A boat travels 25 miles per hour in still water. It takes 3 1/3 hours to travel 40 miles up a river and then return by the same route. What is the speed of the river?

Question

Solving equations with rational expressions

Answer 1

Let Vr be the speed of the river water in mph. Going upriver the boat speed is 25 - Vr relative to the shore. Going downriver it is 25 + Vr. Let Tu be the travel time upriver and Td the travel time downriver. Then:

Tu * (25 - Vr) = Td * (25 + Vr) = 20 miles

Tu + Td = 3.333 hrs

Or

(20 / (25 - Vr)) + (20 / (25 + Vr)) = 3.333

Solving for Vr:

20 * (25 -+ Vr) + 20 * (25 - Vr) = 3.333 * (25 + Vr) * (25 - Vr)

500 + 500 = 3.333 * (625 - Vr^2)

Or

Vr = 18.03 mph

Note that

Tu = 20 / (25 - Vr) = 2.8685

Td = 20 / (25 + Vr) = 0.4648

And

Tu + Td = 3.333

Answer 2

Please when you post these make it very clear what these numbers represent.

It turns out that 40 miles is the distance ONE WAY, not round trip. Please make this

clear in the statement of the problem.

x is the speed of the current.

t is the time traveling upstream against the current.

40 = (25-x)*t = (25 + x)(10/3 - t)

25t - xt = 250/3 - 25t + (10/3)x - xt

adds xt to both sides, so it cancels

25t = 250/3 - 25t + (10/3) x <--- adds xt to both sides

50t = 250/3 + (10/3) x <--- adds 25t to both sides

150t = 250 + 10x <--- multiplies everything by 3

15t = 25 + x <--- divides everything by 10

15t - 25 = x <--- subtracts 25 from both sides

Substitutes into the first equation:

40 = (25 - x)t

= (25 - 15t + 25)t

= (50 -15t)t

= 50t - 15t^2

15t^2 - 50t + 40 = 0

3t^2 - 10t + 8 = 0

quadratic formula says:

t = [ 10 +or- sqrt( 100 - 4*3*8)]/ 6

= [ 10 +or- sqrt( 100 - 96)]/6

= [ 10 +or- sqrt(4)]/6

= [ 10 +or- 2]/6

t = 12/6 = 2 or t = 8/6 = 4/3

Notice that the total of these times is 3 and 1/3 = 10/3

x = 15*t - 25 = 15*2 - 25 = 30 - 25 = 5

the speed of the current is 5.

upstream: (25-5)*2 = 20*2 = 40

downstream (25+5)* 4/3 = 30*4/3 = 120/3 = 40

the speed of the river is 5