Final answer:
The next time both trains will stop on their respective tracks at the same time is at 8:36 am, determined by calculating the least common multiple of their stopping intervals: 9 minutes for one train and 12 minutes for the other.
Step-by-step explanation:
The question at hand involves finding a common time when two trains stopping at different intervals will stop at the same time again. This is a classic least common multiple (LCM) problem in mathematics. One train stops every 9 minutes and the other every 12 minutes. To find when they will both stop at the same time again, calculate the LCM of 9 and 12.
The prime factorization of 9 is 3², and for 12 it's 2² × 3. The LCM is then the highest power of each prime number present in either factorization, which gives us 2² × 3² = 4 × 9 = 36. Therefore, the trains will both stop at the same time every 36 minutes.
Since the trains both stop at 8 am, adding 36 minutes to this time gives us the next stopping time at the same station, which is at 8:36 am.