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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:

1 2 size 12{ { {1} over {2} } } {} , 4 3 size 12{ { {4} over {3} } } {} , 763 1, 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:

3 4 5 6 size 12{ { { { {3} over {4} } } over { { {5} over {6} } } } } {} , 1 3 2 size 12{ { { { {1} over {3} } } over {2} } } {} , 6 9 10 size 12{ { {6} over { { {9} over {"10"} } } } } {} , 4 + 3 8 7 5 6 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 [link] 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 [link] .

Sample set a

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

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

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

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

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4 9 6 Write 6 as 6 1 and divide.

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

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5 + 3 4 46 Simplify the numerator.

4 5 + 3 4 46 = 20 + 3 4 46 = 23 4 46 size 12{ { { { {4 cdot 5+3} over {4} } } over {"46"} } = { { { {"20"+3} over {4} } } over {"46"} } = { { { {"23"} over {4} } } over {"46"} } } {} Write 46 as 46 1 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

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1 4 + 3 8 1 2 + 13 24 = 2 8 + 3 8 12 24 + 13 24 = 2 + 3 8 12 + 13 24 = 5 8 25 24 = 5 8 ÷ 25 24 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 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} } } {}

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

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11 + 3 10 4 4 5 = 11 10 + 3 10 4 5 + 4 5 = 110 + 3 10 20 + 4 5 = 113 10 24 5 = 113 10 ÷ 24 5 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 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"} } } {}

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Practice set a

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

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

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

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

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

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5 + 2 5 3 + 3 5 size 12{ { {5+ { {2} over {5} } } over {3+ { {3} over {5} } } } } {}

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

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

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

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

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

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

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

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Exercises

Simplify each fraction.

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

1

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

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

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

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

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6 + 1 4 11 + 1 4 size 12{ { {6+ { {1} over {4} } } over {"11"+ { {1} over {4} } } } } {}

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

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

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

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

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

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

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

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

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

7

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

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

1

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

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

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

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1 3 + 2 5 3 5 + 17 45 size 12{ { { { {1} over {3} } + { {2} over {5} } } over { { {3} over {5} } + { {"17"} over {"45"} } } } } {}

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

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

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

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3 20 + 11 12 19 7 - 1 11 35 size 12{ { { { {3} over {"20"} } + { {"11"} over {"12"} } } over { { {"19"} over {7} } - 1 { {"11"} over {"35"} } } } } {}

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

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

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

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

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1 1 2 + 15 5 1 4 3 5 12 8 1 3 4 1 2 11 2 3 5 11 12 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"} } } } } } } {}

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5 3 4 + 3 1 5 2 1 5 + 15 7 10 9 1 2 4 1 6 1 8 + 2 1 120 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"} } } } } } } {}

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

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Exercises for review

( [link] ) Find the prime factorization of 882.

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( [link] ) Convert 62 7 size 12{ { {"62"} over {7} } } {} to a mixed number.

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

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( [link] ) Reduce 114 342 size 12{ { {"114"} over {"342"} } } {} to lowest terms.

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( [link] ) Find the value of 6 3 8 4 5 6 size 12{6 { {3} over {8} } - 4 { {5} over {6} } } {} .

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

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( [link] ) Arrange from smallest to largest: 1 2 size 12{ { {1} over {2} } } {} , 3 5 size 12{ { {3} over {5} } } {} , 4 7 size 12{ { {4} over {7} } } {} .

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Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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