Add $ \dfrac{a}{b} $ and $ \dfrac{c}{d} $ to get $ \dfrac{ \color{purple}{ ad+bc } }{ bd }$.

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}{ d }$ and the second by $\color{blue}{ b }$.

$$ \begin{aligned} \frac{a}{b} + \frac{c}{d} & = \frac{ a \cdot \color{blue}{ d }}{ b \cdot \color{blue}{ d }} + \frac{ c \cdot \color{blue}{ b }}{ d \cdot \color{blue}{ b }} = \\[1ex] &=\frac{ \color{purple}{ ad } }{ bd } + \frac{ \color{purple}{ bc } }{ bd }=\frac{ \color{purple}{ ad+bc } }{ bd } \end{aligned} $$