My permanent residence is in the USA and almost all of the money to name is in US dollars (USD) in US accounts. This month I'm in India and may end up having withdrawn more rupees (INR) than I need as a tourist here. From India I'll stopover in Germany for one week, before finally returning to the USA. Naturally, I will need to get some euros (EUR) for this short trip, and I already know that amount is more than the equivalent value of the extra INR I'm holding.
I could either convert my extra INR to EUR, or to USD. In either case the (maybe additional) euros I need would be converted from my USD account, being careful not to withdraw more EUR than I'll need for Germany. I'm wondering if there are any considerations that might favor one or the other conversions, in terms of my overall remaining US funds?
I know there are a lot of factors here, such as different places' exchange fees and that and we cannot predict future exchange rates. I'm not trying to "game" the market's future activity. I think I'm not expecting a straight answer from SE to "please decide for me which currency to covert my rupees" but rather what I really wonder is more theoretical about how foreign exchange markets work:
If 1 INR = p EUR and 1 EUR = q USD, then does 1 INR = pq USD? (assuming fair rates, no fees, and p, q, & pq being measured at the same time.) Would this also be true for any trio (or more, by extension) of world currencies? If not, why not?
(If there is a mathematical name for this property I'm asking about, please enlighten us!)