This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to add and subtract mixed numbers. By the end of the module students should be able to add and subtract mixed numbers.
Section overview
The Method of Converting to Improper Fractions
To add or subtract mixed numbers, convert each mixed number to an improper fraction, then add or subtract the resulting improper fractions.
Sample set a
Find the following sums and differences.
8
3
5
+
5
1
4
size 12{8 { {3} over {5} } +5 { {1} over {4} } } {} . Convert each mixed number to an improper fraction.
8
3
5
=
5
⋅
8
+
3
5
=
40
+
3
5
=
43
5
size 12{8 { {3} over {5} } = { {5 cdot 8+3} over {5} } = { {"40"+3} over {5} } = { {"43"} over {5} } } {}
5
1
4
=
4
⋅
5
+
1
4
=
20
+
1
4
=
21
4
size 12{5 { {1} over {4} } = { {4 cdot 5+1} over {4} } = { {"20"+1} over {4} } = { {"21"} over {4} } } {} Now add the improper fractions
43
5
and
21
4
size 12{ { {"43"} over {5} } " and " { {"21"} over {4} } } {} .
43
5
+
21
4 The LCD = 20.
43
5
+
21
4
=
43
⋅
4
20
+
21
⋅
5
20
=
172
20
+
105
20
=
172
+
105
20
=
277
20
Convert this improper fraction to a mixed number.
=
13
17
20
Thus,
8
3
5
+
5
1
4
=
13
17
20
size 12{8 { {3} over {5} } +5 { {1} over {4} } ="13" { {"17"} over {"20"} } } {} .
3
1
8
−
5
6
size 12{3 { {1} over {8} } - { {5} over {6} } } {} . Convert the mixed number to an improper fraction.
3
1
8
=
3
⋅
8
+
1
8
=
24
+
1
8
=
25
8
size 12{3 { {1} over {8} } = { {3 cdot 8+1} over {8} } = { {"24"+1} over {8} } = { {"25"} over {8} } } {}
25
8
−
5
6
size 12{ { {"25"} over {8} } - { {5} over {6} } } {} The LCD = 24.
25
8
−
5
6
=
25
⋅
3
24
−
5
⋅
4
24
=
75
24
−
20
24
=
75
−
20
24
=
55
24
Convert his improper fraction to a mixed number.
=
2
7
24
Thus,
3
1
8
−
5
6
=
2
7
24
size 12{3 { {1} over {8} } - { {5} over {6} } =2 { {7} over {"24"} } } {} .
Practice set a
Find the following sums and differences.
1
5
9
+
3
2
9
size 12{1 { {5} over {9} } +3 { {2} over {9} } } {}
4
7
9
size 12{4 { {7} over {9} } } {}
10
3
4
−
2
1
2
size 12{"10" { {3} over {4} } - 2 { {1} over {2} } } {}
8
1
4
size 12{8 { {1} over {4} } } {}
2
7
8
+
5
1
4
size 12{2 { {7} over {8} } +5 { {1} over {4} } } {}
8
1
8
size 12{8 { {1} over {8} } } {}
8
3
5
−
3
10
size 12{8 { {3} over {5} } - { {3} over {"10"} } } {}
8
3
10
size 12{8 { {3} over {"10"} } } {}
16
+
2
9
16
size 12{"16"+2 { {9} over {"16"} } } {}
18
9
16
size 12{"18" { {9} over {"16"} } } {}
Exercises
For the following problems, perform each indicated operation.
