This problem involves calculating the total time taken by a boat to travel a certain distance downstream and then return the same distance upstream. We are given the speed of the boat in still water and the speed of the stream.

Understanding Speeds in Water

When a boat travels in water, its speed relative to the bank depends on the direction of travel relative to the stream's current.

• Downstream Travel: The boat travels in the same direction as the stream. The effective speed is the sum of the boat's speed in still water and the stream's speed.
• Upstream Travel: The boat travels against the direction of the stream. The effective speed is the difference between the boat's speed in still water and the stream's speed.

Given Information

We are provided with the following details:

• Speed of the boat in still water ($S_b$) = 6 km/hr
• Speed of the stream ($S_s$) = 3 km/hr
• Distance ($D$) = 27 km (for both downstream and upstream journeys)

Calculating Downstream Speed

The speed of the boat when traveling downstream ($S_{down}$) is calculated by adding the speed of the boat in still water to the speed of the stream:

$$ S_{down} = S_b + S_s $$

Substituting the given values:

$$ S_{down} = 6 \text{ km/hr} + 3 \text{ km/hr} = 9 \text{ km/hr} $$

Calculating Time Taken Downstream

The time taken to cover a distance is given by the formula: Time = Distance / Speed.

Time taken downstream ($T_{down}$) is:

$$ T_{down} = \frac{D}{S_{down}} $$

Substituting the values:

$$ T_{down} = \frac{27 \text{ km}}{9 \text{ km/hr}} = 3 \text{ hr} $$

Calculating Upstream Speed

The speed of the boat when traveling upstream ($S_{up}$) is calculated by subtracting the speed of the stream from the speed of the boat in still water:

$$ S_{up} = S_b - S_s $$

Substituting the given values:

$$ S_{up} = 6 \text{ km/hr} - 3 \text{ km/hr} = 3 \text{ km/hr} $$

Calculating Time Taken Upstream

Similarly, the time taken upstream ($T_{up}$) is:

$$ T_{up} = \frac{D}{S_{up}} $$

Substituting the values:

$$ T_{up} = \frac{27 \text{ km}}{3 \text{ km/hr}} = 9 \text{ hr} $$

Calculating Total Time

The total time taken for the entire journey (downstream and upstream) is the sum of the time taken for each part:

$$ T_{total} = T_{down} + T_{up} $$

$$ T_{total} = 3 \text{ hr} + 9 \text{ hr} = 12 \text{ hr} $$

Summary of Calculations

Here's a summary of the calculated speeds and times:

Therefore, the total time taken to cover 27 km distance downstream and return on it upstream is 12 hours.