# 4.2 Proper fractions, improper fractions, and mixed numbers

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses proper fractions, improper fractions, and mixed numbers. By the end of the module students should be able to distinguish between proper fractions, improper fractions, and mixed numbers, convert an improper fraction to a mixed number and convert a mixed number to an improper fraction.

## Section overview

• Positive Proper Fractions
• Positive Improper Fractions
• Positive Mixed Numbers
• Relating Positive Improper Fractions and Positive Mixed Numbers
• Converting an Improper Fraction to a Mixed Number
• Converting a Mixed Number to an Improper Fraction

Now that we know what positive fractions are, we consider three types of positive fractions: proper fractions, improper fractions, and mixed numbers.

## Positive proper fraction

Fractions in which the whole number in the numerator is strictly less than the whole number in the denominator are called positive proper fractions . On the number line, proper fractions are located in the interval from 0 to 1. Positive proper fractions are always less than one. The closed circle at 0 indicates that 0 is included, while the open circle at 1 indicates that 1 is not included.

Some examples of positive proper fractions are

 $\frac{1}{2}$ , $\frac{3}{5}$ , $\frac{\text{20}}{\text{27}}$ , and $\frac{\text{106}}{\text{255}}$

Note that $\text{1}<\text{2}$ , $\text{3}<\text{5}$ , $\text{20}<\text{27}$ , and $\text{106}<\text{225}$ .

## Positive improper fractions

Fractions in which the whole number in the numerator is greater than or equal to the whole number in the denominator are called positive improper fractions . On the number line, improper fractions lie to the right of (and including) 1. Positive improper fractions are always greater than or equal to 1. Some examples of positive improper fractions are

$\frac{3}{2}$ , $\frac{8}{5}$ , $\frac{4}{4}$ , and $\frac{\text{105}}{\text{16}}$

Note that $3\ge 2$ , $8\ge 5$ , $4\ge 4$ , and $\text{105}\ge \text{16}$ .

## Positive mixed numbers

A number of the form

$\text{nonzero whole number}+\text{proper fraction}$

is called a positive mixed number . For example, $2\frac{3}{5}$ is a mixed number. On the number line, mixed numbers are located in the interval to the right of (and includ­ing) 1. Mixed numbers are always greater than or equal to 1. ## Relating positive improper fractions and positive mixed numbers

A relationship between improper fractions and mixed numbers is suggested by two facts. The first is that improper fractions and mixed numbers are located in the same interval on the number line. The second fact, that mixed numbers are the sum of a natural number and a fraction, can be seen by making the following observa­tions.

Divide a whole quantity into 3 equal parts. Now, consider the following examples by observing the respective shaded areas. In the shaded region, there are 2 one thirds, or $\frac{2}{3}$ .

$2\left(\frac{1}{3}\right)=\frac{2}{3}$ There are 3 one thirds, or $\frac{3}{3}$ , or 1.

$3\left(\frac{1}{3}\right)=\frac{3}{3}\phantom{\rule{8px}{0ex}}\text{or}\phantom{\rule{8px}{0ex}}1$

Thus,

$\frac{3}{3}=1$

Improper fraction = whole number.  There are 4 one thirds, or $\frac{4}{3}$ , or 1 and $\frac{1}{3}$ .

$4\left(\frac{1}{3}\right)=\frac{4}{3}$ or $1\phantom{\rule{8px}{0ex}}\text{and}\phantom{\rule{8px}{0ex}}\frac{1}{3}$

The terms 1 and $\frac{1}{3}$ can be represented as $1+\frac{1}{3}$ or $1\frac{1}{3}$

Thus,

$\frac{4}{3}=1\frac{1}{3}$ .

Improper fraction = mixed number.  There are 5 one thirds, or $\frac{5}{3}$ , or 1 and $\frac{2}{3}$ .

$5\left(\frac{1}{3}\right)=\frac{5}{3}\phantom{\rule{8px}{0ex}}\text{or}\phantom{\rule{8px}{0ex}}1\phantom{\rule{8px}{0ex}}\text{and}\phantom{\rule{8px}{0ex}}\frac{2}{3}$

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
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Rafiq
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Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
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
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Rafiq
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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 ?
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?
what king of growth are you checking .?
Renato
In the number 779,844,205 how many ten millions are there?
From 1973 to 1979, in the United States, there was an increase of 166.6% of Ph.D. social scien­tists to 52,000. How many were there in 1973?
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