We learned on yesterday’s daf that when something non-kosher falls into a pot with kosher food, the mixture will be permitted if the kosher food is at least 60 times the volume of the non-kosher food.
The Gemara on today’s daf tells stories that appear to raise doubts about that ruling. The Gemara relates:
A certain man came before Rabban Gamliel the son of Rabbi with his case. Said Rabban Gamliel: A similar case came before my father where the mixture measured only forty-seven times as much kosher food as non-kosher, and he permitted it. Similarly I will permit with a measure of forty-five-fold.
A certain man came before Rabbi Shimon the son of Rabbi with his case. Said Rabbi Shimon: A similar case came before my father where the mixture measured only forty-five times as much kosher food as non-kosher, and he permitted it. Similarly I will permit the mixture with a measure of forty-three times.
Rashi offers two different explanations for these stories.
According to his first approach, these Sages permitted the mixture even though there was less than the 60 times that was required, since the amount of prohibited material in the mixture was only half the size of an olive. As we have learned (see above, daf 96), ordinarily only eating a full ka-zayit will be considered significant enough for someone to be held liable. Here, too, given the miniscule amount of prohibited material in the mixture, a smaller ratio could be accepted to nullify it (even though half of a ka-zayit remains forbidden on its own on a Biblical level).
Rashi’s second approach suggests that the ruling offered was said rhetorically. Upon being asked the question whether a ratio of 45:1 would suffice to nullify the prohibited food in the mixture Rabban Gamliel responded “My father did not accept 47:1, how could I accept a smaller ratio!?” According to this explanation, these stories reinforce the requirement that we need a minimum amount of 60 times kosher food to permit a mixture.