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### Algebraic Fractions (Economics)

#### Algebraic Fractions

If the numerator and/or the denominator of a fraction is an algebraic expression, then the fraction is called an algebraic fraction.

Examples: $\dfrac{x+1}{x+2},\;\;\;\dfrac{y^2+xy-2}{x^2+xy-1},\;\;\;\dfrac{3x+3}{6},\;\;\;\dfrac{x^2+2x+1}{x^2+3x+2}$

Algebraic fractions do not change when we multiply both the numerator and denominator by the same non-zero expression.

Examples:

\begin{align} \dfrac{y}{y^2+1}&=\dfrac{y \times (y^2+1)}{(y^2+1)\times (y^2+1)}=\dfrac{y(y^2+1)}{(y^2+1)^2}\\ \dfrac{x}{x+1}&=\dfrac{x \times (x-1)}{(x+1)\times (x-1)}=\frac{x^2-x}{x^2-1}\\ \end{align} In the last example we require that $x \neq \pm 1$ as otherwise the fraction is undefined.

#### Simplifying Algebraic Fractions

We simplify algebraic fractions in the same way that we simplify arithmetic fractions. However when simplifying algebraic fractions we must always be aware of the possibility of dividing by zero.

##### Worked Example
###### Worked Example

Simplify $\dfrac{x^2+3x+2}{x^2+4x+3}$.

###### Solution

\begin{align} \dfrac{x^2+3x+2}{x^2+4x+3}&=\dfrac{(x+2)(x+1)}{(x+3)(x+1)}\\ &=\dfrac{x+2}{x+3},\;\;x \neq -1 \end{align} On cancelling the common factor $x+1$. Note that we are careful to point out the value of $x$ at which the expression is not defined.

#### Adding and Subtracting Algebraic Fractions

We add and subtract algebraic fractions in the same way we do with arithmetic fractions.

Example: \begin{align} \frac{y+z-xy}{y+z+1}+\frac{1+xy}{y+z+1}&=\dfrac{(y+z-xy)+(1+xy)}{y+z+1}\\ &=\dfrac{y+z+1}{y+z+1}\\ &=1,\;\;y+z+1 \neq 0 \end{align}

##### Worked Examples
###### Worked Example 1

Write $\dfrac{x+1}{y}+\dfrac{y+1}{x}$ as a single algebraic fraction.

###### Solution

\begin{align} \dfrac{x+1}{y}+\dfrac{y+1}{x}&=\dfrac{(x+1)\times x+(y+1)\times y}{xy}\\ &=\dfrac{x^2+x+y^2+y}{xy} \end{align} The numerator and denominator have no common factors so the fraction is in its simplest form.

###### Worked Example 2

Write $\dfrac{x+y}{xy-x}-\dfrac{xy-1}{xy+x}$ as a single algebraic fraction.

###### Solution

\begin{align} \dfrac{x+y}{xy-x}-\dfrac{xy-1}{y-1}&=\dfrac{(x+y)(xy+x)}{(xy-x)(xy+x)}-\dfrac{(xy-1)(xy-x)}{(xy+x)(xy-x)}\\ &=\dfrac{(x+y)(xy+x)-(xy-1)(xy-x)}{(xy-x)(xy+x)}\\ &=\dfrac{x^2y+x^2+xy^2+xy-x^2y^2+x^2y+xy-x}{(xy-x)(xy+x)}\;\;\;\text{ expanding brackets in the numerator}\\ &=\dfrac{x(xy+x+y^2+y-xy^2+xy+y-1)}{x^2(y-1)(y+1)}\\ &=\dfrac{x(2xy+x+y^2+2y-xy^2-1)}{x^2(y-1)(y+1)}\;\;\;\text{ collecting like terms in the numerator}\\ &=\dfrac{2xy+x+y^2+2y-xy^2-1}{x(y-1)(y+1)}\;\;\;\text{ cancelling the common factor of }x\\ &=\dfrac{2xy+x+y^2+2y-xy^2-1}{x(y^2-1)}\;\;\;\text{ expanding brackets in the denominator} \end{align} The numerator and denominator have no common factors so the fraction is in its simplest form.

#### Multiplying Algebraic Fractions

The rule for multiplying algebraic fractions is the same as that for multiplying arithmetic fractions: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

##### Worked Example
###### Worked Example

Calculate $\dfrac{3x}{y} \times \dfrac{2xy^2}{z}$.

###### Solution

\begin{align} \frac{3x}{y} \times \frac{2xy^2}{z} &= \frac{3x \times 2xy^2}{y \times z}\\ \\ &=\frac{6x^2y^2}{yz}\\ \\ &=\frac{6x^2y}{z}\;\;\;\text{ (on cancelling the common factor of }y)\\ \\ \end{align} The numerator and denominator now have no common factors so the fraction is in its simplest form.

#### Dividing Algebraic Fractions

The rule for dividing algebraic fractions is the same as that for dividing arithmetic fractions: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.$

##### Worked Example
###### Worked Example

Express $\dfrac{4}{1-2x} - 3$ as a single fraction.

###### Solution

First, we put everything over the same denominator $1-2x$, then expand the brackets in the numerator and simplify.

\begin{align} \frac{4}{1-2x} - 3 & = \frac{4}{1-2x} - \frac{3(1-2x)}{1-2x}\\\\ &=\frac{4-3(1-2x)}{1-2x}\\\\ &=\frac{4-3+6x}{1-2x}\\\\ &=\frac{1+6x}{1-2x} \end{align} $1+6x$ and $1-2x$ have no factors in common so the fraction is in its simplest form.

#### Video Examples

Prof. Robin Johnson calculates $\dfrac{7}{60}+\dfrac{2}{5}-\dfrac{5}{12}$ and simplifies $\dfrac{1+2x}{(x+1)(x-2)}+\dfrac{3x-2}{x^2-1}$.

Prof. Robin Johnson simplifies the combines the fractions $\dfrac{2}{1+x} - \dfrac{3}{2-x}$, into a single fraction.

Prof. Robin Johnson combines the fractions $\dfrac{x^3y^4}{z^5}\times\dfrac{z^3}{x^2y^7}$ and $\dfrac{x^2-1}{x+2}\times\dfrac{2x^2+3x-2}{x+1}$.

#### Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.

#### More Support

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