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Areas and Ratios

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

Napoleon's Hat

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?

Plane to See

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.


Age 16 to 18
Challenge Level


This problem explores the consequences of Avogadro's hypothesis that the volume occupied by $1$ mole of gas molecules at a fixed temperature and pressure is the same for any gas. In this problem, work on the assumption that all compounds (including water) are in gaseous form.

$10$cm$^3$ of ethane C$_2$H$_6$ is injected into a large chamber of oxygen and completely combusted at high temperature

2C$_2$H$_6$ + $7$O$_2$ $\rightarrow$ 4CO$_2$+ 6H$_2$O

Does the total volume of gas increase, decrease or stay the same?

Will the same be true if any hydrocarbon is completely combusted? Are there any extreme cases which give rise to very large changes in volume?

Can you find any combustion processes using other compounds for which the total volume of gas is unchanged by the reaction?