Section 2.5
Addition and Subtraction with Mixed
Numbers
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Objectives
o Add mixed numbers.
o Subtract mixed numbers.
o Subtract mixed numbers with borrowing.
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Addition with Mixed Numbers
To Add Mixed Numbers
1. Add the fraction parts.
2. Add the whole numbers.
3. Write the answer as a mixed number with the
fraction part less than 1.
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Example 1: Adding Mixed Numbers
2
5
Find the sum: 5  8
9
9
Solution
We can write each number as a sum and then use the
commutative and associative properties of addition to
treat the whole numbers and fraction parts separately.
2
5
2
5
 2 5
5  8  5   8    5  8    
 9 9
9
9
9
9
7
7
 13   13
9
9
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Example 1: Adding Mixed Numbers (cont.)
Or, vertically,
2
5
9
5
8
9
7
13
9
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Example 2: Adding Mixed Numbers
1
7
Find the sum: 35  22
6
18
Solution
In this case, the fractions do not have the same
denominator.
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Example 2: Adding Mixed Numbers (cont.)
1
7
1
7
35  22  35   22 
6
18
6
18
1 3 7 
  35  22     
 6 3 18 
 3 7
 57    
 18 18 
10
 57
18
5
 57
9
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LCD  18
Reduce the fraction part.
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Example 3: Adding Mixed Numbers
3
3
Find the sum: 5  9  2
4
10
Solution
Since the whole number 2 has no fraction part, we
must be careful to align the whole numbers if we write
the numbers vertically.
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Example 3: Adding Mixed Numbers (cont.)
3
3 5
15 
5 5 
5
4
4 5
20 
 LCD  20
3 2
6 
3
9  9
9
10 2
20 
10
2
2
2
1
1
21
 16  1
 17
16
20
20
20
Fraction is greater than 1.
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Change to a mixed number.
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Example 4: Application of Adding
with Mixed Numbers
1
4
A triangle has sides measuring 3 meters, 6 meters,
5
3
14
and 6
meters. Find the perimeter (total distance
15
around) of the triangle.
Solution
We find the perimeter by adding the lengths of the
three sides.
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Example 4: Application of Adding
with Mixed Numbers (cont.)
1 5
5 
1
3 3  3 
3 5
15
3

4 3
4
12 
6  6   6  LCD  15
5 3
5
15 
14
14
14 
6
6 
 6
15
15 
15
31
1
1
15  15  2  17 meters
15
15
15
1
The perimeter is 17
meters.
15
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Subtraction with Mixed Numbers
To Subtract Mixed Numbers
1. Subtract the fraction parts.
2. Subtract the whole numbers.
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Example 5: Subtracting Mixed Numbers
3
2
Find the difference: 10  6
7
7
Solution
Or, we can subtract vertically.
3
2
 3 2
10  6  10  6    
 7 7
7
7
1
 4
7
1
4
7
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3
10
7
2
 6
7
1
4
7
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Example 6: Subtracting Mixed Numbers
4
1
Find the difference: 13  7
5
3
Solution
4
4 3
12 
13  13   13
5
5 3
15 
LCD  15

1 5
1
5
7  7   7 
3 5
3
15 
7
6
15
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Example 7: Subtracting Mixed Numbers
with Borrowing
1
Find the difference: 6  2
3
Solution
3
Borrow 1 from 6 in the form of as follows:
3
3
6  5
3
1
1
2  2
3
3
2
3
3
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Example 8: Subtracting Mixed Numbers
with Borrowing
5
13
Find the difference: 76  29
12
20
Solution
First, find the LCD and then borrow 1 if necessary.
12  2  2  3
 LCD  2  2  3  5  60
20  2  2  5
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Example 8: Subtracting Mixed Numbers
with Borrowing (cont.)
5

76
12
5 5
76  
12 5
25
76

60
85
75
60
Borrow 1 
60
.
60
13
13 3
39
39
 29
  29    29
  29
20
20 3
60
60
46
23
46
 46
60
30
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Example 9: Subtracting Mixed Numbers
with Borrowing
2
5
Find the difference: 4  1
9 9
Solution
2
4
9
5
1
9
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5
2
is larger than , so "borrow" 1 from 4.
9
9
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Example 9: Subtracting Mixed Numbers
with Borrowing (cont.)
2
2
2
11
Rewrite 4 as 3  1   3  1  3 .
9
9
9
9
2
2
11
4  31  3
9
9
9
5
5
5
 1   1  1
9
9
9
6
2
2 2
9
3
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Practice Problems
Find the following sums.
1
7
1. 11
2.
9
8
2
3
9
1
9
5
1
7
3
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2
3. 1
7
1
8
2
3
2
5
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Practice Problems (cont.)
Find the following differences.
3
4. 3
4
2
1
5. 6
3
1
4
2
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2
6. 5
5
7
3
10
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Practice Problem Answers
1
1. 20
3
97
2. 9
120
27
3. 12
70
3
4. 1
4
5
5. 1
6
7
6. 1
10
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