Anything except policits

I suppose anything could be considered political ;-) But hard for this to.

I have an economy/financial question. I was looking at exchange rates between pound, euro and dollar. (I will use E for Euro and # for pound).

#1 = $1.2816
#1 = E1.2071
E1 = $1.0616

so theoretically, neglecting fees, if you change 100 pounds for dollars you should get the same as if you changed 100 pounds for Euros (120.71) and then changed the 120.71 for dollars. Both those should be $128.16.

Stands to reason that 1.2071 * 1.0616 should be 1.2816 (multiply E/# by $/E). Which it pretty much is. 1.2071 * 1.0616 = 1.281476. Off by .0002

My question is how do they establish exchange rates? If somewhere they establish a rate of # to $, and somewhere else they do the rate # to E, and somewhere E to $, then when you multiply E/# by #/$ you could be somewhat off and you could make money by converting pounds to euros, euros to dollars and the dollars back to pounds, (or converting pounds to dollars, buying euro and then buying pounds), in one of those cases you would end up with more pounds than you started and you could just keep doing that. (Yes, I know you would have to take fees into consideration so the amount off would have to be significant).

But how do they keep everything consistent? Is there just one place where all rates are set? It does not have to be $, #, E. It could be kroner, yen, francs, whatever.