As we have learned, the laws of ḥametz are uniquely strict: even a miniscule amount of ḥametz mixed with a permitted food renders the entire mixture forbidden for consumption or benefit. However, most poskim maintain that if the ḥametz is less than one sixtieth of the mixture, one may salvage its monetary value by selling it to a gentile. For example, if a kilogram of ḥametz falls into a metric ton of another food, one should throw away one kilogram, so as to avoid benefiting from the added ḥametz, and sell the rest to a gentile. This is because when the Sages forbade benefiting from such mixtures, their intention was to forbid benefiting from the ḥametz. Thus, if one disposes of a quantity of the mixture equaling the amount of ḥametz that was added, he does not benefit from it, and he may then sell the mixture to a gentile. If a single wheat grain of ḥametz fell into a large amount of cooked food, it all becomes forbidden to eat, and as long as it remains in a Jew’s hands, it is also forbidden to derive benefit from it. However, one may sell it to a gentile. It is not necessary to dispose of any of the mixture since the wheat did not cause the price to rise (SA 467:10).
Rema (447:1), however, rules stringently in accordance with the view of the few Rishonim who maintain that since it is forbidden to derive benefit from the mixture, it is likewise forbidden to sell it to a gentile. Rather, the entire mixture must be burned. This is the practice of Ashkenazim. However, in a case of a very large loss, even Ashkenazic custom relies upon the view of those who permit selling the mixture to a gentile (MB 447:3).
However, according to Rif and Rosh, if one would discard the value of the ḥametz in the mixture, he would be permitted to sell the mixture to a gentile, since he obtains no benefit from the ḥametz; rather, he merely receives payment for the portion of the mixture that is not ḥametz. The overwhelming majority of poskim agree, and so states SA 467:10. However, Rema writes in Darkhei Moshe 447:2 that Mordechai, Terumat Ha-deshen, and Mahari Brin all adopt a stringent approach – that the entire mixture must be burned. MB 447:3 states in the name of Aḥaronim that one should sell it to a gentile. Moreover, SHT 467:74 states in the name of Beit Meir that if one will suffer a very large monetary loss despite selling the mixture to a gentile, he may even keep the mixture until after Pesaḥ, and then eat it or sell it to a Jew.