Technically speaking "yaşındayım" doesn't make much sense.
It is clear that it's a possessive+locative → yaş → yaşı → yaşında
Why is there a possessive? Certainly we say 3 yaş and NOT *3('ün) yaşı, or something. So where does the possessive come from? Back in the day, it was actually a possessive construction, and you had to say:
Ben 62 yaşIMdayım (I'm in MY 62(nd) year),
Sen 32 yaşINdasın (you are in your 32(nd) year),
O 20 yaşInda (the n is a buffer this time: He is in his 20(th) year),
which actually made more sense. Nowadays, however, this expression has become fixed and we use the 2nd (or 3rd?) person form of it for every person, so you get:
Ben 30 yaşındayım, sen 10 yaşındasın, biz 40 yaşındayız, and so on.
Hamida, i use a computer and before i start a lesson i read the TIPS, before clicking on START. I have found the TIPS for numbers, going back on that skill: https://www.duolingo.com/skill/tr/Numbers/tips-and-notes. You go on the tree and open each skill to revise the grammar rules. However, somes of them do not have TIPS....