multiplication and divisin of fractions

1. Fractions

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

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

4. 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.
p
q
Fractions
3
6

5. 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 .
p
q
3
6
Fractions
3
6

6. 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 .
p
q
3
6
3
6
Fractions

7. 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 .
p
q
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
3
6
Fractions

8. 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 .
p
q
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
3
6
Fractions

9. 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 .
p
q
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
The top number “3” is the
number of parts that we
have and it is called the
numerator.
3
6
Fractions

10. 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 .
p
q
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
The top number “3” is the
number of parts that we
have and it is called the
numerator.
3
6
Fractions
3/6 of a pizza

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

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

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

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

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

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

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

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

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

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

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

22. For larger denominators we can use a pan–pizza for
pictures. For example,
5
8
Fractions
7
12
5/8 of a pizza
Note that or is the same as 1.8
8
12
12
7/12 of a pizza
or

23. For larger denominators we can use a pan–pizza for
pictures. For example,
5
8
Fractions
7
12
5/8 of a pizza
Fact: a
a
Note that or is the same as 1.8
8
12
12
= 1 (provided that a = 0.)
7/12 of a pizza
or

24. Fractions
We may talk about the fractional amount of a group of items.

25. Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.
b. Out of an audience of 72 people at a movie, 7/12 of them
like the show very much. How many people is that?

26. Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.
To calculate such amounts, we always divide the group into
parts indicated by the denominator, then retrieve the number
of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them
like the show very much. How many people is that?

27. Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.
To calculate such amounts, we always divide the group into
parts indicated by the denominator, then retrieve the number
of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them
like the show very much. How many people is that?
3
4 Divide $100 into
4 equal parts.

28. Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.
To calculate such amounts, we always divide the group into
parts indicated by the denominator, then retrieve the number
of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them
like the show very much. How many people is that?
3
4 Divide $100 into
4 equal parts.
100 ÷ 4 = 25
so each part is $25,

29. Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.
To calculate such amounts, we always divide the group into
parts indicated by the denominator, then retrieve the number
of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them
like the show very much. How many people is that?
3
4 Divide $100 into
4 equal parts.
Take 3 parts. 100 ÷ 4 = 25
so each part is $25,

30. Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.
To calculate such amounts, we always divide the group into
parts indicated by the denominator, then retrieve the number
of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them
like the show very much. How many people is that?
3
4 Divide $100 into
4 equal parts.
Take 3 parts. 100 ÷ 4 = 25
so each part is $25,
3 parts make $75.
So ¾ of $100 is $75.

31. Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.
To calculate such amounts, we always divide the group into
parts indicated by the denominator, then retrieve the number
of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them
like the show very much. How many people is that?
3
4 Divide $100 into
4 equal parts.
Take 3 parts. 100 ÷ 4 = 25
so each part is $25,
3 parts make $75.
So ¾ of $100 is $75.
7
12 Divide 72 people
into 12 equal parts.

32. Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.
To calculate such amounts, we always divide the group into
parts indicated by the denominator, then retrieve the number
of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them
like the show very much. How many people is that?
3
4 Divide $100 into
4 equal parts.
Take 3 parts. 100 ÷ 4 = 25
so each part is $25,
3 parts make $75.
So ¾ of $100 is $75.
7
12 Divide 72 people
into 12 equal parts.
72 ÷ 12 = 6
so each part consists of 6 people,

33. Fractions
Example A. a. What is ¾ of $100?
We may talk about the fractional amount of a group of items.
To calculate such amounts, we always divide the group into
parts indicated by the denominator, then retrieve the number
of parts indicated by the numerator.
b. Out of an audience of 72 people at a movie, 7/12 of them
like the show very much. How many people is that?
3
4 Divide $100 into
4 equal parts.
Take 3 parts. 100 ÷ 4 = 25
so each part is $25,
3 parts make $75.
So ¾ of $100 is $75.
7
12 Divide 72 people
into 12 equal parts.
Take 7 parts.
72 ÷ 12 = 6
so each part consists of 6 people,
7 parts make 42 people.
So 7/12 of 92 people is 42 people.

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

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

36. Whole numbers can be viewed as fractions with denominator 1.
Thus 5 = and x = . The fraction = 0, where x  0.5
1
x
1
0
x
Fractions

37. 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.
5
1
x
1
0
x
x
0
Fractions

38. 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.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:

39. 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.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
The denominator (bottom) of a fraction can't
be 0.

40. 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.
5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:
The denominator (bottom) of a fraction can't
be 0. (It's undefined if the denominator is 0.)

41. 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.
5
1
x
1
0
x
x
0
Fractions
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.

