1. Fractions Frank Ma © 2011

2. Fractions p Fractions are numbers of the form (or p/q) where p,q  0 are whole numbers. q

3. Fractions p Fractions are numbers of the form (or p/q) where p,q  0 are whole numbers. q 3 6

4. Fractions p Fractions are numbers of the form (or p/q) where p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items. q 3 6

5. Fractions p Fractions are numbers of the form (or p/q) where p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items. Suppose a pizza is cut into 6 equal slices and we have 3 of them, the fraction that represents this quantity is . q 3 6 3 6

6. Fractions p Fractions are numbers of the form (or p/q) where p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items. Suppose a pizza is cut into 6 equal slices and we have 3 of them, the fraction that represents this quantity is . q 3 6 3 6

7. Fractions p Fractions are numbers of the form (or p/q) where p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items. Suppose a pizza is cut into 6 equal slices and we have 3 of them, the fraction that represents this quantity is . q 3 6 3 6 The bottom number is the number of equal parts in the division and it is called the denominator.

8. Fractions p Fractions are numbers of the form (or p/q) where p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items. Suppose a pizza is cut into 6 equal slices and we have 3 of them, the fraction that represents this quantity is . q 3 6 3 6 The bottom number is the number of equal parts in the division and it is called the denominator.

9. Fractions p Fractions are numbers of the form (or p/q) where p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items. Suppose a pizza is cut into 6 equal slices and we have 3 of them, the fraction that represents this quantity is . q 3 6 The top number “3” is the number of parts that we have and it is called the numerator. 3 6 The bottom number is the number of equal parts in the division and it is called the denominator.

10. Fractions p Fractions are numbers of the form (or p/q) where p,q  0 are whole numbers. Fractions are numbers that measure parts of whole items. Suppose a pizza is cut into 6 equal slices and we have 3 of them, the fraction that represents this quantity is . q 3 6 The top number “3” is the number of parts that we have and it is called the numerator. 3 6 The bottom number is the number of equal parts in the division and it is called the denominator. 3/6 of a pizza

11. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 8

12. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 8 How many slices should we cut the pizza into and how do we do this?

13. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 8 Cut the pizza into 8 pieces,

14. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 8 Cut the pizza into 8 pieces, take 5 of them.

15. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 5/8 of a pizza 8 Cut the pizza into 8 pieces, take 5 of them.

16. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 5/8 of a pizza 8 7 12

17. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 5/8 of a pizza 8 7 12 Cut the pizza into 12 pieces,

18. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 5/8 of a pizza 8 7 12 Cut the pizza into 12 pieces,

19. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 5/8 of a pizza 8 7 12 Cut the pizza into 12 pieces, take 7 of them.

20. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 5/8 of a pizza 8 7 or 12 Cut the pizza into 12 pieces, take 7 of them.

21. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 5/8 of a pizza 8 7/12 of a pizza 7 or 12 Cut the pizza into 12 pieces, take 7 of them.

22. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 5/8 of a pizza 8 7/12 of a pizza 7 or 12 8 12 Note that or is the same as 1. 8 12

23. Fractions For larger denominators we can use a pan–pizza for pictures. For example, 5 5/8 of a pizza 8 7/12 of a pizza 7 or 12 8 12 Note that or is the same as 1. 8 12 a Fact: = 1 (provided that a = 0.) a

24. Fractions Whole numbers can be viewed as fractions with denominator 1.

25. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . x 5 1 1

26. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. x 0 5 1 x 1

27. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. However, does not have any meaning, it is undefined. x 0 5 1 x 1 x 0

28. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. However, does not have any meaning, it is undefined. x 0 5 1 x 1 x 0 The Ultimate No-No of Mathematics:

29. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. However, does not have any meaning, it is undefined. x 0 5 1 x 1 x 0 The Ultimate No-No of Mathematics: The denominator (bottom) of a fraction can't be 0.

30. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. However, does not have any meaning, it is undefined. x 0 5 1 x 1 x 0 The Ultimate No-No of Mathematics: The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.)

31. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. However, does not have any meaning, it is undefined. x 0 5 1 x 1 x 0 The Ultimate No-No of Mathematics: The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.) Fractions that represents the same quantity are called equivalent fractions.

32. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. However, does not have any meaning, it is undefined. x 0 5 1 x 1 x 0 The Ultimate No-No of Mathematics: The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.) Fractions that represents the same quantity are called equivalent fractions. 1 2 = 2 4

33. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. However, does not have any meaning, it is undefined. x 0 5 1 x 1 x 0 The Ultimate No-No of Mathematics: The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.) Fractions that represents the same quantity are called equivalent fractions. 1 2 3 = = 2 4 6

34. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. However, does not have any meaning, it is undefined. x 0 5 1 x 1 x 0 The Ultimate No-No of Mathematics: The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.) Fractions that represents the same quantity are called equivalent fractions. … are equivalent fractions. 1 2 3 4 = = = 2 4 6 8

35. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. However, does not have any meaning, it is undefined. x 0 5 1 x 1 x 0 The Ultimate No-No of Mathematics: The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.) Fractions that represents the same quantity are called equivalent fractions. … are equivalent fractions. The fraction with the smallest denominator of all the equivalent fractions is called the reduced fraction. 1 2 3 4 = = = 2 4 6 8

36. Fractions Whole numbers can be viewed as fractions with denominator 1. Thus 5 = and x = . The fraction = 0, where x  0. However, does not have any meaning, it is undefined. x 0 5 1 x 1 x 0 The Ultimate No-No of Mathematics: The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.) Fractions that represents the same quantity are called equivalent fractions. … are equivalent fractions. The fraction with the smallest denominator of all the equivalent fractions is called the reduced fraction. 1 2 3 4 = = = 2 4 6 8 1 is the reduced one in the above list. 2

37. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. a a a / c = b b b / c

38. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, a a a / c = b b b / c

39. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a*c a*c = b*c b*c

40. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a a*c a*c = = b . b*c b*c

41. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a a*c a*c = = b . b*c b*c (Often we omit writing the 1’s after the cancellation.)

42. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a a*c a*c = = b . b*c b*c (Often we omit writing the 1’s after the cancellation.) To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. What's left is the reduced version.

43. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a a*c a*c = = b . b*c b*c (Often we omit writing the 1’s after the cancellation.) To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. What's left is the reduced version. 78 Example A: Reduce the fraction . 54

44. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a a*c a*c = = b . b*c b*c (Often we omit writing the 1’s after the cancellation.) To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. What's left is the reduced version. 78 Example A: Reduce the fraction . 54 78 = 54

45. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a a*c a*c = = b . b*c b*c (Often we omit writing the 1’s after the cancellation.) To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. What's left is the reduced version. 78 Example A: Reduce the fraction . 54 78 78/2 = 54 54/2

46. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a a*c a*c = = b . b*c b*c (Often we omit writing the 1’s after the cancellation.) To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. What's left is the reduced version. 78 Example A: Reduce the fraction . 54 39 78 78/2 = = 54 54/2 27

47. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a a*c a*c = = b . b*c b*c (Often we omit writing the 1’s after the cancellation.) To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. What's left is the reduced version. 78 Example A: Reduce the fraction . 54 39 78 78/2 39/3 = = 54 54/2 27/3 27

48. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a a*c a*c = = b . b*c b*c (Often we omit writing the 1’s after the cancellation.) To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. What's left is the reduced version. 78 Example A: Reduce the fraction . 54 39 78 78/2 39/3 13 = = = 54 54/2 27/3 9 . 27

49. Fractions Factor Cancellation Rule Given a fraction , then that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction. In other words, a common factor of the numerator and the denominator may be canceled as 1, i.e. a a a / c = b b b / c 1 a a*c a*c = = b . b*c b*c (Often we omit writing the 1’s after the cancellation.) To reduce a fraction, we keep divide the top and bottom by common numbers until no more division is possible. What's left is the reduced version. 78 Example A: Reduce the fraction . 54 39 78 78/2 39/3 13 = = = 54 54/2 27/3 9 . 27 or divide both by 6 in one step.

50. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.

51. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term.

52. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression).

53. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor.

54. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled.

55. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled. 2 + 1 3 = 2 + 3 5

56. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled. 2 + 1 3 = 2 + 3 5 This is addition. Can’t cancel!

57. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled. 2 + 1 3 2 + 1 = = 2 + 3 2 + 3 5 This is addition. Can’t cancel!

58. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled. !? 2 + 1 1 3 2 + 1 = = = 2 + 3 2 + 3 3 5 This is addition. Can’t cancel!

59. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled. !? 2 * 1 1 2 + 1 1 3 2 + 1 = = = = 2 * 3 3 2 + 3 2 + 3 3 5 Yes This is addition. Can’t cancel!

60. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled. !? 2 * 1 1 2 + 1 1 3 2 + 1 = = = = 2 * 3 3 2 + 3 2 + 3 3 5 Yes This is addition. Can’t cancel! Improper Fractions and Mixed Numbers

61. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled. !? 2 * 1 1 2 + 1 1 3 2 + 1 = = = = 2 * 3 3 2 + 3 2 + 3 3 5 Yes This is addition. Can’t cancel! Improper Fractions and Mixed Numbers A fraction whose numerator is the same or more than its denominator (e.g.) is said to be improper. 3 2

62. Fractions One common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator. A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled. !? 2 * 1 1 2 + 1 1 3 2 + 1 = = = = 2 * 3 3 2 + 3 2 + 3 3 5 Yes This is addition. Can’t cancel! Improper Fractions and Mixed Numbers A fraction whose numerator is the same or more than its denominator (e.g.) is said to be improper. We may put an improper fraction into mixed form by division. 3 2

63. Improper Fractions and Mixed Numbers 23 Example B. Put into mixed form. 4

64. Improper Fractions and Mixed Numbers 23 Example B. Put into mixed form. 4 · 23 4 = 5 with remainder 3. ·

65. Improper Fractions and Mixed Numbers 23 Example B. Put into mixed form. 4 23 3 · 23 4 = 5 with remainder 3. Hence, = 5 + · 4 4

66. Improper Fractions and Mixed Numbers 23 Example B. Put into mixed form. 4 23 3 3 · 23 4 = 5 with remainder 3. Hence, 5 = = 5 + · 4 4 . 4

67. Improper Fractions and Mixed Numbers 23 Example B. Put into mixed form. 4 23 3 3 · 23 4 = 5 with remainder 3. Hence, 5 = = 5 + · 4 4 . 4 We may put a mixed number into improper fraction by doing the reverse via multiplication.

68. Improper Fractions and Mixed Numbers 23 Example B. Put into mixed form. 4 23 3 3 · 23 4 = 5 with remainder 3. Hence, 5 = = 5 + · 4 4 . 4 We may put a mixed number into improper fraction by doing the reverse via multiplication. 3 Example C: Put into improper form. 5 4

69. Improper Fractions and Mixed Numbers 23 Example B. Put into mixed form. 4 23 3 3 · 23 4 = 5 with remainder 3. Hence, 5 = = 5 + · 4 4 . 4 We may put a mixed number into improper fraction by doing the reverse via multiplication. 3 Example C: Put into improper form. 5 4 3 4*5 + 3 5 = 4 4

70. Improper Fractions and Mixed Numbers 23 Example B. Put into mixed form. 4 23 3 3 · 23 4 = 5 with remainder 3. Hence, 5 = = 5 + · 4 4 . 4 We may put a mixed number into improper fraction by doing the reverse via multiplication. 3 Example C: Put into improper form. 5 4 3 4*5 + 3 23 5 = = 4 4 4

71. Improper Fractions and Mixed Numbers 23 Example B. Put into mixed form. 4 23 3 3 · 23 4 = 5 with remainder 3. Hence, 5 = = 5 + · 4 4 . 4 We may put a mixed number into improper fraction by doing the reverse via multiplication. 3 Example C: Put into improper form. 5 4 3 4*5 + 3 23 5 = = 4 4 4