Four Equals

The hexagon is made of six triangles, one of which is $\triangle ABO$ . If we set $AB=1$ , the area of the hexagon will be

$A_{hex}$ $=6\times\frac{\sqrt3}4=\frac{3\sqrt3}2$

The height of $\triangle ABO$ , which is $\overline{OF}=\overline {EF}+\overline{OE}=\overline {EF}+\frac 23\overline{EF}=\frac53\overline{EF}$

The $\frac23$ comes in from the fact that $OE$ is distance from the center of $\triangle GHE$ to the vertex, and that is $\frac 23$ of the median.

The sides of the green triangles are all $\frac35\times\overline{AB}=\frac35$ , so the green area is

$A_{4\triangle}$ $=4\times\frac{\sqrt3}4\times\left(\frac35\right)^2=\frac{9\sqrt3}{25}$

$\dfrac{\color{#20A900}{GREEN}}{\color{#CEBB00}{YELLOW}}$ $=\dfrac{A_{4\triangle}}{A_{hex}-A_{4\triangle}}=\dfrac{\frac{9\sqrt3}{25}}{\frac{3\sqrt3}2-\frac{9\sqrt3}{25}}=\dfrac6{25-6}=\dfrac6{19}$

Answer $=6+19=\boxed{25}$

WLOG apothem of hexagon is 1. This is also the median of the six forming triangles. The median of the six triangle forming the hexagonal can be clearly seen to be 1+2/3=5/3 times that of the four triangles.
Also the four and six triangles are similar equilateral triangles.
Thus the ratio of Green to Yellow areas= $\dfrac {4*(3/5)^2}{6*1^2-4*(3/5)^2}=\dfrac{36/25}{6-36/25}=\dfrac{36}{150-36}=6/19=a/b.\\ So~a+b=6+19=25.$