Add $ \dfrac{1}{80} $ and $ \dfrac{1}{150} $ to get $ \dfrac{ \color{purple}{ 23 } }{ 1200 }$.

To add fractions they must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 15 }$ and the second by $\color{blue}{ 8 }$.

$$ \begin{aligned} \frac{1}{80} + \frac{1}{150} & = \frac{ 1 \cdot \color{blue}{ 15 }}{ 80 \cdot \color{blue}{ 15 }} + \frac{ 1 \cdot \color{blue}{ 8 }}{ 150 \cdot \color{blue}{ 8 }} = \\[1ex] &=\frac{ \color{purple}{ 15 } }{ 1200 } + \frac{ \color{purple}{ 8 } }{ 1200 }=\frac{ \color{purple}{ 23 } }{ 1200 } \end{aligned} $$

Multiply $ \dfrac{1}{31} $ by $ i $ to get $ \dfrac{ i }{ 31 } $.

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.

$$ \begin{aligned} \frac{1}{31} \cdot i & \xlongequal{\text{Step 1}} \frac{1}{31} \cdot \frac{i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot i }{ 31 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ i }{ 31 } \end{aligned} $$

Add $ \dfrac{23}{1200} $ and $ \dfrac{i}{31} $ to get $ \dfrac{ \color{purple}{ 1200i+713 } }{ 37200 }$.

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}{ 31 }$ and the second by $\color{blue}{ 1200 }$.

$$ \begin{aligned} \frac{23}{1200} + \frac{i}{31} & = \frac{ 23 \cdot \color{blue}{ 31 }}{ 1200 \cdot \color{blue}{ 31 }} + \frac{ i \cdot \color{blue}{ 1200 }}{ 31 \cdot \color{blue}{ 1200 }} = \\[1ex] &=\frac{ \color{purple}{ 713 } }{ 37200 } + \frac{ \color{purple}{ 1200i } }{ 37200 }=\frac{ \color{purple}{ 1200i+713 } }{ 37200 } \end{aligned} $$

Subtract $6i$ from $ \dfrac{1200i+713}{37200} $ to get $ \dfrac{ \color{purple}{ -222000i+713 } }{ 37200 }$.

Step 1: Write $ 6i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{37200}$.

$$ \begin{aligned} \frac{1200i+713}{37200} -6i & \xlongequal{\text{Step 1}} \frac{1200i+713}{37200} - \frac{6i}{\color{red}{1}} = \frac{ 1200i+713 }{ 37200 } - \frac{ 6i \cdot \color{blue}{ 37200 }}{ 1 \cdot \color{blue}{ 37200 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 1200i+713 } }{ 37200 } - \frac{ \color{purple}{ 223200i } }{ 37200 }=\frac{ \color{purple}{ -222000i+713 } }{ 37200 } \end{aligned} $$

Divide $1$ by $ \dfrac{-222000i+713}{37200} $ to get $ \dfrac{ 37200 }{ -222000i+713 } $.

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.

$$ \begin{aligned} \frac{1}{ \frac{\color{blue}{-222000i+713}}{\color{blue}{37200}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{37200}}{\color{blue}{-222000i+713}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{37200}{-222000i+713} \xlongequal{\text{Step 3}} \frac{ 1 \cdot 37200 }{ 1 \cdot \left( -222000i+713 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 37200 }{ -222000i+713 } \end{aligned} $$