Subtract $ \dfrac{6}{n} $ from $ m $ to get $ \dfrac{ \color{purple}{ mn-6 } }{ n }$.

Step 1: Write $ m $ 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 first fraction by $\color{blue}{ n }$.

$$ \begin{aligned} m- \frac{6}{n} & \xlongequal{\text{Step 1}} \frac{m}{\color{red}{1}} - \frac{6}{n} = \frac{ m \cdot \color{blue}{ n }}{ 1 \cdot \color{blue}{ n }} - \frac{ 6 }{ n } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ mn } }{ n } - \frac{ \color{purple}{ 6 } }{ n }=\frac{ \color{purple}{ mn-6 } }{ n } \end{aligned} $$

Divide $ \dfrac{m}{n} $ by $ 5 $ to get $ \dfrac{ m }{ 5n } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{m}{n} }{5} & \xlongequal{\text{Step 1}} \frac{m}{n} \cdot \frac{\color{blue}{1}}{\color{blue}{5}} \xlongequal{\text{Step 2}} \frac{ m \cdot 1 }{ n \cdot 5 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ m }{ 5n } \end{aligned} $$

Subtract $ \dfrac{6}{n} $ from $ m $ to get $ \dfrac{ \color{purple}{ mn-6 } }{ n }$.

Step 1: Write $ m $ 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 first fraction by $\color{blue}{ n }$.

$$ \begin{aligned} m- \frac{6}{n} & \xlongequal{\text{Step 1}} \frac{m}{\color{red}{1}} - \frac{6}{n} = \frac{ m \cdot \color{blue}{ n }}{ 1 \cdot \color{blue}{ n }} - \frac{ 6 }{ n } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ mn } }{ n } - \frac{ \color{purple}{ 6 } }{ n }=\frac{ \color{purple}{ mn-6 } }{ n } \end{aligned} $$

Multiply $8mn-48$ by $ \dfrac{m}{5n} $ to get $ \dfrac{ 8m^2n-48m }{ 5n } $.

Step 1: Write $ 8mn-48 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 8mn-48 \cdot \frac{m}{5n} & \xlongequal{\text{Step 1}} \frac{8mn-48}{\color{red}{1}} \cdot \frac{m}{5n} \xlongequal{\text{Step 2}} \frac{ \left( 8mn-48 \right) \cdot m }{ 1 \cdot 5n } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8m^2n-48m }{ 5n } \end{aligned} $$

Divide $ \dfrac{mn-6}{n} $ by $ \dfrac{8m^2n-48m}{5n} $ to get $ \dfrac{ 5mn^2-30n }{ 8m^2n^2-48mn } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{mn-6}{n} }{ \frac{\color{blue}{8m^2n-48m}}{\color{blue}{5n}} } & \xlongequal{\text{Step 1}} \frac{mn-6}{n} \cdot \frac{\color{blue}{5n}}{\color{blue}{8m^2n-48m}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( mn-6 \right) \cdot 5n }{ n \cdot \left( 8m^2n-48m \right) } \xlongequal{\text{Step 3}} \frac{ 5mn^2-30n }{ 8m^2n^2-48mn } \end{aligned} $$