Ratio and proportion

Have you ever wondered what it would be like to race against Usain Bolt?

Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?

Two ladders are propped up against facing walls. At what height do the ladders cross?

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Can you work out the fraction of the original triangle that is covered by the inner triangle?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

Can you find an efficent way to mix paints in any ratio?

A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.

Do you have enough information to work out the area of the shaded quadrilateral?