An algebraic fraction is a fraction that contains letters, numbers or both. Simplifying algebraic fractions means writing them in their simplest form by cancelling common factors.

 

Only factors can be cancelled, not terms that are added or subtracted.

 

 
Simplifying Algebraic Fractions by Cancelling Factors

To simplify an algebraic fraction, factorise the numerator and denominator and then cancel any common factors.

 

For example:

$$
\frac{6x}{9}
$$

 

Both \( 6 \) and \( 9 \) have a common factor of \( 3 \):

$$
\frac{6x}{9} = \frac{2x}{3}
$$

 

Another example:

$$
\frac{4a}{8}
$$

$$
= \frac{a}{2}
$$

 

Letters can also be cancelled if they are factors in both the numerator and denominator.

$$
\frac{5x}{x}
$$

$$
= 5
$$

This is only valid if \( x \neq 0 \).

 

 
Fractions with Constant Terms as Denominators

When the denominator is a constant, the same simplification rules apply. The number in the denominator divides every term in the numerator.

 

For example:

$$
\frac{6x}{3}
$$

$$
= 2x
$$

 

 
Adding Algebraic Fractions with Constant Denominators

Algebraic fractions can only be added if they have the same denominator.

 

For example:

$$
\frac{x}{4} + \frac{3}{4}
$$

 

Add the numerators and keep the denominator the same:

$$
\frac{x + 3}{4}
$$

 

Another example:

$$
\frac{2x}{5} + \frac{7}{5}
$$

$$
= \frac{2x + 7}{5}
$$

 

 

Subtracting Algebraic Fractions with Constant Denominators

Subtraction works in the same way as addition, but signs must be handled carefully.

 

For example:

$$
\frac{x}{6} - \frac{5}{6}
$$

$$
= \frac{x - 5}{6}
$$

 

Another example:

$$
\frac{3x}{8} - \frac{2}{8}
$$

$$
= \frac{3x - 2}{8}
$$

 

 

Key Points to Remember

Only cancel factors, not individual terms.
Denominators must be the same before adding or subtracting.
Always simplify the final fraction if possible.