# Types of Fractions

The three types of fractions are :

Proper Fraction

Improper Fraction

Mixed Fraction

A fraction can be classified in three ways proper fraction, improper fraction and mixed fraction.

Let us discuss the three types of fractions with the help of an example.

If Sufi has 3 cookies and she wants to give equal share to Rachel, what share both will get? We divide 3 by 2. It is written as fraction $$\frac{3}{2}$$.

In the above example of sharing 3 cookies among Sufi  and Rachel the fraction $$\frac{3}{2}$$ has 3 as numerator and 2 as denominator. When the numerator is greater than the denominator, the fraction is called the improper fraction. Thus an improper fraction represents a quantity greater than one.

We can represent the share of cookies received by Sufi and Rachel in the following way.

We can write this as 1 $$\frac{1}{2}$$, which is a combination of a whole number and a fraction.

This is called a mixed fraction. Thus, an improper fraction can be expressed as a mixed fraction, where quotient represents the whole number, remainder becomes the numerator and divisor is the denominator. A fraction, where the numerator is less than the denominator is called the proper fraction for example, $$\frac{2}{3}$$, $$\frac{5}{7}$$, $$\frac{3}{5}$$ are proper fractions. A fraction with numerator 1 is called a unit fraction.

Proper fraction:

Fractions whose numerators are less than the denominators are called proper fractions. (Numerator < denominator)

For examples:

$$\frac{2}{3}$$, $$\frac{3}{4}$$, $$\frac{4}{5}$$, $$\frac{5}{6}$$, $$\frac{6}{7}$$, $$\frac{2}{9}$$ $$\frac{5}{8}$$, $$\frac{2}{5}$$, etc are proper fractions.

Two parts are shaded in the above diagram. Total number of equal parts is 3. Therefore, the shaded part can be represented as $$\frac{2}{3}$$ in fraction. The numerator (top number) is less compared to the denominator (bottom number). This type of fraction is called proper fraction.

Similarly,

Three parts are shaded in the above diagram. Total number of equal parts is 4. Therefore, the shaded part can be represented as $$\frac{3}{4}$$ in fraction. The numerator (top number) is less compared to the denominator (bottom number). This type of fraction is called proper fraction.

Note: The value of a proper fraction is always less than 1.

Improper fraction:

Fractions with the numerator either equal to or greater than the denominator are called improper fraction. (Numerator = denominator or, Numerator > denominator)

Fractions like $$\frac{5}{4}$$, $$\frac{17}{5}$$, $$\frac{5}{2}$$ etc. are not proper fractions. These are improper fractions. The fraction $$\frac{7}{7}$$ is an improper fraction.

The fractions $$\frac{5}{4}$$, $$\frac{3}{2}$$, $$\frac{8}{3}$$, $$\frac{6}{5}$$, $$\frac{10}{3}$$, $$\frac{13}{10}$$, $$\frac{15}{4}$$, $$\frac{9}{9}$$, $$\frac{20}{13}$$, $$\frac{12}{12}$$, $$\frac{13}{11}$$, $$\frac{14}{11}$$, $$\frac{17}{17}$$ are the examples of improper fractions. The top number (numerator) is greater than the bottom number (denominator). Such type of fraction is called improper fraction.

Notes:

(i) Every natural number can be written as a fraction in which 1 is it's denominator. For example, 2 = $$\frac{2}{1}$$, 25 = $$\frac{25}{1}$$, 53 = $$\frac{53}{1}$$, etc. So every natural number is an improper fraction.

(ii) The value of an improper fraction is always equal to or greater than 1.

Mixed fraction:

A combination of a proper fraction and a whole number is called a mixed fraction.

1$$\frac{1}{3}$$, 2$$\frac{1}{3}$$, 3$$\frac{2}{5}$$, 4$$\frac{2}{5}$$, 11$$\frac{1}{10}$$, 9$$\frac{13}{15}$$ and 12$$\frac{3}{5}$$ are examples of mixed fraction.

Two $$\frac{1}{2}$$, make a whole. $$\frac{1}{2}$$               $$\frac{1}{2}$$ $$\frac{1}{2}$$ + $$\frac{1}{2}$$ = $$\frac{2}{2}$$ = 1

What will you get if you add one more $$\frac{1}{2}$$ to a whole? $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ = 1 + $$\frac{2}{2}$$ = 1$$\frac{1}{2}$$

Now, you have three half or you can say that you have a whole and a half or $$\frac{1}{2}$$.

Number such as 1$$\frac{1}{2}$$ is a mixed number.

In other words:

A fraction which contains of two parts: (i) a natural number and (ii) a proper fraction, is called a mixed fraction, e.g., 3$$\frac{2}{5}$$, 7$$\frac{3}{4}$$, etc.

In 3$$\frac{2}{5}$$, 3 is the natural number part and $$\frac{2}{5}$$ is the proper fraction part.

In Fact, 3$$\frac{2}{5}$$ means 3 + $$\frac{2}{5}$$.

Note: A mixed number is formed with a whole number and a fraction.

Property 1:

A mixed fraction may always be converted into an improper fraction.

Multiply the natural number by the denominator and add to the numerator. This new numerator over the denominator is the required fraction.

3$$\frac{1}{2}$$ = $$\frac{3 × 2 + 1}{2}$$ = $$\frac{6 + 1}{2}$$ = $$\frac{7}{2}$$ .

Property 2:

An important fraction can be always be converted into a mixed fraction.

Divide the numerator by the denominator to get the quotient and remainder. Then the quotient is the natural number part and the remainder over the denominator is the proper fraction part of the required mixed fraction.

Example: $$\frac{43}{6}$$ can be converted into a mixed fraction as follows:

7
6 |43
- 42
1

Dividing 43 by 6, we get quotient = 7 and remainder = 1.

Therefore, $$\frac{43}{6}$$ = 7 $$\frac{1}{6}$$

Note: Proper fraction is between 0 to 1. Improper fraction is 1 or greater than 1. Mixed fraction is grater than 1.

1. Write $$\frac{37}{4}$$ as mixed fraction.

Solution:

So, Quotient = 9, Remainder = 1 and Divisor = 4

Mixed fraction = Quotient $$\frac{Remainder}{Divisor}$$

So, $$\frac{37}{4}$$ can be expressed as 9$$\frac{1}{4}$$ where 9 is a whole number and $$\frac{1}{4}$$ is a proper fraction.

2. Classify the following as proper fractions, improper fractions or unit fractions.

$$\frac{8}{12}$$, $$\frac{10}{27}$$, $$\frac{17}{12}$$, $$\frac{2}{5}$$, $$\frac{1}{13}$$, $$\frac{5}{12}$$, $$\frac{6}{15}$$, $$\frac{1}{32}$$, $$\frac{31}{12}$$, $$\frac{27}{4}$$

Solution: