Ed’s Infamous Area Problem

Yesterday a colleague asked if I’d seen the maths problem that was going round and featured in the national press. I hadn’t, but was not surprised to see that it was Ed Southall (@solvemymaths) who had posed the problem that had got the world (well, the nation at least) talking maths. My initial thought was that it was great to see a positive discussion of maths in the press. Then I figured I’d need to solve it.

Here is the problem:

What fraction of the area is shaded?

What follows is my solution. Please attempt the problem before reading on, I’d love to see your approach.

Firstly,I did a sketch (of course I’d did. If you didn’t then why on earth not!)

I labelled the base of the rectangle 2x and the height b (it looked like a square, but I didn’t want to assume and figured if it was necessary to Ed would have told us). I realised that I was looking at 2 similar triangles (proof can be made using opposite and alternate angles), with a scale factor of 2 (the base of the bottom is double the base of the top). I know that when working with areas the scale factor is squared. Using an area scale factor of 4, a for the height of the top triangle and (b – a) for the height of the lower triangle I came up with this equation:

Which solved to tell be b was 3a, thus b-a was 2a.

From here it was simple, I worked out the area of the shaded triangle and the whole rectangle put it as a fraction and simplified.

How did you do it?

Edited 28th April 2020 to fix some typos.