In the countryside Syria if you go to someone house during meals times they will invite you and if you don't eat they will consider it as an insult or they might think you are a stingy because there is a saying in Syria that says "Generous people eat when they are invited." On the other hand, if you go in any time they will offer you tea or coffee and if you don't drink then it is an insult. In the cities, it is different you can appologise. 

In our country it is a culture that whenever you are eating and someone is around you. You have to invite them to eat with you or offer them your food even if the food is only good for one person but if you are the person who was been invited to eat and you see that the food is only enough for him or u dont know that person it is your initiative to reply " It's Okay Thanks" or " No Thanks I'm full I justeat awhile ago." ( though u r not really full and u did not eat awhile ago.)

In some countries if you do not share there food with them then they will consider it a kind of insult. I think it depend on the countries. 

Russians call this 'hospitality' and treat with the best  food they prepare in advance if  the guest was  invited, and if a visitor  is unexpected they offer something  as well.

hi dein :) i think that it's an cultural issue here in the arab worlds we prefare to share our food with outhers specially our visitors , we consider it as a kind of generousity that people should have :)

but in forign countries it's not like this at all :)