Does voller end in er because it is dativ and Leute is a plural noun?
No, I believe it’s “voller” here because of genitiv case. In English, we’d say the room is full of people, the “of” there is accomplishing the same thing in English as what genitiv case does in German.
This is extra confusing here because “voller” with the same spelling and same pronunciation could also serve as the comparative form: “dieses Zimmer ist voller als das andere” (this room is fuller than the other one).
These two usages are not the same.
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