Scroll down for a solution to this problem.
The shaded fraction is (9√3-15)/2, which is approximately 0,29.
Start by setting the triangle side to 1 and naming the line segment x as shown. Then recognize two 30-60-90 triangles in the bottom part and two congruent triangles above them, because of the folding.
The line segments shown can now easily be found and x is deducted using the fact that (1+x)/2=√3*(1-x)/2 in the larger 30-60-90 triangle. It gives x=2-√3.
The area of the shaded triangle comes from the rule A=1/2* x√3 * (1+x)/2 * sin60°. If divided by the large triangle area √3/4, one arrives at the desired fraction.
We thank Per Henrik Christiansen for making a nice Desmos graphic animation of this problem.