15 addition and subtraction of fractions

LCM and LCD
Back to Algebra–Ready Review Content.

Definition of LCM
The least common multiple (LCM) of two or more numbers is
the least number that is the multiple of all of these numbers.
LCM and LCD

Definition of LCM
Example A. Find the LCM of 4 and 6.
LCM and LCD

Definition of LCM
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
LCM and LCD

Definition of LCM
The multiples of 6 are 6, 12, 18, 24, 30,…
LCM and LCD

Definition of LCM
The smallest of the common multiples is 12,
LCM and LCD

Definition of LCM
The smallest of the common multiples is 12, so LCM{4, 6 } = 12.
LCM and LCD

Definition of LCM
LCM and LCD
We may improve the above listing-method for finding the LCM.

Definition of LCM
LCM and LCD
Given two or more numbers, we start with the largest number,

Definition of LCM
LCM and LCD
list its multiples in order until we find the one that can divide all
the other numbers.

Definition of LCM
LCM and LCD
Example B. Find the LCM of 8, 9, and 12.
the other numbers.

Definition of LCM
LCM and LCD
The largest number is 12 and the multiples of 12 are 12, 24,
36, 48, 60, 72, 84 …
the other numbers.

Definition of LCM
LCM and LCD
36, 48, 60, 72, 84 … The first number that is also a multiple
of 8 and 9 is 72.
the other numbers.

Definition of LCM
LCM and LCD
36, 48, 60, 72, 84 … The first number that is also a multiple
of 8 and 9 is 72. Hence LCM{8, 9, 12} = 72.
the other numbers.

But when the LCM is large, the listing method is cumbersome.
LCM and LCD

It's easier to find the LCM by constructing it instead.
LCM and LCD

To construct the LCM:
LCM and LCD

a. Factor each number completely
LCM and LCD

b. For each prime factor, take the highest power appearing in
the factorizations.
LCM and LCD

the factorizations. The LCM is their product.
LCM and LCD

Example C. Construct the LCM of {8, 15, 18}.
LCM and LCD

Factor each number completely,
LCM and LCD

8 = 23
LCM and LCD

8 = 23
15 = 3 * 5
LCM and LCD

8 = 23
15 = 3 * 5
18 = 2 * 32
LCM and LCD

8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor:
LCM and LCD

8 = 23
15 = 3 * 5
18 = 2 * 32
factor
LCM and LCD

LCM and LCD
8 = 23
15 = 3 * 5
18 = 2 * 32
factor: 23, 32, 5,

LCM and LCD
8 = 23
15 = 3 * 5
18 = 2 * 32
factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.

LCM and LCD
The LCM of the denominators of a list of fractions is called the
least common denominator (LCD).
8 = 23
15 = 3 * 5
18 = 2 * 32
factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.

8 = 23
15 = 3 * 5
18 = 2 * 32
factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.
LCM and LCD
The LCM of the denominators of a list of fractions is called the
least common denominator (LCD). Following is an
application of the LCM.

Example D. From one pizza, Joe wants 1/3, Mary wants 1/4
and Chuck wants 1/6.
LCM and LCD

and Chuck wants 1/6.
LCM and LCD
Mary Chuck
In picture:
Joe

and Chuck wants 1/6. How many equal slices should we cut
the pizza into and how many slices should each person take?
LCM and LCD
Mary Chuck
In picture:
Joe

What is the fractional amount of the pizza they want in total?
LCM and LCD
Mary Chuck
In picture:
Joe

LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching.
Mary Chuck
In picture:
Joe

LCM and LCD
We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of
6 are 6, 12, 18, 24, …
Mary Chuck
In picture:
Joe

LCM and LCD
6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4,
then 12 is the LCM.
Mary Chuck
In picture:
Joe

LCM and LCD
then 12 is the LCM. Hence we should cut it into 12 slices
Mary Chuck
In picture:
Joe

Joe gets 12*
LCM and LCD
then 12 is the LCM. Hence we should cut it into 12 slices and
Mary Chuck
In picture:
Joe
1
3

Joe gets 12* = 4 slices
LCM and LCD
Mary Chuck
In picture:
Joe
1
3

Joe gets 12* = 4 slices 1
4
Mary gets 12*
LCM and LCD
Mary Chuck
In picture:
Joe
1
3

