Math Doubts

Improper fraction

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.

Introduction

To understand the concept of improper fraction easily, you must know the meaning of “Improper”. According to English language, there are several meanings for improper, in which inappropriate, unacceptable and irregular are some of them but all of them give same meaning closely in this case.

Any quantity can be split as equal number of parts and each part is known as a fraction. Actually, it’s not acceptable to consider all the parts of a whole quantity as a fraction but the whole quantity is in the form of fractions. So, we are improperly taking all the parts of a whole quantity as a fraction and the fraction is called as an improper fraction.

Formula

The quantity of an improper fraction is called by comparing its quantity of total number of selected parts with the number of parts of a whole quantity. It’s actually done by the ratio in mathematics.

$Number \, of \, Selected \, parts$ $\,\geq\,$ $Total \, number \, of \, Parts \, of \, a \, Quantity$

$\therefore \,\,\,$ $Improper \, fraction$ $\,=\,$ $\dfrac{Number \, of \, Selected \, parts}{Total \, number \, of \, Parts \, of \, a \, Quantity}$ $\,\geq\,$ $1$

Examples

An improper fraction is completely opposite to the proper fraction. The following examples help you to understand how fractions are formed improperly.

Whole Quantity as fraction

Take a square and divide it into four equal parts. Actually, it’s not a fraction but all of its parts are fractions. Hence, all the parts of a quantity considers as a fraction improperly. Therefore, the fraction is called as an improper fraction and written in ratio form as $4 : 4$.

improper fraction

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

The value in the numerator is equal to the value in the denominator. Therefore, it’s called as an improper fraction.

Improper fraction = $\dfrac{4}{4}$

The quantity of this fraction is $1$ and it’s a whole number but it is taken as an improper fraction if it’s written as a rational number.

Whole Quantity & One part

improper fraction

Consider two squares. Take, all parts from one square and a part from another square.

Fraction = $\dfrac{4}{4} + \dfrac{1}{4}$ = $\dfrac{5}{4}$

It is written as $5 : 4$ in ratio form. In this case, the value of numerator is greater than its denominator ($5 > 4$). Therefore, it’s known as an improper fraction.

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

Whole Quantity & Two parts

improper fraction

Take, all parts from one square and two parts from another square.

Fraction = $\dfrac{4}{4} + \dfrac{2}{4}$ = $\dfrac{6}{4}$

It’s written as $6 : 4$ in ratio form. The value in the numerator is greater than its denominator ($6 > 4$). So, it’s considered as an improper fraction.

Improper fraction = $\dfrac{6}{4}$

Whole Quantity & Three parts

improper fraction

Consider, all parts from one square and three parts from second square.

Fraction = $\dfrac{4}{4} + \dfrac{3}{4}$ = $\dfrac{7}{4}$

It’s expressed as $7 : 4$ in ratio form mathematically. In this example, the value in the numerator is greater than its denominator ($7 > 4$). Hence, it’s known as an improper fraction.

Improper fraction = $\dfrac{7}{4}$

Whole Quantities

improper fraction

Now, take all parts from two whole quantities.

Fraction = $\dfrac{4}{4} + \dfrac{4}{4}$ = $\dfrac{8}{4}$

It can be written in ratio form as $8 : 4$ in mathematics. The value in the numerator is greater than its denominator ($8 > 4$) and the fraction is known as an improper fraction.

Improper fraction = $\dfrac{8}{4}$

Mathematically, the quantity of this fraction is $2$, which is a whole number but it’s considered as a fraction whenever a whole quantity is written in the form a rational number.

Therefore, the quantities $\dfrac{4}{4}$, $\dfrac{5}{4}$, $\dfrac{6}{4}$, $\dfrac{7}{4}$ and $\dfrac{8}{4}$ are improperly called as fractions and they’re also called as improper fractions due to either same or greater value in the numerator when it’s compared with the value in the denominator.

It’s cleared that the value in the numerator is always greater than or equal to its denominator in the case of an improper fraction. Similarly, any whole number which is expressed in as a rational number, is considered as an improper fraction as well.



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