Understanding Boat and River Speeds

When a boat travels in a river, its speed relative to the land is affected by the river's current. There are two main scenarios:

• Upstream: The boat travels against the current. The effective speed is the boat's speed in still water minus the speed of the current.
• Downstream: The boat travels with the current. The effective speed is the boat's speed in still water plus the speed of the current.

Calculating Upstream Speed

The problem states that the boat sails 15 km upstream in 5 hours.

The formula for speed is:

\( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)

Using this, we can find the upstream speed:

\( \text{Upstream Speed} = \frac{15 \text{ km}}{5 \text{ hours}} = 3 \text{ km/hr} \)

Let:

• \( S_b \) be the speed of the boat in still water (in km/hr).
• \( S_r \) be the speed of the river current (in km/hr).

So, the upstream speed is also given by \( S_b - S_r \).

Therefore, we have the equation:

\( S_b - S_r = 3 \quad \ldots(1) \)

Finding Boat and River Speeds

The problem also states that the speed of the river is one-fourth the speed of the boat in still water.

This gives us another equation:

\( S_r = \frac{1}{4} S_b \quad \ldots(2) \)

Now we can substitute equation (2) into equation (1):

\( S_b - \left(\frac{1}{4} S_b\right) = 3 \)

\( S_b - \frac{1}{4} S_b = 3 \)

Combine the terms involving \( S_b \):

\( \left(1 - \frac{1}{4}\right) S_b = 3 \)

\( \left(\frac{4-1}{4}\right) S_b = 3 \)

\( \frac{3}{4} S_b = 3 \)

Solve for \( S_b \):

\( S_b = 3 \times \frac{4}{3} \)

\( S_b = 4 \text{ km/hr} \)

Now substitute the value of \( S_b \) back into equation (2) to find \( S_r \):

\( S_r = \frac{1}{4} S_b = \frac{1}{4} \times 4 \)

\( S_r = 1 \text{ km/hr} \)

Calculating Downstream Speed

The downstream speed is the sum of the boat's speed in still water and the river's speed.

\( \text{Downstream Speed} = S_b + S_r \)

Using the values we found for \( S_b \) and \( S_r \):

\( \text{Downstream Speed} = 4 \text{ km/hr} + 1 \text{ km/hr} \)

\( \text{Downstream Speed} = 5 \text{ km/hr} \)

Determining Downstream Time

The question asks how long it will take to cover the same distance downstream. The distance is 15 km.

Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

\( \text{Downstream Time} = \frac{15 \text{ km}}{\text{Downstream Speed}} \)

\( \text{Downstream Time} = \frac{15 \text{ km}}{5 \text{ km/hr}} \)

\( \text{Downstream Time} = 3 \text{ hours} \)

Therefore, it will take 3 hours to cover the same distance downstream.