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

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 complex fractions. By the end of the module students should be able to distinguish between simple and complex fractions and convert a complex fraction to a simple fraction.

Section Overview

  • Simple Fractions and Complex Fractions
  • Converting Complex Fractions to Simple Fractions

Simple Fractions and Complex Fractions

Simple Fraction

A simple fraction is any fraction in which the numerator is any whole number and the denominator is any nonzero whole number. Some examples are the following:

12,43,7631,00012 size 12{ { {1} over {2} } } {},43 size 12{ { {4} over {3} } } {},7631,000 size 12{ { {"763"} over {1,"000"} } } {}

Complex Fraction

A complex fraction is any fraction in which the numerator and/or the denomina­tor is a fraction; it is a fraction of fractions. Some examples of complex fractions are the following:

3456,132,6910,4+387563456 size 12{ { { { {3} over {4} } } over { { {5} over {6} } } } } {},132 size 12{ { { { {1} over {3} } } over {2} } } {},6910 size 12{ { {6} over { { {9} over {"10"} } } } } {},4+38756 size 12{ { {4+ { {3} over {8} } } over {7 - { {5} over {6} } } } } {}

Converting Complex Fractions to Simple Fractions

The goal here is to convert a complex fraction to a simple fraction. We can do so by employing the methods of adding, subtracting, multiplying, and dividing fractions. Recall from (Reference) that a fraction bar serves as a grouping symbol separating the fractional quantity into two individual groups. We proceed in simplifying a complex fraction to a simple fraction by simplifying the numerator and the denom­inator of the complex fraction separately. We will simplify the numerator and denominator completely before removing the fraction bar by dividing. This tech­nique is illustrated in problems 3, 4, 5, and 6 of Section 4.

Sample Set A

Convert each of the following complex fractions to a simple fraction.

Example 1

381516381516 size 12{ { { { {3} over {8} } } over { { {"15"} over {"16"} } } } } {}

Convert this complex fraction to a simple fraction by performing the indicated division.

381516 = 38÷1516 The divisor is 1516. Invert 1516 and multiply. = 3 1 8 1 16 2 15 5 = 1 2 1 5 = 2 5 381516 = 38÷1516 The divisor is 1516. Invert 1516 and multiply. = 3 1 8 1 16 2 15 5 = 1 2 1 5 = 2 5

Example 2

496 Write 6 as 61 and divide. 496 Write 6 as 61 and divide.

4 9 6 1 = 4 9 ÷ 6 1 = 4 2 9 1 6 3 = 2 1 9 3 = 2 27 4 9 6 1 = 4 9 ÷ 6 1 = 4 2 9 1 6 3 = 2 1 9 3 = 2 27

Example 3

5+3446 Simplify the numerator. 5+3446 Simplify the numerator.

45+3446=20+3446=23446 Write 46 as 461 . 45+3446=20+3446=23446 size 12{ { { { {4 cdot 5+3} over {4} } } over {"46"} } = { { { {"20"+3} over {4} } } over {"46"} } = { { { {"23"} over {4} } } over {"46"} } } {} Write 46 as 461 size 12{ { {"46"} over {1} } } {} .

23 4 46 1 = 23 4 ÷ 46 1 = 23 1 4 1 46 2 = 1 1 4 2 = 1 8 23 4 46 1 = 23 4 ÷ 46 1 = 23 1 4 1 46 2 = 1 1 4 2 = 1 8

Example 4

14+3812+1324=28+381224+1324=2+3812+1324=582524=58÷252414+3812+1324=28+381224+1324=2+3812+1324=582524=58÷2524 size 12{ { { { {1} over {4} } + { {3} over {8} } } over { { {1} over {2} } + { {"13"} over {"24"} } } } = { { { {2} over {8} } + { {3} over {8} } } over { { {"12"} over {"24"} } + { {"13"} over {"24"} } } } = { { { {2+3} over {8} } } over { { {"12"+"13"} over {"24"} } } } = { { { {5} over {8} } } over { { {"25"} over {"24"} } } } = { {5} over {8} } ¸ { {"25"} over {"24"} } } {}

5 8 ÷ 25 24 = 5 1 8 1 24 3 25 5 = 1 3 1 5 = 3 5 5 8 ÷ 25 24 = 5 1 8 1 24 3 25 5 = 1 3 1 5 = 3 5 size 12{ { {5} over {8} } ¸ { {"25"} over {"24"} } = { { { { {5}}} cSup { size 8{1} } } over { { { {8}}} cSub { size 8{1} } } } cdot { { { { {2}} { {4}}} cSup { size 8{3} } } over { { { {2}} { {5}}} cSub { size 8{5} } } } = { {1 cdot 3} over {1 cdot 5} } = { {3} over {5} } } {}

