2011 AMC 10B Problem 12

Let the inner semicircle radius be $r$. The straight parts of the inside and outside paths have the same total length, so only the semicircular ends change the distance. The two inner semicircles have total length $2\pi r$, while the two outer semicircles have total length $2\pi(r+6)$. The outside path is therefore $12\pi$ meters longer. Keiko takes $36$ more seconds to walk $12\pi$ more meters, so her speed is $\frac{12\pi}{36}=\dfrac\pi3$ meters per second. Thus, A is the correct answer.