### On the Road

Four vehicles travelled on a road with constant velocities. The car overtook the scooter at 12 o'clock, then met the bike at 14.00 and the motorcycle at 16.00. The motorcycle met the scooter at 17.00 then it overtook the bike at 18.00. At what time did the bike and the scooter meet?

### There and Back

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

### Escalator

At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. ... How many steps are there on the escalator?

# In Constantly Passing

##### Stage: 4 Challenge Level:

Thank Justin from Skyview High School, Billings, MT, USA for this solution and well done!

Rate, time and distance are connected by the equation r =d/t .

Call the rate (or speed) of the car r c and the rate of every bus r b . Each bus is a constant distance from the bus preceding it and the bus following it; call this distance d.

For a bus approaching on the other highway and coming towards the car, the rate of the bus relative to the car, considering the bus still, is (r b + r c ). This rate multiplied by the time it takes (three minutes) for the car to close the gap between it and the bus is equal to d, hence:

3(r b + r c ) = d.

The rate of the bus which is travelling in the same direction as the car, relative to the rate of the car (considering the car still) is (r b - r c ). This rate multiplied by the time it takes (six minutes) for the bus to close the gap between it and the car is also equal to d 1 , hence

6(r b - r c ) = d.

Multiplying the first equation by 2 and add the two equations, one obtains

12r b = 3d.

But the distance d between the buses divided by the rate of the bus is equal to the time interval between the buses therefore the time interval = d/r b = 4 minutes.

So the buses leave the depot at intervals of 4 minutes.