# Math Help - Motorboat Speed

1. ## Motorboat Speed

A motorboat heads upstream on a river that has a current of 3 mph. The trip upstream takes 5 hours, while the return trip takes 2.5 hours. What is the speed of the motorboat?

Assume that the motorboat keeps a constant speed relative to the water.

NOTE: What is the meaning of the assumption made above?

I've seen upstream and downstream questions before. However, I always get confused when it comes to setting up the right table or chart that leads to the correct equation that leads to the answer.

2. Originally Posted by symmetry
A motorboat heads upstream on a river that has a current of 3 mph. The trip upstream takes 5 hours, while the return trip takes 2.5 hours. What is the speed of the motorboat?

Assume that the motorboat keeps a constant speed relative to the water.

NOTE: What is the meaning of the assumption made above?

I've seen upstream and downstream questions before. However, I always get confused when it comes to setting up the right table or chart that leads to the correct equation that leads to the answer.
The direction of flow of the water is opposite that of the motorboat for the first leg of the trip, so according to an observer on land the boat will move with a constant speed of v - 3 mph. Thus it takes 5 hours to cover an unknown distance of x miles, and follows the relationship:
$v - 3 = \frac{x}{5}$

On the return trip the boat covers the same distance x in a time of 2.5 hours, now moving with a speed of v + 3 according to a landlubber.
$v + 3 = \frac{x}{2.5} = \frac{x}{\frac{5}{2}} =$ $\frac{2x}{5} = 2 \cdot \frac{x}{5}$

(I've fiddled with the last equation to make it slightly more useful.)

Now, solve the top equation for $\frac{x}{5}$ (rather than for just x, simply for convenience. If you are unsure about this step, solve it for x.) Since this is already done, this is easy!
$\frac{x}{5} = v - 3$

Now insert this into the second equation:
$v + 3 = 2 \cdot (v - 3)$

$v + 3 = 2v - 6$

$-v = -9$

So v = 9 mph.

-Dan

3. ## ok

I never quiet solved for (x/5) but rather for x, y or z.

However, your steps and explanation is very clear and simplistic.