Boat travelling in river without current

[armolinasf]

Homework Statement

This is a question from Gelfand's Algebra:

A boat traveling upstream from A to B can cover the distance in a hours. The boat can travel downstream from B to A in b hours. How long would it take to travel from A to B if there was no current?

The Attempt at a Solution

To make it a little more concrete I plugged in numbers, a=3 and b=2

So, my thinking is that if the boat can cover x distance in 3 hours and the same x distance in 2 hours, then the answer would simply be the average of the two speeds:

((x/2)+(x/3))/2 = 5x/12

But this answer dosen't make sense since the time it would take to travel the distance without the current must be between two and three hours

Im a little bit at a loss of how to think about this one.

thanks for the help

 

[symbolipoint]

A better analysis would go something lik this:

Let c equal the current of the river, as a speed.
Let the speed of the boat if the river had no current, be r.

The upstream speed of the boat in the river, meaning going against the current, r-c.
The downstream speed of the boat in the river is r+c.

 

[armolinasf]

Alright, so basically it's just an algebra problem where you're solving for the speed where the river has no current. That's pretty straight forward thanks

 

[armolinasf]

But just to make sure I'm getting this:

if d=vt, then r+c and r-c will the velocities of the boat in both directions. Therefore, with time a and b, a(r-c)=b(r+c) since the distance traveled is the same.

Then solving for r,the speed of the river without the current, we get r=c(a+b)/(a-b). So if we know the velocity of the river without the current, we can then find the travel time it would have taken to go from A to B without the current? that is, if the distance is one ( can we assume the distance is 1 unit?) all we would have to do would be solve for t in t=d/v which would just be the reciprocal of r since v=r and d=1, or t=(a-b)/c(a+b).

Please let me know if this is right

 

[symbolipoint]

You seem to understand, based on your description in post #4. I did not check your work in the solution process too carefully, but apparently from the given information in your original problem description, the best you can find is a symbolic form for the speed of the boat if no current. You could then use the reciprocal of the speed to get your result involving the distance.

One thing you might want to know is that "velocity" and "speed" are not the same thing. Still, you method of equating the distances is good and is done correctly; from that, you solve for "r". Still as unknown variables are a, b, and c. That is why I say, that your final answer will need to remain in symbolic form.