Math 10 - Unit 3 Final Review - Numbers
Multiple Choice
Identify the choice that best answers the question.
____ 1. Write the prime factorization of 630.
a. 2⋅5⋅7⋅9 b. 2⋅5⋅63 c. 2⋅32⋅5⋅7 d. 2⋅3⋅5⋅7 ____ 2. Write the prime factorization of 4116.
a. 23⋅3⋅72 b. 22⋅3⋅73 c. 22⋅3⋅72 d. 2⋅32⋅73 ____ 3. Determine the greatest common factor of 56 and 88.
a. 77 b. 616 c. 7 d. 8
____ 4. Determine the greatest common factor of 280 and 360.
a. 9 b. 63 c. 2520 d. 40
____ 5. Determine the greatest common factor of 84, 210, and 336.
a. 14 b. 1680 c. 21 d. 42
____ 6. Determine the least common multiple of 10 and 22.
a. 2 b. 55 c. 220 d. 110
____ 7. Determine the least common multiple of 78 and 102.
a. 1326 b. 6 c. 2652 d. 7956
____ 8. Determine the least common multiple of 48, 72, and 108.
a. 432 b. 216 c. 31 104 d. 12
____ 9. A developer wants to subdivide a rectangular plot of land measuring 600 m by 750 m into congruent square lots. What is the side length of the largest possible square?
a. 75 m b. 30 m c. 150 m d. 50 m
____ 10. What is the side length of the smallest square that could be tiled using a 6-cm by 15-cm tile? Assume the tiles cannot be cut.
a. 10 cm b. 90 cm c. 30 cm d. 3 cm
____ 11. One neighbour cuts his lawn every 8 days. Another neighbour cuts her lawn every 10 days. Suppose both neighbours cut their lawns today. How many days will pass before both neighbours cut their lawns on the same day again?
a. 80 days b. 60 days c. 2 days d. 40 days
____ 12. What is the side length of the largest square that could be used to tile a rectangle that measures 6 ft. by 34 ft.? Assume the squares cannot be cut.
a. 6 ft. b. 2 ft. c. 102 ft. d. 4 ft.
____ 13. There are 16 male students and 20 female students in a Grade 10 math class. The teacher wants to divide the class into groups with the same number of males and the same number of females in each group. What is the greatest number of groups the teacher can make?
____ 14. A fruit grower wants to plant 64 apple seedlings and 108 pear seedlings in rows. Each row is to have the same number of each type of seedling. What is the greatest number of rows the grower can plant?
a. 8 b. 16 c. 4 d. 2
____ 15. List the first 4 multiples of 2.
a. 4, 6, 8, 10 c. 2, 4, 6, 8
b. 2, 3, 4, 5 d. 1, 2, 4, 6
____ 16. Which numbers in the list below are multiples of 13? 70, 52, 65, 53, 41, 29, 39
a. 52, 53, 65, 70 c. 52, 65, 70
b. 29, 41, 53, 70 d. 39, 52, 65
____ 17. List all the factors of 33.
a. 1, 3, 11, 33 c. 3, 11
b. 1, 33 d. 3, 3
____ 18. Which numbers in the list below are factors of 75? 2, 3, 4, 5, 6, 8, 9, 10
a. 3 c. 3, 5
b. 5 d. 2, 3, 4, 5
____ 19. Find all the factors of 33 that are prime.
a. 3, 11 c. 1, 33
b. 1, 3, 11, 33 d. 3, 3
____ 20. Complete this statement:
A prime number has exactly ____ factors.
