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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.

Reductant Ratios

Age 16 to 18
Challenge Level


A very uniform grade of iron ore is known to be entirely formed from a mixture of different oxides of iron (iron oxide FeO, magnetite Fe$_3$O$_4$ and hematite Fe$_2$O$_3$), although their ratios are not known.

$1$g of this mixture is fully reduced, leaving $0.748$g of pure iron.

Must all three of the oxides indicated be present in the mixture?

Is there a smallest or largest amount of each oxide present?

What possibilities for the percentage composition of the mixture by mass does the experiment indicate?

Can you create a yield of iron which would prove the absence of at least one of the oxides of iron from the ore?