Addition of the Unlike fractions

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

Introduction

In mathematics, a plus sign appears between two or more unlike fractions in a mathematical expression. It expresses us to calculate the sum of the unlike fractions mathematically.

There are three simple steps for evaluating the sum of the unlike fractions.

1. Write the unlike fractions in a row by placing a plus sign between every two unlike fractions.
2. Find the least common multiple of the denominators and write it as a denominator. Later, divide the lcm by each denominator of the fractions and multiply the quotients with the respective numerators of the fractions. Write the products in a row in the numerator position by connecting every two products by a plus sign.
3. Calculate the sum of the quantities in the numerator and write it in the numerator position. Later, write the denominator in the fraction.

Example

Evaluate sum of the fractions $\dfrac{1}{2}$, $\dfrac{2}{3}$ and $\dfrac{3}{4}$

In this case, the quantities in the denominator position of the fractions are different. So, they are unlike fractions. The addition of two or more unlike fractions can be calculated by following three simple steps.

Addition form of the Unlike fractions

Write the three unlike fractions in a row by displaying a plus sign between every two unlike fractions. It expresses the sum form of the unlike fractions in mathematical form.

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

Express the sum of the unlike fractions

The denominators of the unlike fractions are different. So, it is not possible to add them directly, same as adding the like fractions. However, it can be resolved by calculating the lowest common multiple (LCM) for the denominators.

In this example, the denominators of the fractions are $2$, $3$ and $4$. The LCM of them is $12$. You can find lcm of the numbers by any one of three methods. Now, write the smallest common multiple as the denominator for the sum of the unlike fractions.

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

Now, divide the lcm by each denominator of the fractions to calculate their quotients.

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

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

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

Now, multiply each quotient by the respective numerator of the fraction and then write the products in the numerator position by placing a plus sign between every two products.

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

Evaluate the sum of unlike fractions

Lastly, calculate the sum of the products in the numerator and write it in the numerator position along with denominator for evaluating the sum of the unlike fractions.

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

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

Thus, we can evaluate the sum of the two or more unlike fractions by following above three steps.

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