Add $r$ and $ \dfrac{i}{i+s} $ to get $ \dfrac{ \color{purple}{ ir+rs+i } }{ i+s }$.

Step 1: Write $ r $ 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 $\color{blue}{ i+s }$.

$$ \begin{aligned} r+ \frac{i}{i+s} & \xlongequal{\text{Step 1}} \frac{r}{\color{red}{1}} + \frac{i}{i+s} = \frac{ r \cdot \color{blue}{ \left( i+s \right) }}{ 1 \cdot \color{blue}{ \left( i+s \right) }} + \frac{ i }{ i+s } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ ir+rs } }{ i+s } + \frac{ \color{purple}{ i } }{ i+s }=\frac{ \color{purple}{ ir+rs+i } }{ i+s } \end{aligned} $$