Final answer:
The British did not want to give representation to their colonists in the Americas due to the concept of virtual representation, in which the members of Parliament believed they represented the interests of all groups including the colonies. However, the colonists argued for actual representation to protect their rights and interests.
Step-by-step explanation:
The British did not want to give their colonists in the Americas representation in Parliament because they believed in the concept of virtual representation. According to this theory, the members of Parliament represented the interests of all groups in England and the British colonies, even if the colonists did not directly elect them. The colonists, on the other hand, argued for actual representation, where they would have elected representatives in Parliament to protect their rights and interests. They believed that being taxed by representatives they had not voted for was a violation of their rights as English subjects.