Division of Rational numbers

A mathematical operation of dividing a rational number by another rational numbers is called the division of rational numbers.

Introduction

A division sign is often displayed between two rational numbers and it expresses that the first rational number has to divide by the second rational number. In fact, a rational number cannot be divided directly by another one due to their complex expression. Hence, we have to use a special procedure to find the quotient of them mathematically.

Steps

There are three basic steps to find the quotient of any two rational numbers.

1. Express the division of the rational numbers in fraction form.
2. Keep the rational number in numerator position as it is but multiply it by the reciprocal of the rational number in the denominator.
3. Multiply the rational numbers, then find the product of them and it is equal to the quotient of the division of the given rational numbers.

Example

Simplify $\dfrac{3}{5} \div \dfrac{2}{7}$

Write the division of the rational numbers $\dfrac{3}{5}$ and $\dfrac{2}{7}$ in fraction form.

$= \,\,\,$ $\dfrac{\dfrac{3}{5}}{\dfrac{2}{7}}$

Write the division of the rational numbers in multiplication form as per the property of reciprocal or multiplicative inverse.

$= \,\,\,$ $\dfrac{\dfrac{3}{5} \times 1}{\dfrac{2}{7}}$

$= \,\,\,$ $\dfrac{3}{5} \times \dfrac{1}{\dfrac{2}{7}}$

$= \,\,\,$ $\dfrac{3}{5} \times \dfrac{7}{2}$

Now, find the product of the rational numbers by the multiplication of rational numbers.

$= \,\,\,$ $\dfrac{3 \times 7}{5 \times 2}$

$= \,\,\,$ $\dfrac{21}{10}$

$\therefore \,\,\,$ $\dfrac{3}{5} \div \dfrac{2}{7}$ $\,=\,$ $\dfrac{21}{10}$

In this way, one rational number is divided by another rational number mathematically.

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