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

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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?

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

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

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

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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.

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

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

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

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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?

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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. . . .

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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?

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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. . . .

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

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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. . . .

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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.

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

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Build a scaffold out of drinking-straws to support a cup of water

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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?

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

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My recipe is for 12 cakes - how do I change it if I want to make a different number of cakes?

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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?

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The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?

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Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?

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

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

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Which dilutions can you make using only 10ml pipettes?

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

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

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In the ancient city of Atlantis a solid rectangular object called a Zin was built in honour of the goddess Tina. Your task is to determine on which day of the week the obelisk was completed.

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Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

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The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

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Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

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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.

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

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

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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.

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

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What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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What's the most efficient proportion for a 1 litre tin of paint?

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Construct a line parallel to one side of a triangle so that the triangle is divided into two equal areas.

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Can you work out the fraction of the original triangle that is covered by the inner triangle?

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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. . . .

Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.

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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. . . .