a. 0 c. 3
b. 2 d. 1
____ 21. Find all the common multiples of 4 and 18 that are less than 100.
a. 1, 2, 4, 18 c. 36, 72
b. 72 d. 36
____ 22. Find all the common factors of 27 and 63.
a. 1, 9 c. 1, 3, 9
b. 9 d. 1, 3
____ 23. Determine the square root of 250 000 without using your calculator.
a. 100 b. 63 c. 500 d. 200
____ 24. Determine the cube root of 42 875 without using your calculator.
a. 1225 b. 4763.9 c. 207.1 d. 35
____ 25. A cube has volume 15 625 cm3. What is the surface area of the cube?
____ 26. Determine the side length of this square.
a. 63 cm b. 15.83 cm c. 992.25 cm d. 441 cm
____ 27. Determine the edge length of this cube.
a. 301.87 cm b. 45 cm c. 6.71 cm d. 3375 cm
____ 28. A cube has surface area 3750 square feet. What is its volume? a. 5625 cubic feet c. 1448 cubic feet
b. 25 cubic feet d. 15 625 cubic feet
____ 29. How many perfect square whole numbers are between 5000 and 6000?
a. 6 b. 8 c. 1 d. 7
____ 30. How many perfect cube whole numbers are between 6000 and 8500?
a. 3 b. 2 c. 1 d. 15
____ 31. A rectangular prism with a square base has height 12 ft. The volume of the prism is 3468 cubic feet. Determine the side length of the base of the prism to the nearest foot.
a. 289 ft. b. 72 ft. c. 24 ft. d. 17 ft.
____ 32. A cube with volume 729 m3 is to be painted. Each can of paint covers 32 m2. How many cans of paint are needed to paint the cube?
a. 16 b. 23 c. 13 d. 15
____ 33. An aquarium approximates a cube with surface area 1014 square feet. Water was poured into the empty aquarium at a rate of 6 cubic feet per minute. To the nearest minute, how long did it take to fill the aquarium? a. 2 h 48 min b. 6 h 6 min c. 6 h 5 min d. 2 h 49 min
____ 34. Use prime factorization to determine which of the following numbers is not both a perfect square and a perfect cube.
____ 35. Find the area of this square.
a. 20 square units c. 10 square units b. 25 square units d. 5 square units ____ 36. Find the side length of this square.
a. 49 units c. 1 unit b. 28 units d. 7 units ____ 37. Find 62. a. 24 b. 12 c. 36 d. 6 ____ 38. Find 8 9 Ê Ë ÁÁÁÁ ÁÁ ˆ ¯ ˜˜˜˜ ˜˜ 2 . a. 16 81 b. 8 9 c. 64 9 d. 64 81 ____ 39. Find 49 81. a. 8 3 b. 7 9 c. 7 81 d. 49 81 ____ 40. Find 0.04 . a. 0.2 b. 0.02 c. 8 d. 0.01
____ 41. Identify the index of 3 27.
a. 27 b. 3 c. 7 d. 2
____ 42. Identify the index of 7 53.
a. 53 b. 7 c. 3 d. 5
____ 43. Identify the radicand of 6 48.
____ 44. Identify the radicand of 3 910.
a. 9 b. 910 c. 3 d. 10
____ 45. Evaluate 4 16 without a calculator.
a. 2 b. 2.6 c. 16 d. 1.41
____ 46. Evaluate 3 −64 without a calculator.
a. –4 b. impossible c. –12.8 d. 4 ____ 47. Evaluate 256 625 4 without a calculator. a. 4 5 b. 4 25 c. 16 25 d. 16 5 ____ 48. Write an equivalent form of 9 as a cube root.
a. 3 6561 b. 3 729 c. 3 9 81 d. 81
____ 49. Write an equivalent form of 4
9 as a square root. a. 16 18 b. 64 729 3 c. 8 81 d. 16 81 ____ 50. Determine which of these roots lies between 3 and 4 without using your calculator.
28 3
, 3 −172, 17 , 3 −118
a. 3 −118 b. 17 c. 3 28 d. 3 −172
____ 51. Estimate the value of 3 40 to one decimal place.
a. –0.3 b. 3.4 c. 0.9 d. 5.7
____ 52. Which of these numbers is rational? 4 169, 48 , −16 3 , 8.1 a. 48 b. 8.1 c. 3 −16 d. 4 169 ____ 53. Which of these numbers is irrational?
48 , 3 216, 49 16, −68
a. −68 b. 48 c. 3 216 d. 49
16
____ 54. Order these numbers from greatest to least by estimating each root: 3 99, 170 , 3 3050, 18 , 3 51 a. 170 , 3 99, 3 3050, 18 , 3 51 c. 3 3050, 170 , 3 99, 18 , 3 51
____ 55. Order these numbers from least to greatest by estimating each root: 3 75, 14 , 3 100, 17 , 3 30 a. 3 75, 3 100, 14 , 3 30, 17 c. 3 100, 3 30, 14 , 17 , 3 75
b. 3 30, 14 , 17 , 3 75, 3 100 d. 17 , 3 75, 3 100, 14 , 3 30 ____ 56. Determine which of these numbers is the least by estimating each root.
