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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.


  • Equivalent Fractions
  • Reducing Fractions To Lowest Terms
  • Raising Fractions To Higher Terms

Equivalent fractions

Equivalent fractions

Fractions that have the same value are called equivalent fractions.

For example, 2 3 and 4 6 represent the same part of a whole quantity and are therefore equivalent. Several more collections of equivalent fractions are listed below.

7 6 , 14 12 , 21 18 , 28 24 , 35 30

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Reducing fractions to lowest terms

Reduced to lowest terms

It is often useful to convert one fraction to an equivalent fraction that has reduced values in the numerator and denominator. When a fraction is converted to an equivalent fraction that has the smallest numerator and denominator in the collection of equivalent fractions, it is said to be reduced to lowest terms. The conversion process is called reducing a fraction.

We can reduce a fraction to lowest terms by

  1. Expressing the numerator and denominator as a product of prime numbers. (Find the prime factorization of the numerator and denominator. See Section ( [link] ) for this technique.)
  2. Divide the numerator and denominator by all common factors. (This technique is commonly called “cancelling.”)

Sample set a

Reduce each fraction to lowest terms.

6 18 = 2 · 3 2 · 3 · 3 = 2 · 3 2 · 3 · 3 2 and 3 are common factors . = 1 3

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16 20 = 2 · 2 · 2 · 2 2 · 2 · 5 = 2 · 2 · 2 · 2 2 · 2 · 5 2 is the only common factor . = 4 5

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56 70 = 2 · 4 · 7 2 · 5 · 7 = 2 · 4 · 7 2 · 5 · 7 2 and 7 are common factors . = 4 5

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8 15 = 2 · 2 · 2 3 · 5 There are no common factors . Thus , 8 15  is reduced to lowest terms .

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Raising a fraction to higher terms

Equally important as reducing fractions is raising fractions to higher terms. Raising a fraction to higher terms is the process of constructing an equivalent fraction that has higher values in the numerator and denominator. The higher, equivalent fraction is constructed by multiplying the original fraction by 1.

Notice that 3 5 and 9 15 are equivalent, that is 3 5 = 9 15 . Also,

The product of three over five and one is equal to the product of three over five and three over three. This is equal to the product of three and three over the product of five and three, that in turn is equal to nine over fifteen. There is an arrow pointing towards one and three over three, indicating that one and three over three are equal.

This observation helps us suggest the following method for raising a fraction to higher terms.

Raising a fraction to higher terms

A fraction can be raised to higher terms by multiplying both the numerator and denominator by the same nonzero number.

For example, 3 4 can be raised to 24 32 by multiplying both the numerator and denominator by 8, that is, multiplying by 1 in the form 8 8 .

3 4 = 3 · 8 4 · 8 = 24 32

How did we know to choose 8 as the proper factor? Since we wish to convert 4 to 32 by multiplying it by some number, we know that 4 must be a factor of 32. This means that 4 divides into 32. In fact, 32 ÷ 4 = 8. We divided the original denominator into the new, specified denominator to obtain the proper factor for the multiplication.

Sample set b

Determine the missing numerator or denominator.

3 7 = ? 35 . Divide the original denominator ,  7 ,  into the new denominator , 35. 35 ÷ 7 = 5. Multiply the original numerator by 5 . 3 7 = 3 · 5 7 · 5 = 15 35

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5 6 = 45 ? . Divide the original numerator ,  5 ,  into the new numerator , 45. 45 ÷ 5 = 9. Multiply the original denominator by 9 . 5 6 = 5 · 9 6 · 9 = 45 54

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For the following problems, reduce, if possible, each fraction lowest terms.

For the following problems, determine the missing numerator or denominator.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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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