EXERCISE 1 WORKED EXAMPLES: If x = 9, y = 6 and z = -3 find the value of the following. x 9 3 1 x 9 9 3 b. = = =1 = = = y 6 2 2 2y 2 ´ 6 12 4 2x 2 ´ 9 18 x -1 9 -1 8 4 1 d. = = =3 = = = =1 y 6 6 y 6 6 3 3 y 6 2 y -1 6 -1 5 1 f. = = = = = x 9 3 x +1 9 + 1 10 2 z - 3 -1 z - 3 -1 = = h. = = y 6 2 x 9 3 z-1 3- 1 - 3+- 1 - 4 - 2 2z 2´ 3 6 2 = = = = j. = = = y 6 6 6 3 x 9 9 3 a. c. e. g. i. WRITTEN EXAMPLES: If = 4, y = 6 and z = -2 find the value of the following fractions in their simplest form. 1. 5. 9. 13. 17. 21. 25. 29. 33. x y y x 2x + 1 y 2 x y 3x - y 3x + y z x z+ 4 x z+ x x+ z z+ x x+ z 2. 6. 10. 14. 18. 22. 26. 30. 34. x 2y y 3x y 3x - 3 x y2 2y - x 2y + x z y z-1 y x- y y- x z- x x-z 3. 7. 11. 15. 19. 23. 27. 31. 35. 2x y 3y x 3x - 2 y -1 y- x y+ x y+ x y 2 - x2 3z x 4-z x y+ z z+ y x+ z y+ z 4. 8. 12. 16. 20. 24. 28. 32. 36. x -1 y y+2 x y +1 3x + 2 x+ y xy y- x y 2 - x2 2z y 4z- 1 y y-z z- y x´z y´z DS/GRBJMH: апрель 11, 2020 page 1 of 11 EXERCISE 2 WORKED EXAMPLES: a. b. c. d. e. g. i. Simplify these fractions. 10 2/ ´5 5 2 2x 2/ ´ x x = = = 1 and so = = 2y 2/ ´ y y 6 2/ ´ 3 3 3 10x 2/ ´ 5 ´ x/ 5 2 = = =1 6x 2/ ´ 3 ´ x/ 3 3 10x 2/ ´ 5 ´ x 5 ´ x 5x 2 (which is not usually written as 1 x) = = = 6 2/ ´ 3 3 3 3 10 2/ ´ 5 5 5 = = = 6y 2/ ´ 3 ´ y 3 ´ y 3y 4x 4/ ´ x/ 10x 2/ ´ 5 ´ x 5 ´ x 5x f. = =1 = = = 4x 4/ ´ x/ 6y 2/ ´ 3 ´ y 3 ´ y 3y 2x 2/ ´ x x 2 2/ ´ 1 1 1 h. = = = = = 6y 2/ ´ 3 ´ y 3 ´ y 3y 6 2/ ´ 3 3 ab a/ ´ b/ 1 1 ab a´ b/ a j. = = = 2 2 = 2 = a b a/ ´ a´ b´ b/ a´b ab b b ´ b/ b WRITTEN EXAMPLES: Simplify the following fractions. 5x 2x 3y 1. 2. 3. 2y 3y 5 7x 2y 5x 5. 6. 7. 5y 7x 2 6x 4x 8y 9. 10. 11. 2y 2y 2 3x 3y 3x 13. 14. 15. 6y 12 x 9 8x 12 x 20y 17. 18. 19. 4y 4y 4 5x 5y 5x 21. 22. 23. 15 y 25x 10 8x 4x 12 y 25. 26. 27. 6y 9y 10 15x 20y 12 x 29. 30. 31. 8y 20x 16 ab ab a 33. 34. 35. a b ab bc ca a2 37. 38. 39. ac bc a 3 b a a3 41. 42. 43. b3 a a2 a b ab 45. 46. 47. 2 2 ab ab a2 b bc a2 c 2a 49. 50. 51. c2 ab ca 2 4. 8. 12. 16. 20. 24. 28. 32. 36. 40. 44. 48. 52. 7 7y 3 3x 10 2y 3 15 x 16 4y 5 20x 18 15 y 25 30x b ab b b2 b2 b3 ab ab2 a3b 2c ab2 c3 DS/GRBJMH: апрель 11, 2020 page 2 of 11 EXERCISE 3 WORKED EXAMPLES: Find these products and quotients. Simplify the answers where possible. 