Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

The simplified form of the rational expression $\frac{x}{x+y}\div \frac{y}{x+y}$ is$\frac{x}{y}$.
$\frac{x}{x+y}\div \frac{y}{x+y}$ $\frac{A}{B}\div \frac{C}{D}=\frac{A}{B}\cdot \frac{D}{C}$ The reciprocal of $\frac{y}{x+y}$ is$\frac{x+y}{y}$ So, multiply the reciprocal of the divisor, \begin{align} & \frac{x}{x+y}\div \frac{y}{x+y}=\frac{x}{x+y}\cdot \frac{x+y}{y} \\ & =\frac{\left( x \right)\left( x+y \right)}{\left( x+y \right)\left( y \right)} \end{align} Regroup and remove the factor equal to 1, \begin{align} & \frac{x}{x+y}\div \frac{y}{x+y}=\frac{\left( x \right)\left( x+y \right)}{\left( x+y \right)\left( y \right)} \\ & =\frac{\left( x+y \right)\left( x \right)}{\left( x+y \right)\left( y \right)} \\ & =1\cdot \frac{\left( x \right)}{\left( y \right)} \\ & =\frac{x}{y} \end{align}