Add $ \dfrac{1}{2} $ and $ \dfrac{1}{1+3i} $ to get $ \dfrac{ \color{purple}{ 3i+3 } }{ 6i+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}{ 3i+1 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{1}{2} + \frac{1}{1+3i} & = \frac{ 1 \cdot \color{blue}{ \left( 3i+1 \right) }}{ 2 \cdot \color{blue}{ \left( 3i+1 \right) }} + \frac{ 1 \cdot \color{blue}{ 2 }}{ \left( 1+3i \right) \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 3i+1 } }{ 6i+2 } + \frac{ \color{purple}{ 2 } }{ 6i+2 }=\frac{ \color{purple}{ 3i+3 } }{ 6i+2 } \end{aligned} $$

Divide $ \, 3+3i \, $ by $ \, 2+6i \, $ to get $\,\, \dfrac{6-3i}{10} $.
( view steps )

Add $ \dfrac{6-3i}{10} $ and $ \dfrac{i}{4} $ to get $ \dfrac{ \color{purple}{ -i+12 } }{ 20 }$.

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}{ 2 }$ and the second by $\color{blue}{ 5 }$.

$$ \begin{aligned} \frac{6-3i}{10} + \frac{i}{4} & = \frac{ \left( 6-3i \right) \cdot \color{blue}{ 2 }}{ 10 \cdot \color{blue}{ 2 }} + \frac{ i \cdot \color{blue}{ 5 }}{ 4 \cdot \color{blue}{ 5 }} = \\[1ex] &=\frac{ \color{purple}{ 12-6i } }{ 20 } + \frac{ \color{purple}{ 5i } }{ 20 }=\frac{ \color{purple}{ -i+12 } }{ 20 } \end{aligned} $$