Answer:
To determine when Bus A and Bus B will leave the station at the same time, we need to find the least common multiple (LCM) of their time intervals.
The time intervals for Bus A and Bus B are:
Bus A leaves every 45 minutes (i.e. intervals of 45 minutes)
Bus B leaves every 60 minutes (i.e. intervals of 60 minutes)
To find the LCM of 45 and 60, we can first factor each number into its prime factors:
45 = 3 x 3 x 5
60 = 2 x 2 x 3 x 5
The LCM is the product of the highest power of each prime factor, so:
LCM(45, 60) = 2 x 2 x 3 x 3 x 5 = 180
Therefore, Bus A and Bus B will leave the station at the same time every 180 minutes, or every 3 hours.
Since the buses begin their routes at 7:30 am, we can add multiples of 3 hours to this time to find when Bus A and Bus B will leave together.
The first time after 7:30 am when Bus A and Bus B leave together is:
7:30 am + 3 hours = 10:30 am
So Bus A and Bus B will leave the station together at 10:30 am, and then every 3 hours after that.