## 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 figures 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 squares so using an area scale factor of 4, a for the height if the top triangle and (b – a) for the height of the lower triangle intake 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?

Using the similar triangles relationships: Let the shaded area be x and the whole area be 1

x +1/4 +1/2 – (1/4)x = 1

(3/4)x = 1/4

x = 1/3

I did it just as you did, finding the intersection of the diagonal lines algebraically and working from that.

What gets me is that before calculating I tried to think what I expected the answer to be, and came out with one-third, and I don’t know why that sounded right. I’m curious whether I had encountered the problem before and remembered the answer without remembering it, or whether there’s some obvious way to see it that my conscious mind isn’t noticing. Or if the lazy part of my mind figured the answer must be some nice simple fraction, and 1/2 is right out, and 1/4 looks too small, so 1/3 is the only plausible choice because this kind of setup never gives you a 2/5th or a 3/7th or something like that.

My initial thought was 1 third when o looked at it, but then I thought it would come out as 2/5