3
1
8
+
4
3
8
size 12{3 { {1} over {8} } +4 { {3} over {8} } } {}
7
1
2
size 12{7 { {1} over {2} } } {}
5
1
3
+
6
1
3
size 12{5 { {1} over {3} } +6 { {1} over {3} } } {}
10
5
12
+
2
1
12
size 12{"10" { {5} over {"12"} } +2 { {1} over {"12"} } } {}
12
1
2
size 12{"12" { {1} over {2} } } {}
15
1
5
−
11
3
5
size 12{"15" { {1} over {5} } -"11" { {3} over {5} } } {}
9
3
11
+
12
3
11
size 12{9 { {3} over {"11"} } +"12" { {3} over {"11"} } } {}
21
6
11
size 12{"21" { {6} over {"11"} } } {}
1
1
6
+
3
2
6
+
8
1
6
size 12{1 { {1} over {6} } +3 { {2} over {6} } +8 { {1} over {6} } } {}
5
3
8
+
1
1
8
−
2
5
8
size 12{5 { {3} over {8} } +1 { {1} over {8} } -2 { {5} over {8} } } {}
3
7
8
size 12{3 { {7} over {8} } } {}
3
5
+
5
1
5
size 12{ { {3} over {5} } +5 { {1} over {5} } } {}
2
2
9
−
5
9
size 12{2 { {2} over {9} } - { {5} over {9} } } {}
1
2
3
size 12{1 { {2} over {3} } } {}
6
+
11
2
3
size 12{6+"11" { {2} over {3} } } {}
17
−
8
3
14
size 12{"17"-8 { {3} over {"14"} } } {}
8
11
14
size 12{8 { {"11"} over {"14"} } } {}
5
1
3
+
2
1
4
size 12{5 { {1} over {3} } +2 { {1} over {4} } } {}
6
2
7
−
1
1
3
size 12{6 { {2} over {7} } -1 { {1} over {3} } } {}
4
20
21
size 12{4 { {"20"} over {"21"} } } {}
8
2
5
+
4
1
10
size 12{8 { {2} over {5} } +4 { {1} over {"10"} } } {}
1
1
3
+
12
3
8
size 12{1 { {1} over {3} } +"12" { {3} over {8} } } {}
13
17
24
size 12{"13" { {"17"} over {"24"} } } {}
3
1
4
+
1
1
3
−
2
1
2
size 12{3 { {1} over {4} } +1 { {1} over {3} } -2 { {1} over {2} } } {}
4
3
4
−
3
5
6
+
1
2
3
size 12{4 { {3} over {4} } -3 { {5} over {6} } +1 { {2} over {3} } } {}
2
7
12
size 12{"2" { {7} over {12} } } {}
3
1
12
+
4
1
3
+
1
1
4
size 12{3 { {1} over {"12"} } +4 { {1} over {3} } +1 { {1} over {4} } } {}
5
1
15
+
8
3
10
−
5
4
5
size 12{5 { {1} over {"15"} } +8 { {3} over {"10"} } -5 { {4} over {5} } } {}
7
17
30
size 12{7 { {"17"} over {"30"} } } {}
7
1
3
+
8
5
6
−
2
1
4
size 12{7 { {1} over {3} } +8 { {5} over {6} } -2 { {1} over {4} } } {}
19
20
21
+
42
6
7
−
5
14
+
12
1
7
size 12{"19" { {"20"} over {"21"} } +"42" { {6} over {7} } - { {5} over {"14"} } +"12" { {1} over {7} } } {}
74
25
42
size 12{"74" { {"25"} over {"42"} } } {}
1
16
+
4
3
4
+
10
3
8
−
9
size 12{ { {1} over {"16"} } +4 { {3} over {4} } +"10" { {3} over {8} } -9} {}
11
−
2
9
+
10
1
3
−
2
3
−
5
1
6
+
6
1
18
size 12{"11"- { {2} over {9} } +"10" { {1} over {3} } - { {2} over {3} } -5 { {1} over {6} } +6 { {1} over {"18"} } } {}
21
1
3
size 12{"21" { {1} over {3} } } {}
5
2
+
2
1
6
+
11
1
3
−
11
6
size 12{ { {5} over {2} } +2 { {1} over {6} } +"11" { {1} over {3} } - { {"11"} over {6} } } {}
1
1
8
+
9
4
−
1
16
−
1
32
+
19
8
size 12{1 { {1} over {8} } + { {9} over {4} } - { {1} over {"16"} } - { {1} over {"32"} } + { {"19"} over {8} } } {}
5
21
32
size 12{5 { {"21"} over {"32"} } } {}
22
3
8
−
16
1
7
size 12{"22" { {3} over {8} } -"16" { {1} over {7} } } {}
