Add $ \dfrac{m}{2} $ and $ \dfrac{2}{5} $ to get $ \dfrac{ \color{purple}{ 5m+4 } }{ 10 }$.

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

$$ \begin{aligned} \frac{m}{2} + \frac{2}{5} & = \frac{ m \cdot \color{blue}{ 5 }}{ 2 \cdot \color{blue}{ 5 }} + \frac{ 2 \cdot \color{blue}{ 2 }}{ 5 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 5m } }{ 10 } + \frac{ \color{purple}{ 4 } }{ 10 }=\frac{ \color{purple}{ 5m+4 } }{ 10 } \end{aligned} $$

Add $ \dfrac{4}{5} $ and $ \dfrac{4}{25} $ to get $ \dfrac{ \color{purple}{ 24 } }{ 25 }$.

To add fractions they must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 5 }$.

$$ \begin{aligned} \frac{4}{5} + \frac{4}{25} & = \frac{ 4 \cdot \color{blue}{ 5 }}{ 5 \cdot \color{blue}{ 5 }} + \frac{ 4 }{ 25 } = \\[1ex] &=\frac{ \color{purple}{ 20 } }{ 25 } + \frac{ \color{purple}{ 4 } }{ 25 }=\frac{ \color{purple}{ 24 } }{ 25 } \end{aligned} $$

Divide $ \dfrac{5m+4}{10} $ by $ \dfrac{24}{25} $ to get $ \dfrac{ 125m+100 }{ 240 } $.

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.

$$ \begin{aligned} \frac{ \frac{5m+4}{10} }{ \frac{\color{blue}{24}}{\color{blue}{25}} } & \xlongequal{\text{Step 1}} \frac{5m+4}{10} \cdot \frac{\color{blue}{25}}{\color{blue}{24}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 5m+4 \right) \cdot 25 }{ 10 \cdot 24 } \xlongequal{\text{Step 3}} \frac{ 125m+100 }{ 240 } \end{aligned} $$