Understanding the Boat and Stream Problem

This problem involves calculating the speed of a boat in still water given its travel times over a fixed distance downstream and upstream. We need to use the concepts of relative speed in water.

Calculating Downstream Speed

The boat travels downstream, meaning it moves with the direction of the stream. The effective speed is the sum of the boat's speed and the stream's speed.

• Distance Covered = 72 km
• Time Taken Downstream = 6 hours

The formula for speed is distance divided by time.

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

Speed Downstream = $$ \frac{72 \text{ km}}{6 \text{ hours}} $$

Speed Downstream = 12 km/h

Calculating Upstream Speed

The boat travels upstream, meaning it moves against the direction of the stream. The effective speed is the difference between the boat's speed and the stream's speed.

• Distance Covered = 72 km
• Time Taken Upstream = 12 hours

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

Speed Upstream = $$ \frac{72 \text{ km}}{12 \text{ hours}} $$

Speed Upstream = 6 km/h

Finding the Boat Speed

Let the speed of the boat in still water be $b$ km/h and the speed of the stream be $s$ km/h.

From the downstream and upstream calculations, we have two equations:

1. Speed Downstream: $$ b + s = 12 $$
2. Speed Upstream: $$ b - s = 6 $$

To find the speed of the boat ($b$), we can add these two equations together:

$$ (b + s) + (b - s) = 12 + 6 $$

$$ 2b = 18 $$

$$ b = \frac{18}{2} $$

$$ b = 9 \text{ km/h} $$

We can also find the speed of the stream ($s$) by subtracting the second equation from the first:

$$ (b + s) - (b - s) = 12 - 6 $$

$$ 2s = 6 $$

$$ s = \frac{6}{2} $$

$$ s = 3 \text{ km/h} $$

Conclusion

The question asks for the speed of the boat. Based on our calculations, the speed of the boat in still water is 9 km/h.