123a-1-f5 addition and subtraction of fractions

Addition and Subtraction of Fractions

Suppose a pizza is cut into 4 equal slices

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

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

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

1
2
3
+
=
4
4
4
of the entire pizza.
1
2
4
4

1
2
3
+
=
4
4
4
of the entire pizza. In picture:
=
+
1
2
3
4
4
4

1
2
3
+
=
4
4
4
=
+
1
2
3
4
4
4
Addition and Subtraction of Fractions With the Same Denominator

1
2
3
+
=
4
4
4
=
+
1
2
3
4
4
4
To add or subtract fractions of the same denominator, keep the same denominator, add or subtract the numerators

1
2
3
+
=
4
4
4
=
+
1
2
3
4
4
4
a
b
a ± b
±
=
d
d
d

1
2
3
+
=
4
4
4
=
+
1
2
3
4
4
4
,then simplify the result.
a
b
a ± b
±
=
d
d
d

Example A:
a.
7
11
+
12
12

Example A:
a.
7
7 + 11
11
+
=
=
12
12
12

Example A:
a.
7
7 + 11
18
11
+
=
=
12
12
12
12

Example A:
a.
7
7 + 11
18
11
18/6
=
+
=
=
=
12
12
12
12/6
12

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12

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

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

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
8
4
2
10
b.
+
=
=
–
15
15
15
15
15

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
2
8
4
2
10
b.
+
=
=
=
–
3
15
15
15
15
15

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

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

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
2
8
4
2
10
b.
+
=
=
=
–
3
15
15
15
15
15
+
1
1
3
2

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
2
8
4
2
10
b.
+
=
=
=
–
3
15
15
15
15
15
=
+
?
1
1
3
2
?

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
2
8
4
2
10
b.
+
=
=
=
–
3
15
15
15
15
15
=
+
?
1
1
3
2
?
To add them, first find the LCD of ½ and 1/3, which is 6.

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
2
8
4
2
10
b.
+
=
=
=
–
3
15
15
15
15
15
=
+
?
1
1
3
2
?
We then cut each pizza into 6 slices.

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
2
8
4
2
10
b.
+
=
=
=
–
3
15
15
15
15
15
=
+
?
1
1
3
2
?
We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6.

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
2
8
4
2
10
b.
+
=
=
=
–
3
15
15
15
15
15
=
+
?
1
1
3
2
?
We then cut each pizza into 6 slices. Both fractions may be converted to have the denominator 6. Specifically,
1
3
1
2
=
=
2
6
3
6

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
2
8
4
2
10
b.
+
=
=
=
–
3
15
15
15
15
15
=
+
?
1
3
3
6
?
1
3
1
2
=
=
2
6
3
6

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
2
8
4
2
10
b.
+
=
=
=
–
3
15
15
15
15
15
=
+
?
2
3
6
6
?
1
3
1
2
=
=
2
6
3
6

Example A:
a.
7
7 + 11
18
3
11
18/6
=
+
=
=
=
12
12
12
2
12/6
12
8 + 4 – 2
2
8
4
2
10
b.
+
=
=
=
–
3
15
15
15
15
15
=
+
5
2
3
6
6
6
1
3
1
2
1
1
3
2
5
=
=
Hence,
+
=
+
=
2
6
3
6
2
3
6
6
6

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.

The easiest common denominator to use is the LCD, the least common denominator. We list the steps below.

Addition and Subtraction of Fractions With the Different Denominator

1. Find their LCD

1. Find their LCD
2. Convert all the different-denominator-fractions to the have
the LCD as the denominator.

1. Find their LCD
3. Add and subtract the adjusted fractions then simplify the result.

1. Find their LCD
5
3
a.
Example B:
+
6
8

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

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

1. Find their LCD
5
3
a.
Example B:
+
6
8
Step 1: To find the LCD, list the multiples of 8 which are
8, 16, 24, .. we see that the LCD is 24.

Step 2: Convert each fraction to have 24 as the denominator.

5
5
For , the new numerator is 24 * = 20,
6
6

5
5
5
20
For , the new numerator is 24 * = 20, hence
=
6
6
6
24

5
5
5
20
=
6
6
6
24
3
3
For , the new numerator is 24 * = 9,
8
8

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
Step 3: Add the converted fractions.

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
+
=
+
6
8
24
24

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
12
8
16

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
12
8
16

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16
7
= 28
48 *
12

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16
7
7
28
= 28
48 *
so
=
12
12
48

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16
7
7
28
= 28
48 *
so
=
12
12
48
5
= 30
48 *
8

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16
7
7
28
= 28
48 *
so
=
12
12
48
5
5
30
= 30
48 *
so
=
8
8
48

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16
7
7
28
= 28
48 *
so
=
12
12
48
5
5
30
= 30
48 *
so
=
8
8
48
9
= 27
48 *
16

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16
7
7
28
= 28
48 *
so
=
12
12
48
5
5
30
= 30
48 *
so
=
8
8
48
9
9
27
= 27
48 *
so
=
16
16
48

