# 4.3 Equivalent fractions, reducing fractions to lowest terms, and

 Page 1 / 3
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.

Notice that both $\frac{2}{3}$ and $\frac{4}{6}$ 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.

## A test for equivalent fractions using the cross product

These pairs of products are called cross products .

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

Thus, $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent, that is, $\frac{2}{3}=\frac{4}{6}$ .

## Sample set a

Determine if the following pairs of fractions are equivalent.

$\frac{3}{4}\text{and}\frac{6}{8}$ . Test for equality of the cross products.

The cross products are equals.

The fractions $\frac{3}{4}$ and $\frac{6}{8}$ are equivalent, so $\frac{3}{4}=\frac{6}{8}$ .

$\frac{3}{8}\text{and}\frac{9}{\text{16}}$ . Test for equality of the cross products.

The cross products are not equal.

The fractions $\frac{3}{8}$ and $\frac{9}{\text{16}}$ are not equivalent.

## Practice set a

Determine if the pairs of fractions are equivalent.

$\frac{1}{2}$ , $\frac{3}{6}$

, yes

$\frac{4}{5}$ , $\frac{\text{12}}{\text{15}}$

, yes

$\frac{2}{3}$ , $\frac{8}{\text{15}}$

$\mathrm{30}\ne \mathrm{24}$ , no

$\frac{1}{8}$ , $\frac{5}{\text{40}}$

, yes

$\frac{3}{\text{12}}$ , $\frac{1}{4}$

, yes

## 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 $\frac{9}{\text{15}}$ and $\frac{3}{5}$ . First, divide both the numerator and denominator of $\frac{9}{\text{15}}$ by 3. The fractions $\frac{9}{\text{15}}$ and $\frac{3}{5}$ are equivalent.

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

$\frac{9÷3}{\text{15}÷3}=\frac{3}{5}$

The fraction $\frac{9}{\text{15}}$ is converted to $\frac{3}{5}$ .

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

$\frac{9}{\text{15}}=\frac{3\cdot 3}{5\cdot 3}$

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.

Consider the collection of equivalent fractions

$\frac{5}{\text{20}}$ , $\frac{4}{\text{16}}$ , $\frac{3}{\text{12}}$ , $\frac{2}{8}$ , $\frac{1}{4}$

## Reduced to lowest terms

Notice that each of the first four fractions can be reduced to the last fraction, $\frac{1}{4}$ , 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 $\frac{1}{4}$ , $\frac{3}{8}$ , $\frac{2}{5}$ , and $\frac{7}{\text{10}}$ are all reduced to lowest terms.

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket 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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
7hours 36 min - 4hours 50 min
Tanis Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of mathematics' conversation and receive update notifications?

 By Sam Luong By Rhodes By By