Addition of Rational numbers

A mathematical operation of adding two or more rational numbers is called the addition of rational numbers.

Introduction

In mathematics, two or more rational numbers are connected by plus sign in some cases and it expresses that we have to perform the addition of them. Actually, it is not possible to add the rational numbers directly due to their expression. So, we have to use a special procedure to find the sum of two or more rational numbers mathematically.

Required knowledge

For adding the rational numbers, you have to learn the following two mathematical concepts.

1. Least Common Multiple (L.C.M)
2. Methods of Finding Lowest Common Multiple (L.C.M)

Steps

The sum of two or more rational numbers can be calculated in three simple steps.

1. Find the least common multiple of denominators (consequents) of all rational numbers. Then, write the L.C.M in denominator position.
2. Now, divide the lowest common multiple (L.C.M) by the denominator of each rational number and then multiply the quotient by the numerator (antecedent) of the respective rational number. Finally, write all the products in numerator by connecting them using a plus sign.
3. Simplify the expression in the numerator to get the sum of the rational numbers.

Example

Find $1+\dfrac{2}{3}+\dfrac{7}{4}$

In this example, $1$ is an integer but it can be written as a rational number $\dfrac{1}{1}$ and it is connected by plus sign with two rational numbers $\dfrac{2}{3}$ and $\dfrac{7}{4}$

$= \,\,\,$ $\dfrac{1}{1}+\dfrac{2}{3}+\dfrac{7}{4}$

$1, 3$ and $4$ are denominators and find the L.C.M for them. It is calculated that the least common multiple of them is $12$ and write it in the denominator position.

$= \,\,\,$ $\dfrac{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}{12}$

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

$(1). \,\,\,\,\,\,$ $\dfrac{12}{1} \,=\, 12$

$(2). \,\,\,\,\,\,$ $\dfrac{12}{3} \,=\, 4$

$(3). \,\,\,\,\,\,$ $\dfrac{12}{4} \,=\, 3$

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

$= \,\,\,$ $\dfrac{12 \times 1+4 \times 2+3 \times 7}{12}$

Now, simplify the expression in the numerator.

$= \,\,\,$ $\dfrac{12+8+21}{12}$

$\therefore \,\,\,$ $1+\dfrac{2}{3}+\dfrac{7}{4}$ $\,=\,$ $\dfrac{41}{12}$

Thus, the rational numbers are added in the arithmetic mathematics.

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