addition and subtraction of fractions

1. Addition and Subtraction of Fractions

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

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

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

5. 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,
Addition and Subtraction of Fractions
1
4
2
4

6. 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
+
Addition and Subtraction of Fractions
1
4
2
4
1
4
2
4

7. 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
+
Addition and Subtraction of Fractions
1
4
2
4
= 3
4
of the entire pizza.
1
4
2
4

8. 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
+
Addition and Subtraction of Fractions
1
4
2
4
= 3
4
of the entire pizza. In picture:
+ =
1
4
2
4
3
4

9. 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
+
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions With the Same
Denominator
1
4
2
4
= 3
4
of the entire pizza. In picture:
+ =
1
4
2
4
3
4

10. 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
+
Addition and Subtraction of Fractions
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
1
4
2
4
= 3
4
of the entire pizza. In picture:
+ =
1
4
2
4
3
4

11. 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
+
Addition and Subtraction of Fractions
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
1
4
2
4
= 3
4
of the entire pizza. In picture:
±
a
d
b
d
= a ± b
d
+ =
1
4
2
4
3
4

12. 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
+
Addition and Subtraction of Fractions
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.
1
4
2
4
= 3
4
of the entire pizza. In picture:
±
a
d
b
d
= a ± b
d
+ =
1
4
2
4
3
4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

34. Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ = 8 + 4 – 2
15
= 2
3
=b.
Addition and Subtraction of Fractions
8
15
4
15
– 2
15
10
15
11
12 =
Fractions with different denominators can’t be added directly
since the “size” of the fractions don’t match. For example
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,
+
3
6
2
6
=
1
2
=
3
6
1
3
=
2
6
Hence,
1
2
+
1
3
=
3
6
+
2
6
=
5
6
5
6

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

36. 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.
Addition and Subtraction of Fractions

37. 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

38. 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
Addition and Subtraction of Fractions With the Different
Denominator

39. 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
Addition and Subtraction of Fractions With the Different
Denominator
1. Find their LCD

40. 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
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. 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
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. 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
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.
Example B:
5
6
3
8
+a.

43. 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
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.
Example B:
5
6
3
8
+a.
Step 1: To find the LCD, list the multiples of 8

44. 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
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.
Example B:
5
6
3
8
+a.
Step 1: To find the LCD, list the multiples of 8 which are
8, 16, 24, ..

45. 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
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.
Example B:
5
6
3
8
+a.
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.
For , the new numerator is 24 * = 20,
5
6
5
6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

68. 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 the quantity
x by a, then divide by a, we get back x.
Addition and Subtraction of Fractions

69. 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 the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5
Addition and Subtraction of Fractions

70. 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 the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2,
Addition and Subtraction of Fractions

71. 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 the quantity
x by a, then divide by a, we get back x.
For example, 2 * 5 / 5 = 10/5 = 2, 3 * 8 / 8 =
Addition and Subtraction of Fractions

72. 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 the 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.
Addition and Subtraction of Fractions

73. 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 the 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.
Addition and Subtraction of Fractions
Multiplier Method (for adding and subtracting fractions)

74. 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 the 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.
Addition and Subtraction of Fractions
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. Example C. a.
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 the 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.
Addition and Subtraction of Fractions
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
6
3
8
+

76. Example C. a.
The LCD is 24.
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 the 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.
Addition and Subtraction of Fractions
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
6
3
8
+

77. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
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 the 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.
Addition and Subtraction of Fractions
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
6
3
8
+

78. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
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 the 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.
Addition and Subtraction of Fractions
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
6
3
8
+

79. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
4
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 the 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.
Addition and Subtraction of Fractions
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
6
3
8
+

80. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24
4 3
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 the 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.
Addition and Subtraction of Fractions
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
6
3
8
+

81. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24 = (4*5 + 3*3) / 24
4 3
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 the 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.
Addition and Subtraction of Fractions
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
6
3
8
+

82. Example C. a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
5
6
3
8+( ) * 24 / 24 = (4*5 + 3*3) / 24 = 29/24 =
4 3 29
24
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 the 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.
Addition and Subtraction of Fractions
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
6
3
8
+

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

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

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

86. ( ) * 48 / 48
7
12
5
8+ – 16
94
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9

87. ( ) * 48 / 48
67
12
5
8+ – 16
94
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9

88. ( ) * 48 / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9

89. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9

90. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9

91. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
48
31

92. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
48
31

93. ( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
The LCD is 48. Multiply the problem by 48, expand the
multiplication, then divide the result by 48.
Addition and Subtraction of Fractions
b.
7
12
5
8
+ –
16
9
48
31
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.

94. We obtain are two products.
a
b
c
d
Cross Multiplication
A useful eyeballing-procedure with two fractions is the cross-
multiplication.
a*d b*c
! Make sure that the denominators cross over and up with
the numerators stay put.
a
b
c
d
adbc
Addition and Subtraction of Fractions
take their denominators and multiply them diagonally across.
To cross-multiply two given fractions as shown,
Do not cross downward as shown here.

95. Cross–Multiplication for Addition or Subtraction
We may cross multiply to add or subtract two fractions with
the product of the denominators as the common denominator.
a
b
c
d
Afterwards we reduce if necessary for the simplified answer.
Example D. Calculate
± =
a*d ±b*c
bd
3
5
5
6
– =
5*5 – 6*3
6*5
7
30
=a.
5
12
5
9
– =
5*12 – 9*5
9*12
15
108
=b. 5
36
=
In a. the LCD = 30 = 6*5 so the crossing method is the same
as the Multiplier Method. However in b. the crossing method
yielded an answer that needed to be reduced. we need both
methods.
Addition and Subtraction of Fractions

96. Exercise A. Calculate and simplify the answers.
1
2
3
2
+1. 2. 3. 4.5
3
1
3+
5
4
3
4
+
5
2
3
2
+
5
5
3
5–5. 6. 7. 8.6
6
5
6
–
9
9
4
9– 1
4
7–
B. Calculate by the Multiplier Method and simplify the
answers.
1
2
1
3
+17. 18. 19. 20.1
2
1
3
–
2
3
3
2
+
3
4
2
5
+
5
6
4
7–21. 22. 23. 24.7
10
2
5
–
5
11
3
4+
5
9
7
15–
Addition and Subtraction of Fractions
9. 1
2
9
– 10. 1
3
8
– 11. 4
3
4– 12. 8
3
8
–
13. 11
3
5– 14. 9
3
8
– 15.14
1
6
– 16. 21
3
11
–5 6 8

97. Addition and Subtraction of FractionsC. Addition and Subtraction of Fractions
6
1
4
5
25. 26. 27. 28.
29. 30. 31. 32.
33. 34. 35.
36. 38.37.
39. 40.
6
5
4
7

12
7
9
5

12
5
8
3

16
5
24
7

18
5
12
7

20
3
24
11

15
7
18
5

9
4
6
1
4
3

10
7
6
1
4
5

12
5
6
1
8
3

12
1
9
5
8
7

9
2
16
1
24
5

18
7
12
1
4
5

12
7
16
1
18
5

10
7
18
5
24
7


98. Addition and Subtraction of Fractions
7
18
443. + 15
7
12
11
+44.
18
19
24
7
– 15
5
4
541. – 6
4
9
5
– 6
3
8
5
+ 42.
12
+
–
5
745. – 12
7
9
1
– 6
5
8
5
+ 46.
8
–3