jmc

For a circle of radius $r$ and an arc of $\theta$ degrees, the arc length is $(2\pi r)\frac{\theta }{360}$. Thus, for the same arc length, the arc angle is inversely proportional to the radius, so the ratio of the radius of circle $A$ to the radius of circle $B$ is $40:55$, or $8:11$. Since the ratio of the areas of two circles is the square of the ratio of their radii, the ratio of the area of circle $A$ to the area of circle $B$ is $(8/11{)}^2=\boxed{\frac{64}{121}}$.