5 2 5 ´ 2 10 a c a´ c ac a. and so ´ = ´ = = = 7 3 7 ´ 3 21 b d b´ d bd 5 2 5 3 5 ´ 3 15 a c a d a´ d ad b. and so ¸ = ´ = ¸ = ´ = = = 7 3 7 2 7 ´ 2 14 b d b c b´ c bc 5a 2b 5 ´ a´ 2/ ´ b/ 5a a b a´b/ a c. ´ = = ´ = = and 6b 3c 3 6/ ´ b/ ´ 3´ c 9c b c b/ ´c c a a a/ c c 5a 2a 5a 3c 5 ´ a/ ´ /3 ´ c 5c d. ¸ = ´ = and ¸ = ´ = = 6b 3c 6b 2a 2 6/ ´ b´ 2 ´ a/ 4b b c b a/ b WRITTEN EXAMPLES: Find the following products and quotients. Simplify your answers where possible. x y x y x z x z 1. 2. 3. 4. ´ ¸ ´ ¸ y z y z y y y y x z x z x x x x 5. 6. 7. 8. ´ ¸ ´ ¸ y x y x y z y z x x x x x y x y 9. 10. 11. 12. ´ ¸ ´ ¸ y y y y y x y x 1 1 1 1 13. x ´ 14. x ¸ 15. 16. ´y ¸y y y x x 2a b 2a b 3a c 3a c 17. 18. 19. 20. ´ ¸ ´ ¸ 3b 2c 3b 2c 4b 3b 4b 3b 2a b 2a b 3a a 3a a 21. 22. 23. 24. ´ ¸ ´ ¸ 3b 6a 3b 6a 4b 6c 4b 6c 5a 2a 5a 2a 5a 3b 5a 3b 25. 26. 27. 28. ´ ¸ ´ ¸ 6b 3b 6b 3b 6b 4a 6b 4a 3a 2 3a 2 3 3b 3 3b 29. 30. 31. 32. ´ ¸ ´ ¸ 4 3a 4 3a 4b 2 4b 2 x y z x y z x y z x y z *33. *34. *35. *36. ´ ´ ´ ¸ ¸ ´ ¸ ¸ y z x y z x y z x y z x 3a 2b c 3a 2b c 3a 2b c 3a 2b c *37. *38. *39. *40. ´ ´ ´ ¸ ¸ ´ ¸ ¸ 4b 3c 2a 4b 3c 2a 4b 3c 2a 4b 3c 2a EXERCISE 4 WORKED EXAMPLES: Find these sums and differences. Simplify your answers where possible. 5 1 5 +1 6 a b a+ b 5 1 5 +1 6 and so and + = = + = + = = 7 7 7 7 7 7 7 c c c c 5 1 5 -1 4 a b a-b 5 1 5 -1 4 b. and so and - = = - = - = = 7 7 7 7 7 7 7 c c c c 5a a 5a+ a 6a 3a c. + = = = 8 8 8 8 4 5a a 5a- a 4a a d. - = = = 8 8 8 8 2 a b a+ b a b a- b In general, and + = - = c c c c c c a. DS/GRBJMH: апрель 11, 2020 page 3 of 11 WRITTEN EXAMPLES: Find the following sums and differences. Simplify your answers where possible. x y x y x y x y 1. 2. 3. 4. + + 2 2 3 3 4 4 5 5 x y z x y z x y z x y z 5. 6. 7. 8. + + + - + - 2 2 2 3 3 3 4 4 4 5 5 5 3 2 3 2 4 1 4 1 9. 10. 11. 12. + + y y y y x x x x 3 1 3 1 4 2 4 2 13. 14. 15. 16. + + y y y y x x x x 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 17. 18. 19. 20. + + + + - - + - - + y y y y y y y y x x x x x x x x x x x x x y y y y y 21. 22. 23. 24. + + + + + + 2 2 3 3 3 4 4 6 6 6 3x x 3x x 3y y 3y y 25. 26. 27. 28. + + 2 2 2 2 4 4 4 4 5x 3x 5x 3x 5y y 5y y 29. 30. 31. 32. + + 4 4 4 4 6 6 6 6 7x 5x 7x 5x 5y y 5y y 33. 34. 35. 36. + + 8 8 8 8 8 8 8 8 7 5 7 5 5 1 5 1 37. 38. 39. 40. + + 8y 8y 8y 8y 8x 8x 8x 8x 5 1 5 3 4 2 7 5 41. 42. 43. 44. + + 3y 3y 6y 6y 2x 2x 4x 4x 11x 7x 5x x 11x 7x 5x x 11y 7y 5y y 45. 