Ratio and proportion

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?

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

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

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

Scientists often require solutions which are diluted to a particular concentration. In this problem, you can explore the mathematics of simple dilutions

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?