Let's define the following:

• $v_b$: the speed of the boat in still water
• $v_c$: the speed of the current
• Speed upstream = 10 km/h
• Speed downstream = 15 km/h

Step 1: Set up the equations

When the boat travels upstream, the current opposes the boat, so its effective speed is $v_b - v_c$. When the boat travels downstream, the current assists the boat, so its effective speed is $v_b + v_c$.

We can write these two relationships as:

$v_b - v_c = 10 \quad \text{(upstream speed)}$ $v_b + v_c = 15 \quad \text{(downstream speed)}$

Step 2: Solve the system of equations

To find $v_c$, we can add the two equations:

$(v_b - v_c) + (v_b + v_c) = 10 + 15$ $2v_b = 25$ $v_b = \frac{25}{2} = 12.5 \, \text{km/h}$

Now, substitute $v_b = 12.5 \, \text{km/h}$ into one of the original equations to find $v_c$. Using $v_b - v_c = 10$:

$12.5 - v_c = 10$ $v_c = 12.5 - 10 = 2.5 \, \text{km/h}$

Final Answer:

The speed of the current is 2.5 km/h.

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Would you like more details on how to solve similar problems or need clarification on any steps?

Follow-up Questions:

1. How would the current's speed change if the boat's speed in still water was 20 km/h?
2. What would be the boat's time to travel 50 km upstream and 50 km downstream?
3. How do you derive the effective speed in both upstream and downstream cases?
4. How can you solve a similar problem if the boat's speed in still water is unknown?
5. What other factors could influence the boat’s speed in a real-world scenario?

Tip:

When solving speed problems involving currents, always remember to break down the problem into separate upstream and downstream equations to easily isolate the current's speed.