Four squares inside another square. What fraction is yellow?

Solution

The yellow square covers 1/5 of the total area.

First, scale the diagram such that the red squares have unit sides and call the yellow side x. Comparing bottom and left side of the large square, it is straightforward to fill in the lengths as shown, also using congruency of the two left-side triangles.

Now consider the small triangle in the middle. It is similar to the larger ones and this gives (x-1)/1=1/x. This has one positive solution x=(1+√5)/2, also know as the Golden Ratio. The required fraction is x²/(2+x)²=1/5.

Poem

The corner stone
Not a heart being alone
The stony heart is speaking
Cannot stop skipping
Cause the lonely girl is crying
She realizes she is suffering
Stones are hard and bleeding
And secrets conversing
Some of them are holding
And try to hide something
Stones are fighting

Bella