference between k and 1 8f is the same as the positive difference between 1 f 1 and f. Which of the following could be the value 2 3 of k? (A) 1 f 7 (B) 1 f 6 (C) 7 24 f (D) 23 f 24 (E) 4 f f 0.0N x 1f 0.00P y The fractions 1 f 1 and f can be written as decimals as shown above, x y where N and P represent different digits. Which of the following is equal to 1 f if S equals N times P and S is a digit? xy (A) S (B) 0.000S (C) 0.00S (D) 0.0S (E) 0.S

25 b c b c 58. T B ` a U B 3 V B 3 W Each letter in the equation above represents a digit that is less than or equal to 2. What four digit number does TUVW represent? In the figure above, any path from A to B must follow the connected line segments in the direction shown by the arrows. How any different paths are there from A to B? (A) Five (B) Six (C) Seven (D) Eight (E) Nine

26 60. Two cities n miles apart are located s inches apart on a certain map that is drawn to scale. What is the distance, in inches, on the map between two cities that are n + 1 miles apart? (A) (B) (C) nf s ` a n 1 f s s f ` a n 1 (D) s ` n 1 n a f (E) n f ` a s n 1 0, then x 61. If x and y are integers and y f is an integer. y Which of the following values of x and y proves that the statement above is not true? (A) x 9, y 3 (B) x 10, y 4 (C) x 6, y 3 (D) x 5, y 5 (E) x 7, y What is the least positive integer n for which 12n is the cube of an integer? R +R2 19T R and T represent digits in the correctly worked addition problem above. What digit does T represent? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8

27 64. If n is a positive number, which of the following is equal to 8n? (A) p64n w w (B) q 8n 2 (C) q w 16n 2 (D) 2 p4n w (E) 4q w 4n How many three-digit numbers have the hundreds digit equal to 3 and the units digit equal to 4? (A) 10 (B) 19 (C) 20 (D) 190 (E) The scenic route to Mia's office is 5 kilometers longer than the direct route. When she goes by the scenic route and returns by the direct route, the round rip is 35 kilometers. How many kilometers is the direct route? (A) 5 (B) 12 1 f 2 (C) 15 (D) 20 (E) 22 1 f 2

28 67. A certain scale for weighing food registers only weights above 6 pounds. A person who wanted to know the weight of 1 package each of chicken, beef and turkey weighed every possible pair of these packages and got the following results: The chicken and the beef weighed 7 pounds. The chicken and the turkey weighed 8 pounds. The beef and the turkey weighed 9 pounds. What is the weight of the package of turkey? (A) 2 pounds (B) 3 pounds (C) 4 pounds (D) 5 pounds (E) 6 pounds 68. In a game, all tokens of the same color are worth the same number of points. If one player won 2 red tokens and 4 blue tokens for a total score of 24 points, and another player won 3 red tokens and 2 blue tokens for a total score of 16 points, how many points is a blue token worth? (A) 2 (B) 4 (C) 5 (D) 6 (E) For every 1,000 cubic meters of air that comes through a filtering system, 0.05 gram of dust is removed. How many grams of dust are removed 7 when 10 cubic meters of air have been filtered? (A) 5 (B) 50 (C) 500 (D) 5,000 (E) 50,000

29 70. In the figure above, one of the nine integers from 1 to 9 is to be placed in each of the nine small squares so that the sums of the three integers in all rows, columns, and diagonals are equal. What is the value of x? (A) 1 (B) 4 (C) 6 (D) 7 (E) Airplane P and airplane Q fly along parallel paths each 3,000 miles long. If they start at the same time and if P flies at a constant speed of 600 miles per hour and Q flies at a constant speed of 580 miles per hour, how many miles will Q have left to fly after P completes is flight? (A) 20 (B) 60 (C) 100 (D) 120 (E) If m and n are both negative numbers, m is less 1, and n is greater 1, which of the following gives all possible values of the product mn? (A) All negative numbers (B) All negative numbers less 1 (C) All negative numbers greater 1 (D) All positive numbers (E) All positive numbers less than 1

30 73. In the figure above, the large rectangle is divided into 6 identical small squares. If the perimeter of the large rectangle is 30, what is the perimeter of one of the small squares? (A) 5 (B) 8 (C) 9 (D) 10 (E) A right circular cylinder has height 6 and volume 54. What is the circumference of its base? (A) 2 (B) 3 (C) 6 (D) 9 (E) The stopping distance of a car is the number of feet that the car travels after the driver starts applying the brakes. The stopping distance of a certain car is directly proportional to the square of the speed of the car, in miles per hour, at the time the brakes are first applied. If the car's stopping distance for an initial speed of 20 miles per hour is 17 feet, what is the stopping distance for an initial speed of 40 miles per hour? (A) 34 feet (B) 51 feet (C) 60 feet (D) 68 feet (E) 85 feet 76. A printing press produces 4,200 posters per hour. At this rate, in how many minutes can the printing press produce 840 posters? (A) 0.2 (B) 1.5 (C) 5 (D) 12 (E) 70

31 77. A bucket holds 4 quarts of popcorn. If 1 3f cup of corn kernels makes 2 quarts of popcorn, how many buckets can be filled with the popcorn made from 4 cups of kernels? (A) 96 (B) 24 (C) 6 (D) 3 (E) 1 1 f 2 78, 79, 80. A positive integer is called a palindrome if it reads the same forward as it does backward. For example, 959 and 8228 are palindromes, whereas 1332 is not. Neither the first nor the last digit of a palindrome can be 0. Which of the following integers is a palindrome? (A) 550 (B) 2255 (C) 2525 (D) 2552 (E) 5002 How many three-digit palindromes are there? (A) 19 (B) 20 (C) 90 (D) 100 (E) 810

