Ratio and proportion

problem
Golden thoughts

Age

14 to 16

Challenge level

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.
problem
Mixing more paints

Age

14 to 16

Challenge level

Can you find an efficent way to mix paints in any ratio?
problem
Dilution series calculator

Age

14 to 16

Challenge level

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

Age

14 to 16

Challenge level

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

Age

16 to 18

Challenge level

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?
problem
Chord

Age

16 to 18

Challenge level

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.
problem
Halving the triangle

Age

16 to 18

Challenge level

Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.
problem
Extreme dissociation

Age

16 to 18

Challenge level

In this question we push the pH formula to its theoretical limits.
problem
Another triangle in a triangle

Age

16 to 18

Challenge level

Can you work out the fraction of the original triangle that is covered by the green triangle?
problem
Areas and ratios

Age

16 to 18

Challenge level

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