What was the reasoning behind the addition of the 3/5 Compromise? What did the South want and why?

The Three-Fifths Compromise seemed to guarantee that the South would be strongly represented in the House of Representatives and would have disproportionate power in electing Presidents.

Southern states had wanted representation apportioned by population

Explanation:

a social contract.

"a social contract is an agreement between citizens and their leader where the citizens give up certain freedoms in exchange for protection. the constitution of the united states is an example of a social contract."