Types of fractions

In our chapter What's a Unit Fraction?, we discovered that to identify a shaded fraction, or what fraction something is of another, we need to know:

1. How many pieces the original shape is cut up into. This number goes on the bottom of the fraction. And,
2. How many parts do we have, or are shaded. This number goes on the top of the fraction.

Definitions

We also learned that a unit fraction is a fraction with a $1$1 in the numerator.  

Let's now look at some formal mathematical names for parts of the fraction, and types of fractions. 

The number on the bottom of the fraction is called the DENOMINATOR. The denominator tells you how many pieces the original shape is cut up into.

The number on the top of the fraction is called the NUMERATOR. The numerator tells you how many parts we are interested in.  

We also already know that a unit fraction is a fraction with a $1$1 in the numerator.  

A proper fraction is a fraction whose numerator is less than the denominator. The top is less than the bottom. This means its total size is less than $1$1. These are all proper fractions, $\frac{1}{2}$12​, $\frac{3}{8}$38​, $\frac{11}{25}$1125​ and $\frac{158}{236}$158236​.

An improper fraction is a fraction whose numerator is greater than the denominator. The bottom is less than the top. This means its size is more than $1$1. These are all improper fractions $\frac{16}{7}$167​, $\frac{23}{10}$2310​ and $\frac{9}{8}$98​.

A mixed number is a fraction that contains both whole and fractional parts. It also has a size larger than $1$1. These are all mixed numbers: $1\frac{4}{7}$147​, $16\frac{1}{2}$1612​ and $3\frac{7}{8}$378​.

 

Conversions

We need to convert between mixed numbers and improper fractions.  

These two fractions have the same value:

$\frac{4}{3}$43​ and $1\frac{1}{3}$113​, they are just written in two different forms.

$\frac{4}{3}$43​ is $4$4 thirds, we know that $3$3 thirds is one whole, and this $1$1 gets written out the front of our mixed fraction. After we take the $3$3 thirds from the $4$4 thirds we have $1$1 third remaining, and this last piece we write next to the whole number.

 

Some more conversions:

 

Worked Examples

Question 1

Question 2

Question 3

Question 4