Commentary for Berakhot 78:5
היכא דאיכא כהן כולי עלמא לא פליגי דחשוב עדיף כי פליגי דליכא כהן דתנן כל מקום שיש כהן תורם מן היפה וכל מקום שאין כהן תורם מן המתקיים ר' יהודה אומר אין תורם אלא מן היפה
Where there is a priest present, there is no difference of opinion that what is of greater value is to be preferred, the disagreement arising where there is no priest. For there is a Mishnaic teaching : Wherever there is a priest present, the Terumah is to be taken from the best, but where there is no priest, it is to be taken from that which is most durable ; R. Judah says : It is only to be taken from the best.
Abraham Cohen Footnotes to the English Translation of Masechet Berakhot
To receive the Terumah.
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Abraham Cohen Footnotes to the English Translation of Masechet Berakhot
In that case, the Terumah would have to be stored until it could be handed to a priest, and the half-onion would decay more quickly than a whole one.
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