Join right lower corner to midpoint of top line. A trapezium is formed. The area ratios for the 4 triangles in the trapezium are 1:2:2:4 with total of 9 parts. The area ratio of (square minus trapezium) to (trapezium) is 1:3 hence (square minus trapezium) has 9/3 = 3 parts. Hence fraction of red triangle is 4/(1+2+2+4 +3) = 4/12 = 1/3. The trapezium area ratio can be derived from properties of similar triangles between 2 parallel lines. The general form is T^2:TB:TB:B^2 where T (1 in this problem) is top side ratio and B (2 in this problem) is bottom side ratio, T^2 (1 in this problem) is top triangle area ratio, TB (2 in this problem) is lateral triangles area ratio (equal on both sides), B^2 (4 in this problem) is bottom triangle area ratio. These ratios are very useful in solving area ratio problems.

The area of the triangle is BH/2. We know that the Base is s. The Height of triangle B is half the height of triangle A, by the similar triangle argument you put forward. Combine with the fact that the heights of A and B add up to S, we get that the height of triangle A is 2S/3. So (S)(2S/3)/2 = S^2/3, and therefore, area of triangle / area of square = 1/3.