# Math Equation: Can anyone help me?

Hello! I'm really horrible at math. I'm working hard on this problem, but can't seem to get it right!

The current of a river is 2 miles per hour. A boat travels to a point 8 miles upstream and back again in 3 hours. What is the speed of the boat in still water?

Please explain it so I can understand every point of the problem! I would really appreciate it! Thanks again!

### 3 Answers

- 8 years agoBest Answer
The key equation you need is D = R * T, where D is the distance traveled, R is the rate of travel (i.e. the speed), and T is the amount of time traveled at that speed. You will need to use this formula twice, once upstream, and once downstream. You will end up with an equation for each in two variables, which turns out in this problem to be R and T. The fact you have two equations in R and T, will allow you to solve them for each variable; although the variable we're only interested in in this problem is R.

Upstream Analysis: We know D = 8 miles, and the rate of travel is R - 2, where R is the speed of the boat in still water. It is R - 2, because we are going upstream, and the current is slowing us down by 2 miles per hour. For time, T, we don't know how much time it takes to only go upstream (we only now how long the round trip takes), so call it T, for now. So our equation that results from our upstream analysis is 8 = (R-2)*T. [Equation #1]

Downstream Analysis: We know D = 8 miles again, and this time the current is working with us to move us faster, so the speed is R+2. Once again, we do not know the time, but now we can use the fact the round trip took 3 hours, and the trip upstream took T. So time to go downstream is 3 - T. So our equation that results from our downstream analysis is: 8 = (R+2)*(3 - T). [Equation #2]

So now you have two equations in two unknowns, and are able to solve for R, the rate in still water.

Also, be careful in these problems that all your units are the same. In this problem, miles and hours are the only units of distance and time used, so it is not an issue. Some problems, however, will give you mixed units such as speeds in miles per hour, and then tell you that the trip took 20 minutes. In that case, you will have to pick one (e.g. hours) to which you will convert the others (e.g. minutes).

If you need more help, please clarify where you are in the process and what's giving you trouble. I'd be more than happy to continue to assist.

- 8 years ago
Note that the above answer is incorrect due to the current in the water. If the water was still, the above method would be applicable, but it isn't.

Brace yourself for a long problem!

We'll need to use the distance equation:

v = x/t

or

vt = x

We know that it travels 8 miles upstream at a speed of some v - 2 (we subtract 2 because of the current in the river opposing the craft). v represents the boat's speed in still water. Therefore:

(v - 2)t1 = 8

Note that I used t1 instead of t because there will be different times for travelling the same distance upriver and downriver.

When the boat turns around, its speed will no longer be v - 2, but v + 2 due to the river's current now flowing in the same direction as the boat. So the equation representing the trip back downriver is

(v + 2)t2 = 8

Again, I used t2 because it won't be the same amount of time as t1, so we need a different variable name. Lastly we need a relationship between t1 and t2. We know that the total time of the trip was 3 hours, so that means

t1 + t2 = 3

Now we have the three equations:

(v - 2)t1 = 8

(v + 2)t2 = 8

t1 + t2 = 3

I can see that, using the first two equations, I can find t1 and t2 both in terms of v, so I will do just that:

(v - 2)t1 = 8

t1 = 8/(v - 2)

and

(v + 2)t2 = 8

t2 = 8/(v + 2)

Now we can substitute these into the third equation to get an equation with just v in it:

t1 + t2 = 3

(8/(v - 2)) + (8/(v + 2)) = 3

[8(v - 2) + 8(v + 2)] / (v - 2)(v + 2) = 3

8(v - 2) + 8(v + 2) = 3(v - 2)(v + 2)

8v - 16 + 8v + 16 = 3v^2 - 12

16v = 3v^2 - 12

3v^2 - 16v - 12 = 0

Factor:

(v - 6)(3v + 2) = 0

v = 6, -2/3

(If you aren't familiar with techniques to factor polynomials, please review the helpful tutorials on purplemath.com. You'll also find other useful topics if you get stuck on something else.)

We'll discard the negative answer, since it doesn't make sense in this problem, leaving us with

v = 6

If we needed to solve for t1 and t2, we could also do that now:

(v - 2)t1 = 8

((6) - 2)t1 = 8

4t1 = 8

t1 = 2

and finally

t1 + t2 = 3

(2) + t2 = 3

t2 = 1

Substituting these values into the earlier equations shows that the answers are all correct. I hope this was helpful!

Source(s): Purplemath.com - 8 years ago
So in total, it travels 16 miles in 3 hours.

In still water, it would of traveled 10 miles to the point and 6 miles from the point away - so in total again 16 miles in 3 hours.

Now, you must use a physics equation:

speed=distance/time

16/3 = speed

It travels at 5,3333333333333333333333333333333... miles per hour without the current.

Source(s): personal knowlage