Challah, Chapter 3, Mishnah 9. In the previous mishnah we dealt with a mixture of leaven from which challah had not been separated, which came from a dough that had not had challah taken from it and became mixed into a dough from which challah had already been separated, and with the question of how challah is separated from the small quantity that fell in and became mixed. Our mishnah continues along the same line, and discusses a mixture of produce that is obligated in Terumah and tithes with produce that is not obligated in them.
The cases in the mishnah:
Olives: Olives picked by the owner, which are obligated in tithes, that became mixed with "zeisei nikuf" - olives gathered by the poor as part of the gifts to the poor, which are not obligated in Terumah and tithes. The word "nikuf" means striking, since the poor would beat the tree and knock down the olives left on it; and in general the term refers to olives exempt from Terumah and tithes.
Grapes: Grapes harvested by the owner, which are obligated in Terumah and tithes, that became mixed with "olelos" - one of the gifts to the poor, grapes that do not grow in a cluster.
How does one separate from such a mixture?
He has a supply from elsewhere: If he has other tevel in his possession that is obligated in Terumah and tithes, he separates from it on behalf of the portion of the mixture that is obligated in Terumah and tithes, according to the proportion of the mixture that is obligated.
He has no supply from elsewhere: He separates Terumah and terumas maaser from the entire mixture and gives them to a kohen, while the remaining gifts - "hamaaser umaaser sheini lefi cheshbon" - the tithe and the second tithe are given only according to the proportion of the mixture that is obligated in Terumah and tithes.
It emerges that the stringency applies specifically to Terumah and terumas maaser. In order to take terumas maaser, one must first separate maaser, ten percent of the entire mixture, and from it a tenth is given to the kohen. However, there is no obligation to give the Levi that same tithe itself, the full ten percent of the entire mixture, but only according to the calculation.
In other words, the stringency concerns giving to the party fit to receive - the kohen - with regard to Terumah and terumas maaser, which must be separated from the entire mixture: the amount that the whole mixture would have been obligated in as Terumah, say one fiftieth of the entire mixture, and likewise one percent which is terumas maaser. The tithe and the second tithe, by contrast, are given according to the mixed-in quantity that is obligated in tithes alone, and not according to the entire quantity.
In summary: In this mishnah we learned the law of a mixture of produce obligated in Terumah and tithes with produce of the gifts to the poor that is exempt from them - zeisei nikuf and olelos. If he has a supply from elsewhere, he separates on their behalf according to the calculation; and if not, he separates Terumah and terumas maaser from the entire mixture and gives them to a kohen, while the tithe and the second tithe are given according to the proportion of the mixture that is obligated.