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Addition and Subtraction of Fractions, Comparing Fractions, and Complex Fractions: Addition and Subtraction of Mixed Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: 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.

Example 1

835+514835+514 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 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} } } {}

514=45+14=20+14=214514=45+14=20+14=214 size 12{5 { {1} over {4} } = { {4 cdot 5+1} over {4} } = { {"20"+1} over {4} } = { {"21"} over {4} } } {} Now add the improper fractions 435 and 214435 and 214 size 12{ { {"43"} over {5} } " and " { {"21"} over {4} } } {}.

435+214435+214 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. = 131720 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. = 131720

Thus, 835+514=131720835+514=131720 size 12{8 { {3} over {5} } +5 { {1} over {4} } ="13" { {"17"} over {"20"} } } {}.

Example 2

3185631856 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 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} } } {}

2585625856 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 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, 31856=272431856=2724 size 12{3 { {1} over {8} } - { {5} over {6} } =2 { {7} over {"24"} } } {}.

Practice Set A

Find the following sums and differences.

Exercise 1

159+329159+329 size 12{1 { {5} over {9} } +3 { {2} over {9} } } {}

Solution

479479 size 12{4 { {7} over {9} } } {}

Exercise 2

10342121034212 size 12{"10" { {3} over {4} } - 2 { {1} over {2} } } {}

Solution

814814 size 12{8 { {1} over {4} } } {}

Exercise 3

278+514278+514 size 12{2 { {7} over {8} } +5 { {1} over {4} } } {}

Solution

818818 size 12{8 { {1} over {8} } } {}

Exercise 4

835310835310 size 12{8 { {3} over {5} } - { {3} over {"10"} } } {}

Solution

83108310 size 12{8 { {3} over {"10"} } } {}

Exercise 5

16+291616+2916 size 12{"16"+2 { {9} over {"16"} } } {}

Solution

1891618916 size 12{"18" { {9} over {"16"} } } {}

Exercises

For the following problems, perform each indicated opera­tion.

Exercise 6

318+438318+438 size 12{3 { {1} over {8} } +4 { {3} over {8} } } {}

Solution

712712 size 12{7 { {1} over {2} } } {}

Exercise 7

513+613513+613 size 12{5 { {1} over {3} } +6 { {1} over {3} } } {}

Exercise 8

10512+211210512+2112 size 12{"10" { {5} over {"12"} } +2 { {1} over {"12"} } } {}

Solution

12121212 size 12{"12" { {1} over {2} } } {}

Exercise 9

1515113515151135 size 12{"15" { {1} over {5} } -"11" { {3} over {5} } } {}

Exercise 10

9311+123119311+12311 size 12{9 { {3} over {"11"} } +"12" { {3} over {"11"} } } {}

Solution

2161121611 size 12{"21" { {6} over {"11"} } } {}

Exercise 11

116+326+816116+326+816 size 12{1 { {1} over {6} } +3 { {2} over {6} } +8 { {1} over {6} } } {}

Exercise 12

538+118258538+118258 size 12{5 { {3} over {8} } +1 { {1} over {8} } -2 { {5} over {8} } } {}

Solution

378378 size 12{3 { {7} over {8} } } {}

Exercise 13

35+51535+515 size 12{ { {3} over {5} } +5 { {1} over {5} } } {}

Exercise 14

2295922959 size 12{2 { {2} over {9} } - { {5} over {9} } } {}

Solution

123123 size 12{1 { {2} over {3} } } {}

Exercise 15

6+11236+1123 size 12{6+"11" { {2} over {3} } } {}

Exercise 16

178314178314 size 12{"17"-8 { {3} over {"14"} } } {}

Solution

8111481114 size 12{8 { {"11"} over {"14"} } } {}

Exercise 17

513+214513+214 size 12{5 { {1} over {3} } +2 { {1} over {4} } } {}

Exercise 18

627113627113 size 12{6 { {2} over {7} } -1 { {1} over {3} } } {}

Solution

4202142021 size 12{4 { {"20"} over {"21"} } } {}

Exercise 19

825+4110825+4110 size 12{8 { {2} over {5} } +4 { {1} over {"10"} } } {}

Exercise 20

113+1238113+1238 size 12{1 { {1} over {3} } +"12" { {3} over {8} } } {}

Solution

131724131724 size 12{"13" { {"17"} over {"24"} } } {}

Exercise 21

314+113212314+113212 size 12{3 { {1} over {4} } +1 { {1} over {3} } -2 { {1} over {2} } } {}

Exercise 22

434356+123434356+123 size 12{4 { {3} over {4} } -3 { {5} over {6} } +1 { {2} over {3} } } {}

Solution

27122712 size 12{"2" { {7} over {12} } } {}

Exercise 23

3112+413+1143112+413+114 size 12{3 { {1} over {"12"} } +4 { {1} over {3} } +1 { {1} over {4} } } {}

Exercise 24

5115+83105455115+8310545 size 12{5 { {1} over {"15"} } +8 { {3} over {"10"} } -5 { {4} over {5} } } {}

