Ratio and proportion

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.

Using an understanding that 1:2 and 2:3 were good ratios, start with a length and keep reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range.

The Pythagoreans noticed that nice simple ratios of string length made nice sounds together.

Move the point P to see how P' moves. Then use your insights to calculate a missing length.

Show that it is rare for a ratio of ratios to be rational.

Why is the modern piano tuned using an equal tempered scale and what has this got to do with logarithms?

Weekly Problem 3 - 2007
What is the ratio of the area of the table covered twice, to the uncovered area?

A square is divided into four rectangles and a square. Can you work out the ratio of the side lengths of the rectangles?

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

We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.