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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

  • 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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Exercises

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

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

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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