# Math 10 - Unit 3 Final Review - Numbers

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1 Class: Date: Math 10 - Unit Final Review - Numbers Multiple Choice Identify the choice that best answers the question. 1. Write the prime factorization of 60. a b. 2 6 c d Write the prime factorization of 116. a b c d Determine the greatest common factor of 6 and 88. a. 77 b. 616 c. 7 d. 8. Determine the greatest common factor of 280 and 60. a. 9 b. 6 c. 220 d. 0. Determine the greatest common factor of 8, 210, and 6. a. 1 b c. 21 d Determine the least common multiple of 10 and 22. a. 2 b. c. 220 d Determine the least common multiple of 78 and 102. a. 126 b. 6 c. 262 d Determine the least common multiple of 8, 72, and 108. a. 2 b. 216 c d A developer wants to subdivide a rectangular plot of land measuring 600 m by 70 m into congruent square lots. What is the side length of the largest possible square? a. 7 m b. 0 m c. 10 m d. 0 m 10. What is the side length of the smallest square that could be tiled using a 6-cm by 1-cm tile? Assume the tiles cannot be cut. a. 10 cm b. 90 cm c. 0 cm d. 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. 0 days 12. What is the side length of the largest square that could be used to tile a rectangle that measures 6 ft. by ft.? Assume the squares cannot be cut. a. 6 ft. b. 2 ft. c. 102 ft. d. ft. 1. 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? a. 12 b. c. 8 d. 16 1

2 1. A fruit grower wants to plant 6 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. d List the first multiples of 2. a., 6, 8, 10 c. 2,, 6, 8 b. 2,,, d. 1, 2,, Which numbers in the list below are multiples of 1? 70, 2, 6,, 1, 29, 9 a. 2,, 6, 70 c. 2, 6, 70 b. 29, 1,, 70 d. 9, 2, List all the factors of. a. 1,, 11, c., 11 b. 1, d., 18. Which numbers in the list below are factors of 7? 2,,,, 6, 8, 9, 10 a. c., b. d. 2,,, 19. Find all the factors of that are prime. a., 11 c. 1, b. 1,, 11, d., 20. Complete this statement: A prime number has exactly factors. a. 0 c. b. 2 d Find all the common multiples of and 18 that are less than 100. a. 1, 2,, 18 c. 6, 72 b. 72 d Find all the common factors of 27 and 6. a. 1, 9 c. 1,, 9 b. 9 d. 1, 2. Determine the square root of without using your calculator. a. 100 b. 6 c. 00 d Determine the cube root of 2 87 without using your calculator. a. 122 b c d. 2. A cube has volume 1 62 cm. What is the surface area of the cube? a cm 2 b. 70 cm 2 c. 2 cm 2 d cm 2 2

3 26. Determine the side length of this square. a. 6 cm b. 1.8 cm c cm d. 1 cm 27. Determine the edge length of this cube. a cm b. cm c cm d. 7 cm 28. A cube has surface area 70 square feet. What is its volume? a. 62 cubic feet c. 18 cubic feet b. 2 cubic feet d cubic feet 29. How many perfect square whole numbers are between 000 and 6000? a. 6 b. 8 c. 1 d How many perfect cube whole numbers are between 6000 and 800? a. b. 2 c. 1 d A rectangular prism with a square base has height 12 ft. The volume of the prism is 68 cubic feet. Determine the side length of the base of the prism to the nearest foot. a. 289 ft. b. 72 ft. c. 2 ft. d. 17 ft. 2. A cube with volume 729 m is to be painted. Each can of paint covers 2 m 2. How many cans of paint are needed to paint the cube? a. 16 b. 2 c. 1 d. 1. An aquarium approximates a cube with surface area 101 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 8 min b. 6 h 6 min c. 6 h min d. 2 h 9 min. Use prime factorization to determine which of the following numbers is not both a perfect square and a perfect cube. a. 1 1 b. 12 c d. 1 62

4 . Find the area of this square. a. 20 square units c. 10 square units b. 2 square units d. square units 6. Find the side length of this square. a. 9 units c. 1 unit b. 28 units d. 7 units 7. Find 6 2. a. 2 b. 12 c. 6 d. 6 Ê 8ˆ 8. Find Ë Á 9 16 a b. 8 9 c. 6 9 d Find a b. 7 9 c d Find 0.0. a. 0.2 b c. 8 d Identify the index of 2 7. a. 2 7 b. c. 7 d Identify the index of 7. a. b. 7 c. d.. Identify the radicand of 8 6. a. b. 8 c. 6 d. 8

