### In Constantly Passing

A car is travelling along a dual carriageway at constant speed. Every 3 minutes a bus passes going in the opposite direction, while every 6 minutes a bus passes the car travelling in the same direction. Buses leave the depot at regular intervals; they travel along the dual carriageway and back to the depot at a constant speed. At what interval do the buses leave the depot?

### Take a Message Soldier

A messenger runs from the rear to the head of a marching column and back. When he gets back, the rear is where the head was when he set off. What is the ratio of his speed to that of the column?

# Escalator

##### Stage: 4 Challenge Level:

Will of The Ridgeway School; Xing Cong of Raffles Institution, Singapore; Elizabeth of Ipswich High School; and Alexandra of Bancroft's School, Woodford Green, Essex; all used very similar methods. This is Alexandra's beautifully clear solution:

There are 100 steps. When the taller man reaches the top the shorter man will have climbed one third of 75 steps i.e. 25 steps. This means that the shorter man is half way up the escalator when the taller man has reached the top. At this moment, there are 50 steps between the men (the difference between 75 and 25) which means that half the escalator is equal to 50 steps. The whole escalator is therefore double this, that is 100 steps.

Although it takes longer and more work, this problem can be solved by supposing there are k steps on the escalator, that the speed of the escalator is v steps per minute and the speeds of the two men are u and 3u steps per minute. Now you need four equations to find k. You can write down these four equations from the distance, speed and time for the smaller man relative to the ground, for the smaller man relative to the escalator, for the taller man relative to the ground and for the taller man relative to the escalator.