14 , 3 30, 4 100, 3 75, 17
a. 4 100 b. 3 30 c. 14 d. 3 75
____ 57. Between which two consecutive integers on a number line would you locate 3 −18? Don’t use your calculator!
a. –2 and –3 b. –3 and –4 c. 2 and 3 d. –1 and –2 ____ 58. Which of these numbers is an integer, but not a whole number?
–9, 0, 1, 5
a. 0 b. –9 c. 5 d. 1
____ 59. Which of these numbers is a natural number? 9, 0, –1, 1.8
a. 9 b. 0 c. 1.8 d. –1
____ 60. Which of these numbers is a whole number, but not a natural number? 0, –3, 1, 8.1
a. 8.1 b. 1 c. 0 d. –3
____ 61. For which number will the fifth root be irrational? 625, 7776, 16 807, –32
a. 7776 b. –32 c. 16 807 d. 625
____ 62. To which set(s) of numbers does − 25 belong?
a. II and III only b. III only c. I, II and III only d. IV only ____ 63. Write 108 in simplest form.
a. 3 12 b. 6 3 c. 36 3 d. 3 6
____ 64. Write 675 in simplest form.
a. 3 75 b. 15 3 c. 225 3 d. 3 15
____ 65. Write 200 in simplest form.
____ 66. Write 3 80 in simplest form.
a. 103 2 b. 23 10 c. 83 10 d. 43 5
____ 67. Write 3 1250 in simplest form.
a. 103 5 b. 53 10 c. 1253 10 d. 253 2
____ 68. Write 3 1875 in simplest form.
a. 153 5 b. 53 15 c. 1253 15 d. 253 3
____ 69. Write 4 405 in simplest form.
a. 34 5 b. 814 5 c. 94 5 d. 54 3
____ 70. Write 6 5 as an entire radical.
a. 30 b. 150 c. 180 d. 900
____ 71. Write 3 2 as an entire radical.
a. 6 b. 12 c. 18 d. 36
____ 72. Write 33 4 as an entire radical.
a. 3 108 b. 3 144 c. 3 36 d. 3 192
____ 73. Write 53 9 as an entire radical.
a. 3 1125 b. 3 2025 c. 3 225 d. 3 3645
____ 74. Write 34 2 as an entire radical.
a. 4 48 b. 4 18 c. 4 162 d. 4 36
____ 75. Write 98 in simplest form.
a. 7 14 b. 7 2 c. 2 7 d. 49 2
____ 76. Write 3 1372 in simplest form.
a. 73 28 b. 43 7 c. 143 7 d. 73 4
____ 77. Write 4 160 in simplest form.
a. 24 10 b. 44 10 c. 104 2 d. 24 20
____ 78. Write 7 14 as an entire radical.
a. 9604 b. 98 c. 686 d. 1372
____ 79. Write 163 3 as an entire radical.
a. 3 768 b. 3 12 288 c. 3 2304 d. 3 432
____ 80. Write 45 12 as an entire radical.
____ 81. A square has an area of 12 square inches. Determine the side length of the square as a radical in simplest form.
a. 4 3 in. b. 2 6 in. c. 3 2 in. d. 2 3 in.
____ 82. A cube has a volume of 7290 cm3. Determine the edge length of the cube as a radical in simplest form. a. 93 90 cm b. 93 10 cm c. 813 10 cm d. 103 9 cm
____ 83. Order these numbers from greatest to least: 2 30 , 3 3 , 2 7 , 5 5 , 2 13
a. 2 13 , 2 7 , 3 3 , 5 5 , 2 30 c. 5 5 , 2 30 , 3 3 , 2 13 , 2 7 b. 5 5 , 2 30 , 2 13 , 2 7 , 3 3 d. 3 3 , 5 5 , 2 30 , 2 13 , 2 7 ____ 84. Between which 2 consecutive whole numbers is 207 ?