42. 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.
5
1
x
1
0
x
x
0
Fractions
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

43. 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.
5
1
x
1
0
x
x
0
Fractions
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
=
3
6

44. 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.
5
1
x
1
0
x
x
0
Fractions
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
=
2
4
=
3
6
=
4
8

45. 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.
5
1
x
1
0
x
x
0
Fractions
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
=
2
4
=
3
6
=
4
8

46. 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.
5
1
x
1
0
x
x
0
Fractions
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
=
2
4
=
3
6
=
4
8
is the reduced one in the above list.
1
2

47. 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
b
a
b =
a / c
Fractions
b / c

48. 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
b
a
b =
a / c
Fractions
b / c

49. 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
b
a
b =
a / c
=
a*c
b*c
a*c
b*c
1
Fractions
b / c

50. 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
b
a
b =
a / c
a
b .
=
a*c
b*c
=
a*c
b*c
1
Fractions
b / c

51. 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
b
a
b =
a / c
a
b .
=
a*c
b*c
=
a*c
b*c
1
Fractions
b / c
(Often we omit writing the 1’s after the cancellation.)

52. 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
b
a
b =
a / c
a
b .
=
a*c
b*c
=
a*c
b*c
1
Fractions
b / c
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.
(Often we omit writing the 1’s after the cancellation.)

53. 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
b
a
b =
a / c
a
b .
=
a*c
b*c
=
a*c
b*c
1
Fractions
b / c
Example B. Reduce the fraction .78
54
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.
(Often we omit writing the 1’s after the cancellation.)

54. 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
b
a
b =
a / c
a
b .
=
a*c
b*c
=
a*c
b*c
1
Fractions
b / c
Example B. Reduce the fraction .78
54
78
54
=
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.
(Often we omit writing the 1’s after the cancellation.)

55. 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
b
a
b =
a / c
a
b .
=
a*c
b*c
=
a*c
b*c
1
Fractions
b / c
Example B. Reduce the fraction .78
54
78
54
=
78/2
54/2
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.
(Often we omit writing the 1’s after the cancellation.)

56. 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
b
a
b =
a / c
a
b .
=
a*c
b*c
=
a*c
b*c
1
Fractions
b / c
Example B. Reduce the fraction .78
54
78
54
=
78/2
54/2
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.
=
39
27
(Often we omit writing the 1’s after the cancellation.)

57. 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
b
a
b =
a / c
a
b .
=
a*c
b*c
=
a*c
b*c
1
Fractions
b / c
Example B. Reduce the fraction .78
54
78
54
=
78/2
54/2
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.
=
39/3
27/3
39
27
(Often we omit writing the 1’s after the cancellation.)

58. 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
b
a
b =
a / c
a
b .
=
a*c
b*c
=
a*c
b*c
1
Fractions
b / c
Example B. Reduce the fraction .78
54
78
54
=
78/2
54/2
= 13
9 .
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.
=
39/3
27/3
39
27
(Often we omit writing the 1’s after the cancellation.)

59. 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
b
a
b =
a / c
a
b .
=
a*c
b*c
=
a*c
b*c
1
Fractions
b / c
Example B. Reduce the fraction .78
54
78
54
=
78/2
54/2
= 13
9 .
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.
=
39/3
27/3
or divide both by 6 in one step.
39
27
(Often we omit writing the 1’s after the cancellation.)

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.

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.

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).

63. 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.

64. 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.

65. 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.
2 + 1
2 + 3
3
5
=
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.

66. 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.
2 + 1
2 + 3
3
5
=
This is addition. Can’t cancel!
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.

67. 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.
2 + 1
2 + 3
= 2 + 1
2 + 3
3
5
=
This is addition. Can’t cancel!
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.

68. 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.
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
This is addition. Can’t cancel!
!?
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.

69. 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.
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
This is addition. Can’t cancel!
!? 2 * 1
2 * 3
=
1
3
Yes
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.

70. 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.
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
This is addition. Can’t cancel!
!?
Improper Fractions and Mixed Numbers
2 * 1
2 * 3
=
1
3
Yes
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.

71. 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.
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
This is addition. Can’t cancel!
!?
A fraction whose numerator is the same or more than its
denominator (e.g. ) is said to be improper .
Improper Fractions and Mixed Numbers
3
2
2 * 1
2 * 3
=
1
3
Yes
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.

72. 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.
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
This is addition. Can’t cancel!
!?
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.
Improper Fractions and Mixed Numbers
3
2
2 * 1
2 * 3
=
1
3
Yes
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.