32 The next two palindromes greater than are m and p, where m < p. What is the value of m? (A) 10 (B) 90 (C) 100 (D) 110 (E) What is the number that satisfies the following three conditions? It is an integer greater than 999 and less than 1,234. The sum of its digits is 14. Its tens and units digits are the same. 82. In the exact middle of a certain book, when the page numbers on the facing pages, x and x + 1, are multiplied together, the product is 210. If all of the pages are numbered in order, how many numbered pages are in the book? (A) 24 (B) 26 (C) 28 (D) 32 (E) Chuck is writing the page number on the bottom of each page of a 25- page book report, starting with 1. How many digits will he have written after he has written the number 25? (A) 35 (B) 40 (C) 41 (D) 49 (E) 50

33 84. 1A + A 2B In the correctly solved addition problem above, A and B represent digits. If A is not equal to B, how many different digits from 0 through 9 could A represent? (A) Two (B) Three (C) Five (D) Seven (E) Nine b c 85. The quantity 3 B 8 12 is how many times the quantity 3 B 8 5? (A) 7 (B) 8 (C) 21 7 (D) 8 (E) 3 B If the ratio of two positive integers is 3 to 2, which of the following statements about these integers CANNOT be true? (A) Their sum is an odd integer. (B) Their sum is an even integer. (C) Their product is divisible by 6. (D) Their product is an even integer. (E) Their product is an odd integer. 87. A flock of geese on a pond were being observed continuously. At 1:00 P.M., 1 f 1 of the geese flew away. At 2:00 P.M., f of the geese that 5 8 remained flew away. At 3:00 P.M., 3 times as many geese as had flown away at 1:00 P.M., flew away, leaving 28 geese on the pond. At no other times did any geese arrive or fly away. How many geese were in the original flock? b c

34 88., 89. The diagram above represents six building lots along a street. There are no other residential sites in the area. Five families - v, w, x, y, and z - are each interested in purchasing a lot, with the following restrictions. v will occupy lot 6. y and z will live on different sides of the street. w and x will live on the same side of the street, and x will be the only next-door neighbor that w has. One lot will remain unsold. If all five families purchased lots and fulfilled all the restrictions, which of the following pairs of lots could be the ones purchased by y and z? (A) 1 and 2 (B) 1 and 3 (C) 2 and 3 (D) 3 and 5 (E) 3 and 6 If all five families purchased lots and fulfilled all the restrictions and if y purchased lot 3, which of the following must be true? (A) I only (B) II only (C) III only (D) I and III (E) II and III I. w purchased lot 1. II. x purchased lot 4. III. z purchased lot 5.

35 90. If the symbols above represent the facts that a > g and b > f, which of the following is true?

36 91. By 7:00 P.M., 1 3f of the junior class had arrived at a school dance. By 8:00 P.M., 30 more juniors had arrived, raising attendance to 1 2f of the junior class. How many people are in the junior class? (A) 30 (B) 90 (C) 120 (D) 180 (E) A recipe for making 10 loaves of bread requires 24 cups of flour and 4 tablespoons of baking powder. If the proportions in this recipe are to be used to make 3 loaves of bread, how many cups of flour will be needed? (Do not round your answer.) 93. The total cost of a taxicab ride is the sum of (1) a basic fixed charge for using the taxicab, and (2) an additional charge for each 1 4f of a mile that is traveled. If the total cost to ride 3 f 1 mile is $4.00 and the total cost to ride 1 f miles 4 2 is $5.50, what is the total cost, in dollars, of a 3- mile ride. (Disregard the $ sign when gridding your answer. If for example, your answer is $1.37, grid 1.37) 94. If n is 3 less than w and w is 1 more than z, what is the value of n when z = 1? 1 (B) 0 (C) 1 (D) 2 (E) 3

37 95. What was the initial weight, in pounds, of a person who now weighs x pounds and who lost y pounds and then gained 5 pounds? (A) x 5 (B) x 5 (C) x y 5 (D) 5 (E) y When a number x is subtracted from 36 and the difference is divided by x, the result is 2. What is the value of x? (A) 12 (B) 17 (C) 18 (D) 21 (E) A class has twice as many boys as girls. The students in the class stand in one line, with a girl at the front of the line. Which of the following must be true? (A) The last person in line is a girl. (B) The last person in line is a boy. (C) There are more girls than boys in the class. (D) There are at least two girls standing next to each other, (E) There are at least two boys standing next to each other. 98. A certain fraction is equivalent to 1 2f. If 3 is added to its numerator and 2 is added to its denominator, the resulting fraction is equivalent to 3 4f. What is the numerator of the original fraction? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

38 99. On the staircase shown above, both the depth and the height of each step are s, and each step forms a right angle. What is the value of t in terms of s? (A) 10s (B) 14s (C) 7s 2 (D) 7s 3 (E) 2 s 2 p w p w

39 100. A person selects a value for a and then follows the steps shown in the diagram above to result in a value of f. Which of the following statements must be true for all positive integer values of a? I. f is even. II. f has 5 as a factor. III. b < c < f (A) I only (B) II only (C) III only (D) II and III only (E) I, II, and III