The reason why you need 15 barracks

[toey]

With 15 barracks, for each one, you can split the time of your training, to 1/15th of the time, for EACH barracks.

I think this is not totally correct.
When you have multiple barracks, it splits the number of training troops. It doesn't split the time. Therefore, when you train 1 troops, it will take the same amount of time either you have 1 or 15 barracks.