Ratio and proportion

problem
Equal temperament

Age

14 to 16

Challenge level

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.
problem
Pythagoras' comma

Age

14 to 16

Challenge level

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.
problem
Six notes all nice ratios

Age

14 to 16

Challenge level

The Pythagoreans noticed that nice simple ratios of string length made nice sounds together.
problem
Points in pairs

Age

14 to 16

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

11 to 14

Challenge level

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

Age

14 to 16

Challenge level

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

Age

14 to 16

Challenge level

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

Age

14 to 16

Challenge level

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.