If three people ate together – regardless of whether they began eating together, or if two began eating together and were then joined by a third – they are required to recite a Zimun before Birkat Ha’mazon. Zimun is so important that if two people ate together, it is a Misva for them to try to find a third person to join them in order to obligate themselves to recite a Zimun. The Mishna Berura says that even if seven people ate together, it is a Misva for them to try to find three more people, so they can recite Zimun with Hashem’s Name (which is included only if ten people ate together). If fewer than seven people ate together, there is no Misva to try to find more people so Hashem’s Name can be included.
A common case that arises is when three people ate together and one needs to leave before the other two have finished eating. Halacha does not require the other two people to stop eating to recite a Zimun, though clearly, if they agree, then a Zimun is recited and then the one who needs to leave may do so. The question becomes what this person should do if the other two refuse to interrupt their meal for a Zimun. Must he wait for them to finish eating, or may he leave, which will result in none of them being able to recite a Zimun?
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Korah, rules that this individual may not leave, regardless of the circumstances. Even if he will incur a financial loss by remaining – such as if he will miss time from work, or an important meeting – nevertheless, it is forbidden to leave without a Zimun once a Zimun is required.
Hacham David Yosef, however, in Halacha Berura, rules more leniently, writing that although this person must remain and wait for the other two to finish, nevertheless, he may leave if a financial loss is at stake. If he stands to lose money by remaining, he may recite Birkat Ha’mazon without a Zimun and then leave. Hacham David adds, parenthetically, that the other two have no right to protest the fact that the third is taking away their Misva of Zimun, since they had the option of interrupting their meal for just a few moments in order to recite a Zimun, which would have avoided this entire problem.
Preferably, Hacham David writes, if a person knows he will need to leave a meal early, he should sit down initially with the intention of leaving early, without a Zimun. According to some Halachic authorities, this intention absolves the group of the requirement of Zimun, since from the outset, they sat down knowing that one would leave before the others.
Of course, the simplest solution is for the other two to acquiesce and agree to interrupt their meal for the few moments needed to recite a Zimun.
Summary: If three people sit down to eat together, and one of them knows he will need to leave early, he should preferably have in mind from the outset that he will not be joining the others for a Zimun. If he did not, and the other two refuse to interrupt their meal for a Zimun, he must wait for them to finish eating, and he may not leave, unless this will cause him a financial loss, in which case he may recite Birkat Ha’mazon and leave.