Solution

7173071730 size 12{7 { {"17"} over {"30"} } } {}

Exercise 25

713+856214713+856214 size 12{7 { {1} over {3} } +8 { {5} over {6} } -2 { {1} over {4} } } {}

Exercise 26

192021+4267514+1217192021+4267514+1217 size 12{"19" { {"20"} over {"21"} } +"42" { {6} over {7} } - { {5} over {"14"} } +"12" { {1} over {7} } } {}

Solution

742542742542 size 12{"74" { {"25"} over {"42"} } } {}

Exercise 27

116+434+10389116+434+10389 size 12{ { {1} over {"16"} } +4 { {3} over {4} } +"10" { {3} over {8} } -9} {}

Exercise 28

1129+101323516+61181129+101323516+6118 size 12{"11"- { {2} over {9} } +"10" { {1} over {3} } - { {2} over {3} } -5 { {1} over {6} } +6 { {1} over {"18"} } } {}

Solution

21132113 size 12{"21" { {1} over {3} } } {}

Exercise 29

52+216+111311652+216+1113116 size 12{ { {5} over {2} } +2 { {1} over {6} } +"11" { {1} over {3} } - { {"11"} over {6} } } {}

Exercise 30

118+94116132+198118+94116132+198 size 12{1 { {1} over {8} } + { {9} over {4} } - { {1} over {"16"} } - { {1} over {"32"} } + { {"19"} over {8} } } {}

Solution

5213252132 size 12{5 { {"21"} over {"32"} } } {}

Exercise 31

2238161722381617 size 12{"22" { {3} over {8} } -"16" { {1} over {7} } } {}

Exercise 32

1549+49161549+4916 size 12{"15" { {4} over {9} } +4 { {9} over {"16"} } } {}

Solution

201144201144 size 12{"20" { {1} over {"144"} } } {}

Exercise 33

41788+5911041788+59110 size 12{4 { {"17"} over {"88"} } +5 { {9} over {"110"} } } {}

Exercise 34

61112+2361112+23 size 12{6 { {"11"} over {"12"} } + { {2} over {3} } } {}

Solution

77127712 size 12{7 { {7} over {"12"} } } {}

Exercise 35

891679891679 size 12{8 { {9} over {"16"} } - { {7} over {9} } } {}

Exercise 36

52111125211112 size 12{5 { {2} over {"11"} } - { {1} over {"12"} } } {}

Solution

513132513132 size 12{5 { {"13"} over {"132"} } } {}

Exercise 37

18151633341815163334 size 12{"18" { {"15"} over {"16"} } - { {"33"} over {"34"} } } {}

Exercise 38

18911221561891122156 size 12{1 { {"89"} over {"112"} } - { {"21"} over {"56"} } } {}

Solution

147212147212 size 12{1 { {"47"} over {"212"} } } {}

Exercise 39

1111247131811112471318 size 12{"11" { {"11"} over {"24"} } -7 { {"13"} over {"18"} } } {}

Exercise 40

527843542+1121527843542+1121 size 12{5 { {"27"} over {"84"} } -3 { {5} over {"42"} } +1 { {1} over {"21"} } } {}

Solution

314314 size 12{3 { {1} over {4} } } {}

Exercise 41

1614816196+11441614816196+1144 size 12{"16" { {1} over {"48"} } -"16" { {1} over {"96"} } + { {1} over {"144"} } } {}

Exercise 42

A man pours 258258 size 12{2 { {5} over {8} } } {} gallons of paint from a bucket into a tray. After he finishes pouring, there are 114114 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.

Solution

258gallons258gallons

Exercise 43

A particular computer stock opened at 37383738 size 12{"37" { {3} over {8} } } {} and closed at 38143814 size 12{"38" { {1} over {4} } } {} . What was the net gain for this stock?

Exercise 44

A particular diet program claims that 43164316 size 12{4 { {3} over {"16"} } } {} pounds can be lost the first month, 314314 size 12{3 { {1} over {4} } } {} pounds can be lost the second month, and 112112 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?

Solution

81516pounds81516pounds

Exercise 45

If a person who weighs 1453414534 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?

Exercise 46

If the diet program described in the problem above makes the additional claim that from the fourth month on, a person will lose 118118 size 12{1 { {1} over {8} } } {} pounds a month, how much will a person who begins the program weighing 2083420834 size 12{"208" { {3} over {4} } } {} pounds weight after 8 months?

Solution

194316pounds194316pounds

Exercises for Review

Exercise 47

((Reference)) Use exponents to write 444444 size 12{4 cdot 4 cdot 4} {}.

Exercise 48

((Reference)) Find the greatest common factor of 14 and 20.

Solution

2

Exercise 49

((Reference)) Convert 165165 size 12{ { {"16"} over {5} } } {} to a mixed number.

Exercise 50

((Reference)) Find the sum. 49+19+2949+19+29 size 12{ { {4} over {9} } + { {1} over {9} } + { {2} over {9} } } {}.

Solution

7979 size 12{ { {7} over {9} } } {}

Exercise 51

((Reference)) Find the difference. 15263101526310 size 12{ { {"15"} over {"26"} } - { {3} over {"10"} } } {}.

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