Example 5

4 + 56 7 13 = 4 6 + 5 6 7 3 1 3 = 29 6 20 3 = 29 6 ÷ 20 3 = 29 6 2 3 1 20 = 29 40 4 + 56 7 13 = 4 6 + 5 6 7 3 1 3 = 29 6 20 3 = 29 6 ÷ 20 3 = 29 6 2 3 1 20 = 29 40

Example 6

11+310445=1110+31045+45=110+31020+45=11310245=11310÷24511+310445=1110+31045+45=110+31020+45=11310245=11310÷245 size 12{ { {"11"+ { {3} over {"10"} } } over {4 { {4} over {5} } } } = { { { {"11" cdot "10"+3} over {"10"} } } over { { {4 cdot 5+4} over {5} } } } = { { { {"110"+3} over {"10"} } } over { { {"20"+4} over {5} } } } = { { { {"113"} over {"10"} } } over { { {"24"} over {5} } } } = { {"113"} over {"10"} } ¸ { {"24"} over {5} } } {}

113 10 ÷ 24 5 = 113 10 2 5 1 24 = 113 1 2 24 = 113 48 = 2 17 48 113 10 ÷ 24 5 = 113 10 2 5 1 24 = 113 1 2 24 = 113 48 = 2 17 48 size 12{ { {"113"} over {"10"} } ¸ { {"24"} over {5} } = { {"113"} over { { { {1}} { {0}}} cSub { size 8{2} } } } cdot { { { { {5}}} cSup { size 8{1} } } over {"24"} } = { {"113" cdot 1} over {2 cdot "24"} } = { {"113"} over {"48"} } =2 { {"17"} over {"48"} } } {}

Practice Set A

Convert each of the following complex fractions to a simple fraction.

Exercise 1

4981549815 size 12{ { { { {4} over {9} } } over { { {8} over {"15"} } } } } {}

Solution

5656 size 12{ { {5} over {6} } } {}

Exercise 2

7102871028 size 12{ { { { {7} over {"10"} } } over {"28"} } } {}

Solution

140140 size 12{ { {1} over {"40"} } } {}

Exercise 3

5+253+355+253+35 size 12{ { {5+ { {2} over {5} } } over {3+ { {3} over {5} } } } } {}

Solution

3232 size 12{ { {3} over {2} } } {}

Exercise 4

18+78631018+786310 size 12{ { { { {1} over {8} } + { {7} over {8} } } over {6- { {3} over {"10"} } } } } {}

Solution

10571057 size 12{ { {"10"} over {"57"} } } {}

Exercise 5

16+58591416+585914 size 12{ { { { {1} over {6} } + { {5} over {8} } } over { { {5} over {9} } - { {1} over {4} } } } } {}

Solution

2132221322 size 12{2 { {"13"} over {"22"} } } {}

Exercise 6

16102311567761610231156776 size 12{ { {"16"-"10" { {2} over {3} } } over {"11" { {5} over {6} } -7 { {7} over {6} } } } } {}

Solution

15111511 size 12{1 { {5} over {"11"} } } {}

Exercises

Simplify each fraction.

Exercise 7

3591535915 size 12{ { { { {3} over {5} } } over { { {9} over {"15"} } } } } {}

Solution

1

Exercise 8

13191319 size 12{ { { { {1} over {3} } } over { { {1} over {9} } } } } {}

Exercise 9

1451214512 size 12{ { { { {1} over {4} } } over { { {5} over {"12"} } } } } {}

Solution

3535 size 12{ { {3} over {5} } } {}

Exercise 10

8941589415 size 12{ { { { {8} over {9} } } over { { {4} over {"15"} } } } } {}

Exercise 11

6+1411+146+1411+14 size 12{ { {6+ { {1} over {4} } } over {"11"+ { {1} over {4} } } } } {}

Solution

5959 size 12{ { {5} over {9} } } {}

Exercise 12

2+127+122+127+12 size 12{ { {2+ { {1} over {2} } } over {7+ { {1} over {2} } } } } {}

Exercise 13

5+132+2155+132+215 size 12{ { {5+ { {1} over {3} } } over {2+ { {2} over {"15"} } } } } {}

Solution

5252 size 12{ { {5} over {2} } } {}

Exercise 14

9+121+8119+121+811 size 12{ { {9+ { {1} over {2} } } over {1+ { {8} over {"11"} } } } } {}

