The blue line arrives every 10 minutes the red line arrives every 8 minutes and the yellow line arrives every 12 minutes, how long until they arrive at the station at the same time

They all arrive at the station at the same time after 120 minutes.

Step-by-step explanation:

For solving these types of question we use Least Common Multiples (L. C. M.).

Since for getting a common time they will meet, we interested in a multiplier.

Now, We will take L. C. M. of all the numbers:

10 = 2 * 5

8 = 2 * 2 * 2

12 = 2 * 2 * 3

∴ L. C. M. = 2 * 2 * 2 * 3 * 5 = 120 {taking smallest common multiples)

Thus, they all lines will meet after 120 minutes.