a. 206 and 208 b. 51 and 52 c. 14 and 15 d. 196 and 225 ____ 85. What is the greatest whole number less than 39 ?
a. 10 b. 7 c. 6 d. 5
____ 86. What is the least whole number greater than 47 ?
a. 6 b. 7 c. 13 d. 25
____ 87. Estimate the value of 13 to the nearest tenth.
a. 3 b. 4.1 c. 4 d. 3.6
____ 88. Find the approximate side length of a square with area 31 cm2. Give your answer to the nearest tenth.
a. 5.6 cm b. 3.9 cm c. 7.8 cm d. 15.5 cm
Short Answer
89. Write the prime factorization of 35 700.
90. Determine the greatest common factor of 735 and 1715. 91. Determine the least common multiple of 450 and 180.
92. A builder wants to cover a wall measuring 9 ft. by 15 ft. with square pieces of plywood. a) What is the side length of the largest square that could be used to cover the wall?
Assume the squares cannot be cut.
b) How many square pieces of plywood would be needed?
93. Bill and Betty do chores at home. Bill mows the lawn every 8 days, and Betty bathes the dog every 14 days. Suppose Bill and Betty do their chores today. How many days will pass before they both do their chores on the same day again?
95. A cube has volume 9261 cubic inches. Determine the area of one face of the cube. 96. Determine the square root of 308 025.
97. Which of these numbers is not a perfect square: 16, 17, 36, or 64? 98. Which of these numbers is a perfect square: 30, 49, 60, or 84? 99. Evaluate 4 625.
100. Evaluate 3 −1728.
101. Estimate the value of 35 to one decimal place.
102. Between which 2 consecutive integers on a number line would you locate 4 220? 103. Which of these numbers are rational numbers, but not integers?
3.12, –4, 5, 4 7, 2.4, −8 3 , 0, 51 2, 16 4
104. Which of these numbers are irrational? −1024
5
, 72 , 3 125, 6.314, 4 64, –12.8, 196 , 8.121 121 112 111… 105. Determine the side length of a square with area 72 cm2.
Write your answer to the nearest tenth of a centimetre. 106. Determine the edge length of a cube with volume 55 cm3.
Write your answer to the nearest tenth of a centimetre. 107. Write 1694 in simplest form.
108. Write 8 19 as an entire radical. 109. Write 5 28 125 in simplest form. 110. Which of these numbers is the greatest?
Problem
111. A rectangle is divided into 2 smaller rectangles. The area of the rectangle on the left is 209 square inches, and the area of the rectangle on the right is 319 square inches. Determine the greatest possible measure of the side that the two rectangles share.
112. A cube has surface area 2646 m2. What is its volume?
113. Use factoring to determine whether 4913 is a perfect square, a perfect cube, or neither.
114. Germaine wants to paint a cube with volume 2744 m3. Each tub of paint covers 79 m2. How many tubs of paint does Germaine need to paint the cube?
115. Find the number whose square root is 13. Explain your strategy. 116. The factors of 180 are listed in ascending order.
180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 Is 180 a perfect square?
How do you know?
117. A square gymnasium floor has area 81 m2. Find the perimeter of the gymnasium floor. 118. Is the cube root of 250 rational or irrational?
Use 2 different strategies to justify your answer.
119. Order these numbers from least to greatest: 38 , 3 515, 13
3 , 2 , 128 3
Math 10 - Unit 3 Final Review - Numbers
Answer Section
MULTIPLE CHOICE 1. C 2. B 3. D 4. D 5. D 6. D 7. A 8. A 9. C 10. C 11. D 12. B 13. B 14. C 15. C 16. D 17. A 18. C 19. A 20. B 21. C 22. C 23. C 24. D 25. B 26. A 27. B 28. D 29. D 30. B 31. D 32. A 33. B 34. B 35. B 36. D 37. C 38. D 39. B40. A 41. B 42. B 43. B 44. B 45. A 46. A 47. A 48. B 49. D 50. C 51. B 52. D 53. B 54. C 55. B 56. B 57. A 58. B 59. A 60. C 61. D 62. A 63. B 64. B 65. B 66. B 67. B 68. B 69. A 70. C 71. C 72. A 73. A 74. C 75. B 76. D 77. A 78. C 79. B 80. D 81. D 82. B
85. C 86. B 87. D 88. A SHORT ANSWER 89. 2⋅2⋅3⋅5⋅5⋅7⋅17, or 22⋅3⋅52⋅7⋅17 90. 245 91. 900 92. a) 3 ft. b) 15 93. 56 days 94. 82 95. 441 square inches 96. 555 97. 17 98. 49 99. 5 100. −12 101. 5.9 102. 3 and 4 103. 3.12, 4 7, 2.4, and 5 1 2 104. 72 , 4 64, and 8.121 121 112 111… 105. 8.5 cm 106. 3.8 cm 107. 11 14 108. 1216 109. 55 9 110. 133 90 PROBLEM