5 . Identify the radicand of a. 9 b c. d. 10. Evaluate 16 without a calculator. a. 2 b. 2.6 c. 16 d Evaluate 6 without a calculator. a. b. impossible c d. 7. Evaluate a. 26 without a calculator. 62 b. 2 c d Write an equivalent form of 9 as a cube root. a. 661 b. 729 c d Write an equivalent form of as a square root. 9 a b. 6 c d Determine which of these roots lies between and without using your calculator. 28, 172, 17, 118 a. 118 b. 17 c. 28 d Estimate the value of 0 to one decimal place. a. 0. b.. c. 0.9 d Which of these numbers is rational? 169, 8, 16, 8.1 a. 8 b. 8.1 c. 16. Which of these numbers is irrational? 8, 216, 9 16, 68 a. 68 b. 8 c. 216 d. d Order these numbers from greatest to least by estimating each root: 99 a. 170, 99 b. 00, 00, 18, 1, 18, 1, 170, 99 c. 00 d. 00, 170, 00, 18, 1, 170, 99, 18, 1, 170, 18, 1, 99

6 . Order these numbers from least to greatest by estimating each root: 7 a. 7 b. 0, 100, 1, 0, 1, 17, 7, 100, 17 c. 100 d. 17, 7 6. Determine which of these numbers is the least by estimating each root. 1, 0 a. 100, 100, 7, 17 b. 0, 0, 1, 17, 7, 1, 100, 17, 0, 100, 1, 0 c. 1 d Between which two consecutive integers on a number line would you locate 18? Don t use your calculator! a. 2 and b. and c. 2 and d. 1 and 2 8. Which of these numbers is an integer, but not a whole number? 9, 0, 1, a. 0 b. 9 c. d Which of these numbers is a natural number? 9, 0, 1, 1.8 a. 9 b. 0 c. 1.8 d Which of these numbers is a whole number, but not a natural number? 0,, 1, 8.1 a. 8.1 b. 1 c. 0 d. 61. For which number will the fifth root be irrational? 62, 7776, , 2 a b. 2 c d To which set(s) of numbers does 2 belong? a. II and III only b. III only c. I, II and III only d. IV only 6. Write 108 in simplest form. a. 12 b. 6 c. 6 d Write 67 in simplest form. a. 7 b. 1 c. 22 d Write 200 in simplest form. a. 2 0 b c d

7 66. Write 80 a Write 120 a Write 187 a Write 0 a. in simplest form. b in simplest form. b. 10 in simplest form. b. 1 in simplest form. b. 81 c c c c. 9 d. d. 2 2 d. 2 d. 70. Write 6 as an entire radical. a. 0 b. 10 c. 180 d Write 2 as an entire radical. a. 6 b. 12 c. 18 d Write a Write 9 a Write 2 a. 8 as an entire radical. b. 1 as an entire radical. b. 202 as an entire radical. b. 18 c. 6 c. 22 c. 162 d. 192 d. 6 d Write 98 in simplest form. a. 7 1 b. 7 2 c. 2 7 d Write 172 a Write 160 a in simplest form. b. 7 in simplest form. b. 10 c. 1 7 c d. 7 d Write 7 1 as an entire radical. a. 960 b. 98 c. 686 d Write 16 a Write 12 a. 192 as an entire radical. b as an entire radical. b. 20 c. 20 c d. 2 d

8 81. A square has an area of 12 square inches. Determine the side length of the square as a radical in simplest form. a. in. b. 2 6 in. c. 2 in. d. 2 in. 82. A cube has a volume of 7290 cm. Determine the edge length of the cube as a radical in simplest form. a cm b cm c cm d Order these numbers from greatest to least: 2 0,, 2 7,, 2 1 a. 2 1, 2 7,,, 2 0 c., 2 0,, 2 1, 2 7 b., 2 0, 2 1, 2 7, d.,, 2 0, 2 1, Between which 2 consecutive whole numbers is 207? a. 206 and 208 b. 1 and 2 c. 1 and 1 d. 196 and What is the greatest whole number less than 9? a. 10 b. 7 c. 6 d. 86. What is the least whole number greater than 7? a. 6 b. 7 c. 1 d Estimate the value of 1 to the nearest tenth. a. b..1 c. d Find the approximate side length of a square with area 1 cm 2. Give your answer to the nearest tenth. a..6 cm b..9 cm c. 7.8 cm d. 1. cm cm Short Answer 89. Write the prime factorization of Determine the greatest common factor of 7 and Determine the least common multiple of 0 and A builder wants to cover a wall measuring 9 ft. by 1 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? 9. Bill and Betty do chores at home. Bill mows the lawn every 8 days, and Betty bathes the dog every 1 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? 9. Determine the cube root of

9 9. A cube has volume 9261 cubic inches. Determine the area of one face of the cube. 96. Determine the square root of Which of these numbers is not a perfect square: 16, 17, 6, or 6? 98. Which of these numbers is a perfect square: 0, 9, 60, or 8? 99. Evaluate Evaluate Estimate the value of to one decimal place Between which 2 consecutive integers on a number line would you locate 220? 10. Which of these numbers are rational numbers, but not integers?.12,,, 7, 2., 8, 0, 1 2, Which of these numbers are irrational? 102, 72, 12, 6.1, 6, 12.8, 196, Determine the side length of a square with area 72 cm 2. Write your answer to the nearest tenth of a centimetre Determine the edge length of a cube with volume cm. Write your answer to the nearest tenth of a centimetre Write 169 in simplest form Write 8 19 as an entire radical Write in simplest form Which of these numbers is the greatest? 7, 9 2, 11 2, 1 90,