46. 47. + + + + + 12 12 12 12 12 12 12 12 12 12 12 12 11y 7y 5y y 11 7 5 1 11 7 5 1 48. 49. 50. - + + + + + 12 12 12 12 12x 12x 12x 12x 12x 12x 12x 12x 11 7 5 1 11 7 5 1 51. 52. + + 12y 12y 12y 12y 12y 12y 12y 12y x y x y x y z x y z 53. 54. 55. 56. + + - + z z z z a a a a a a EXERCISE 5 WORKED EXAMPLES: a. c. Find these sums and differences. Simplify your answer where possible. 3x x 6x x 6x + x 7x + = + = = 2 4 4 4 4 4 3x x 9x 2x 9x + 2x 11x + = + = = 4 6 12 12 12 12 b. d. 4y 5y 8y 5y 8y -5y 3y y = = = = 3 6 6 6 6 6 2 3x 2y 9x 4y 9x - 4y = = 2 3 6 6 6 Note: Always choose your common denominator to be the lowest one possible. DS/GRBJMH: апрель 11, 2020 page 4 of 11 WRITTEN EXAMPLES: Find the following sums and differences. Simplify your answers where possible. x x x x 3y y 3y y 1. 2. 3. 4. + + 2 4 2 4 4 2 4 2 x x x x 5y y 5y y 5. 6. 7. 8. + + 2 6 2 6 6 2 6 2 x x 2x x 5y 2y 5y y 9. 10. 11. 12. + + 3 6 3 6 6 3 6 3 x x 3x x 5y 3y 5y y 13. 14. 15. 16. + + 4 6 4 6 6 4 6 4 x y x y 2x y 2x y 17. 18. 19. 20. + + 2 3 2 3 3 2 3 2 x y 2x y 3x y 3x 2y 21. 22. 23. 24. + + 3 4 3 4 4 3 4 3 5x 2x x 5x 2x x 5x 2x x 5x 2x x 25. 26. 27. 28. + + + - + - 6 3 2 6 3 2 6 3 2 6 3 2 5x 2y z 5x 3y 2z 5x 3y z 3x 2y z 29. 30. 31. 32. + + + - + - 6 3 2 6 4 3 6 4 2 4 3 2 EXERCISE 6 WORKED EXAMPLES: Find these sums and differences. Simplify your answers where possible. 3 1 3 2 3+2 5 5 1 10 1 10 -1 9 3 a. b. + = + = = = = = = 3y 6y 6y 6y 6y 6y 2y 4x 2x 4x 4x 4x 4x c. 3 5 9 10 9 + 10 19 + = + = = 4y 6y 12y 12y 12y 12y d. 2 4 2y 12x 2y - 12x - = = 3x y 3xy 3xy 3xy Note: Always choose your common denominator to be the lowest one possible. WRITTEN EXAMPLES: Find the following sums and differences. Simplify your answers where possible. 3 1 3 1 5 3 5 3 1. 2. 3. 4. + + 2y 4 y 2y 4y 2x 4 x 2x 4x 3 1 3 1 3 5 3 5 5. 6. 7. 8. + + 2y 6y 2y 6y 2x 6x 2x 6x 4 1 4 5 7 4 11 4 9. 10. 11. 12. + + 6y 3y 6y 3y 3x 6x 3x 6x 3 1 3 1 5 1 5 1 13. 14. 15. 16. + + 6y 4 y 6y 4y 4x 6x 4x 6x 1 1 4 1 1 1 5 2 17. 18. 19. 20. + + 2x 3x 3x 2x 3x 4 x 4x 3x 1 1 3 2 3 4 2 3 21. 22. 23. 24. + + x y x y 2x y x 4y 5 4 4 5 2 5 5 2 25. 26. 27. 28. + + 2x 3y 3x 2y 3x 4 y 4x 3y 5 2 1 5 2 1 5 2 1 5 2 1 29. 30. 31. 32. + + + - + - 6x 3x 2x 6x 3x 2x 6x 3x 2x 6x 3x 2x 5 2 1 5 3 2 5 3 1 3 2 1 *33. *34. *35. *36. + + + + 6x 3y 2z 6x 4 y 3z 6x 4y 2z 4x 3y 2z DS/GRBJMH: апрель 11, 2020 page 5 of 11 EXERCISE 7 ORAL EXAMPLES: 1 1. 