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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses multiplication of fractions. By the end of the module students should be able to understand the concept of multiplication of fractions, multiply one fraction by another, multiply mixed numbers and find powers and roots of various fractions.

Section overview

  • Fractions of Fractions
  • Multiplication of Fractions
  • Multiplication of Fractions by Dividing Out Common Factors
  • Multiplication of Mixed Numbers
  • Powers and Roots of Fractions

Fractions of fractions

We know that a fraction represents a part of a whole quantity. For example, two fifths of one unit can be represented by

A rectangle equally divided into five parts. Each part is labeled one-fifth. Two of the parts are shaded. 2 5 size 12{ { {2} over {5} } } {} of the whole is shaded.

A natural question is, what is a fractional part of a fractional quantity, or, what is a fraction of a fraction? For example, what 2 3 size 12{ { {2} over {3} } } {} of 1 2 size 12{ { {1} over {2} } } {} ?

We can suggest an answer to this question by using a picture to examine 2 3 size 12{ { {2} over {3} } } {} of 1 2 size 12{ { {1} over {2} } } {} .

First, let’s represent 1 2 size 12{ { {1} over {2} } } {} .

A rectangle equally divided into two parts. Both parts are labeled one-half. One of the parts is shaded. 1 2 size 12{ { {1} over {2} } } {} of the whole is shaded.

Then divide each of the 1 2 size 12{ { {1} over {2} } } {} parts into 3 equal parts.

A rectangle divided into six equal parts in a gridlike fashion, with three rows and two columns. Each part is labeled one-sixth. Below the rectangles are brackets showing that each column of sixths is equal to one-half. Each part is 1 6 size 12{ { {1} over {6} } } {} of the whole.

Now we’ll take 2 3 size 12{ { {2} over {3} } } {} of the 1 2 size 12{ { {1} over {2} } } {} unit.

A rectangle divided into six equal parts in a gridlike fashion, with three rows and two columns. Each part is labeled one-sixth. Below the rectangles are brackets showing that each column of sixths is equal to one-half. The first and second boxes in the left column are shaded. 2 3 size 12{ { {2} over {3} } } {} of 1 2 size 12{ { {1} over {2} } } {} is 2 6 size 12{ { {2} over {6} } } {} , which reduces to 1 3 size 12{ { {1} over {3} } } {} .

Multiplication of fractions

Now we ask, what arithmetic operation (+, –, ×, ÷) will produce 2 6 size 12{ { {2} over {6} } } {} from 2 3 size 12{ { {2} over {3} } } {} of 1 2 size 12{ { {1} over {2} } } {} ?

Notice that, if in the fractions 2 3 size 12{ { {2} over {3} } } {} and 1 2 size 12{ { {1} over {2} } } {} , we multiply the numerators together and the denominators together, we get precisely 2 6 size 12{ { {2} over {6} } } {} .

2 1 3 2 = 2 6 size 12{ { {2 cdot 1} over {3 cdot 2} } = { {2} over {6} } } {}

This reduces to 1 3 size 12{ { {1} over {3} } } {} as before.

Using this observation, we can suggest the following:

  1. The word "of" translates to the arithmetic operation "times."
  2. To multiply two or more fractions, multiply the numerators together and then multiply the denominators together. Reduce if necessary.

numerator 1 denominator 1 numerator 2 denominator 2 = numerator 1 denominator 1 numerator 2 denominator 2

Sample set a

Perform the following multiplications.

3 4 1 6 = 3 1 4 6 = 3 24 Now, reduce.

= 3 1 24 8 = 1 8 size 12{ {}= { { { { {3}}} cSup { size 8{1} } } over { { { {2}} { {4}}} cSub { size 8{8} } } } = { {1} over {8} } } {}

Thus

3 4 1 6 = 1 8 size 12{ { {3} over {4} } cdot { {1} over {6} } = { {1} over {8} } } {}

This means that 3 4 size 12{ { {3} over {4} } } {} of 1 6 size 12{ { {1} over {6} } } {} is 1 8 size 12{ { {1} over {8} } } {} , that is, 3 4 size 12{ { {3} over {4} } } {} of 1 6 size 12{ { {1} over {6} } } {} of a unit is 1 8 size 12{ { {1} over {8} } } {} of the original unit.

