A fraction whose numerator is less than its denominator is called a proper fraction.

Firstly, you have to know the meaning of “Proper” for understanding the concept of proper fraction clearly. As per English language, there are some meanings for proper, which are right, acceptable, true, appropriate, and so on. All these are giving same meaning closely.

Any quantity can be divided as equal number of parts and each part is a fraction. It is acceptable to take either one or more parts from whole quantity as a fraction and it’s not right to consider all parts of whole quantity as a fraction because the sum of all parts represents a whole quantity. Therefore, it is proper that the quantity of a fraction is always less than the quantity of whole quantity, and the fraction is called a proper fraction.

The quantity of a proper fraction can be calculated by comparing its quantity with the actual quantity and it can be done mathematically by the ratio mathematically.

$Number \, of \, Selected \, parts$ $\,<\,$ $Total \, number \, of \, Parts$

$\therefore \,\,\,$ $Proper \, fraction$ $\,=\,$ $\dfrac{Number \, of \, Selected \, parts}{Total \, number \, of \, Parts}$ $\,<\,$ $1$

Take one circle and divide it into five equal parts.

Take one part from five parts and compare them. One part can’t be a circle. Actually, it is one part from five parts. Mathematically, the relationship can be expressed by ratio between them. Therefore $1:5$.

Fraction = $\dfrac{1}{5}$

The value in the numerator is less than the value in the denominator ($1 < 5$). Therefore, it's called as a proper fraction.

Proper fraction = $\dfrac{1}{5}$

Take two parts and compare it with actual quantity. Two parts cannot be one circle and it can be understood from their ratio. Therefore, $2 : 5$.

Fraction = $\dfrac{2}{5}$

In this fraction, the value in the numerator is less than the value in the denominator ($2 < 5$). Therefore, the fraction called as a proper fraction.

Proper fraction = $\dfrac{2}{5}$

Now, take three parts and compare them, also express relation between them in a ratio. Therefore, their ratio is $3 : 5$.

Fraction = $\dfrac{3}{5}$

It clears that the value in the numerator is less than the value in the denominator ($3 < 5$). Therefore, it is obviously called as a proper fraction.

Proper fraction = $\dfrac{3}{5}$

Consider four parts from five parts. You can understand that the quantity of four parts is not equal to the quantity of the whole circle. It can be understood from ration between them. In other words, $4 : 5$.

Fraction = $\dfrac{4}{5}$

You can understand from the fraction that the value in the numerator is less than the value in the denominator ($4 < 5$). Therefore, the fraction is known as a proper fraction.

Proper fraction = $\dfrac{4}{5}$

Now, take all parts and compare its quantity with original quantity. It can be written in ratio form as $5 : 5$.

Quantity = $\dfrac{5}{5}$ = $1$

In this case, the values in the numerator and denominators are equal. Therefore, its quantity is $1$, which represents that they both are same.

Therefore, it’s inappropriate to consider the whole quantity as a fraction mathematically.

This example clears that the quantities $\dfrac{1}{5}$, $\dfrac{2}{5}$, $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are called fractions basically and they can also be called as proper fractions because the value in the numerator is less than its denominator.

Whenever you see a rational number, in which the numerator is less than the denominator, then you have to understand that the rational number represents a proper fraction mathematically.

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