Add $ \dfrac{1}{3} $ and $ v $ to get $ \dfrac{ \color{purple}{ 3v+1 } }{ 3 }$.

Step 1: Write $ v $ 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 second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{1}{3} +v & \xlongequal{\text{Step 1}} \frac{1}{3} + \frac{v}{\color{red}{1}} = \frac{ 1 }{ 3 } + \frac{ v \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 1 } }{ 3 } + \frac{ \color{purple}{ 3v } }{ 3 }=\frac{ \color{purple}{ 3v+1 } }{ 3 } \end{aligned} $$

Add $ \dfrac{3v+1}{3} $ and $ 3 $ to get $ \dfrac{ \color{purple}{ 3v+10 } }{ 3 }$.

Step 1: Write $ 3 $ 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 second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{3v+1}{3} +3 & \xlongequal{\text{Step 1}} \frac{3v+1}{3} + \frac{3}{\color{red}{1}} = \frac{ 3v+1 }{ 3 } + \frac{ 3 \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 3v+1 } }{ 3 } + \frac{ \color{purple}{ 9 } }{ 3 }=\frac{ \color{purple}{ 3v+10 } }{ 3 } \end{aligned} $$