Step 1 :Given that trains on one route stop at the station every 9 minutes and trains on the other route stop there every 6 minutes, we need to find the least common multiple (LCM) of 9 and 6. The LCM of two numbers is the smallest number that is a multiple of both numbers.
Step 2 :Let's denote the interval of the first train as \(train1_{interval}\) and the interval of the second train as \(train2_{interval}\). So, \(train1_{interval} = 9\) and \(train2_{interval} = 6\).
Step 3 :We calculate the LCM of \(train1_{interval}\) and \(train2_{interval}\), which is 18.
Step 4 :\(\boxed{18}\) minutes after midday, trains on both routes will stop at the station at the same time.