Sims 3 torrent expansions not working with authentic Sims 3 game?

I have a hard copy of Sims 3 which works fine (I upgraded to a Mac recently so had to get a new serial for my game but otherwise there's no probs with it, be it software or disc issues. All 100% perfect. I did, however, lose my Sims 3 Ambitions disc recently - this was the original reason I downloaded "The Sims 3: Ambitions [FULL] * Games4theworld *".). I downloaded the .rar files of several other Sims 3 expansions as per instructions, followed the instructions to make them .dmg files, but the installer won't open the installer from the .dmg and most of the instructions fall back to a cracked Sims 3 that I should have but didn't download since I have the real thing.

I've been getting multiple issues trying to open the files (even with The Unarchiver app, which refuses to open half the damn files and claims it can't open certain extensions even though the extensions are in the app's list of extractable extensions.)

Does anyone know if there's anyway I can get the expansions to install for use with a real copy of Sims 3 and not a cracked version? All I really wanted was just a replacement of my lost disc without having to buy it again and now I've spent days downloading torrents and even making my Sims 3 glitch a couple of times. =/