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Both mean 'both' but båda is the pronoun 'both' as in vi båda 'the both of us' or båda är bra 'both are good'. både is the conjunction 'both' and is only ever used together with an och. E.g. både du och jag 'both you and me', hon kan både sjunga och spela 'she can both sing and play'.
The two of us both heard the first word begin with a "B" at normal speed but a "G" when played slowly. Anyone have a theory as to why?