Rada

Rada literally means "she who is happy" in Russian, nothing to do with care.
Pronounced RAH dah, accent on first syllable, sounds much softer in Russian then in English. [noted -ed]

"Rada" also means "happy" in Russian.

Other names with the root "rad", which are used in Bulgaria, are the masculine "Radoslav" (the stress is on the 2nd "a"; the root "rad" and the root "slav" - the root of "glory"; there is also a masculine name "Slav"); its feminine form "Radoslava" (the stress is on the 2nd "a"; there is also a feminine name "Slava"' "slava" means "glory" in Bulgarian); the masculine name "Radostin" (the stress is on the "i"; "radost" means "joy", "gladness" in Bulgarian); its feminine form "Radostina" (the stress is on the "i"); the feminine name "Radost"; the masculine name "Radomir" (the stress is on the "i"; "mir" means "peace" in Bulgarian and it's the root of "peace"); its feminine form "Radomira" (the stress is on the "i"; there is also a feminine name "Mira"); the masculine name "Radoi" (the stress is on the diphthong "oi").

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Also the masculine name "Radko", its feminine form "Radka" and the name "Rado", which is the masculine form of "Rada".

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And the feminine name "Radina" (the stress is on the "i") and its masculine form "Radin" (the stress is on the "i").

Sometimes used as a short form of Radmila.

Also a Bulgarian name, a very common at that. I have never met a Russian named Rada, but plenty of Bulgarians.