Mishnah 2 in the fifth chapter of Maseches Terumos also deals with a mixture of medumah. The case: one se'ah of tamei Terumah - a unit that has already become tamei, whose only remaining benefit to the kohen is at most as a source of fuel - that fell into a hundred se'ah of tahor chullin. Since there are a hundred ordinary units against the one, the amount required for nullification, the main problem has apparently been resolved, and all that remains is a monetary question: returning to the kohanim the property that belongs to them, and giving back to the kohen one unit of Terumah.
Two further problems arise here:
'Laundering' the Terumah: this is the easier of the two problems. When the se'ah is returned to the kohen, he receives food that he is apparently permitted to eat. It turns out that the Terumah entered the mixture tamei and fit only for burning as fuel, and emerges from it as brand new Terumah fit to be eaten. Is it right to allow this?
The element of tumah embedded in the mixture: ultimately, what comes out of the mixture contains within it a certain element of tumah. True, the laws of nullification establish that when there is a majority, or the amount required for nullification, the status of the entire mixture follows the majority, and it is tahor. But there is no denying that the tamei element exists and is embedded within it, and there is reason to be concerned that under certain circumstances the tumah will reawaken and begin to spread.
The view of Rabbi Eliezer:
The Mishnah says: "Rabbi Eliezer omer: terum v'tisaref" - Rabbi Eliezer says: it should be taken out and burned. We remove that one unit that fell into the mixture and return it to the kohen, since there is nothing to be gained by withholding money from the kohen. But what had served as a source of fuel is now returned to him only to be burned.
The word "tisaref" needs precision: in the context of Terumah it usually means burning while deriving benefit, such as using it as fuel, whereas in other places it means burning with no benefit at all. The Bartenura explains explicitly that here, according to Rabbi Eliezer, the obligation to burn is without benefit, and this is equivalent to "let it rot." The Terumah is removed from the mixture so that the farmer will have no financial incentive in its falling in - that he should profit and receive more than he had at the outset - but the kohen may not derive benefit from it.
Rabbi Eliezer's reasoning is stated explicitly in the Mishnah: "she'ani omer: se'ah shenaflah hi se'ah she'altah" - for I say: the se'ah that fell in is the se'ah that came up. It is as though that very unit that fell in is the one being returned to the kohen. Of course he does not mean this literally; Rabbi Eliezer knows that less than one percent of the mixture is Terumah, and no miracle occurred whereby precisely the grains of Terumah were removed. Rather, he wishes to treat the matter stringently, and to assume that it is all still Terumah and that all the Terumah is tamei. And although a kohen may generally burn tamei Terumah and derive benefit from it, here Rabbi Eliezer is more stringent: since people are liable not to take the status of the Terumah seriously, the kohen might even come to eat it in its tamei state. Rabbi Eliezer therefore requires burning with no benefit derived at all.
The view of the Chachamim, which is the halachah:
The Chachamim are far more lenient and say "ta'aleh" - it should be brought up. One se'ah is removed and given to the kohen, and the kohen may eat it and not merely burn it. It turns out that the Terumah has indeed been 'laundered': it entered tamei and emerges fit to be eaten. Nevertheless, the concern remains that particles of tumah are present within it, and under the wrong conditions this tumah could spread.
The two conditions for tumah to spread through food:
Hechsher: in order for food to be susceptible to tumah, it must first be prepared for that by water or by one of the seven liquids that render food susceptible to tumah.
The volume of a k'beitzah: in order to transmit tumah, the food must have at least the volume of an egg.
One point deserves emphasis because it is not intuitive: even though nullification has taken place here, and the halachic status of the entire mixture is tahor, we are aware that tamei components are absorbed within it, and there is a possibility of 'chozer v'ni'or' - that the very minority we are halachically ignoring will reawaken and become halachically significant later on. Therefore, as in the previous Mishnah, care must be taken that the tumah not be able to spread: either by keeping everything dry, or by keeping it in units of less than a k'beitzah.
The se'ah is given to the kohen in full, and he is told: this is your produce, and you may eat it as Terumah; but since there is a concern of tumah here, and it must not be allowed to mix with the rest of his pure foods, you must make sure not to spread the tumah hidden within it. Therefore the Mishnah says: "tei'achel nikudim, o kelayos, o tilosh b'mei peiros, o tis'chalek l'isos, u'vilvad shelo yehei b'makom echad k'beitzah" - it should be eaten as nikudim, or as roasted kernels, or kneaded with fruit juice, or divided among several doughs, provided that there is not the volume of an egg in any one place:
"nikudim" - that it be eaten dry. According to other explanations, and this is also the view of the Rambam, 'nikudim' means small portions of less than the volume of an egg.
"o kelayos" - roasted, like roasted wheat.
"o tilosh b'mei peiros" - one may even knead dough from it, provided that the liquid used for the kneading is not actual water but fruit juice, which does not enable food to become susceptible to tumah.
"o tis'chalek l'isos" - to divide the se'ah among many doughs, so that no single dough contains a full egg's volume of what might be impure in the mixture.
The calculation itself is more refined: in the se'ah removed from the mixture there is only about one percent of tumah against ninety nine percent of pure food - in fact one part in a hundred and one, and for the sake of a round number we say one percent. It is therefore enough to be careful not to put more than a hundred and one eggs' volume of this flour into a single dough, in which everything could join together and transmit tumah. As long as it is used in portions smaller than that, no single dough contains the measure of one egg of impure particles that could combine and transmit tumah. It is worth remembering that a se'ah is a large measure - one hundred and forty four eggs.
In summary: In this Mishnah, Rabbi Eliezer and the Sages disagree about a se'ah of impure Terumah that fell into a hundred se'ah of pure unconsecrated produce. Rabbi Eliezer is stringent and says "terum v'tisaref" - it should be lifted out and burned, since "the se'ah that fell is the se'ah that came up," and no benefit at all may be derived from it. The Sages are lenient and say "ta'aleh" - it is lifted out and the mixture is permitted, and the kohen may even eat it, provided that he guards against spreading the tumah hidden in the mixture - by eating it as nikudim, as roasted kernels, by kneading it with fruit juice, or by dividing it among doughs none of which contains an egg's volume. The halachah follows the Sages.