A boat covers 22.4 km in downstream in 48 minutes and the speed of the stream is 40% of the speed of the boat in still water. Find the ratio of time taken by boat to cover 54 km in upstream to the time taken by boat to cover 210 km in downstream respectively?

The correct option is 3 : 5.

Let us solve the problem step-by-step.

Step 1: Calculate the downstream speed of the boat.

The boat covers a distance of 22.4 km downstream in 48 minutes.
First, convert the time from minutes to hours:
Time = 48 / 60 hours = 4 / 5 hours = 0.8 hours.

Downstream speed ($v_d$) is given by the formula:
Speed = Distance / Time

$v_d=\frac{22.4}{0.8}=28\text{km/h}$

Step 2: Find the speed of the boat in still water and the speed of the stream.

Let the speed of the boat in still water be $u$ km/h.
Given that the speed of the stream ($v$) is 40% of the speed of the boat in still water:
$v=0.4u$

The downstream speed is the sum of the speed of the boat in still water and the speed of the stream:
$v_d=u+v$

Substitute $v=0.4u$ and $v_d=28$ into the equation:

$u+0.4u=28$

$1.4u=28$

$u=\frac{28}{1.4}=20\text{km/h}$

Now, calculate the speed of the stream:
$v=0.4\times 20=8\text{km/h}$

Step 3: Calculate the upstream speed of the boat.

Upstream speed ($v_u$) is the difference between the speed of the boat in still water and the speed of the stream:
$v_u=u-v$

$v_u=20-8=12\text{km/h}$

Step 4: Find the times taken for the given upstream and downstream distances.

Time taken to cover 54 km upstream ($T_u$):

$T_u=\frac{54}{12}=4.5\text{hours}$

Time taken to cover 210 km downstream ($T_d$):

$T_d=\frac{210}{28}=7.5\text{hours}$

Step 5: Find the required ratio of times.

The ratio of time taken upstream to time taken downstream is:

$\text{Ratio}=\frac{T_u}{T_d}=\frac{4.5}{7.5}=\frac{45}{75}=\frac{3}{5}$

Thus, the ratio is 3 : 5.