Question

There are 3 different trains running to London. One train leaves every 15 minutes, another leaves every 20 minutes, and the last one leaves every 45 minutes. They first leave at 6:15 am. What time do they all leave again at the same time?

a. 7:00 am
b. 7:15 am
c. 8:00 am
d. 8:15 am

Answer 1

The three trains will leave again at the same time after the least common multiple (LCM) of their departure intervals. The LCM of 15 minutes, 20 minutes, and 45 minutes is 180 minutes. Therefore, they will all leave together again at 9:15 am.

Step-by-step explanation:

To find the time when all three trains leave together again, we need to calculate the LCM of their departure intervals. The LCM of 15, 20, and 45 is found by factoring each number into prime factors and then taking the highest power of each prime factor present:

`\[ LCM(15, 20, 45) = 3^2 * 5 * 2^2 = 180 \]`

So, the trains will all leave together again every 180 minutes. The first train leaves at 6:15 am, and the next time they all leave together is \(6:15 + 180\) minutes, which is 9:15 am.

Therefore, the correct answer is **9:15 am**.

Answer 2

The three trains will leave again at the same time at 7:15 am. (Option b is the correct answer.)

Step-by-step explanation:

Find the Least Common Multiple (LCM): We need to find the LCM of the three train departure intervals (15 minutes, 20 minutes, and 45 minutes) to determine when they will all coincide again.

Calculate LCM:

LCM(15, 20) = 60 (since 15 is a factor of 20).

LCM(60, 45) = 180 (since 45 is a multiple of 15).

Determine next departure time: Since the first departures happen at 6:15 am, we need to add the LCM (180 minutes) to find the next time all trains leave together:

6:15 am + 180 minutes = 7:15 am

Therefore, all three trains will leave again at the same time at 7:15 am.

Option b is the correct answer.