# Dividing algebraic fractions

In this post we will learn to divide two algebraic fractions using solved examples.

To understand this chapter you should have basic knowledge about factorization of polynomials.

## How to divide algebraic fractions ?

Two fractions can be easily divided by converting division sign into multiplication by taking reciprocal of divisor and then multiplying the fractions.

To divide two fractions, follow the below step;

(a) Convert division into multiplication by taking reciprocal of divisor.

(b) Now multiply numerator and denominator separately

(c) If possible, simplify the fraction by cancelling the common terms.

Generally, the above division can be expressed as;

\mathtt{\Longrightarrow \ \frac{Numerator\ 1}{Denominator\ 1} \ \div \frac{Numerator\ 2}{Denominator\ 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{Numerator\ 1}{Denominator\ 1} \times \frac{Denominator\ 2}{Numerator\ 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{Numerator\ 1\ \times Denominator\ 2}{Denominator\ 1\times Numerator\ 2}}

I hope you understand the above concept. Let us solve some examples for further clarity.

## Dividing algebraic fractions – Solved examples

Example 01
Divide the below algebraic fraction

\mathtt{\Longrightarrow \ \frac{x}{y} \ \div \frac{7}{3x}}

Solution
Take reciprocal of divisor to convert division into multiplication.

\mathtt{\Longrightarrow \ \frac{x}{y} \ \div \frac{7}{3x}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x}{y} \times \frac{3x}{7}}

Now multiply the numerator and denominator separately.

\mathtt{\Longrightarrow \ \frac{x\times 3x}{y\times 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3x^{2}}{7y}}

Example 02
Divide the algebraic fractions.

\mathtt{\Longrightarrow \ \frac{3xy}{2z} \ \div \frac{5x}{7z}}

Solution
Take reciprocal of divisor to convert division into multiplication.

\mathtt{\Longrightarrow \ \frac{3xy}{2z} \ \div \frac{5x}{7z}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3xy}{2z} \times \frac{7z}{5x}}

Cancelling the common terms from numerator and denominator.

\mathtt{\Longrightarrow \ \frac{3\ \cancel{x} \ y}{2\ \cancel{z}} \times \frac{7\ \cancel{z}}{5\ \cancel{x}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3y\times 7}{2\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{21y}{10}}

Hence, the above expression is the solution.

Example 03
Divide the below algebraic fractions.

\mathtt{\Longrightarrow \ \frac{5\left( x^{2} -y^{2}\right)}{11x^{2} y} \ \div \frac{3( x+y)}{2y}}

Solution
Take reciprocal of divisor to convert division into multiplication.

\mathtt{\Longrightarrow \frac{5\left( x^{2} -y^{2}\right)}{11x^{2} y} \times \frac{2y}{3\ ( x+y)}}

Referring formula;
\mathtt{\left( a^{2} -b^{2}\right) =( a-b)( a+b)}

Using the formula in above expression, we get;

\mathtt{\Longrightarrow \ \frac{5( x-y)( x+y)}{11x^{2} y} \ \times \frac{2y}{3( x+y)}}

Cancelling out common term from numerator and denominator.

\mathtt{\Longrightarrow \ \frac{5( x-y) \ \cancel{( x+y)}}{11x^{2} \ \cancel{y}} \ \times \frac{2\ \cancel{y}}{3\ \cancel{( x+y)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10\ ( x-y)}{33\ x^{2}}}

Hence, the above expression is the solution.

Example 04
Divide the below algebraic fractions.

\mathtt{\Longrightarrow \ \frac{x^{2} -6x+9}{( x+5)} \ \div \frac{x^{2} -9}{x^{2} +10x+25}}

Solution
Take reciprocal of divisor to convert division into multiplication.

\mathtt{\Longrightarrow \ \frac{x^{2} -6x+9}{( x+5)} \ \times \frac{x^{2} +10x+25}{x^{2} -9}}

Now try to factorize the given polynomial into simple terms.

\mathtt{\Longrightarrow \frac{x^{2} -2.x.3+3^{2}}{( x+5)} \times \frac{x^{2} +2.5.x+5^{2}}{x^{2} -3^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{( x-3)^{2}}{( x+5)} \times \frac{( x+5)^{2}}{( x-3)( x+3)}}

Cancel the common terms from numerator and denominator.

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

Hence, the above expression is the solution.

Example 05
Divide the below algebraic fractions.

\mathtt{\Longrightarrow \ \frac{x^{2} +11x}{6xy} \div \frac{\left( x^{2} -121\right)}{3xy}}

Solution
Take reciprocal of divisor to convert division into multiplication.

\mathtt{\Longrightarrow \frac{x^{2} +11x}{6xy} \times \frac{3xy}{\left( x^{2} -121\right)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x( x+11)}{6xy} \times \frac{3xy}{( x-11)( x+11)}}

Cancel out the common terms.

\mathtt{\Longrightarrow \ \frac{x\ \cancel{( x+11)}}{\mathbf{2} \ \cancel{6xy}} \times \frac{\cancel{3xy}}{( x-11)\cancel{( x+11)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x}{2( x-11)}}

Hence, the above expression is the solution.