### 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 18Challenge 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?