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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.

Overview

  • Multiplication of Fractions
  • Division of Fractions
  • Addition and Subtraction of Fractions

Multiplication of fractions

Multiplication of fractions

To multiply two fractions, multiply the numerators together and multiply the denominators together. Reduce to lowest terms if possible.

For example, multiply 3 4 · 1 6 .

3 4 · 1 6 = 3 · 1 4 · 6 = 3 24 Now reduce . = 3 · 1 2 · 2 · 2 · 3 = 3 · 1 2 · 2 · 2 · 3 3 is the only common factor . = 1 8
Notice that we since had to reduce, we nearly started over again with the original two fractions. If we factor first, then cancel, then multiply, we will save time and energy and still obtain the correct product.

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Sample set a

Perform the following multiplications.

1 4 · 8 9 = 1 2 · 2 · 2 · 2 · 2 3 · 3 = 1 2 · 2 · 2 · 2 · 2 3 · 3 2 is a common factor . = 1 1 · 2 3 · 3 = 1 · 2 1 · 3 · 3 = 2 9

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3 4 · 8 9 · 5 12 = 3 2 · 2 · 2 · 2 · 2 3 · 3 · 5 2 · 2 · 3 = 3 2 · 2 · 2 · 2 · 2 3 · 3 · 5 2 · 2 · 3 2 and 3 are common factors . = 1 · 1 · 5 3 · 2 · 3 = 5 18

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Division of fractions

Reciprocals

Two numbers whose product is 1 are reciprocals of each other. For example, since 4 5 · 5 4 = 1 , 4 5 and 5 4 are reciprocals of each other. Some other pairs of reciprocals are listed below.

2 7 , 7 2 3 4 , 4 3 6 1 , 1 6

Reciprocals are used in division of fractions.

Division of fractions

To divide a first fraction by a second fraction, multiply the first fraction by the reciprocal of the second fraction. Reduce if possible.

This method is sometimes called the “invert and multiply” method.

Sample set b

Perform the following divisions.

1 3 ÷ 3 4 . The divisor is  3 4 . Its reciprocal is  4 3 . 1 3 ÷ 3 4 = 1 3 · 4 3 = 1 · 4 3 · 3 = 4 9

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3 8 ÷ 5 4 . The divisor is  5 4 . Its reciprocal is  4 5 . 3 8 ÷ 5 4 = 3 8 · 4 5 = 3 2 · 2 · 2 · 2 · 2 5 = 3 2 · 2 · 2 · 2 · 2 5 2 is a common factor . = 3 · 1 2 · 5 = 3 10

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5 6 ÷ 5 12 . The divisor is  5 12 . Its reciprocal is  12 5 . 5 6 ÷ 5 12 = 5 6 · 12 5 = 5 2 · 3 · 2 · 2 · 3 5 = 5 2 · 3 · 2 · 2 · 3 5 = 1 · 2 1 = 2

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Addition and subtraction of fractions

Fractions with like denominators

To add (or subtract) two or more fractions that have the same denominators, add (or subtract) the numerators and place the resulting sum over the common denominator. Reduce if possible.

CAUTION

Add or subtract only the numerators. Do not add or subtract the denominators!

Sample set c

Find the following sums.

3 7 + 2 7 . The denominators are the same .  Add the numerators and place the sum over 7 . 3 7 + 2 7 = 3 + 2 7 = 5 7

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7 9 4 9 . The denominators are the same .  Subtract 4 from 7 and place the difference over 9 . 7 9 4 9 = 7 4 9 = 3 9 = 1 3

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Fractions can only be added or subtracted conveniently if they have like denominators.

Fractions with unlike denominators

To add or subtract fractions having unlike denominators, convert each fraction to an equivalent fraction having as the denominator the least common multiple of the original denominators.

The least common multiple of the original denominators is commonly referred to as the least common denominator (LCD). See Section ( [link] ) for the technique of finding the least common multiple of several numbers.

Sample set d

Find each sum or difference.

1 6 + 3 4 . The denominators are not alike .  Find the LCD of 6 and 4 . { 6 = 2 · 3 4 = 2 2 The LCD is  2 2 · 3 = 4 · 3 = 12. Convert each of the original fractions to equivalent fractions having the common denominator 12 . 1 6 = 1 · 2 6 · 2 = 2 12 3 4 = 3 · 3 4 · 3 = 9 12 Now we can proceed with the addition . 1 6 + 3 4 = 2 12 + 9 12 = 2 + 9 12 = 11 12

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5 9 5 12 . The denominators are not alike .  Find the LCD of 9 and 12 . { 9 = 3 2 12 = 2 2 · 3 The LCD is  2 2 · 3 2 = 4 · 9 = 36. Convert each of the original fractions to equivalent fractions having the common denominator 36 . 5 9 = 5 · 4 9 · 4 = 20 36 5 12 = 5 · 3 12 · 3 = 15 36 Now we can proceed with the subtraction . 5 9 5 12 = 20 36 15 36 = 20 15 36 = 5 36

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Exercises

For the following problems, perform each indicated operation.

9 16 · 20 27

5 12

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21 25 · 15 14

9 10

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3 7 · 14 18 · 6 2

1

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14 15 · 21 28 · 45 7

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16 20 + 1 20 + 2 20

19 20

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11 16 + 9 16 5 16

15 16

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25 36 7 10

1 180

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8 3 1 4 + 7 36

47 18

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Questions & Answers

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
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
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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