73. 23
4
Improper Fractions and Mixed Numbers
Example C. Put into mixed form.

74. 23
4
23 4 = 5 with remainder 3.·
·
Improper Fractions and Mixed Numbers
Example C. Put into mixed form.

75. 23
4
23 4 = 5 with remainder 3. Hence,·
·
23
4
= 5 +
Improper Fractions and Mixed Numbers
Example C. Put into mixed form.
3
4

76. 23
4
23 4 = 5 with remainder 3. Hence,·
·
23
4
= 5 + 5 3
4 .
Improper Fractions and Mixed Numbers
Example C. Put into mixed form.
3
4
=

77. 23
4
23 4 = 5 with remainder 3. Hence,·
·
23
4
= 5 + 5 3
4 .
Improper Fractions and Mixed Numbers
Example C. Put into mixed form.
3
4
=
We may put a mixed number into improper fraction by doing
the reverse via multiplication.

78. 23
4
23 4 = 5 with remainder 3. Hence,·
·
23
4
= 5 + 5 3
4 .
Improper Fractions and Mixed Numbers
Example C. Put into mixed form.
3
4
=
We may put a mixed number into improper fraction by doing
the reverse via multiplication.
Example D. Put into improper form.5 3
4

79. 23
4
23 4 = 5 with remainder 3. Hence,·
·
23
4
= 5 + 5 3
4 .
5 3
4
= 4*5 + 3
4
Improper Fractions and Mixed Numbers
Example C. Put into mixed form.
3
4
=
We may put a mixed number into improper fraction by doing
the reverse via multiplication.
Example D. Put into improper form.5 3
4

80. 23
4
23 4 = 5 with remainder 3. Hence,·
·
23
4
= 5 + 5 3
4 .
5 3
4
= 4*5 + 3
4
23
4
=
Improper Fractions and Mixed Numbers
Example C. Put into mixed form.
3
4
=
We may put a mixed number into improper fraction by doing
the reverse via multiplication.
Example D. Put into improper form.5 3
4

81. 23
4
23 4 = 5 with remainder 3. Hence,·
·
23
4
= 5 + 5 3
4 .
5 3
4
= 4*5 + 3
4
23
4
=
Improper Fractions and Mixed Numbers
Example C. Put into mixed form.
3
4
=
We may put a mixed number into improper fraction by doing
the reverse via multiplication.
Example D. Put into improper form.5 3
4

82. Rule for Multiplication of Fractions
Multiplication and Division of Fractions

83. c
d
=
a*c
b*d
a
b
*
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

84. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
12
25
15
8
*a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

85. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

86. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

87. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

88. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
=
3*3
2*5
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

89. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

90. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
b.
8
9
7
8
*
10
11
9
10
**
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

91. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** =
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

92. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** =
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

93. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** =
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

94. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** =
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Each set of cancellation
produces a “1”, which
does not affect final the
product.
Multiplication and Division of Fractions

95. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** = =
7
11
a.
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

96. c
d
=
a*c
b*d
a
b
*
Example E. Multiply by reducing first.
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** = =
7
11
a.
Can't do this for addition and subtraction, i.e.
c
d
=
a c
b d
a
b
±
±
±
Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.
Multiplication and Division of Fractions

97. a
b
d
a
b
d
d
1
The fractional multiplications are important.or* *
Often in these problems the denominator b can be cancelled
against d = .
Multiplication and Division of Fractions

98. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18a.
The fractional multiplications are important.or* *
*
Often in these problems the denominator b can be cancelled
against d = .
Multiplication and Division of Fractions

99. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18a.
The fractional multiplications are important.
6
or* *
*
Often in these problems the denominator b can be cancelled
against d = .
Multiplication and Division of Fractions

100. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6a.
The fractional multiplications are important.
6
or* *
* *
Often in these problems the denominator b can be cancelled
against d = .
Multiplication and Division of Fractions

101. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
or* *
* *
Often in these problems the denominator b can be cancelled
against d = .
Multiplication and Division of Fractions

102. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
11
16
48b.
or* *
* *
*
Often in these problems the denominator b can be cancelled
against d = .
Multiplication and Division of Fractions

103. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
11
16
48b.
3
or* *
* *
*
Often in these problems the denominator b can be cancelled
against d = .
Multiplication and Division of Fractions

104. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
11
16
48b.
3
or* *
* *
* = 3 * 11
Often in these problems the denominator b can be cancelled
against d = .
Multiplication and Division of Fractions

105. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
11
16
48b.
3
or* *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelled
against d = .
Multiplication and Division of Fractions

106. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
11
16
48b.
3
or* *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelled
against d = .
The often used phrases " (fraction) of .." are translated to
multiplications correspond to this kind of problems.
Multiplication and Division of Fractions

107. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
11
16
48b.
3
or* *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelled
against d = .
The often used phrases " (fraction) of .." are translated to
multiplications correspond to this kind of problems.
Example G. a. What is of $108?2
3
Multiplication and Division of Fractions

108. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
11
16
48b.
3
or* *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelled
against d = .
The often used phrases " (fraction) of .." are translated to
multiplications correspond to this kind of problems.
Example G. a. What is of $108?2
3
* 108
2
3
The statement translates into
Multiplication and Division of Fractions

109. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
11
16
48b.
3
or* *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelled
against d = .
The often used phrases " (fraction) of .." are translated to
multiplications correspond to this kind of problems.
Example G. a. What is of $108?2
3
* 108
2
3
36
The statement translates into
Multiplication and Division of Fractions

110. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
11
16
48b.
3
or* *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelled
against d = .
The often used phrases " (fraction) of .." are translated to
multiplications correspond to this kind of problems.
Example G. a. What is of $108?2
3
* 108 = 2 * 36
2
3
36
The statement translates into
Multiplication and Division of Fractions

111. a
b
d
a
b
d
d
1
Example F. Multiply by cancelling first.
2
3
18 = 2 6 = 12a.
The fractional multiplications are important.
6
11
16
48b.
3
Multiplication and Division of Fractions
or* *
* *
* = 3 * 11 = 33
Often in these problems the denominator b can be cancelled
against d = .
The often used phrases " (fraction) of .." are translated to
multiplications correspond to this kind of problems.
Example G. a. What is of $108?2
3
* 108 = 2 * 36 = 72 $.
2
3
36
The statement translates into

112. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
Multiplication and Division of Fractions

113. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
For chocolate, ¼ of 48 is
1
4
* 48
Multiplication and Division of Fractions

114. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
Multiplication and Division of Fractions

115. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

116. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
1
3
* 48
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

117. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
1
3
* 48
16
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

118. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
1
3
* 48
16
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

119. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
1
3
* 48
16
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops.
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

120. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
1
3
* 48
16
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

121. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
1
3
* 48
16
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48
=
20/4
48/4
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

122. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
1
3
* 48
16
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48
=
20/4
48/4
=
5
12
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

123. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
1
3
* 48
16
c. A class has x students, ¾ of them are girls, how many girls
are there?
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48
=
20/4
48/4
=
5
12
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

124. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
1
3
* 48
16
c. A class has x students, ¾ of them are girls, how many girls
are there?
3
4
* x.
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48
=
20/4
48/4
=
5
12
It translates into multiplication as
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

125. b. A bag of mixed candy contains 48 pieces of chocolate,
caramel and lemon drops. 1/4 of them are chocolate, 1/3 of
them are caramel. How many pieces of each are there? What
fraction of the candies are lemon drops?
1
3
* 48
16
c. A class has x students, ¾ of them are girls, how many girls
are there?
3
4
* x.
For chocolate, ¼ of 48 is
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,
so there are 16 pieces of caramel candies.
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48
=
20/4
48/4
=
5
12
It translates into multiplication as
so there are 12 pieces of chocolate candies.
Multiplication and Division of Fractions

126. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions

127. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
So the reciprocal of is ,
2
3
3
2

128. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5

129. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
the reciprocal of is 3,1
3

130. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3

131. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3

132. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3

133. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3
2
3
3
2* = 1,

134. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3
2
3
3
2* = 1, 5 1
5* = 1,

135. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3
2
3
3
2* = 1, 5 1
5* = 1, x 1
x* = 1,

136. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 1
5* = 1, x 1
x* = 1,
1
x

137. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 1
5* = 1, x 1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 ,*
1
2

138. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 1
5* = 1, x 1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 , both yield 5.*
1
2

139. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 1
5* = 1, x 1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 , both yield 5.*
1
2
Rule for Division of Fractions
To divide by a fraction x, restate it as multiplying by the
reciprocal 1/x , that is,

140. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 1
5* = 1, x 1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 , both yield 5.*
1
2
Rule for Division of Fractions
To divide by a fraction x, restate it as multiplying by the
reciprocal 1/x , that is, d
c
a
b
*
c
d
=
a
b
÷
reciprocate

