Final answer:
The least amount of time for both trains A and B to arrive at their starting points at the same time is 36 seconds, calculated using the least common multiple (LCM) of their return times.
Step-by-step explanation:
The least amount of time when both trains A and B will arrive at their starting points at the same time can be found by computing the least common multiple (LCM) of their individual times to return to the starting point, which are 12 seconds and 9 seconds respectively. To calculate the LCM, we could list multiples of each time until we find the smallest multiple they have in common. However, a more efficient method would be to use the prime factorization of 12 and 9 and then multiply the highest power of all the primes together to get the LCM.
Train A: 12 seconds (which is 22 × 3)
Train B: 9 seconds (which is 32)
LCM: 22 × 32 = 4 × 9 = 36 seconds
Therefore, 36 seconds is the least amount of time after which both trains will arrive at their starting points simultaneously.
To determine the least amount of time when both trains will arrive at their starting points simultaneously, we need to find the least common multiple (LCM) of the two times it takes for each train to complete one full rotation
Train A returns to its starting point every 12 seconds, while train B returns to its starting point every 9 seconds.
Using the LCM, we find that the least amount of time when both trains will arrive at their starting points at the same time is 36 seconds.