   SEARCH HOME Math Central Quandaries & Queries  Question from amanda, a student: a tugboat must travel 24 miles against a 4 mile per hour current on the Potomac River and return. At what constant speed must the tugboat travel to make the trip in 12 hours. Round answer to the nearest tenth mph. Hi Amanda.

Let b = the speed of the boat relative to still water. This is the constant speed you are trying to find. Then b - 4 is its speed going upstream and b + 4 is its speed returning downstream.

Let t = the time it takes to get to the far point where the boat turns around. Then 12 - t = the time it takes for it to return (because the total time is 12 hours).

Now we know that speed = distance / time. So we can write the two legs of the journey as follows:

Upstream: (b - 4) = 24 / t
Downstream: (b + 4) = 24 / (12 - t)

So we have two equations with two unknown variables t and b. Since we really only care about b, we should solve one of the equations for t and then substitute. Let's use the upstream equation:

(b - 4) = 24 / t
t = 24 / (b - 4)

Next, substitute this expression for t into the second equation wherever you see t:

(b + 4) = 24 / (12 - t)
(b + 4) = 24 / (12 - (24 / (b - 4)) )

Can you solve the equation for b now?
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.