15
4
9
+
4
9
16
size 12{"15" { {4} over {9} } +4 { {9} over {"16"} } } {}
20
1
144
size 12{"20" { {1} over {"144"} } } {}
4
17
88
+
5
9
110
size 12{4 { {"17"} over {"88"} } +5 { {9} over {"110"} } } {}
6
11
12
+
2
3
size 12{6 { {"11"} over {"12"} } + { {2} over {3} } } {}
7
7
12
size 12{7 { {7} over {"12"} } } {}
8
9
16
−
7
9
size 12{8 { {9} over {"16"} } - { {7} over {9} } } {}
5
2
11
−
1
12
size 12{5 { {2} over {"11"} } - { {1} over {"12"} } } {}
5
13
132
size 12{5 { {"13"} over {"132"} } } {}
18
15
16
−
33
34
size 12{"18" { {"15"} over {"16"} } - { {"33"} over {"34"} } } {}
1
89
112
−
21
56
size 12{1 { {"89"} over {"112"} } - { {"21"} over {"56"} } } {}
1
47
212
size 12{1 { {"47"} over {"212"} } } {}
11
11
24
−
7
13
18
size 12{"11" { {"11"} over {"24"} } -7 { {"13"} over {"18"} } } {}
5
27
84
−
3
5
42
+
1
1
21
size 12{5 { {"27"} over {"84"} } -3 { {5} over {"42"} } +1 { {1} over {"21"} } } {}
3
1
4
size 12{3 { {1} over {4} } } {}
16
1
48
−
16
1
96
+
1
144
size 12{"16" { {1} over {"48"} } -"16" { {1} over {"96"} } + { {1} over {"144"} } } {}
A man pours
2
5
8
size 12{2 { {5} over {8} } } {} gallons of paint from a bucket into a tray. After he finishes pouring, there are
1
1
4
size 12{1 { {1} over {4} } } {} gallons of paint left in his bucket. How much paint did the man pour into the tray?
Hint Think about the wording.
A particular computer stock opened at
37
3
8
size 12{"37" { {3} over {8} } } {} and closed at
38
1
4
size 12{"38" { {1} over {4} } } {} . What was the net gain for this stock?
A particular diet program claims that
4
3
16
size 12{4 { {3} over {"16"} } } {} pounds can be lost the first month,
3
1
4
size 12{3 { {1} over {4} } } {} pounds can be lost the second month, and
1
1
2
size 12{1 { {1} over {2} } } {} pounds can be lost the third month. How many pounds does this diet program claim a person can lose over a 3-month period?
If a person who weighs
145
3
4
size 12{"145" { {3} over {4} } } {} pounds goes on the diet program described in the problem above, how much would he weigh at the end of 3 months?
If the diet program described in the problem above makes the additional claim that from the fourth month on, a person will lose
1
1
8
size 12{1 { {1} over {8} } } {} pounds a month, how much will a person who begins the program weighing
208
3
4
size 12{"208" { {3} over {4} } } {} pounds weight after 8 months?
Exercises for review
(
[link] ) Use exponents to write
4
⋅
4
⋅
4
size 12{4 cdot 4 cdot 4} {} .
(
[link] ) Find the greatest common factor of 14 and 20.
(
[link] ) Convert
16
5
size 12{ { {"16"} over {5} } } {} to a mixed number.
(
[link] ) Find the sum.
4
9
+
1
9
+
2
9
size 12{ { {4} over {9} } + { {1} over {9} } + { {2} over {9} } } {} .
7
9
size 12{ { {7} over {9} } } {}
(
[link] ) Find the difference.
15
26
−
3
10
size 12{ { {"15"} over {"26"} } - { {3} over {"10"} } } {} .