In this problem the second train leaves 10 mins. after the first.
In the answer provided, T for train k to T is t+1/6 and T for train t to k is T. I was wondering if there is a rule that we should follow in adding the minutes instead of subtracting. Why don't we instead do T for K to T train and t-1/6 for T to K train. In the second case however we would be stuck in solving an equation 240t+160t-26.66=300. This will provide the answer 49min, which will solve the problem in terms of the first train (49 min. after 12:00). The original solution is in terms of the second train (39min after 12:10). In this case how do we keep in mind that the answer will be in terms of the second train, which left at 12:10. The train with variable T is the one that the equation is solved for? I am sure that there will be a trap answer in case we are not sure in terms of which train we solved for.
The same about problem 15. The equation is solved in terms of Nicky 18 sec(+12 head start) = 30 sec. In case we want to solve in terms of Christina we have to do t-12 for her and t for Nicky. That will yiled t=30
So is there a particular rule in terms of which object (or person) in these poroblems we should solve for. Is it more acceptable to add time (do t+12; t+1/6, etc) instead of subtract time for the other person/object (t-12; t-1/6).
I am kind of confused here.

Hi there,
The rule I always follow is to let the person/train/car/etc. who leaves LATER have the "t" for that later time, and then to add the time to t that the earlier person has already used.

So if train X leaves at noon and train Y leaves ten minutes later(which is 1/6 of an hour), I say:

train Y has "t" time
train X has "t + 1/6" time

I think it is easier to always set things up the same way, and this is the way that I and my students have generally found to be the easiest.

_________________
Jamie Nelson
ManhattanGMAT Instructor