If anyone could help with the following question it would be much appreciated:

A and B are 2 villages on the banks of a river. By river they are 30km apart. The maximum speed of the boat is 6km/h.

The journey from A to B and back to A takes a total time of 13.5h. What is the speed of flow of the river?

I think there may be a mistake in the question as I can get to 13.33...h by using a river speed of 3km/h. On way from A to B upstream resultant speed would be 3km/h hence taking 10 hours and from B to A resultant speed would be 9km/h hence taking 3.333.. hours.

Any help would be gratefully received

First I guess we need to assume that the boat always travels at its maximum speed.

If the speed of the river flow is x, then the actual speed of the boat for each leg will be 6 + x (when travelling downstream) and 6 - x (when travelling upstream)

Time = distance/speed.

Total time = 30 / (6 + x) + 30 / (6 - x) = 13.5 hours

You then need to rearrange this to find x. (which is a bit tricky and beyond 11+ I think?)

I think it eventually gives x = sqrt (36 - 360/13.5) = 3.055 km/h

Thanks surfer fish, I agree beyond 11+ but was in an old past paper. I think there may have been a typo and it should have been 13 and a third hours as this would give x as 3km/h so I agree with your calculations

Might it be that for an 11+ question they don't expect the algebraic solution? Instead the equation could be solved iteratively by assuming values for x and adjusting until the answer reaches 13.5?

It's a poor question as you need to assume so much ie you can use the boat adjacent to the bank, river is straight, ...

It did say in the question to try different values for speed but it seems like trial and error would take quite a while for an 11yo who probably would struggle to understand it anyway.



I agree, it was on the entrance exam for a very popular school but it is a very old paper