### Calculating with letters: Fractions with letters

### Simplification of fractions with letters

Just as with ordinary fractions, fractions with letters can sometimes be simplified. If the numerator and denominator have a common denominator, you can divide both of them by this divisor. This can be a number, but also an expression with variables.

\[\begin{aligned}\frac{2a+4}{4a+2b}&=\frac{a+2}{2a+b}\quad (\mathrm{division\;by\;}2)\\ \\ \frac{a}{a^2+a}&=\frac{1}{a+1} \quad (\mathrm{division\;by\;}a)\end{aligned}\]

In the last example, there is a caveat: this is only allowed if the expression that you want to use as divisor is not equal to 0. More precisely, you should write here: \[\frac{a}{a^2+a}=\frac{1}{a+1}\quad\mathrm{if}\quad a\ne 0\]

Two more of such examples:

Two examples \[\begin{aligned}\frac{a-1}{a^2-1}=\frac{a-1}{(a-1)(a+1)}\;& =\frac{1}{a+1}\quad\mathrm{if}\quad a\ne 1\\ \\ \\ \frac{a^4-b^4}{a^3+a^2b+ab^2+b^3}&=\frac{(a^2-b^2)(a^2+b^2)}{(a+b)(a^2+b^2)}\\ \\ &= \frac{a^2-b^2}{a+b}\\ \\ &= \frac{(a-b)(a+b)}{a+b}\\ \\ &= a-b \quad\mathrm{if}\quad a+b\ne 0\;\;\mathrm{and}\;\;a^2+b^2\ne 0\end{aligned}\]

Mathcentre video clips

Simplification of Algebraic Fractions (17:33)