# Adding algebraic fraction

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
\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

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