jmc

Let the radius of the large semicircle be $6x$. The diameter of the smallest semicircle is $\frac{1}{1+2+3}=\frac{1}{6}$ of the diameter of the largest semicircle, so the radius of the smallest semicircle is $x$. Similarly, the radius of the next smallest semicircle is $2x$, and the radius of the next semicircle is $3x$. The unshaded area then is the sum of the areas of the three smallest semicircles: $$\frac{1}{2}(x{)}^2\pi +\frac{1}{2}(2x{)}^2\pi +\frac{1}{2}(3x{)}^2\pi =\frac{1}{2}(x^2+4x^2+9x^2)\pi =(7x^2)\pi .$$ The largest semicircle has area $\frac{1}{2}(6x{)}^2\pi =18x^2\pi$, so the shaded area is $$18x^2\pi -7x^2\pi =11x^2\pi .$$ Therefore, the desired ratio is $$\frac{11x^2\pi }{7x^2\pi }=\boxed{\frac{11}{7}}.$$