In this chapter we will learn to add two or more algebraic fractions with solved examples.

To understand the chapter, you should have basic knowledge of fractions and LCM.

## How to add algebraic fractions ?

In order to add algebraic fraction, **follow the below steps;**

(a) **Factorize the denominator of each fraction into simple components**.

(b) **Find LCM of the denominators**.

(c) **Multiply each algebraic fraction to make denominator equal to LCM**.

Now we have algebraic fraction with common denominator.

(d) **Simply add the numerator** and retain the denominator.

(e) If possible, **reduce the resulting fraction into simple terms.**

I hope you understood the above 5 steps. Let us now some problems for further clarity.

## Adding algebraic fractions – Solved examples

**Example 01**

Add the below algebraic fractions.

\mathtt{\Longrightarrow \ \frac{1}{x} +\frac{2}{3x^{2}} \ }

**Solution****(a) Find LCM of denominators.**

\mathtt{LCM\left( x,\ 3x^{2}\right) =3x^{2}}

**(b) Multiply each fractions** to make denominator equals \mathtt{3x^{2}}

**Fraction** \mathtt{\frac{1}{x} \ }

Multiply numerator and denominator by \mathtt{\frac{1}{3x}} \\\ \\

\mathtt{\Longrightarrow \ \frac{1\times 3x}{x\times 3x}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3x}{3x^{2}}}

**Fraction** \mathtt{\frac{2}{3x^{2}}}

Here denominator is already equal to \mathtt{3x^{2}} . So don’t need to do anything here.

Now we **have got fractions with same denominator, simply add the numerator and retain the denominator**.

\mathtt{\Longrightarrow \ \frac{3x}{3x^{2}} +\frac{2}{3x^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3x+2}{3x^{2}}}

Hence, \mathtt{ \ \frac{3x+2}{3x^{2}}} is the solution of given addition.

**Example 02**

Add the below algebraic fractions.

\mathtt{\Longrightarrow \ \frac{4}{x+y} +\frac{7}{x-y}}

**Solution**

Note that the given fractions have different denominators.

We will **first try to make the fractions with same denominator and then do the addition**.

(a) **Find LCM of denominators.**

LCM (x + y, x – y) = (x + y).(x – y)

(b) **Multiply each fraction to make denominator equals (x + y).(x – y)**

**Fraction** \mathtt{\frac{4}{x+y}}

Multiply numerator and denominator by (x – y).

\mathtt{\Longrightarrow \ \frac{4}{x+y} \times \frac{x-y}{x-y}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4( x-y)}{( x+y)( x-y)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4x-4y}{x^{2} -y^{2}}}

**Fraction** \mathtt{\frac{7}{x-y}}

Multiply numerator and denominator by (x + y)

\mathtt{\Longrightarrow \ \frac{7}{x-y} \times \frac{x+y}{x+y}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7( x+y)}{( x-y)( x+y)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7x+7y}{x^{2} -y^{2}}}

Now we have fractions with same denominator.

**Simply add the numerator and retain the denominator.**

\mathtt{\Longrightarrow \ \frac{4x-4y}{x^{2} -y^{2}} +\frac{7x+7y}{x^{2} -y^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4x-4y+7x+7y}{x^{2} -y^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11x+3y}{x^{2} -y^{2}}}

Hence, \mathtt{\frac{11x+3y}{x^{2} -y^{2}}} is the solution.

**Example 03****Add the fractions.**

\mathtt{\Longrightarrow \ \frac{9}{x^{2} +xy} +\frac{2}{( x+y)^{2}}}

**Solution**

(a) **Factorize each of the algebraic fraction into smaller components**.

**Fraction** \mathtt{\frac{9}{x^{2} +xy}}

\mathtt{\Longrightarrow \ \frac{9}{x^{2} +xy}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9}{x( x+y)}}

**Fraction** \mathtt{\frac{2}{( x+y)^{2}}}

Already factorized to smaller components. No need to do anything.

(b) **Find LCM of denominator.**

\mathtt{\Longrightarrow \ LCM\ \left( \ x( x+y) \ \&\ ( x+y)^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ x\ ( x+y)^{2} \ }

(c) **Multiply each fraction to made denominator equal to LCM.****Fraction** \mathtt{\frac{9}{x( x+y)}}

Multiply numerator and denominator by (x + y)

\mathtt{\Longrightarrow \ \frac{9\ \times ( x+y)}{x( x+y) \times ( x+y)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9x+9y}{x( x+y)^{2}}}

**Fraction** \mathtt{\frac{2}{( x+y)^{2}}}

Multiply numerator and denominator by x.

\mathtt{\Longrightarrow \ \frac{2\times x}{( x+y)^{2} \times x}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2x}{x( x+y)^{2}}}

**Now we have fractions with same denominator. **

**Simply add the numerator and retain the denominator.**

\mathtt{\Longrightarrow \frac{9x+9y}{x( x+y)^{2}} +\frac{2x}{x( x+y)^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9x+9y+2x}{x( x+y)^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11x+9y}{x( x+y)^{2}}}

Hence, the above term is the solution.

**Example 04**

Add the below algebraic fractions

\mathtt{\Longrightarrow \ \frac{x}{( x+3)} +\frac{x+9}{( x+3)( x-5)}}

**Solution**

All the fractions are present in factorized form so don’t need to do any simplification.

To add these fractions, we have to make the denominators equal.**Find LCM of denominators.**

\mathtt{\Longrightarrow \ LCM\ ( \ ( x+3) ,\ ( x+3)( x-5) \ )}\\\ \\ \mathtt{\Longrightarrow \ ( x+3)( x-5)}

**Multiply each fraction to make denominator equals (x + 3) (x – 5).**

Fraction \mathtt{\frac{x}{( x+3)}}

Multiply numerator and denominator by (x – 5)

\mathtt{\Longrightarrow \ \frac{x.\ ( x-5)}{( x+3) .( x-5)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x^{2} -5x}{( x+3)( x-5)}}

Fraction \mathtt{\frac{x+9}{( x+3)( x-5)}}

The denominator is already equal to LCM, hence don’t need to do anything.

**Now we have fractions with same denominator, simply add the numerator and retain the denominator.**

\mathtt{\Longrightarrow \ \frac{x^{2} -5x}{( x+3)( x-5)} +\frac{x+9}{( x+3)( x-5)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x^{2} -5x+x+9}{( x+3)( x-5)}}\\\ \\ \mathtt{\Longrightarrow \frac{x^{2} -4x+9}{( x+3)( x-5)} \ }

Hence, the above algebraic fraction is the solution.

**Example 05**

Add the algebraic fraction

\mathtt{\Longrightarrow \frac{5}{x} +\frac{3x}{y}} **Solution****Find the LCM of denominator.**

LCM (x, y) = xy

Multiply each fraction to make denominator xy.

**Fraction 5/x**

Multiply numerator and denominator by y.

\mathtt{\Longrightarrow \ \frac{5\times y}{x\times y}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5y}{xy}}

**Fraction 3x/y**

Multiply numerator and denominator by x.

\mathtt{\Longrightarrow \ \frac{3x\times x}{y\times x}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3x^{2}}{xy}}

Now we have fractions with same denominator. Simply add the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{5y}{xy} +\frac{3x^{2}}{xy}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5y+3x^{2}}{xy}}

Hence, the above fraction is the solution