Multiply $ \dfrac{1}{100} $ by $ i $ to get $ \dfrac{ i }{ 100 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{100} \cdot i & \xlongequal{\text{Step 1}} \frac{1}{100} \cdot \frac{i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot i }{ 100 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ i }{ 100 } \end{aligned} $$

Add $ \dfrac{i}{100} $ and $ \dfrac{1}{9+99i} $ to get $ \dfrac{ \color{purple}{ 99i^2+9i+100 } }{ 9900i+900 }$.

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}{ 99i+9 }$ and the second by $\color{blue}{ 100 }$.

$$ \begin{aligned} \frac{i}{100} + \frac{1}{9+99i} & = \frac{ i \cdot \color{blue}{ \left( 99i+9 \right) }}{ 100 \cdot \color{blue}{ \left( 99i+9 \right) }} + \frac{ 1 \cdot \color{blue}{ 100 }}{ \left( 9+99i \right) \cdot \color{blue}{ 100 }} = \\[1ex] &=\frac{ \color{purple}{ 99i^2+9i } }{ 9900i+900 } + \frac{ \color{purple}{ 100 } }{ 9900i+900 }=\frac{ \color{purple}{ 99i^2+9i+100 } }{ 9900i+900 } \end{aligned} $$

$$ 99i^2 = 99 \cdot (-1) = -99 $$

Simplify numerator

$$ \color{blue}{-99} +9i+ \color{blue}{100} = 9i+ \color{blue}{1} $$

Divide $ \, 1+9i \, $ by $ \, 900+9900i \, $ to get $\,\, \dfrac{50-i}{54900} $.
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