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A, B, and C denotes the areas of the color each letter is located in. So, for example, C denotes the small yellow square.

Let T be the area of the total big square.

Consider the triangle which is formed by A and B together. The area of that triangle is clearly T/4.

There are 4 of the A and B triangles, so their total area is T, i.e. T=4(A+B). But since each B is overlapped by an A the 4 (A+B) triangles do not cover the whole big square, but leaves square C free in the middle. Therefore C=4A.

Notice also that, as indicated by the red lines, the area of B is 4 times the area of A, i.e. B=4A.

So

A+B = A+4A

A+B = 5A

A = (A+B)/5

Multiply by 4.

4A = 4(A+B)/5

We know that

T=4(A+B)

So we put that into what we got, and thus

4A = T/5

And we also know that C=4A

Therefore

C = T/5

So the area of the yellow square is a 5th of the big square.