Add $1$ and $ \dfrac{x}{y} $ to get $ \dfrac{ \color{purple}{ x+y } }{ y }$.

Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ y }$.

$$ \begin{aligned} 1+ \frac{x}{y} & \xlongequal{\text{Step 1}} \frac{1}{\color{red}{1}} + \frac{x}{y} = \frac{ 1 \cdot \color{blue}{ y }}{ 1 \cdot \color{blue}{ y }} + \frac{ x }{ y } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ y } }{ y } + \frac{ \color{purple}{ x } }{ y }=\frac{ \color{purple}{ x+y } }{ y } \end{aligned} $$

Divide $1$ by $ \dfrac{x+y}{y} $ to get $ \dfrac{ y }{ x+y } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Write $ 1 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{ \frac{\color{blue}{x+y}}{\color{blue}{y}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{y}}{\color{blue}{x+y}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{y}{x+y} \xlongequal{\text{Step 3}} \frac{ 1 \cdot y }{ 1 \cdot \left( x+y \right) } \xlongequal{\text{Step 4}} \frac{ y }{ x+y } \end{aligned} $$