The Torah in Parashat Shelach (15:17-21) introduces the mitzva of challa, which requires removing a piece of dough from one’s batter and giving it to a kohen. The portion given to a kohen must be treated as teruma, which means it may be eaten only in a state of ritual purity, and it must not come in contact with ritual impurity. As these conditions cannot be met nowadays, the portion of dough removed and designated as challa is generally burned, or wrapped and discarded.
As challa shares the halakhic properties of teruma, it is subject to a unique stringency relevant to bittul – negation by the majority in a mixture. If a piece of dough designated as challa is mixed with regular dough, the entire mixture becomes forbidden unless the challa comprises a proportion of less than 1/100th. If the challa comprises 1/100th or more of the mixture, then the entire mixture has the status of challa and is thus forbidden for consumption. This ruling is codified and discussed by the Shulchan Arukh (Y.D. 323).
The Rama, citing earlier sources, writes that in such a case, there is a way of circumventing the prohibition and preserving the batter. Namely, one can perform hatarat nedarim – the annulment of his “vow” – to retroactively annul his designation of the portion of dough as challa. Just as one who takes a vow can, in most circumstances, have the vow annulled by approaching a scholar or Beit Din and explaining that he regrets taking the vow, similarly, one can annul his designation of a portion of food as teruma or challa through the same procedure. And thus if a portion of challa mixes with a large batter, one can avoid having to discard the entire batter by retroactively annulling his designation of that portion as challa. Of course, he must then remove a new portion to fulfill the challa obligation.
However, the Rama specifies that this solution is effective only if nobody had yet partaken of the mixture. Once the batter was baked and some of it was eaten, one may not have the challa designation annulled. The reason, as the Rama explains elsewhere (in Darkhei Moshe), is because if the designation is retroactively annulled after one has partaken of the food, then he will have ended up eating tevel – food from which the required tithes and other gifts were not all taken. Since the dough will now not have had challa removed, it retroactively assumes the status of tevel, which is forbidden for consumption. It is preferable to be in violation of eating a mixture containing challa than to be in violation of eating tevel, because, as the Shakh (323:6) explains, there are some halakhic authorities who permit a mixture containing a minority component of challa, even if the proportion exceeds 1:101. Although Halakha does not follow this view, nevertheless, the fact that such a position exists makes it preferable to eat a mixture containing a small quantity of challa than to be in violation of tevel.
Several centuries later, however, another halakhic scholar found a solution even for situations where one partook of the mixture before the challa designation was annulled. Rav Chaim Elazar Wachs, in his Nefesh Chaya (Y.D. 91), advises that instead of annulling altogether the designation of a portion of dough as challa, one can annul the designation of the vast majority of that portion as challa. The result of this annulment would be that most of the piece that had been designated as challa retroactively loses its status as challa, but a small percentage of that piece retains its challa status. Therefore, since a small portion of the dough has, even in the end, been designated as challa, the batter is not considered tevel. And, since the portion designated as challa is minuscule, it certainly comprises less than 1/100th of the mixture, and so the entire mixture is permissible for consumption.