Why did the colonists and british fight in the american revolution?

The colonists thought they were being taxed for problems that they did not create, and in wars that they did not participate in. They thought George III was a tyrant, which he was.  Their reasons are clearly laid out in the declaration of Independence. They have (I would say) about 2 dozen grievances and their attitude was enough was enough.

The British fought because their empire was being threatened. They were losing colonies which they thought were rightfully theirs. They came very close (in the beginning) to winning. Maybe George was a little insane, but he was their king and the government flowed from him.

in what process is there a multiple choice answer?

the density would be a smaller number.