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16
7
7
28
30
27
28
= 28
=
48 *
so
=
+
–
12
12
48
48
48
48
5
5
30
= 30
48 *
so
=
8
8
48
9
9
27
= 27
48 *
so
=
16
16
48

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16
7
7
28
30
27
28
= 28
=
48 *
so
=
+
–
12
12
48
48
48
48
5
5
28 + 30 – 27
30
= 30
48 *
so
=
=
48
8
8
48
9
9
27
= 27
48 *
so
=
16
16
48

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16
7
7
28
30
27
28
= 28
=
48 *
so
=
+
–
12
12
48
48
48
48
5
5
28 + 30 – 27
30
= 30
48 *
so
=
=
48
8
8
48
31
9
9
27
=
= 27
48 *
so
=
48
16
16
48

5
5
5
20
=
6
6
6
24
3
3
3
9
=
8
8
8
24
5
3
20
9
29
+
=
+
=
6
8
24
24
24
7
5
9
b.
+
–
The LCD is 48.
Convert:
12
8
16
7
7
28
30
27
28
= 28
=
48 *
so
=
+
–
12
12
48
48
48
48
5
5
28 + 30 – 27
30
= 30
48 *
so
=
=
48
8
8
48
31
9
9
27
=
= 27
48 *
so
=
48
16
16
48
HW. pg. 25 61 – 81 all

We introduce the following Multiplier-Method to add or subtract fractions to reduce the amount of repetitive copying.

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

For example, 2* 5 / 5

For example, 2* 5 / 5 = 10/5 = 2,

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 =

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.
Multiplier Method(for adding and subtracting fractions)

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.
To add or subtract fractions, multiply the problem by the LCD (expand it distributive using law), then divide by the LCD.

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.
5
3
Example C: a.
+
6
8

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.
5
3
Example C: a.
The LCD is 24.
+
6
8

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.
5
3
Example C: a.
The LCD is 24. Multiply the problem by 24, then divide by 24.
+
6
8

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.
5
3
Example C: a.
+
6
8
5
3
24 / 24
(
)
+
*
6
8

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.
5
3
Example C: a.
+
6
8
4
5
3
24 / 24
(
)
+
*
6
8

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.
5
3
Example C: a.
+
6
8
4
3
5
3
24 / 24
(
)
+
*
6
8

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.
5
3
Example C: a.
+
6
8
4
3
5
3
24 / 24 = (4*5 + 3*3) / 24
(
)
+
*
6
8

For example, 2* 5 / 5 = 10/5 = 2, 3* 8 / 8 = 24/8 = 3.
5
3
Example C: a.
+
6
8
29
4
3
5
3
24 / 24 = (4*5 + 3*3) / 24 = 29/24 =
(
)
+
*
24
6
8

7
5
9
b.
+
–
12
8
16

7
5
9
b.
+
–
12
8
16
The LCD is 48. Multiply the problem by 48, expand the multiplication, divide the result by 48.

7
5
9
b.
+
–
12
8
16
5
9
7
+
–
(
)
* 48 / 48
12
8
16

7
5
9
b.
+
–
12
8
16
4
5
9
7
+
–
(
)
* 48 / 48
12
8
16

7
5
9
b.
+
–
12
8
16
6
4
5
9
7
+
–
(
)
* 48 / 48
12
8
16

7
5
9
b.
+
–
12
8
16
6
3
4
5
9
7
+
–
(
)
* 48 / 48
12
8
16

7
5
9
b.
+
–
12
8
16
6
3
4
5
9
7
+
–
(
)
* 48 / 48
12
8
16
= (4*7 + 6*5 –3*9) / 48

7
5
9
b.
+
–
12
8
16
6
3
4
5
9
7
+
–
(
)
* 48 / 48
12
8
16
= (4*7 + 6*5 –3*9) / 48
= (28 + 30 – 27) / 48

7
5
9
b.
+
–
12
8
16
6
3
4
5
9
7
+
–
(
)
* 48 / 48
12
8
16
= (4*7 + 6*5 –3*9) / 48
= (28 + 30 – 27) / 48
=
31
48

Exercise A. Calculate and simplify the answers.
1
3
5
3
5
3
5
1
1.
2.
3.
4.
+
+
+
+
2
2
4
4
2
2
3
3
5
3
9
4
4
6
5
5.
6.
7.
8.
–
–
–
–
1
5
5
9
9
7
6
6
3
2
3
3
11.
9.
–
12.
–
10.
–
–
4
1
8
1
4
9
8
8
1
3
3
3
15.
6
8
–
16.
–
14
21
5
14.
–
13.
–
9
11
6
11
8
5
B. Calculate by the Multiplier Method and simplify the answers.
1
1
2
3
3
2
1
1
17.
18.
19.
20.
+
+
+
–
2
3
3
2
4
5
2
3
5
4
5
3
5
7
7
2
21.
22.
23.
24.
–
+
–
–
6
7
11
4
9
15
10
5

C. Addition and Subtraction of Fractions

123a-1-f5 addition and subtraction of fractions

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$123a-1-f5 addition and subtraction of fractions$ 123a-1-f5 addition and subtraction of fractions Presentation Transcript