Correct Answer: B Let $S_A$ be the speed of Train A and $S_B$ be the speed of Train B. Let $D$ be the distance between station P and station Q. Let M be the point where the two trains meet for the first time. Let $t_1$ be the time elapsed until the first meeting. **Step 1: Determine the ratio of speeds ($S_A : S_B$) using the information about the first meeting.** When two trains start simultaneously from two points and travel towards each other, and after meeting, they take $T_A$ and $T_B$ hours respectively to reach their destinations, then the ratio of their speeds is given by the formula: $\frac{S_A}{S_B} = \sqrt{\frac{T_B}{T_A}}$ In this problem, $T_A = 4$ hours (time taken by Train A to reach Q after meeting) and $T_B = 9$ hours (time taken by Train B to reach P after meeting). Substituting these values into the formula: $\frac{S_A}{S_B} = \sqrt{\frac{9}{4}}$ $\frac{S_A}{S_B} = \frac{3}{2}$ So, the ratio of the speed of Train A to Train B is 3:2. **Step 2: Understand the dynamics of the second meeting.** When two objects start simultaneously from opposite ends, travel towards each other, and upon reaching their respective destinations, immediately turn back and continue traveling at their constant speeds, the following properties hold for their meetings: * For the first meeting, the total distance covered by both objects combined is $D$ (the distance between their starting points). * For the second meeting, the total distance covered by both objects combined is $3D$. * For the third meeting, the total distance covered by both objects combined is $5D$. * In general, for the $n$-th meeting, the total distance covered by both objects combined is $(2n-1)D$. In this problem, we are interested in the second meeting. Thus, the total distance covered by Train A and Train B combined until their second meeting will be $3D$. **Step 3: Calculate the ratio of total distances traveled until the second meeting.** Let $t_2$ be the total time elapsed from the start until the second meeting. Since both trains start simultaneously and travel for the same duration $t_2$ until they meet for the second time, the total distance traveled by each train will be directly proportional to its speed. Total distance traveled by Train A ($Dist_A$) = $S_A \times t_2$ Total distance traveled by Train B ($Dist_B$) = $S_B \times t_2$ The ratio of their total distances traveled is: $\frac{Dist_A}{Dist_B} = \frac{S_A \times t_2}{S_B \times t_2}$ $\frac{Dist_A}{Dist_B} = \frac{S_A}{S_B}$ From Step 1, we know that $\frac{S_A}{S_B} = \frac{3}{2}$. Therefore, the ratio of the total distance traveled by Train A to the total distance traveled by Train B when they meet for the second time is $3:2$. The final answer is $\boxed{\text{3:2}}$