Since the perimeter of the square is 16cm, each side is 4cm.
If we subtract the area of the two unshaded triangles from the area of the square we will know the area of the shaded portion...
The small inverted triangle on the left has a height of 4 and a base of 1. The are of the bottom triangle has a height of 2 and a base of 4...
So the area of the two triangles is 4*1/2+2*4/2=2+4=6
The area of the square is just 4*4=16 so:
The shaded area is 16-6=10cm^2
The proportion of the shaded area to the total area is then:
10cm^2:16cm^2
5:8