Add $12$ and $ \dfrac{1}{a-8} $ to get $ \dfrac{ \color{purple}{ 12a-95 } }{ a-8 }$.

Step 1: Write $ 12 $ 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 $\color{blue}{ a-8 }$.

$$ \begin{aligned} 12+ \frac{1}{a-8} & \xlongequal{\text{Step 1}} \frac{12}{\color{red}{1}} + \frac{1}{a-8} = \frac{ 12 \cdot \color{blue}{ \left( a-8 \right) }}{ 1 \cdot \color{blue}{ \left( a-8 \right) }} + \frac{ 1 }{ a-8 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 12a-96 } }{ a-8 } + \frac{ \color{purple}{ 1 } }{ a-8 }=\frac{ \color{purple}{ 12a-95 } }{ a-8 } \end{aligned} $$

Divide $ \dfrac{a}{a-8} $ by $ \dfrac{12a-95}{a-8} $ to get $ \dfrac{ a }{ 12a-95 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Cancel $ \color{red}{ a-8 } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{a}{a-8} }{ \frac{\color{blue}{12a-95}}{\color{blue}{a-8}} } & \xlongequal{\text{Step 1}} \frac{a}{a-8} \cdot \frac{\color{blue}{a-8}}{\color{blue}{12a-95}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{a}{\color{red}{1}} \cdot \frac{\color{red}{1}}{12a-95} \xlongequal{\text{Step 3}} \frac{ a \cdot 1 }{ 1 \cdot \left( 12a-95 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ a }{ 12a-95 } \end{aligned} $$