Exercise 15

4+101312394+10131239 size 12{ { {4+ { {"10"} over {"13"} } } over { { {"12"} over {"39"} } } } } {}

Solution

312312 size 12{ { {"31"} over {2} } } {}

Exercise 16

13+27262113+272621 size 12{ { { { {1} over {3} } + { {2} over {7} } } over { { {"26"} over {"21"} } } } } {}

Exercise 17

56141125614112 size 12{ { { { {5} over {6} } - { {1} over {4} } } over { { {1} over {"12"} } } } } {}

Solution

7

Exercise 18

310+4121990310+4121990 size 12{ { { { {3} over {"10"} } + { {4} over {"12"} } } over { { {"19"} over {"90"} } } } } {}

Exercise 19

916+7313948916+7313948 size 12{ { { { {9} over {"16"} } + { {7} over {3} } } over { { {"139"} over {"48"} } } } } {}

Solution

1

Exercise 20

128889316128889316 size 12{ { { { {1} over {"288"} } } over { { {8} over {9} } - { {3} over {"16"} } } } } {}

Exercise 21

2742951111327429511113 size 12{ { { { {27} over {"429"} } } over { { {5} over {11} } - { {1} over {"13"} } } } } {}

Solution

1616 size 12{ { {1} over {6} } } {}

Exercise 22

13+2535+174513+2535+1745 size 12{ { { { {1} over {3} } + { {2} over {5} } } over { { {3} over {5} } + { {"17"} over {"45"} } } } } {}

Exercise 23

970+5421330-121970+5421330-121 size 12{ { { { {9} over {"70"} } + { {5} over {"42"} } } over { { {"13"} over {"30"} } - { {1} over {"21"} } } } } {}

Solution

52815281 size 12{ { {"52"} over {"81"} } } {}

Exercise 24

116+114231360116+114231360 size 12{ { { { {1} over {"16"} } + { {1} over {"14"} } } over { { {2} over {3} } - { {"13"} over {"60"} } } } } {}

Exercise 25

320+1112197-11135320+1112197-11135 size 12{ { { { {3} over {"20"} } + { {"11"} over {"12"} } } over { { {"19"} over {7} } - 1 { {"11"} over {"35"} } } } } {}

Solution

16211621 size 12{ { {"16"} over {"21"} } } {}

Exercise 26

22311214+111622311214+1116 size 12{ { {2 { {2} over {3} } -1 { {1} over {2} } } over { { {1} over {4} } +1 { {1} over {"16"} } } } } {}

Exercise 27

315+313651563315+313651563 size 12{ { {3 { {1} over {5} } +3 { {1} over {3} } } over { { {6} over {5} } - { {"15"} over {"63"} } } } } {}

Solution

686101686101 size 12{ { {"686"} over {"101"} } } {}

Exercise 28

112+155143512813412112351112112+155143512813412112351112 size 12{ { { { {1 { {1} over {2} } +"15"} over {5 { {1} over {4} } -3 { {5} over {"12"} } } } } over { { {8 { {1} over {3} } -4 { {1} over {2} } } over {"11" { {2} over {3} } -5 { {"11"} over {"12"} } } } } } } {}

Exercise 29

534+315215+1571091241618+21120534+315215+1571091241618+21120 size 12{ { { { {5 { {3} over {4} } +3 { {1} over {5} } } over {2 { {1} over {5} } +"15" { {7} over {"10"} } } } } over { { {9 { {1} over {2} } - 4 { {1} over {6} } } over { { {1} over {8} } +2 { {1} over {"120"} } } } } } } {}

Solution

1313 size 12{ { {1} over {3} } } {}

Exercises for Review

Exercise 30

((Reference)) Find the prime factorization of 882.

Exercise 31

((Reference)) Convert 627627 size 12{ { {"62"} over {7} } } {} to a mixed number.

Solution

867867 size 12{8 { {6} over {7} } } {}

Exercise 32

((Reference)) Reduce 114342114342 size 12{ { {"114"} over {"342"} } } {} to lowest terms.

Exercise 33

((Reference)) Find the value of 638456638456 size 12{6 { {3} over {8} } - 4 { {5} over {6} } } {}.

Solution

1132411324 size 12{1 { {"13"} over {"24"} } } {} or 37243724 size 12{ { {"37"} over {"24"} } } {}

Exercise 34

((Reference)) Arrange from smallest to largest: 1212 size 12{ { {1} over {2} } } {}, 3535 size 12{ { {3} over {5} } } {}, 4747 size 12{ { {4} over {7} } } {}.

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