Ratio and proportion

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Two Ladders

Age

14 to 16

Challenge level

2 out of 3

Two ladders are propped up against facing walls. At what height do the ladders cross?
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Sitting Pretty

Age

14 to 16

Challenge level

2 out of 3

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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Triangle in a Triangle

Age

14 to 16

Challenge level

2 out of 3

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

Age

14 to 16

Challenge level

2 out of 3

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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Speeding Boats

Age

14 to 16

Challenge level

2 out of 3

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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Golden Thoughts

Age

14 to 16

Challenge level

3 out of 3

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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Mixing More Paints

Age

14 to 16

Challenge level

3 out of 3

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

Age

14 to 16

Challenge level

3 out of 3

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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Areas and Ratios

Age

16 to 18

Challenge level

2 out of 3

Do you have enough information to work out the area of the shaded quadrilateral?
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Plane to See

Age

16 to 18

Challenge level

1 out of 3

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.