The problem describes a motorboat traveling upstream and downstream, and we are tasked with finding two values:

1. The rate of the boat in still water.
2. The rate of the current.

Let's break it down:

Given:

• The boat travels 260 km upstream in 5 hours.
• The boat travels 340 km downstream in 5 hours.

Let:

• $b$ = speed of the boat in still water (in km/h).
• $c$ = speed of the current (in km/h).

Upstream motion:

When the boat moves upstream, the current slows it down, so its effective speed is $b - c$. Using the formula for speed, $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$, we have: $\frac{260}{5} = b - c$ 52 = b - c \tag{1}

Downstream motion:

When the boat moves downstream, the current adds to the boat's speed, so its effective speed is $b + c$. Similarly, we have: $\frac{340}{5} = b + c$ 68 = b + c \tag{2}

Solving the system of equations:

We have two equations:

1. $b - c = 52$
2. $b + c = 68$

Adding these two equations: $(b - c) + (b + c) = 52 + 68$ $2b = 120$ $b = 60 \, \text{km/h}$

Now, substitute $b = 60$ into equation (2): $60 + c = 68$ $c = 8 \, \text{km/h}$

Conclusion:

• The rate of the boat in still water is 60 km/h.
• The rate of the current is 8 km/h.

Would you like further details or clarifications?

Here are 5 related questions:

1. What happens to the boat's speed if the current's speed increases?
2. How can we generalize this problem for different distances and times?
3. What is the importance of relative speed in upstream and downstream problems?
4. How would the solution change if the boat traveled different distances upstream and downstream?
5. Can you find an alternative method to solve this problem using a different approach?

Tip: In problems involving upstream and downstream speeds, remember that the current always affects the boat's speed differently depending on direction. Upstream subtracts the current's speed, while downstream adds it.