Ratio of the downstream speed of boat to the upstream speed of boat is 3: 1 and the time taken by a boat to cover 144 km in downstream is 8 hours. Find the time taken by the boat to cover same distance in still water.

The correct option is 12 hours.

Let's solve the problem step-by-step:

Step 1: Find the downstream speed of the boat.
We are given that the boat covers a distance of 144 km downstream in 8 hours.
The formula for speed is:

$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$

Therefore, the downstream speed of the boat ($S_d$) is:

$S_d=\frac{144\text{km}}{8\text{hours}}$

Step 2: Find the upstream speed of the boat.
The ratio of the downstream speed to the upstream speed of the boat is given as 3 : 1.
Let the downstream speed be
$3x$
and the upstream speed be
$1x$.
Since the downstream speed is 18 km/h:

$3x=18\text{km/h}$

Solving for $x$:

$x=\frac{18}{3}=6\text{km/h}$

Thus, the upstream speed of the boat ($S_u$) is:

$S_u=6\text{km/h}$

Step 3: Calculate the speed of the boat in still water.
The speed of a boat in still water ($S_b$) is the average of its downstream and upstream speeds:

$S_b=\frac{S_d+S_u}{2}$

Substitute the values:

$S_b=\frac{18+6}{2}=\frac{24}{2}=12\text{km/h}$

Step 4: Find the time taken to cover the same distance in still water.
The distance to be covered is the same, which is 144 km.
The formula for time is:

$\text{Time}=\frac{\text{Distance}}{\text{Speed in still water}}$

Substitute the values:

$\text{Time}=\frac{144\text{km}}{12\text{km/h}}$