111. The area of a rectangle is the product of its dimensions.
List the factors of 209 and 319. The factors represent possible lengths of a side of each rectangle. Check to see which factors of 209 are also factors of 319.
The greatest common factor will be the greatest possible measure of the side that the two rectangles share. The factors of 209 are: 1, 11, 19, 209
The factors of 319 are: 1, 11, 29, 319
The greatest common factor of 209 and 319 is 11.
112. To calculate the volume, first determine the edge length of the cube.
The surface area of a cube is the sum of the areas of its 6 congruent square faces. So, the area, A, of one face is:
A= 2646 6
A=441
The edge length, e, of the cube is the square root of the area of one square face.
e= 441
e=21
So, the volume, V, of the cube is the cube of its edge length.
V=213
V=9261
The volume of the cube is 9261 m3.
113. Write 4913 as a product of its prime factors. 4913=17⋅17⋅17
Since 4913 is the product of three equal factors, it is a perfect cube. It is not possible to rearrange the factors in 2 equal groups, so 4913 is not a perfect square.
114. To calculate how many tubs of paint are needed, first determine the surface area of the cube. The edge length, e, of a cube is equal to the cube root of its volume.
e= 3 2744
e=14
The surface area, SA, of a cube is the sum of the areas of its 6 congruent square faces.
SA=6(14⋅14)
SA=6(196)
SA=1176
Calculate how many tubs of paint are needed: 1176
79 =14.8860. . .
Germaine needs 15 tubs of paint to paint the cube.
115. To find the number whose square root is 13, find the square of 13. The square of 13 is: 132 =13×13
=169
116. Any perfect square can be written as the product of two equal factors, and this factor is listed only once in the list of factors. So, a perfect square has an odd number of factors.
180 has 18 factors. 18 is an even number.
So, 180 is not a perfect square.
117. Find the side length of the gymnasium floor:
Find a number which, when multiplied by itself, gives 81. 9 × 9 = 81
So, the gymnasium floor has side length 9 m.
Perimeter is the distance around the gymnasium floor. A square has 4 equal sides, so the perimeter of the floor is: 9 m + 9 m + 9 m + 9 m = 36 m
The perimeter of the gymnasium floor is 36 m.
118. 250 is not a perfect cube, so the cube root of 250 is irrational. 250
3
= 6.299 605 249 474. . .
6.299 605 249 474. . . does not appear to terminate or repeat. So, the cube root of 250 is likely irrational.
119. 38 is between the perfect squares 36 and 49, so 38 is between 6 and 7. Use a calculator: 38 =Ö 6.2
515 is between the perfect cubes 512 and 729, so 3 515 is between 8 and 9. Use a calculator: 3 515 =Ö 8
Write 13 3 as:
13 3 = 4.3
2 is between the perfect squares 1 and 4, so 2 is between 1 and 2. Use a calculator: 2 =Ö 1.4
128 is between the perfect cubes 125 and 216, so 3 128 is between 5 and 6. Use a calculator: 3 128 =Ö 5
From least to greatest: 2 , 13
3 , 128 3
120. Use the Pythagorean Theorem in ∆ABD to determine BD. 102 =52+BD2 BD2 =102−52 BD2 =75 BD= 75 BD=5 3 BD= 1 2BC So, BC=2⋅BD BC=2 5Ê 3 Ë ÁÁÁ ˆ¯˜˜˜ =10 3 The length of BC is 10 3 ft.
