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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses equivalent fractions, reducing fractions to lowest terms, and raising fractions to higher terms. By the end of the module students should be able to recognize equivalent fractions, reduce a fraction to lowest terms and be able to raise a fraction to higher terms.

Section overview

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

Equivalent fractions

Let's examine the following two diagrams.

A rectangle divided equally into three parts, each marked one-third. The left two parts are shaded. To the right of the box is the caption, two-thirds of the whole is shaded. Below this is a rectangle equally divided into six part, with the leftmost four part shaded. to the right of this rectangle is the caption, four-sixths of the whole is shaded.

Notice that both 2 3 size 12{ { {2} over {3} } } {} and 4 6 size 12{ { {4} over {6} } } {} represent the same part of the whole, that is, they represent the same number.

Equivalent fractions

Fractions that have the same value are called equivalent fractions . Equiva­lent fractions may look different, but they are still the same point on the number line.

There is an interesting property that equivalent fractions satisfy.

two-thirds and four-sixths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

A test for equivalent fractions using the cross product

These pairs of products are called cross products .

Is two time six equal to three times four? Yes.

If the cross products are equal, the fractions are equivalent. If the cross products are not equal, the fractions are not equivalent.

Thus, 2 3 size 12{ { {2} over {3} } } {} and 4 6 size 12{ { {4} over {6} } } {} are equivalent, that is, 2 3 = 4 6 size 12{ { {2} over {3} } = { {4} over {6} } } {} .

Sample set a

Determine if the following pairs of fractions are equivalent.

3 4 and 6 8 size 12{ { {3} over {4} } `"and " { {6} over {8} } } {} . Test for equality of the cross products.

three-fourths and six-eigths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

Is three times eight equal to six times four? yes. The cross products are equals.

The fractions 3 4 and 6 8 are equivalent, so 3 4 = 6 8 .

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3 8 and 9 16 size 12{ { {3} over {8} } " and " { {9} over {"16"} } } {} . Test for equality of the cross products.

Three-eights and nine-sixteenths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

is three times sixteen equal to nine times eight? No. forty-eight does not equal seventy-two. The cross products are not equal.

The fractions 3 8 size 12{ { {3} over {8} } } {} and 9 16 size 12{ { {9} over {"16"} } } {} are not equivalent.

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Practice set a

Determine if the pairs of fractions are equivalent.

1 2 size 12{ { {1} over {2} } } {} , 3 6 size 12{ { {3} over {6} } } {}

Six equals six. , yes

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4 5 size 12{ { {4} over {5} } } {} , 12 15 size 12{ { {"12"} over {"15"} } } {}

Sixty equals sixty. , yes

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2 3 size 12{ { {2} over {3} } } {} , 8 15 size 12{ { {8} over {"15"} } } {}

30 24 , no

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1 8 size 12{ { {1} over {8} } } {} , 5 40 size 12{ { {5} over {"40"} } } {}

Forty equals forty. , yes

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3 12 size 12{ { {3} over {"12"} } } {} , 1 4 size 12{ { {1} over {4} } } {}

Twelve equals twelve. , yes

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Reducing fractions to lowest terms

It is often very useful to conver t one fraction to an equivalent fraction that has reduced values in the numerator and denominator. We can suggest a method for doing so by considering the equivalent fractions 9 15 size 12{ { {9} over {"15"} } } {} and 3 5 size 12{ { {3} over {5} } } {} . First, divide both the numerator and denominator of 9 15 size 12{ { {9} over {"15"} } } {} by 3. The fractions 9 15 size 12{ { {9} over {"15"} } } {} and 3 5 size 12{ { {3} over {5} } } {} are equivalent.

(Can you prove this?) So, 9 15 = 3 5 size 12{ { {9} over {"15"} } = { {3} over {5} } } {} . We wish to convert 9 15 size 12{ { {9} over {"15"} } } {} to 3 5 size 12{ { {3} over {5} } } {} . Now divide the numerator and denominator of 9 15 size 12{ { {9} over {"15"} } } {} by 3, and see what happens.

9 ÷ 3 15 ÷ 3 = 3 5 size 12{ { {9 div 3} over {"15" div 3} } = { {3} over {5} } } {}

The fraction 9 15 size 12{ { {9} over {"15"} } } {} is converted to 3 5 size 12{ { {3} over {5} } } {} .

A natural question is "Why did we choose to divide by 3?" Notice that

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

We can see that the factor 3 is common to both the numerator and denominator.

Reducing a fraction

From these observations we can suggest the following method for converting one fraction to an equivalent fraction that has reduced values in the numerator and denominator. The method is called reducing a fraction .

A fraction can be reduced by dividing both the numerator and denominator by the same nonzero whole number.

Nine-twelfths is equal to nine divided by three, over nine divided by three, which is equal to three-fourths. Sixteen thirtieths is equal to sixteen divided by two, over thirty divided by 2, which is equal to eight-fifteenths. Notice that three over three and two over two are both equal to 1.

Consider the collection of equivalent fractions

5 20 size 12{ { {5} over {"20"} } } {} , 4 16 size 12{ { {4} over {"16"} } } {} , 3 12 size 12{ { {3} over {"12"} } } {} , 2 8 size 12{ { {2} over {8} } } {} , 1 4 size 12{ { {1} over {4} } } {}

Reduced to lowest terms

Notice that each of the first four fractions can be reduced to the last fraction, 1 4 size 12{ { {1} over {4} } } {} , by dividing both the numerator and denominator by, respectively, 5, 4, 3, and 2. When a fraction is converted to the fraction that has the smallest numerator and denomi­nator in its collection of equivalent fractions, it is said to be reduced to lowest terms . The fractions 1 4 size 12{ { {1} over {4} } } {} , 3 8 size 12{ { {3} over {8} } } {} , 2 5 size 12{ { {2} over {5} } } {} , and 7 10 size 12{ { {7} over {"10"} } } {} are all reduced to lowest terms.

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..
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Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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Application of nanotechnology in medicine
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
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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
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What is meant by 'nano scale'?
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scanning tunneling microscope
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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
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
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biomolecules are e building blocks of every organics and inorganic materials.
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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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