Answer: A

Small Hint:

Let the faster and slower speeds be $u$ and $v$, and write the same-direction and opposite-direction relative-speed equations

Big Hint:

Use $(u-v)r=k$ and $(u+v)t=k$, then solve for $\frac{u}{v}$

Solution:

Let the speeds be $u\gt v.$ The two meeting conditions give $$u-v=\frac kr,\qquad u+v=\frac kt.$$ Adding and subtracting these equations yields $$\begin{aligned} u&=\frac k2\left(\frac1r+\frac1t\right),\\ v&=\frac k2\left(\frac1t-\frac1r\right). \end{aligned}$$ Therefore $$\frac uv=\frac{r+t}{r-t}.$$

Thus, the correct answer is A.