A boat takes 19 hours for travelling downstream from point A to point B and…

Concept

In still water a boat moves at its own speed. A stream adds its speed to the boat going downstream and takes it away going upstream, so downstream speed = speed in still water + speed of the stream, and upstream speed = speed in still water - speed of the stream.

The time for any leg of a journey = distance for that leg divided by the effective speed on that leg, and the time for a journey made of several legs is the sum of the individual leg times.

Application

Here the speed in still water is 14 kmph and the stream is 4 kmph, so the downstream speed is 14 + 4 = 18 kmph and the upstream speed is 14 - 4 = 10 kmph. Let the distance AB be d km.

1. The outward leg from A to B is with the stream: it covers d km at 18 kmph, taking d/18 hours.

2. C is the midpoint of AB, so the return leg from B to C covers d/2 km.

3. That return leg is against the stream at 10 kmph, so it takes (d/2) / 10 = d/20 hours.

4. The whole journey is given as 19 hours, so d/18 + d/20 = 19.

5. Taking the LCM of 18 and 20, which is 180: 10d/180 + 9d/180 = 19d/180 = 19.

6. Therefore 19d = 19 x 180, which gives d = 180.

Cross-check

Putting d = 180 km back into the journey: the downstream leg takes 180 / 18 = 10 hours and the 90 km upstream return takes 90 / 10 = 9 hours. Together that is 10 + 9 = 19 hours, exactly the time stated in the question, so the value is consistent.

Contrast

Total time here is directly proportional to AB, since the total works out to 19d/180 hours, so a shorter AB always gives a shorter total journey:

• AB = 120 km gives a total of 12 2/3 hours.

• AB = 140 km gives a total of 14 7/9 hours.

• AB = 160 km gives a total of 16 8/9 hours.

The distance between A and B is 180 km.