2 ´ = ? 2 1 ´3=? 3 2. 19 ´ 2 ´ 1 =? 2 3. Given that 25 x 4 = 100 1 then 25 x 4 x 2 = ? ´3=? 3 37 x 4 x ? = 37 and 25 x 4 x Ошибка!= ? 4 ´? =1 Ошибка!x ? = 1 46 x Ошибка!x ? = 46 and 25 x 4 x Ошибка!= ? Ошибка!x ? = 1 64 x Ошибка!x ? = 64 4. Given that 125 x 8 = 1000 Ошибка!x ? = 1 73 x Ошибка!x ? = 73 then 125 x 8 x 3 = ? Ошибка!x ? = 1 82 x Ошибка!x ? = 82 and 125 x 8 x Ошибка!= ? Ошибка!x ? = 1 91 x Ошибка!x ? = 91 and 125 x 8 x Ошибка!= ? 5. Hence we can multiply one side of an equation by a number provided we also multiply the WORKED EXAMPLES: Solve these equations for x. 3 1 a. b. c. 4x = 4x = 4x = 3 2 2 1 1 1 1 3 Þ ´ 4x = ´ 3 Þ ´ 4x = ´ 1 1 -1 4 4 4 4 2 Þ ´ 4x = ´ 4 4 2 1 4x 1 3 1 4x 1 3 1 4x 1 -1 i.e. ´ i.e. ´ i.e. ´ = ´ = ´ = ´ 4 1 4 1 4 1 4 2 4 1 4 2 28 ´ 3 Þ x= 4 d. g. x = 3 4 x Þ 4 ´ = 4´- 3 4 4 x i.e. ´ = 4´- 3 1 4 Þ x= 12 5x =3 6 6 5x 6 3 Þ ´ = ´ 5 6 5 1 18 5 3 i.e. x = 3 5 i.e. x = Þ x= e. - 1 8 x 1 = 4 2 x 1 Þ4´ =4´ 4 2 4 x 4 1 i.e. ´ = ´ 1 4 1 2 Þ x= f. Þ x=2 h. 5x -1 = 6 2 6 5x 6 -1 Þ ´ = ´ 5 6 5 2 6 i.e. x = 10 3 i.e. x = 5 i. 3 8 x -3 = 4 2 x 3 Þ4´ =4´ 4 2 4 x 4 -3 i.e. ´ = ´ 1 4 1 2 Þ x= 6 5x 3 = 6 2 6 5x 6 3 Þ ´ = ´ 5 6 5 2 18 10 4 i.e. x = 1 5 i.e. x = DS/GRBJMH: апрель 11, 2020 page 6 of 11 WRITTTEN EXAMPLES: Solve these equations for x (x Q). 1. 2x =1 2. 2x=- 3 3. 2x = 5 3 4 5 5. 2x = 6. 2x = 7. 2x = 4 3 6 x x x 9. 10. 11. =1 = 3 =5 2 2 2 x 4 x 3 x -6 13. 14. 15. = = = 2 3 2 4 2 5 17. 3x =1 18. 3x=- 2 19. 3x = 4 1 3 4 21. 3x = 22. 3x = 23. 3x = 2 4 5 x x x 25. 26. 27. =1 = 2 =4 3 3 3 x 1 x 4 x -4 29. 30. 31. = = = 3 2 3 3 3 5 33. 6x =1 34. 6x=- 2 35. 6x = 4 2 3 3 37. 6x = 38. 6x = 39. 6x = 3 4 2 x x x 41. 42. 43. =1 = 2 =4 6 6 6 x 2 x 3 x -3 45. 46. 47. = = = 6 3 6 4 6 2 2x 3x 2x 1 49. 50. 51. =6 =6 = 3 2 3 4 2x 4 2x 5 3x - 4 53. 54. 55. = = = 3 5 3 6 2 5 3x 4x 3x 1 57. 58. 59. =6 =6 = 4 3 4 2 3x 5 3x 2 4x - 5 61. 62. 63. = = = 4 6 4 5 3 6 6x 6x 3 5x -1 65. 66. 67. =4 = = 5 5 4 6 3 - 4. 2x= 7 8. 2x = 12. 16. 20. 24. 28. 32. 36. 40. 44. 48. 52. 56. 60. 64. 68. - 6 5 x = 7 2 x -5 = 2 6 3x= 5 6 3x = 5 x = 5 3 x -6 = 3 5 6x= 3 4 6x = 3 x = 3 6 x 4 = 6 3 3x - 1 = 2 4 3x 5 = 2 6 4x - 1 = 3 2 4x - 2 = 3 5 5x 3 = 6 4 DS/GRBJMH: апрель 11, 2020 page 7 of 11 REVISION EXERCISE If x = 8 and y = -6 find the value of the following fractions in their simplest form. y y x+ y x- y 1. 