A motorboat takes 5 hours to travel 100 miles going upstream. The return trip takes 2 hours going downstream. What is the rate of the boat in still water and what is the rate of the current?

Answer 1

When the boat is travelling upstream, the total velocity of the boat is its velocity in still water MINUS the velocity of the current. When travelling downstream, the boat is travelling faster, so the total velocity is the velocity of the boat in still water PLUS the velocity of the current. We can express these facts as two equations:

1. (v_B - v_c)*5 = 100
2. (v_B + v_c)*2 = 100

Here, v_B is the velocity of the boat and v_c is the velocity of the current. We can simplify these two equations a little bit to rewrite them as

1. v_B - v_c = 20
2. v_B + v_c = 50

Now, we can solve this system of equations by adding the two together. This gives

1. 2*v_B = 70 => v_B = 35 miles/hour

Now that we know the velocity of the boat in the still water, we can plug it back into either of our equations to find the velocity of the current. I will use equation 2:

1. 35 + v_c = 50 => v_c = 15 miles/hour

To recap, what we did here was break the problem up into two different equations. In the first equation, we know that the boat is travelling slower because it is fighting the current. In this case we subtract the velocity of the current. In the other case, the current is helping the boat move faster, so we add the two velocities.