What was the goal behind the 9 state approval of the United States constitution?

Answer 1

The short answer is that by having nine states the Constitution would guarantee a majority of the citizens of the United States would be agreeing to it. So any combination of nine states would add up to over 50 per cent of the population.

They also required more than a simple majority (e.g. 7) so that states of varying interests would be satisfied. The biggest conflict at that point of our history was between large states and small states, this was also the reason for two bodies of legislature. One of those bodies was by population (House) the other was two per state (Senate).

Lastly, the framers argued that they did not need all thirteen states (although all thirteen states eventually did ratify) because in the declaration of independence the thirteen states presented themselves as one people, not thirteen individual peoples.

For more information, suggested reading would include The Federalist Paper and "Miracle at Philadelphia" by Catherine Drinker Bowen. Or if you are more of a visual learner: https://www.youtube.com/watch?v=bO7FQsCcbD8&list=PL8dPuuaLjXtMwmepBjTSG593eG7ObzO7s&index=8