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.

Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track?

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. . . .

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

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

In a race the odds are: 2 to 1 against the rhinoceros winning and 3 to 2 against the hippopotamus winning. What are the odds against the elephant winning if the race is fair?

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?

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

Some of the numbers have fallen off Becky's number line. Can you figure out what they were?

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. . . .

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

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. . . .

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

Can you work out how to produce different shades of pink paint?

Four jewellers share their stock. Can you work out the relative values of their gems?

The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?

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

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the. . . .

A farmer is supplying a mix of seeds, nuts and dried apricots to a manufacturer of crunchy cereal bars. What combination of ingredients costing £5 per kg could he supply?

An article for teachers which discusses the differences between ratio and proportion, and invites readers to contribute their own thoughts.

Can you fill in the mixed up numbers in this dilution calculation?

Is there a temperature at which Celsius and Fahrenheit readings are the same?

A garrison of 600 men has just enough bread ... but, with the news that the enemy was planning an attack... How many ounces of bread a day must each man in the garrison be allowed, to hold out 45. . . .

How long will it take Mary and Nigel to wash an elephant if they work together?

A trapezium is divided into four triangles by its diagonals. Can you work out the area of the trapezium?

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?

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.

One night two candles were lit. Can you work out how long each candle was originally?

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

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.

Which dilutions can you make using only 10ml pipettes?

Is it cheaper to cook a meal from scratch or to buy a ready meal? What difference does the number of people you're cooking for make?

Andy is desperate to reach John o'Groats first. Can you devise a winning race plan?

When Charlie retires, he's looking forward to the quiet life, whereas Alison wants a busy and exciting retirement. Can you advise them on where they should go?

Which exact dilution ratios can you make using only 2 dilutions?

Which dilutions can you make using 10ml pipettes and 100ml measuring cylinders?

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?

A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

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