Stupid currency question

let’s say 2009 JPY/USD@90 2010 JPD/USD@100 from “complicated calculation,” we can see USD appreciate by 11.11%. And we will say JPY depreciate by the same extent 11.11%, right. But actually, from strict calculation, yen only depreciate by 10%. One currency appreciate by A% against another currency, which does not mean another currency depreciate by the exact A%(approximately depreciate by A%.) right?

inverse your ratios i.e. 1/90 = 0.0111 and do the same for 1/10

I think your question can be answered by this analogy. Say you have $100. If it drops 50% it becomes $50. What does it become if it then rises 50%? back to $100? No, it goes to $75. Same thing with your example. 100 rises by 10% to 110, but when it drops 10% it doesn’t become 90.