Simplification of Fractions

We have studied some procedures for working with fractions in earlier tutorials.

\(\dfrac{a}{b}\times \dfrac{c}{d}=\dfrac{ac}{bd} \qquad (b\ne 0; d\ne 0)\)
\(\dfrac{a}{b}+\dfrac{c}{b}=\dfrac{a+c}{b} \qquad (b\ne 0)\)
\(\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times \dfrac{d}{c}=\dfrac{ad}{bc} \qquad (b\ne 0; c\ne 0; d\ne 0)\)

Note: dividing by a fraction is the same as multiplying by the reciprocal of the fraction.

In some cases of simplifying an algebraic expression, the expression will be a fraction. For example,

\[\cfrac{x^{2} + 3x}{x + 3}\]

has a quadratic binomial in the numerator and a linear binomial in the denominator. We have to apply the different factorisation methods in order to factorise the numerator and the denominator before we can simplify the expression.

\begin{align*} \cfrac{{x}^{2} + 3x}{x + 3} & = \cfrac{x(x + 3)}{x + 3}\\ & =x \qquad \qquad (x\ne -3) \end{align*}

If \(x=-3\) then the denominator, \(x + 3 = 0\) and the fraction is undefined.

This video shows some examples of simplifying fractions.

Example

Question

Simplify: \[\cfrac{ax-b+x-ab}{a{x}^{2}-abx}, \quad (x\ne 0;x\ne b)\]

Use grouping to factorise the numerator and take out the common factor \(ax\) in the denominator

\[\cfrac{(ax - ab) + (x - b)}{a{x}^{2} - abx} = \cfrac{a(x - b) + (x - b)}{ax(x - b)}\]

Take out common factor \((x-b)\) in the numerator

\[=\cfrac{(x - b)(a + 1)}{ax(x - b)}\]

Cancel the common factor in the numerator and the denominator to give the final answer

\[= \cfrac{a + 1}{ax}\]

Example

Question

Simplify: \[\cfrac{{x}^{2}-x-2}{{x}^{2}-4}\div\cfrac{{x}^{2}+x}{{x}^{2}+2x}, \quad (x\ne 0;x\ne ±2)\]

Factorise the numerator and denominator

\[= \cfrac{(x + 1)(x - 2)}{(x + 2)(x - 2)} \div \cfrac{x(x + 1)}{x(x + 2)}\]

Change the division sign and multiply by the reciprocal

\[= \cfrac{(x + 1)(x - 2)}{(x + 2)(x - 2)}\times \cfrac{x(x + 2)}{x(x + 1)}\]

Write the final answer

\[=1\]

Example

Question

Simplify: \[\cfrac{x - 2}{{x}^{2} - 4} + \cfrac{{x}^{2}}{x - 2} - \cfrac{{x}^{3} + x - 4}{{x}^{2} - 4}, \quad (x\ne ±2)\]

Factorise the denominators

\[\cfrac{x - 2}{(x + 2)(x - 2)} + \cfrac{{x}^{2}}{x - 2} - \cfrac{{x}^{3} + x - 4}{(x + 2)(x - 2)}\]

Make all denominators the same so that we can add or subtract the fractions

The lowest common denominator is \((x-2)(x+2)\).

\[\cfrac{x - 2}{(x + 2)(x - 2)} + \cfrac{({x}^{2})(x + 2)}{(x + 2)(x - 2)} - \cfrac{{x}^{3} + x - 4}{(x + 2)(x - 2)}\]

Write as one fraction

\[\cfrac{x - 2 + ({x}^{2})(x + 2) - ({x}^{3} + x - 4)}{(x + 2)(x - 2)}\]

Simplify

\[\cfrac{x - 2 + {x}^{3} + 2{x}^{2} - {x}^{3} - x + 4}{(x + 2)(x - 2)} = \cfrac{2{x}^{2} + 2}{(x + 2)(x - 2)}\]

Take out the common factor and write the final answer

\[\cfrac{2({x}^{2} + 1)}{(x + 2)(x - 2)}\]

Example

Question

Simplify: \[\cfrac{2}{{x}^{2} - x} + \cfrac{{x}^{2} + x + 1}{{x}^{3} - 1} - \cfrac{x}{{x}^{2} - 1}, \quad (x\ne 0;x\ne ±1)\]

Factorise the numerator and denominator

\[\cfrac{2}{x(x - 1)} + \cfrac{({x}^{2} + x + 1)}{(x - 1)({x}^{2} + x + 1)} - \cfrac{x}{(x - 1)(x + 1)}\]

Simplify and find the common denominator

\[\cfrac{2(x + 1) + x(x + 1) - {x}^{2}}{x(x - 1)(x + 1)}\]