A motorboat whose speed is 20 km/h in still water takes 30 minutes more to go 24 km upstream than to cover the same distance downstream. If the speed of the boat in still water is increased by 2 km/h, then how much time will it take to go 39 km downstream and 30 km upstream?

To find the total time taken for the downstream and upstream journeys under the new conditions, we can break the problem down into step-by-step calculations.

Step 1: Find the speed of the stream
Let the speed of the stream be represented by $s$ km/h.
Given that the speed of the boat in still water is $20$ km/h:
The speed of the boat upstream (against the stream) is $(20-s)$ km/h.
The speed of the boat downstream (with the stream) is $(20+s)$ km/h.

The distance covered in both directions is $24$ km, and the difference in time is $30$ minutes, which is equal to $\frac{1}{2}$ hour.
Using the formula $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$, we can write the equation as:

$\frac{24}{20-s}-\frac{24}{20+s}=\frac{1}{2}$

Now, let us solve for $s$:
Take $24$ common on the left-hand side:

$24\left[\frac{(20+s)-(20-s)}{(20-s)(20+s)}\right]=\frac{1}{2}$

Simplify the expression inside the brackets:

$24\left[\frac{2s}{400-s^2}\right]=\frac{1}{2}$

$\frac{48s}{400-s^2}=\frac{1}{2}$

Cross-multiplying gives:

$96s=400-s^2$

Rearranging this into a standard quadratic equation:

$s^2+96s-400=0$

Factoring the quadratic equation:

$(s+100)(s-4)=0$

Since the speed of the stream cannot be negative, we choose the positive value:
$s=4$ km/h.

Step 2: Calculate the new speeds
The speed of the boat in still water is increased by $2$ km/h:
New speed in still water = $20+2=22$ km/h.
Speed of the stream = $4$ km/h.
New downstream speed = $22+4=26$ km/h.
New upstream speed = $22-4=18$ km/h.

Step 3: Calculate the total time required
Time taken to cover $39$ km downstream:

$T_{\text{down}}=\frac{39}{26}=1.5\text{hours}$

Time taken to cover $30$ km upstream:

$T_{\text{up}}=\frac{30}{18}=\frac{5}{3}\text{hours}=1\text{hour}40\text{minutes}$

Total time required for both journeys is:
$\text{Total Time}=1\text{hour}30\text{minutes}+1\text{hour}40\text{minutes}=3\text{hours}10\text{minutes}$.

Therefore, the correct option is 3 h 10 minutes.