2. 3. 4. y x+ y x 3x 2y y+2 x y2 5. 6. 7. 8. y-2 x x x Simplify the following fractions. 12 x 8x 6 2x 9. 10. 11. 12. 6y 12 y 2y 8 3x 6y 15 x 4 13. 14. 15. 16. 8y 4x 3 10 x 10 x xy 20 y 17. 18. 19. 20. 15 y xy 20 x 2 2 x y y xy x y 21. 22. 23. 24. 2 2 xy xy x 2y 2 x Find the following sums, differences, products and quotients. Simplify your answers where possible. 1 x 1 x x x 25. 26. 27. y ´ 28. x ¸ ´ ¸ x y y y y y 3x 2y 3x 9x 3x 12z 3x 12z 29. 30. 31. 32. ´ ¸ ´ ¸ 4y 3z 4y 4z 4y 5x 4y 5y 5x 3x 5x 3x 5 3 5 3 33. 34. 35. 36. + + 4y 4 y 4y 4y 4 4 4 4 7x 5x 3x x 7x 5x 3x x 7y 5y 3y y 7y 5y 3y y 37. 38. 39. 36. + + + + + - + 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 7 5 3 1 7 5 3 1 7 5 3 1 7 5 3 1 41. 42. 43. 44. + + + + - + 8y 8y 8y 8y 8y 8y 8y 8y 8x 8x 8x 8x 8x 8x 8x 8x 3x x 4x 5x 3y y 2x y 45. 46. 47. 48. + + 2 6 3 6 4 6 3 4 3 5 4 5 3 4 2 1 49. 50. 51. 52. + + 2x y 3x 4y 2x 4 x 3x 6x 5x 3x 2x 5 3 2 5y 3y 2y 5 3 2 53. 54. 55. 56. + + + + - + + 6y 4y 3y 6 4 3 6x 4 x 3x 6 4 3 Solve these equations for x. 5x 2 x -1 6x - 2 57. 4x = 6 58. 59. 60. = = = 6 3 4 6 5 3 DS/GRBJMH: апрель 11, 2020 page 8 of 11 EXTRA EXERCISE 1. Consider the following fractions Ошибка!, Ошибка!, Ошибка!, Ошибка! a. Which ones are equal to each other? b. Which one is undefined if x = 0? c. What can you say about y in all them? 2. In this question consider any 2 natural numbers (e.g. 4 and 3) a. i. Find the sum of the numbers and hence find the reciprocal of their sum. ii. Find the reciprocal of each number and hence find the sum of their reciprocals. iii. (True or False) The reciprocal of their sum is equal to the sum of their reciprocals. b. i. Find the product of the numbers and hence find the reciprocal of their product. ii. Find the reciprocal of each number and hence find the product of their reciprocals. iii. (True or False) The reciprocal of their product is equal to the product of their reciprocals. c. Rewrite the statements in (a) (iii) and (b) (iii) which are true in terms of x and y (where x and y are natural numbers). d. Is this statement in (c) true if x and y are integers? 3. Copy down and check the first four statements. Copy down and complete the next four statements. 