Ratio and proportion

problem
Timber!

Age

7 to 11

Challenge level

How can the school caretaker be sure that the tree would miss the school buildings if it fell?
problem
Cherry buns

Age

7 to 11

Challenge level

Sam's grandmother has an old recipe for cherry buns. She has enough mixture to put 45 grams in each of 12 paper cake cases. What was the weight of one egg?
problem
Hello again

Age

11 to 14

Challenge level

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?
problem
ArRh!

Age

14 to 16

Challenge level

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. What is the value of r/R?
problem
A shade crossed

Age

14 to 16

Challenge level

Find the area of the shaded region created by the two overlapping triangles in terms of a and b?
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
Conical bottle

Age

14 to 16

Challenge level

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?
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
Golden ratio

Age

16 to 18

Challenge level

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.
problem
Rati-o

Age

11 to 14

Challenge level

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?