
mehrheitlich
by majority
Supported by the majority of a group or applying to the largest part of something; for the most part.
Der Vorschlag wurde vom Ausschuss mehrheitlich angenommen.
The proposal was accepted by majority vote by the committee.
by majority / predominantly
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by majority
Supported by the majority of a group or applying to the largest part of something; for the most part.
Der Vorschlag wurde vom Ausschuss mehrheitlich angenommen.
The proposal was accepted by majority vote by the committee.

predominantly
For the most part; mainly. Describes that something consists mainly of a certain thing or is characterized by a certain quality.
Die Bevölkerung in dieser Region ist mehrheitlich katholisch.
The population in this region is predominantly Catholic.
The word is derived from 'die Mehrheit' (the majority). The ending '-lich' turns the noun into an adjective or adverb with the meaning 'like the majority' or 'for the most part'.
Die Mehrheit der Bürger hat für den Kandidaten gestimmt.
The majority of citizens voted for the candidate.
'Mehrheitlich' (by majority / predominantly) can be used as an adverb without an ending or as a declined adjective before a noun. As an adverb, it describes the manner of an action.
Die Aktionäre stimmten mehrheitlich zu.
The shareholders agreed by majority vote.
In the meaning 'von der Mehrheit' (by the majority), the word is often combined with verbs that express a decision or opinion: 'annehmen' (to accept), 'ablehnen' (to reject), 'beschließen' (to decide), or 'zustimmen' (to agree).
Der Antrag wurde vom Parlament mehrheitlich abgelehnt.
The motion was rejected by majority vote by parliament.
mainly
Hauptsächlich and mehrheitlich can both describe the largest part of a group or something as a whole. Hauptsächlich identifies the main focus, whereas mehrheitlich suggests a share of more than half. Use hauptsächlich for general weighting and mehrheitlich when the majority is explicitly important.

Das Publikum bestand mehrheitlich aus Familien, die deutlich mehr als die Hälfte ausmachten.
The audience consisted predominantly of families, who made up clearly more than half of it.
Das Publikum bestand hauptsächlich aus Familien, aber auch viele Einzelpersonen waren dabei.
The audience consisted mainly of families, but many individuals were also there.