3 8 4 size 12{ { {3} over {8} } cdot 4} {} . Write 4 as a fraction by writing 4 1 size 12{ { {4} over {1} } } {}

3 8 4 1 = 3 4 8 1 = 12 8 = 12 3 8 2 = 3 2 size 12{ { {3} over {8} } cdot { {4} over {1} } = { {3 cdot 4} over {8 cdot 1} } = { {"12"} over {8} } = { { { { {1}} { {2}}} cSup { size 8{3} } } over { { { {8}}} cSub { size 8{2} } } } = { {3} over {2} } } {}

3 8 4 = 3 2 size 12{ { {3} over {8} } cdot 4= { {3} over {2} } } {}

This means that 3 8 size 12{ { {3} over {8} } } {} of 4 whole units is 3 2 size 12{ { {3} over {2} } } {} of one whole unit.

2 5 5 8 1 4 = 2 5 1 5 8 4 = 10 1 160 16 = 1 16 size 12{ { {2} over {5} } cdot { {5} over {8} } cdot { {1} over {4} } = { {2 cdot 5 cdot 1} over {5 cdot 8 cdot 4} } = { { { { {1}} { {0}}} cSup { size 8{1} } } over { { { {1}} { {6}} { {0}}} cSub { size 8{"16"} } } } = { { {1} cSup {} } over { {"16"} cSub {} } } } {}

This means that 2 5 size 12{ { {2} over {5} } } {} of 5 8 size 12{ { {5} over {8} } } {} of 1 4 size 12{ { {1} over {4} } } {} of a whole unit is 1 16 size 12{ { {1} over {"16"} } } {} of the original unit.

Practice set a

Perform the following multiplications.

2 5 1 6 size 12{ { {2} over {5} } cdot { {1} over {6} } } {}

1 15 size 12{ { {1} over {"15"} } } {}

1 4 8 9 size 12{ { {1} over {4} } cdot { {8} over {9} } } {}

2 9 size 12{ { {2} over {9} } } {}

4 9 15 16 size 12{ { {4} over {9} } cdot { {15} over {16} } } {}

5 12 size 12{ { {5} over {"12"} } } {}

2 3 2 3 size 12{ left ( { {2} over {3} } right ) left ( { {2} over {3} } right )} {}

4 9 size 12{ { {4} over {9} } } {}

7 4 8 5 size 12{ left ( { {7} over {4} } right ) left ( { {8} over {5} } right )} {}

14 5 size 12{ { {"14"} over {5} } } {}

5 6 7 8 size 12{ { {5} over {6} } cdot { {7} over {8} } } {}

35 48 size 12{ { {"35"} over {"48"} } } {}

2 3 5 size 12{ { {2} over {3} } cdot 5} {}

10 3 size 12{ { {"10"} over {3} } } {}

3 4 10 size 12{ left ( { {3} over {4} } right ) left ("10" right )} {}

15 2 size 12{ { {"15"} over {2} } } {}

3 4 8 9 5 12 size 12{ { {3} over {4} } cdot { {8} over {9} } cdot { {5} over {"12"} } } {}

5 18 size 12{ { {5} over {"18"} } } {}

Multiplying fractions by dividing out common factors

We have seen that to multiply two fractions together, we multiply numerators together, then denominators together, then reduce to lowest terms, if necessary. The reduction can be tedious if the numbers in the fractions are large. For example,

9 16 10 21 = 9 10 16 21 = 90 336 = 45 168 = 15 28 size 12{ { {9} over {"16"} } cdot { {"10"} over {"21"} } = { {9 cdot "10"} over {"16" cdot "21"} } = { {"90"} over {"336"} } = { {"45"} over {"168"} } = { {"15"} over {"28"} } } {}

We avoid the process of reducing if we divide out common factors before we multi­ply.

9 16 10 21 = 9 3 16 8 10 5 21 7 = 3 5 8 7 = 15 56 size 12{ { {9} over {"16"} } cdot { {"10"} over {"21"} } = { { { { {9}}} cSup { size 8{3} } } over { { { {1}} { {6}}} cSub { size 8{8} } } } cdot { { { { {1}} { {0}}} cSup { size 8{5} } } over { { { {2}} { {1}}} cSub { size 8{7} } } } = { {3 cdot 5} over {8 cdot 7} } = { {"15"} over {"56"} } } {}

Divide 3 into 9 and 21, and divide 2 into 10 and 16. The product is a fraction that is reduced to lowest terms.

The process of multiplication by dividing out common factors

To multiply fractions by dividing out common factors, divide out factors that are common to both a numerator and a denominator. The factor being divided out can appear in any numerator and any denominator.

Questions & Answers

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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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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
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Professor
I think
Professor
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Rafiq
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Damian
How we are making nano material?
LITNING Reply
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What is meant by 'nano scale'?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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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
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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
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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Bob Reply
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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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Renato
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Stoney Reply
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Adin Reply
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Kyle
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
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sciencedirect big data base
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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