1 a. If x ´ (1+1) = y then y ´ (1- ) = x 2 1 1 If x ´ (1+ ) = y then y ´ (1- ) = x 2 3 1 1 If x ´ (1+ ) = y then y ´ (1- ) = x 3 4 1 1 If x ´ (1+ ) = y then y ´ (1- ) = x 4 5 1 If x ´ (1+ ) = y then 5 1 If x ´ (1+ ) = y then 6 1 If x ´ (1+ ) = y then 10 1 If x ´ (1+ ) = y then n b. Prove the last statement in (a). c. Copy down and complete this statement. 1 If x ´ (1- ) = y then y ´ (1+ ?) = x n DS/GRBJMH: апрель 11, 2020 page 9 of 11 4. a. c. d. e. 5. Copy down and check these statements. 1 1 If x ´ (1+ ) = y then y ´ (1- ) = x 5 6 2 2 If x ´ (1+ ) = y then y ´ (1- ) = x 5 7 3 3 If x ´ (1+ ) = y then y ´ (1- ) = x 5 8 4 4 If x ´ (1+ ) = y then y ´ (1- ) = x 5 9 Copy down and complete this statement. m If x ´ (1+ ) = y then y ´ (1- ?) = x n Prove the statement in (c). b. Copy down and complete these statements. 1 If x ´ (1+ ) = y then y ´ (1- ?) = x n 2 If x ´ (1+ ) = y then y ´ (1- ?) = x n 3 If x ´ (1+ ) = y then y ´ (1- ?) = x n 4 If x ´ (1+ ) = y then y ´ (1- ?) = x n Copy down and complete this statement. m If x ´ (1- ) = y then y ´ (1+ ?) = x n Copy down and check the first four statements. Copy down and complete the next four statements. a. Ошибка! = 1 - Ошибка! Ошибка!+ (Ошибка!)Ошибка! = 1 - (Ошибка!)Ошибка! Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка! = 1-( Ошибка!)Ошибка! Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка! = 1-( Ошибка!)Ошибка! Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!) 5 = 10 = n = Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!) 5 + (Ошибка!)Ошибка! = Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ ... + (Ошибка!) Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ ... + (Ошибка!) b. Ошибка! = Ошибка!- Ошибка!x Ошибка! Ошибка!+ (Ошибка!)Ошибка! = Ошибка!- Ошибка!x (Ошибка!)Ошибка! Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка! = Ошибка!- Ошибка!x (Ошибка!) 3 Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка! = Ошибка!Ошибка!x (Ошибка!)Ошибка! Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!) 5 = 10 = n = Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!) 5 + (Ошибка!)Ошибка! = Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ ... + (Ошибка!) Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ (Ошибка!)Ошибка!+ ... + (Ошибка!) DS/GRBJMH: апрель 11, 2020 page 10 of 11 DS/GRBJMH: апрель 11, 2020 page 11 of 11