Why is the modern piano tuned using an equal tempered scale and
what has this got to do with logarithms?
Show that it is rare for a ratio of ratios to be rational.
The Pythagoreans noticed that nice simple ratios of string length
made nice sounds together.
The Earth is further from the Sun than Venus, but how much further?
Twice as far? Ten times?
A new problem posed by Lyndon Baker who has devised many NRICH
problems over the years.
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.
Can you work out the fraction of the original triangle that is
covered by the inner triangle?
What's the most efficient proportion for a 1 litre tin of paint?
Scientists often require solutions which are diluted to a
particular concentration. In this problem, you can explore the
mathematics of simple dilutions
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
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.
In the diagram the radius length is 10 units, OP is 8 units and OQ
is 6 units. If the distance PQ is 5 units what is the distance P'Q'
Which exact dilution ratios can you make using only 2 dilutions?
What does the empirical formula of this mixture of iron oxides tell
you about its consituents?
A Sudoku with clues as ratios.
Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .
A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
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?
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
Two motorboats travelling up and down a lake at constant speeds
leave opposite ends A and B at the same instant, passing each
other, for the first time 600 metres from A, and on their return,
400. . . .
Straight lines are drawn from each corner of a square to the mid
points of the opposite sides. Express the area of the octagon that
is formed at the centre as a fraction of the area of the square.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Take any rectangle ABCD such that AB > BC. The point P is on AB
and Q is on CD. Show that there is exactly one position of P and Q
such that APCQ is a rhombus.
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
The diagram shows a regular pentagon with sides of unit length.
Find all the angles in the diagram. Prove that the quadrilateral
shown in red is a rhombus.
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.
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
A Sudoku with clues as ratios or fractions.
Match pairs of cards so that they have equivalent ratios.