10 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 19 square inches. Determine the greatest possible measure of the side that the two rectangles share A cube has surface area 266 m 2. What is its volume? 11. Use factoring to determine whether 91 is a perfect square, a perfect cube, or neither. 11. Germaine wants to paint a cube with volume 27 m. Each tub of paint covers 79 m 2. How many tubs of paint does Germaine need to paint the cube? 11. Find the number whose square root is 1. Explain your strategy The factors of 180 are listed in ascending order. 180: 1, 2,,,, 6, 9, 10, 12, 1, 18, 20, 0, 6,, 60, 90, 180 Is 180 a perfect square? How do you know? 117. A square gymnasium floor has area 81 m 2. Find the perimeter of the gymnasium floor Is the cube root of 20 rational or irrational? Use 2 different strategies to justify your answer Order these numbers from least to greatest: 8, 1, 1, 2, In isosceles ABC, what is the length of BC? Write your answer as a mixed radical. 10

11 Math 10 - Unit Final Review - Numbers Answer Section MULTIPLE CHOICE 1. C 2. B. D. D. D 6. D 7. A 8. A 9. C 10. C 11. D 12. B 1. B 1. C 1. C 16. D 17. A 18. C 19. A 20. B 21. C 22. C 2. C 2. D 2. B 26. A 27. B 28. D 29. D 0. B 1. D 2. A. B. B. B 6. D 7. C 8. D 9. B 1

12 0. A 1. B 2. B. B. B. A 6. A 7. A 8. B 9. D 0. C 1. B 2. D. B. C. B 6. B 7. A 8. B 9. A 60. C 61. D 62. A 6. B 6. B 6. B 66. B 67. B 68. B 69. A 70. C 71. C 72. A 7. A 7. C 7. B 76. D 77. A 78. C 79. B 80. D 81. D 82. B 8. B 8. C 2

13 8. C 86. B 87. D 88. A SHORT ANSWER , or a) ft. b) days square inches and , 7, 2., and , 6, and cm cm PROBLEM 111. The area of a rectangle is the product of its dimensions. List the factors of 209 and 19. The factors represent possible lengths of a side of each rectangle. Check to see which factors of 209 are also factors of 19. 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 19 are: 1, 11, 29, 19 The greatest common factor of 209 and 19 is 11. Τhe greatest possible measure of the side that the two rectangles share is 11 in.

14 112. 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 = A = 1 The edge length, e, of the cube is the square root of the area of one square face. e = 1 e = 21 So, the volume, V, of the cube is the cube of its edge length. V = 21 V = 9261 The volume of the cube is 9261 m. 11. Write 91 as a product of its prime factors. 91 = Since 91 is the product of three equalfactors, it is a perfect cube. It is not possible to rearrange the factors in 2 equalgroups, so 91 is not a perfect square. 11. 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 = 27 e = 1 The surface area, SA, of a cube is the sum of the areas of its 6 congruent square faces. SA = 6(1 1) SA = 6(196) SA = 1176 Calculate how many tubs of paint are needed: = Germaine needs 1 tubs of paint to paint the cube. 11. To find the number whose square root is 1, find the square of 1. The square of 1 is: 1 2 = 1 1 = 169 The number whose square root is 1 is 169.

15 116. 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 Find the side length of the gymnasium floor: Find a number which, when multiplied by itself, gives = 81 So, the gymnasium floor has side length 9 m. Perimeter is the distance around the gymnasium floor. A square has equal sides, so the perimeter of the floor is: 9 m + 9 m + 9 m + 9 m = 6 m The perimeter of the gymnasium floor is 6 m is not a perfect cube, so the cube root of 20 is irrational. 20 = does not appear to terminate or repeat. So, the cube root of 20 is likely irrational is between the perfect squares 6 and 9, so 8 is between 6 and 7. Use a calculator: 8 =Ö is between the perfect cubes 12 and 729, so 1 Use a calculator: 1 Write 1 as: 1 =. =Ö 8 is between 8 and 9. 2 is between the perfect squares 1 and, so 2 is between 1 and 2. Use a calculator: 2 =Ö is between the perfect cubes 12 and 216, so 128 Use a calculator: 128 =Ö From least to greatest: 2, 1, 128, 8, 1 is between and 6.

16 120. Use the Pythagorean Theorem in ABD to determine BD = 2 + BD 2 BD 2 = BD 2 = 7 BD = 7 BD = BD = 1 2 BC So, BC = 2 BD Ê ˆ BC = 2 Ë Á = 10 The length of BC is 10 ft. 6

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