Subtraction of Rational numbers

A mathematical operation of subtracting a rational number from another rational number is called the subtraction of rational numbers.

Introduction

Two rational numbers are connected by a negative sign (minus) in some cases and it expresses that we have to subtract a rational number from another. In mathematics, it is not possible to subtract any rational number from another rational number directly due to their special form. Hence, we must use a special procedure for finding the difference of them mathematically.

Required knowledge

In order to do subtraction of the rational numbers, you must have knowledge on the following two mathematical concepts.

Steps

The difference of the rational numbers can be evaluated in three simple steps.

Find the least common multiple (L.C.M) of denominators (consequents) of rational numbers and then, write it as denominator.
Divide the lowest common multiple (L.C.M) by the denominator of each rational number. Later, multiply the quotient by the numerator (antecedent) of the respective rational number. Finally, write the products in numerator by connecting them using a negative sign.
Finally, simplify the expression in the numerator for obtaining the difference of the rational numbers.

Example

Evaluate $\dfrac{4}{3}-\dfrac{2}{5}$

In this example, a minus sign is displayed between the rational numbers $\dfrac{4}{3}$ and $\dfrac{2}{5}$. It expresses that we have to subtract the rational number $\dfrac{2}{5}$ from another rational number $\dfrac{4}{3}$.

Step – 1

$3$ and $5$ are denominators and calculate the L.C.M for them. It is calculated that the least common multiple of them is $15$. Now, write it as the denominator.

$= \,\,\,$ $\dfrac{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}{15}$

Step – 2

Now, divide the lowest common multiple by each denominator of the rational number.

$(1). \,\,\,\,\,\,$ $\dfrac{15}{3} \,=\, 5$

$(2). \,\,\,\,\,\,$ $\dfrac{15}{5} \,=\, 3$

Now, multiply the quotient by the respective numerator of the rational number and write them in the numerator by connecting them by a minus sign.

$= \,\,\,$ $\dfrac{5 \times 4-3 \times 2}{15}$

Step – 3

Finally, simplify the expression in the numerator.

$= \,\,\,$ $\dfrac{20-6}{15}$

$\therefore \,\,\,$ $\dfrac{4}{3}-\dfrac{2}{5}$ $\,=\,$ $\dfrac{14}{15}$

In this way, the rational numbers are subtracted in the arithmetic mathematics.