# A boat on a river travels downstream between two points, 20 miles apart, in one hour. The return trip against the current takes 2 1/2 hours. what is the boat's speed, and how fast does the current in the river flow?

Sep 3, 2015

The boat's speed is $\text{14 mi/h}$ and the current's speed is $\text{6 mi/h}$.

#### Explanation:

The idea here is that, for the boat's first trip, the speed of the current will add to the speed of the boat. For the return trip, the speed of the current will be subtracted from the speed of the boat.

You can tell that this is what is happening because it takes less time for the boat to travel the same distance, 20 miles, for the first trip, when it's being aided by the current, than for the return trip, when it's moving againt current.

This is of course if you take the direction in which the boat is moving to be the positive direction.

So, for the first trip you have

${v}_{\text{boat" + v_"current}} = \frac{d}{t} _ 1$

and for the return trip you have

${v}_{\text{boat" - v_"current}} = \frac{d}{t} _ 2$

This is equivalent to

${v}_{\text{boat" + v_"current" = "20 miles"/"1 hour" = "20 mi/h}}$

and

${v}_{\text{boat" - v_"current" = "20 miles"/"2.5 hours" = "8 mi/h}}$

Add these two equations to get

v_"boat" + color(red)(cancel(color(black)(v_"current"))) + v_"boat" - color(red)(cancel(color(black)(v_"current"))) = 28

2 * v_"boat" = 28 implies v_"boat" = color(green)("14 mi/h")

This means that the current has a speed of

v_"current" = 20 - 14 = color(green)("6 mi/h")