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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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