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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?

Think about the wording.

2 5 8 gallons

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?

8 15 16 pounds

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?

194 3 16 pounds

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.

2

( [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"} } } {} .

Questions & Answers

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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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