Let's denote the speed of the boat in still water as 'b' and the speed of the current as 'c'.
When the boat travels downstream, it benefits from the speed of the current, so its effective speed is increased. We can set up the equation:
Downstream speed = Boat's speed in still water + Speed of current
Given that the boat travels 2 km downstream in 10 minutes, we can convert the time to hours by dividing by 60:
Downstream speed = 2 km / (10 minutes / 60 minutes/hour) = 2 km / (10/60) hour = 12 km/h
Using the equation above, we can write:
12 km/h = b + c (Equation 1)
Similarly, when the boat travels upstream, it opposes the speed of the current, so its effective speed is decreased. We have:
Upstream speed = Boat's speed in still water - Speed of current
Given that the boat travels 1 km upstream in 5 minutes, we can convert the time to hours:
Upstream speed = 1 km / (5 minutes / 60 minutes/hour) = 1 km / (5/60) hour = 12 km/h
Using the equation above, we can write:
12 km/h = b - c (Equation 2)
We now have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of 'b' and 'c'.
Adding Equation 1 and Equation 2:
12 km/h + 12 km/h = b + c + b - c
24 km/h = 2b
Dividing both sides by 2:
12 km/h = b
So, the speed of the boat in still water is 12 km/h.
Substituting the value of 'b' into Equation 1:
12 km/h = 12 km/h + c
c = 0 km/h
Therefore, the speed of the current is 0 km/h (or no current).
In summary, the speed of the boat in still water is 12 km/h, and there is no current (0 km/h).