141. The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions
Two Important Facts About Reciprocals
I. The product of x with its reciprocal is 1.
So the reciprocal of is ,
2
3
3
2
the reciprocal of 5 is ,
1
5
and the reciprocal of x is .1
xthe reciprocal of is 3,1
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5 1
5* = 1, x 1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 , both yield 5.*
1
2
Rule for Division of Fractions
To divide by a fraction x, restate it as multiplying by the
reciprocal 1/x , that is, d
c
=
a*d
b*c
a
b
*
c
d
=
a
b
÷
reciprocate

142. Example F. Divide the following fractions.
8
15
=
12
25
a. ÷
Reciprocal and Division of Fractions

143. Example F. Divide the following fractions.
15
8
12
25
*
8
15
=
12
25
a. ÷
Reciprocal and Division of Fractions

144. Example F. Divide the following fractions.
15
8
12
25
*
8
15
=
12
25 2
3
a. ÷
Reciprocal and Division of Fractions

145. Example F. Divide the following fractions.
15
8
12
25
*
8
15
=
12
25 5
3
2
3
a. ÷
Reciprocal and Division of Fractions

146. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a. ÷
Reciprocal and Division of Fractions

147. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
÷
÷ =b.
Reciprocal and Division of Fractions

148. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
÷
÷ = *b.
Reciprocal and Division of Fractions

149. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
÷
÷ = *b.
Reciprocal and Division of Fractions

150. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions

151. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
1
65d. ÷

152. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
6
1*1
6 =5d. ÷ 5

153. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
6
1 = 30*1
6 =5d. ÷ 5

154. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
6
1 = 30*1
6 =5d. ÷ 5
Example G. We have ¾ cups of sugar. A cookie recipe calls
for 1/16 cup of sugar for each cookie. How many cookies
can we make?

155. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
6
1 = 30*1
6 =5d. ÷ 5
Example G. We have ¾ cups of sugar. A cookie recipe calls
for 1/16 cup of sugar for each cookie. How many cookies
can we make?
We can make
3
4
÷ 1
16

156. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
6
1 = 30*1
6 =5d. ÷ 5
Example G. We have ¾ cups of sugar. A cookie recipe calls
for 1/16 cup of sugar for each cookie. How many cookies
can we make?
We can make
3
4
÷ 1
16
= 3
4
*
16
1

157. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
6
1 = 30*1
6 =5d. ÷ 5
Example G. We have ¾ cups of sugar. A cookie recipe calls
for 1/16 cup of sugar for each cookie. How many cookies
can we make?
We can make
3
4
÷ 1
16
= 3
4
*
16
1
4

158. Example F. Divide the following fractions.
15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Reciprocal and Division of Fractions
6
1 = 30*1
6 =5d. ÷ 5
Example G. We have ¾ cups of sugar. A cookie recipe calls
for 1/16 cup of sugar for each cookie. How many cookies
can we make?
We can make
3
4
÷ 1
16
= 3
4
*
16
1
= 3 * 4 = 12 cookies.
4
HW: Do the web homework "Multiplication of Fractions"

159. Multiplication and Division of Fractions
Remember to cancel first!

160. Multiplication and Division of Fractions
Exercise. B.
12. In a class of 48 people, 1/3 of them are boys, how many girls are there?
13. In a class of 60 people, 3/4 of them are not boys, how many boys are there?
14. In a class of 72 people, 5/6 of them are not girls, how many boys are there?
15. In a class of 56 people, 3/7 of them are not boys, how many girls are there?
16. In a class of 60 people, 1/3 of them are girls, how many are not girls?
17. In a class of 60 people, 2/5 of them are not girls, how are not boys?
18. In a class of 108 people, 5/9 of them are girls, how many are not boys?
A mixed bag of candies has 72 pieces of colored candies, 1/8 of them are red, 1/3
of them are green, ½ of them are blue and the rest are yellow.
19. How many green ones are there?
20. How many are blue?
21. How many are not yellow?
20. How many are not blue and not green?
21. In a group of 108 people, 4/9 of them adults (aged 18 or over), 1/3 of them are
teens (aged from 12 to 17) and the rest are children. Of the adults 2/3 are females,
3/4 of the teens are males and 1/2 of the children are girls. Complete the following
table.
22. How many females are there and what is the fraction of the females to entire
group?
23. How many are not male–adults and what is the fraction of them to entire
group?

161. Improper Fractions and Mixed Numbers
B. Convert the following improper fractions into mixed
numbers then convert the mixed numbers back to the
improper form.
9
2
11
3
9
4
13
5
37
12
86
11
121
17
1. 2. 3. 4. 5. 6. 7.
Exercise. A. Reduce the following fractions.
4
6 ,
8
12 ,
15
9 ,
24
18 ,
30
42 ,
54
36 ,
60
48 ,
72
108