1. Addition and Subtraction of Fractions Frank Ma © 2011

2. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices

3. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices

4. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, 1 4

5. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, 1 2 4 4

6. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take 1 2 + 4 4 1 2 4 4

7. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take 1 2 3 + = 4 4 4 of the entire pizza. 1 2 4 4

8. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take 1 2 3 + = 4 4 4 of the entire pizza. In picture: = + 1 2 3 4 4 4

9. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take 1 2 3 + = 4 4 4 of the entire pizza. In picture: = + 1 2 3 4 4 4 Addition and Subtraction of Fractions With the Same Denominator

10. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take 1 2 3 + = 4 4 4 of the entire pizza. In picture: = + 1 2 3 4 4 4 Addition and Subtraction of Fractions With the Same Denominator To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators

11. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take 1 2 3 + = 4 4 4 of the entire pizza. In picture: = + 1 2 3 4 4 4 Addition and Subtraction of Fractions With the Same Denominator To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators a b a ± b ± = d d d

12. Addition and Subtraction of Fractions Suppose a pizza is cut into 4 equal slices and Joe takes one slice or ¼ of the pizza, Mary takes two slices or 2/4 of the pizza, altogether they take 1 2 3 + = 4 4 4 of the entire pizza. In picture: = + 1 2 3 4 4 4 Addition and Subtraction of Fractions With the Same Denominator To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators ,then simplify the result. a b a ± b ± = d d d

13. Addition and Subtraction of Fractions Example A: a. 7 11 + 12 12

14. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 11 + = = 12 12 12

15. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 11 + = = 12 12 12 12

16. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 11 18/6 = + = = = 12 12 12 12/6 12

17. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12

18. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 4 2 b. + = – 15 15 15

19. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 8 4 2 b. + = = – 15 15 15 15

20. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 8 4 2 10 b. + = = – 15 15 15 15 15

21. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15

22. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match.

23. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example 1 2

24. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example + 1 1 3 2

25. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example = + ? 1 1 3 2 ?

26. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example = + ? 1 1 3 2 ? To add them, first find the LCD of ½ and 1/3, which is 6.

27. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example = + ? 1 1 3 2 ? To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices.

28. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example = + ? 1 1 3 2 ? To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6.

29. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example = + ? 1 1 3 2 ? To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, 1 3 1 2 = = 2 6 3 6

30. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example = + ? 1 1 3 2 ? To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, 1 3 1 2 = = 2 6 3 6

31. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example = + ? 1 3 3 6 ? To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, 1 3 1 2 = = 2 6 3 6

32. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example = + ? 1 3 3 6 ? To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, 1 3 1 2 = = 2 6 3 6

33. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example = + ? 2 3 6 6 ? To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, 1 3 1 2 = = 2 6 3 6

34. Addition and Subtraction of Fractions Example A: a. 7 7 + 11 18 3 11 18/6 = + = = = 12 12 12 2 12/6 12 8 + 4 – 2 2 8 4 2 10 b. + = = = – 3 15 15 15 15 15 Fractions with different denominators can’t be added directly since the “size” of the fractions don’t match. For example = + 5 2 3 6 6 6 To add them, first find the LCD of ½ and 1/3, which is 6. We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically, 1 3 1 2 1 1 3 2 5 = = Hence, + = + = 2 6 3 6 2 3 6 6 6

35. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them.

36. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator.

37. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.

38. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions With the Different Denominator

39. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions With the Different Denominator 1. Find their LCD

40. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions With the Different Denominator 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator.

41. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions With the Different Denominator 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator. 3. Add and subtract the adjusted fractions then simplify the result.

42. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions With the Different Denominator 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator. 3. Add and subtract the adjusted fractions then simplify the result. 5 3 a. Example B: + 6 8

43. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions With the Different Denominator 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator. 3. Add and subtract the adjusted fractions then simplify the result. 5 3 a. Example B: + 6 8 Step 1: To find the LCD, list the multiples of 8

44. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions With the Different Denominator 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator. 3. Add and subtract the adjusted fractions then simplify the result. 5 3 a. Example B: + 6 8 Step 1: To find the LCD, list the multiples of 8 which are 8, 16, 24, ..

45. Addition and Subtraction of Fractions We need to convert fractions of different denominators to a common denominator in order to add or subtract them. The easiest common denominator to use is the LCD, the least common denominator. We list the steps below. Addition and Subtraction of Fractions With the Different Denominator 1. Find their LCD 2. Convert all the different-denominator-fractions to the have the LCD as the denominator. 3. Add and subtract the adjusted fractions then simplify the result. 5 3 a. Example B: + 6 8 Step 1: To find the LCD, list the multiples of 8 which are 8, 16, 24, .. we see that the LCD is 24.

46. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator.

47. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 For , the new numerator is 24 * = 20, 6 6

48. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24

49. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 For , the new numerator is 24 * = 9, 8 8

50. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24

51. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions.

52. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 + = + 6 8 24 24

53. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24

54. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – 12 8 16

55. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. 12 8 16

56. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16

57. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16 7 = 28 48 * 12

58. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16 7 7 28 = 28 48 * so = 12 12 48

59. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16 7 7 28 = 28 48 * so = 12 12 48 5 = 30 48 * 8

60. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16 7 7 28 = 28 48 * so = 12 12 48 5 5 30 = 30 48 * so = 8 8 48

61. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16 7 7 28 = 28 48 * so = 12 12 48 5 5 30 = 30 48 * so = 8 8 48 9 = 27 48 * 16

62. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16 7 7 28 = 28 48 * so = 12 12 48 5 5 30 = 30 48 * so = 8 8 48 9 9 27 = 27 48 * so = 16 16 48

63. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16 7 7 28 30 27 28 = 28 = 48 * so = + – 12 12 48 48 48 48 5 5 30 = 30 48 * so = 8 8 48 9 9 27 = 27 48 * so = 16 16 48

64. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16 7 7 28 30 27 28 = 28 = 48 * so = + – 12 12 48 48 48 48 5 5 28 + 30 – 27 30 = 30 48 * so = = 48 8 8 48 9 9 27 = 27 48 * so = 16 16 48

65. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16 7 7 28 30 27 28 = 28 = 48 * so = + – 12 12 48 48 48 48 5 5 28 + 30 – 27 30 = 30 48 * so = = 48 8 8 48 31 9 9 27 = = 27 48 * so = 48 16 16 48

66. Addition and Subtraction of Fractions Step 2: Convert each fraction to have 24 as the denominator. 5 5 5 20 For , the new numerator is 24 * = 20, hence = 6 6 6 24 3 3 3 9 For , the new numerator is 24 * = 9, hence = 8 8 8 24 Step 3: Add the converted fractions. 5 3 20 9 29 + = + = 6 8 24 24 24 7 5 9 b. + – The LCD is 48. Convert: 12 8 16 7 7 28 30 27 28 = 28 = 48 * so = + – 12 12 48 48 48 48 5 5 28 + 30 – 27 30 = 30 48 * so = = 48 8 8 48 31 9 9 27 = = 27 48 * so = 48 16 16 48

67. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying.

68. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x.

69. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5

70. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2,

71. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 =

72. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.

73. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3. Multiplier Method(for adding and subtracting fractions)

74. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3. Multiplier Method(for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD.

75. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3. Multiplier Method(for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. 5 3 Example C: a. + 6 8

76. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3. Multiplier Method(for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. 5 3 Example C: a. The LCD is 24. + 6 8

77. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3. Multiplier Method(for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. 5 3 Example C: a. The LCD is 24. Multiply the problem by 24, then divide by 24. + 6 8

78. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3. Multiplier Method(for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. 5 3 Example C: a. The LCD is 24. Multiply the problem by 24, then divide by 24. + 6 8 5 3 24 / 24 ( ) + * 6 8

79. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3. Multiplier Method(for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. 5 3 Example C: a. The LCD is 24. Multiply the problem by 24, then divide by 24. + 6 8 4 5 3 24 / 24 ( ) + * 6 8

80. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3. Multiplier Method(for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. 5 3 Example C: a. The LCD is 24. Multiply the problem by 24, then divide by 24. + 6 8 4 3 5 3 24 / 24 ( ) + * 6 8

81. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3. Multiplier Method(for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. 5 3 Example C: a. The LCD is 24. Multiply the problem by 24, then divide by 24. + 6 8 4 3 5 3 24 / 24 = (4*5 + 3*3) / 24 ( ) + * 6 8

82. Addition and Subtraction of Fractions We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying. This method is based on the fact that if we multiply a quantity x by a, then divide by a, we get back x. For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3. Multiplier Method(for adding and subtracting fractions) To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD. 5 3 Example C: a. The LCD is 24. Multiply the problem by 24, then divide by 24. + 6 8 29 4 3 5 3 24 / 24 = (4*5 + 3*3) / 24 = 29/24 = ( ) + * 24 6 8

83. Addition and Subtraction of Fractions 7 5 9 b. + – 12 8 16

84. Addition and Subtraction of Fractions 7 5 9 b. + – 12 8 16 The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.

85. Addition and Subtraction of Fractions 7 5 9 b. + – 12 8 16 The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48. 5 9 7 + – ( ) * 48 / 48 12 8 16

86. Addition and Subtraction of Fractions 7 5 9 b. + – 12 8 16 The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48. 4 5 9 7 + – ( ) * 48 / 48 12 8 16

87. Addition and Subtraction of Fractions 7 5 9 b. + – 12 8 16 The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48. 6 4 5 9 7 + – ( ) * 48 / 48 12 8 16

88. Addition and Subtraction of Fractions 7 5 9 b. + – 12 8 16 The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48. 6 3 4 5 9 7 + – ( ) * 48 / 48 12 8 16

89. Addition and Subtraction of Fractions 7 5 9 b. + – 12 8 16 The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48. 6 3 4 5 9 7 + – ( ) * 48 / 48 12 8 16 = (4*7 + 6*5 –3*9) / 48

90. Addition and Subtraction of Fractions 7 5 9 b. + – 12 8 16 The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48. 6 3 4 5 9 7 + – ( ) * 48 / 48 12 8 16 = (4*7 + 6*5 –3*9) / 48 = (28 + 30 – 27) / 48

91. Addition and Subtraction of Fractions 7 5 9 b. + – 12 8 16 The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48. 6 3 4 5 9 7 + – ( ) * 48 / 48 12 8 16 = (4*7 + 6*5 –3*9) / 48 = (28 + 30 – 27) / 48 = 31 48

92. Addition and Subtraction of Fractions 7 5 9 b. + – 12 8 16 The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48. 6 3 4 5 9 7 + – ( ) * 48 / 48 12 8 16 = (4*7 + 6*5 –3*9) / 48 = (28 + 30 – 27) / 48 = 31 48 We will learn the cross–multiplication method to + or – two fractions shortly. Together with the above multiplier method, these two methods offer the most efficient ways to handle problems containing + or – of fractions. These two methods extend to operations of the rational (fractional) formulas and will use these two methods extensively.