Upstream downstream problem

Question

A girl sailed downstream from Point A to Point B on a river. Her boat can travel 8km faster downstream than upstream in one hour. She spent a total of 3 hours travelling from A to B and then back to A. She travelled 2 km less in the third hour than in the second hour. What is the distance in km between points A and B?

Answer 1

Let d be the distance between A and B

Let v be the downstream speed

Divide the 3 hour time into 3, one hour slots. Assume that the sailboat turns around sometime in the second hour.

distance/downspeed + distance/upspeed = 3 hours

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

d = (3v²-24v)/(2v-8)

Let t be the fraction of the second hour that the boat travels downstream

Going down stream

d = v(1) + vt

1-t is the time that the boat travels upstream in the second hour

Going up stream

d = (v-8)(1-t) + (v-8)(1) = 2v-vt+8t-16

Set the two distance equations equal

v+vt = 2v-vt+8t-16

simplifying

v = (16-8t)/(1-2t)

Using the difference between the second and third hour distances:

[vt + (v-8)(1-t)] - (v-8) = 2

vt+r-vt-8+8t-v+8=2

8t = 2

t = 1/4

From above:

v = (16-8(1/4))/(1-2(1/4)) = 14/(1/2) = 28 kph

substituting 28 into the equation above for d

d = (3(28)²-24(28))/(2(28)-8) = 35 km

Checking:

d/v + d/(v-8) = 35/28 + 35/20 = 35(20+28)/(28(20)) = 35(48)/560 = 3