Question:
Why for zimun to say Elokeinu you need 7 of 10 to eat bread and for shmonei Esrah only 6 is sufficient ,plus 4 more?
Answer:
This is a very good question. Rabeinu Yona (Brachos 48a) says that it is because by zimun we need, ruba d’minkira- a prominent majority, and when it is seven it is more than two thirds, however six is only a majority but it still isn’t a wide majority.
The question still remains though, why by zimun do we need 7 that ate bread and eating vegetables is not good enough, but by davening 6 that are davening is enough. Why do we need such a large majority here? I saw an idea a few seforim write, that zimun is different than davening, because when we answer for a zimun we say “boruh sheachalnu m’shelo”, and we are thanking the one who we ate from. The people that only ate vegetables didn’t really “eat”, therefore although they can still join, it has to only be a small minority. On the otherhand, by davening, since there is no factor that it looks like a lie, therefore a plain majority will suffice.
Best wishes
Sources:
O:CH 197-3, Chidushei Basra (on Mishna Berura 1-2) page 96, MIli Dbrachos pg. 353 D:H Shuv.
