Multiplication and Division of Algebraic Fractions

Multiplying

When it comes to working with algebraic fractions and applying the four operations, the process is exactly the same as when we worked with numeric fractions.

Let's have a look at a simple example of multiplying two numerical fractions.

Example 1

Simplify $\frac{3}{4}\times\frac{5}{7}$34​×57​

$\frac{3}{4}\times\frac{2}{5}$34​×25​	$=$=	$\frac{3\times5}{4\times7}$3×54×7​     Multiplying numerators and denominators
	$=$=	$\frac{15}{28}$1528​                 Simplifying the numerator

Since $\frac{15}{28}$1528​ doesn't have any common factors between the numerator and denominator, that is the most simplified form of our answer.

 

Now let's apply the same process to multiplying algebraic fractions.

Example 2

Simplify $\frac{y}{5}\times\frac{3}{m}$y5​×3m​

$\frac{y}{5}\times\frac{3}{m}$y5​×3m​	$=$=	$\frac{y\times3}{5m}$y×35m​    Multiplying numerator and denominators
	$=$=	$\frac{3y}{5m}$3y5m​     Simplifying the numerator

Again, since the numerator $3y$3y and the denominator $5m$5m don't have any common factors, $\frac{3y}{5m}$3y5m​ is the simplest form of our answer.

 

 

Practice Questions

Question 1

Question 2

 

 

Dividing

Again, the process for dividing is the same as when we divided numeric fractions. 

Example 3

Simplify $\frac{2}{3}\div\frac{3}{5}$23​÷​35​

$\frac{2}{3}\div\frac{3}{5}$23​÷​35​	$=$=	$\frac{2}{3}\times\frac{5}{3}$23​×53​	Dividing by a fraction is the same as multiplying by its reciprocal.  So invert and multiply.
	$=$=	$\frac{2\times5}{3\times3}$2×53×3​	Multiply numerators and denominators respectively.
	$=$=	$\frac{10}{9}$109​

Since $\frac{10}{9}$109​ doesn't have any common factors between the numerator and denominator, that is the most simplified form of our answer.

 

Now let's apply the same process to dividing algebraic fractions.

Example 2

Simplify $\frac{m}{3}\div\frac{5}{x}$m3​÷​5x​

$\frac{m}{3}\div\frac{5}{x}$m3​÷​5x​	$=$=	$\frac{m}{3}\times\frac{x}{5}$m3​×x5​	Dividing by a fraction is the same as multiplying by its reciprocal.  So invert and multiply.
	$=$=	$\frac{m\times x}{3\times5}$m×x3×5​	Multiply numerators and denominators respectively.
	"="	$\frac{mx}{15}$mx15​

Again, since the numerator $mx$mx and the denominator $15$15 don't have any common factors, $\frac{mx}{15}$mx15​ is the simplest form of our answer.

 

 

Practice Questions

Question 4

Question 5

Question 6