4
Mary gets 12* = 3 slices
LCM and LCD
Mary Chuck
In picture:
Joe
1
3

4
1
6
Chuck gets 12*
LCM and LCD
Mary Chuck
In picture:
Joe
1
3

4
1
6
Chuck gets 12* = 2 slices
LCM and LCD
Mary Chuck
In picture:
Joe
1
3

4
1
6
LCM and LCD
Mary Chuck
In picture:
Joe
1
3
In total, that is 4 + 2 + 3 = 9 slices,

4
1
6
LCM and LCD
Mary Chuck
In picture:
Joe
1
3
In total, that is 4 + 2 + 3 = 9 slices, or of the pizza.
9
12

4
1
6
LCM and LCD
Mary Chuck
In picture:
Joe
1
3
In total, that is 4 + 2 + 3 = 9 slices, or of the pizza.
9
12
=
3
4

Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6
of it and Chuck wants 5/12 of it. How many equal slices should
we cut the pizza and how many slices should each person
take?
LCM and LCD

take?
LCM and LCD
In the above example, we found that is the same .
1
3
4
12

take?
LCM and LCD
1
3
4
12
The following theorem tells us how to convert the denominator
of a fraction to a fraction with a different denominator.

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
To convert the fraction into a fraction with denominator d,
the new numerator is * d.
a
b
a
b

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
Example D: Convert to a fraction with denominator 48.
9
16

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
The new denominator is 48,
9
16

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
The new denominator is 48, then the new numerator is
48*
9
16
9
16

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
48*
9
16
9
16
3

take?
LCM and LCD
1
3
4
12
Multiplier Theorem:
a
b
a
b
48* = 27 so =
9
16
9
16
3
9
16
27
48 .

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

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

one slice or ¼ of the pizza, Mary takes two slices or 2/4 of
the pizza,
1
4
2
4

the pizza, altogether they take
+
1
4
2
4
1
4
2
4

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

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

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

+
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
+ =
1
4
2
4
3
4

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

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

Example A:
a. 7
12
+
11
12

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Example A:
a. 7
12
+ =
7 + 11
12
18
12
= 3
2
18/6
12/6
=
+ =
8 + 4 – 2
15
=
2
3
=b.
8
15
4
15
–
2
15
10
15
11
12 =
converted to have the denominator 6. Specifically,
1
2
= 3
6
1
3
= 2
6
+
1
2
1
3
=
?
?

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

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

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

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.

common denominator. We list the steps below.

Addition and Subtraction of Fractions With the Different
Denominator

Denominator
1. Find their LCD

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

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

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

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

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

Denominator
1. Find their LCD
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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

b.
7
12
5
8+ –
16
9

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

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

( ) * 48 / 48
7
12
5
8+ – 16
94
b.
7
12
5
8+ –
16
9

( ) * 48 / 48
67
12
5
8+ – 16
94
b.
7
12
5
8+ –
16
9

( ) * 48 / 48
67
12
5
8+ – 16
94 3
b.
7
12
5
8+ –
16
9

( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
67
12
5
8+ – 16
94 3
b.
7
12
5
8+ –
16
9

( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
67
12
5
8+ – 16
94 3
b.
7
12
5
8+ –
16
9

( ) * 48 / 48
= (4*7 + 6*5 – 3*9) / 48
= (28 + 30 – 27) / 48
=
67
12
5
8+ – 16
94 3
b.
7
12
5
8+ –
16
9
48
31

LCM and LCD
Exercise A. Find the LCM.
1. a.{6, 8} b. {6, 9} c. {3, 4}
d. {4, 10}
2. a.{5, 6, 8} b. {4, 6, 9} c. {3, 4, 5}
d. {4, 6, 10}
3. a.{6, 8, 9} b. {6, 9, 10} c. {4, 9, 10}
d. {6, 8, 10}
4. a.{4, 8, 15} b. {8, 9, 12} c. {6, 9, 15}
5. a.{6, 8, 15} b. {8, 9, 15} c. {6, 9, 16}
6. a.{8, 12, 15} b. { 9, 12, 15} c. { 9, 12, 16}
7. a.{8, 12, 18} b. {8, 12, 20} c. { 12, 15, 16}
8. a.{8, 12, 15, 18} b. {8, 12, 16, 20}
9. a.{8, 15, 18, 20} b. {9, 16, 20, 24}

B. Convert the fractions to fractions with the given
denominators.
10. Convert to denominator 12.
2
3 ,
3
4 ,
5
6 ,
7
4
1
6 ,
3
4 ,
5
6 ,
3
8
7
12 ,
5
4 ,
8
9 ,
11
6
9
10 ,
7
12 ,
13
5 ,
11
15
LCM and LCD

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–
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

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


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

15 addition and subtraction of fractions

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