How did the Great Compromise and the Three-Fifths Compromise help the convention reach agreements?

Prior to the Great Compromise, there were ongoing discussions about how representation would happen for the new Congress. The Virginia Plan proposed 2 houses based on population, which would favor those states with large populations. Small states were concerned that this system would give them very little say, and so there came another proposal, the New Jersey plan: a single house with equal representation.

Neither proposal made both sides satisfied, and so they borrowed a little from each idea in the Great Compromise to make the system we still have today: 2 houses, with one based on population and one based on equal representation.

Slavery added in another level to the representation debate. Southern states, having large slave populations, wanted to count slaves for representation in Congress which would give them more seats, and therefore more power. The North (which had far less slaves) did not like this idea, but did think that those southern states should pay taxes for their large slave populations. The resulting compromise counted 3 out of 5 slaves for representation and taxation purposes.