Reference for Menachot 100:8

נטמא מחצה של בין הערבים או שאבד יכול יביא חצי עשרון מביתו ערבית ויקריב ת"ל מחציתה בבקר ומחציתה בערב מחצה משלם הוא מביא

Thus he must bring [another] whole tenth and divide it, offering one half and leaving the other half to perish; and so the result is that two halves are offered and two halves are left t perish. Where the High Priest that offered the half in the morning died and they appointed another High Priest in his place, I might say that he may bring a half-tenth from his house or that he may use the remaining half-tenth of the first [High Priest].

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