Golden Thoughts

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.

From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

The Rescaled Map

Stage: 4 Challenge Level:

We aim to scale the countries on the map below to represent population size.

We use the hint given and let the ocean keep its original proportion of the map, which is $\frac{39}{64}\times 100=61$ of the map.

 Country Area (million square kilometres) Proportion of total land area Population (millions) Proportion of total population A 8 0.32 40 0.4 B 5 0.2 36 0.36 C 12 0.48 24 0.24

We then find out the factors by which the country sizes need to change:

A needs to cover $\frac{0.4}{0.32}=1.25$ times more area on the map

B needs to cover $\frac{0.36}{0.2}=1.8$ times more area on the map

C needs to cover $\frac{0.24}{0.48}=0.5$ times less area on the map

So we first stretch each country by its scale factor in the vertical direction. For example, the country B would have new dimensions as below:

And so the overall map becomes (each square has dimensions 0.5 x 0.5)

We now scale in the horizontal direction, and for example the country C (marked in yellow) would have new dimensions as below:

And so the overall map becomes (each square again with dimensions 0.5 x 0.5)

Now we combine the scaling factors in both the horizontal and vertical directions. For example, the country A would have dimensions as below:

Well done to Jack from the King John School, who made a good attempt at resizing the countries in both directions. We leave this part for you to try! Perhaps scale countries B and C next, and then draw them on a map, making sure the ratio of land to ocean is kept constant.