# Addition of the fractions with the different denominators

## Definition

A mathematical operation of adding two or more fractions with the different denominators is called the addition of the unlike fractions.

### Introduction

A plus sign displays between two fractions to express sum of them in mathematics. In some cases, two or more fractions with different denominators are involved in addition and the mathematical process of adding the fractions having different denominators is called the addition of the unlike fractions.

There are four steps to add two or more fractions with the unlike denominators.

1. Write the unlike fractions in a row by placing a plus sign between every two unlike fractions.
2. In this case, the denominators of the fractions are different. So, the denominator for their sum is evaluated by finding the least or lowest common multiple (L.C.M) of their denominators.
3. Now, divide the least common multiple (L.C.M) by the denominator of every fraction and then obtain the products by multiplying the numerator of every fraction with corresponding quotient.
4. Finally, add the products and it will be the numerator for the sum of the unlike fractions.

Now, let’s learn how to use the above four steps to find the sum of the fractions having different denominators.

#### Example

Add the fractions $\dfrac{1}{2}$, $\dfrac{2}{3}$ and $\dfrac{3}{4}$

Firstly, let’s write the three fractions having different denominators in a row by including a plus sign between every two fractions.

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

Look at the denominators of the fractions and we can understand that the denominators of them are different. So, find the smallest or least common multiple (L.C.M) of the different denominators and it will be the denominator for the sum of the unlike fractions. The lowest common multiple (L.C.M) of the denominators $2,$ $3$ and $4$ is $12$.

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

Now, divide the lcm by each denominator of the fraction to find the quotients as follows.

$(1).\,\,\,$ $\dfrac{12}{2} \,=\, 6$

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

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

Now, multiply the numerator by the respective quotient. The fractions that contain different denominators are involved in the addition. So, add the products and the sum of the products is the numerator for the sum of the unlike fractions.

$\implies$ $\dfrac{1}{2}$ $+$ $\dfrac{2}{3}$ $+$ $\dfrac{3}{4}$ $\,=\,$ $\dfrac{1 \times 6+2 \times 4+3 \times 3}{12}$

Finally, add the products in the numerator to find the sum of them.

$\implies$ $\dfrac{1}{2}$ $+$ $\dfrac{2}{3}$ $+$ $\dfrac{3}{4}$ $\,=\,$ $\dfrac{6+8+9}{12}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{1}{2}$ $+$ $\dfrac{2}{3}$ $+$ $\dfrac{3}{4}$ $\,=\,$ $\dfrac{23}{12}$

In this way, the fractions that consist of different denominators are added in mathematics by using the above procedure.

#### Worksheet

The list of questions on adding the fractions having the different denominators for your practice and examples with solutions to learn how to add the two or more fractions with different denominators.

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