What are algebraic fractions?

In mathematical terms, a common fraction is a number that is written in the form of $$\frac{a}{b}$$ or $$a/b$$, where the numerator is $$a$$ and the denominator is $$b$$. On the other hand, an algebraic fraction is one in which the numerator and the denominator are both algebraic expressions. 

The real-life example of algebraic expressions can be seen in how the applications of fractions are employed to divide the pizza slices equally amongst everyone.

E2.3A: Manipulate algebraic fractions

The rational fraction which has a numerator, or a denominator as an algebraic expression, is called an algebraic fraction. The examples of algebraic fractions are $$\frac{2x}{x^{2}+4x-3}$$ and $$ \frac{\sqrt{x+5}}{x^{2}-3}$$. 

Now, the component that needs to be done is  manipulating, or to put it other way, solving or simplifying the algebraic fraction. In this part, one more than one fraction are needed to be solved to get the simplest and most understandable algebraic expression. 

This may be understood with the help of an example. Let us imagine you have two algebraic fractions, $$\frac{2x}{3}$$ and $$\frac{4x}{6}$$ and you want to add them together. To add the fractions, first get the LCM of the denominators of both fractions in the given expression, i.e., $$3$$ and $$6$$. After taking LCM, cross multiply the two fractions to get one fraction, then simplify the expressions by adding the common terms. After completing the task, you will be left with one simple algebraic expression.

Worked examples of algebraic fractions

Example 1: Evaluate $$\frac{3a}{4}\div \frac{9a}{10}$$.

Step 1: Write the given algebraic fraction.

$$\frac{3a}{4}\div \frac{9a}{10}$$

Step 2: Change the sign of divide to the sign of multiply by reversing the second fractions.

$$\frac{3a}{4}\times \frac{10}{9a}$$

Step 3: Solve the fractions further.

$$\frac{3a\times 10}{4\times 9a}$$

Step 4: Simplify.

$$\frac{30a}{36a}$$

Step 5: Eliminate the common terms from the numerator and the denominator.

$$\frac{5}{6}$$

Example 2: Evaluate $$\frac{2}{x-3}+\frac{3}{x-2}$$.

Step 1: Write the given algebraic fraction.

$$\frac{2}{x-3} +\frac{3}{x-2}$$.

Step 2: Take the LCM of the given denominators

$$\left(x-3\right) \left(x-2 \right)$$ which is equal to $$x^{2}-5x+6$$

Step 3: Cross multiply the two fractions.

$$\frac{2}{x-3}+\frac{3}{x-2}=\frac{2\left(x-2 \right)+3\left(x-3 \right)}{x^{2}-5x+6}$$

Step 4: Solve the expression further and find out the answer.

$$\frac{2}{x-3}+\frac{3}{x-2}=\frac{2x-4+3x-9}{x^{2}-5x+6}$$

$$\frac{2}{x-3}+\frac{3}{x-2}=\frac{5x-13}{x^{2}-5x+6}$$

E2.3B: Factorise and simplify rational expressions.

In order to make any algebraic fraction understandable, we have to simplify and factorise the terms of the rational expression. This simplification task makes the whole expression very easily understandable. To solve the expression, we must follow certain steps to ensure that both the mathematic procedure as well the solution are factually correct. Now, as shown in  the picture below, the various steps are described starting from taking the LCM and making the denominator the same, then adding and subtracting the numerator terms, and finally obtaining the simplest rational expression.

Now, in the following picture, we are given an equation to solve  that requires us to divide the two terms. Now, instead of dividing, we can reverse the second equation and perform cross multiplication of the terms. We can then eliminate the similar terms and get the simple expression.

Worked examples of factorising and simplifying algebraic fractions

Example 1: Simplify $$\frac{y^{3}-4y}{\left(y^{2}-4y+4\right) \left(y^{2}+4y+4\right )}$$.

Step 1: Write the given expression.

$$\frac{y^{3}-4y}{\left(y^{2}-4y+4\right) \left(y^{2}+4y+4\right )}$$

Step 2: Factorise the numerator of the given expression.

$$(y^{3}-4y)=y\left (y^{2}-4 \right )$$

$$y\left(y-2 \right ) \left(y+2 \right )$$

Step 3: Factorise the denominator of the given expression.

$$\left(y^{2}-4y+4\right) \left(y^{2}+4y+4\right )= \left(y-2 \right )\left(y-2 \right ) \left(y+2 \right )\left(y+2 \right )$$

$$\left(y-2 \right )^{2} \left(y+2 \right )^{2}$$

Step 4: Substitute the factorised values of numerator and denominator.

$$\frac{y^{3}-4y}{\left(y^{2}-4y+4\right) \left(y^{2}+4y+4\right )}=\frac{y\left(y-2 \right ) \left(y+2 \right )}{\left(y-2 \right )^{2} \left(y+2 \right )^{2}}$$

Step 5: Simplify the above expression.

$$\frac{y^{3}-4y}{\left(y^{2}-4y+4\right)\left(y^{2}+4y+4\right )}=\frac{y}{\left(y-2 \right ) \left(y+2 \right )}$$

$$\frac{y}{y^{2}-4}$$