Final answer:
Sharecroppers indeed paid their rent in shares of crops, making the statement true. However, the claim that Proprietors in a proprietary colony had no responsibilities beyond collecting profits is false; they had administrative and governance duties as well.
Step-by-step explanation:
The statement that sharecroppers were tenant farmers who paid their rent with shares of their crops is true. Sharecropping was a system that became common in the Southern United States after the Civil War, where the landowner allowed a tenant to use the land in return for a share of the crops produced on that land.
Regarding the running of a proprietary colony, the statement that Proprietors have no responsibilities except to collect the profits is false. In a proprietary colony, the Proprietors were granted full sovereign authority by the king and were responsible for the administration and governance of the colony, not merely for collecting profits.