The Constitution solved the large state versus small state problem by establishing a bicameral legislature, with all states being equal in the higher house and small states having an edge in the lower house. It also helped by establishing the electoral college for electing presidents.
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The Great Compromise handled the problems between the large states and the small states in the Constitutional Convention.
Large and small states
The Compromise called for Congress to consist of two houses instead of just one. One would give each state the same number of votes (Senate) the other would have votes determined by population. This tended to even out the voting strengths between the large and small states so that the larger states would not be able to impose their interests on the small states.
The most serious disagreement in the debate between large and small states at the Constitutional Convention was the issue of representation. Small states feared that their voices would be lost if representatives were chosen based on population, while big states didn't think it was fair that the small states would have as much influence as they had.
The Supremacy Clause grants supremacy to the United States Federal Government in any conflict between state and federal law. However, since the Federal Government has a limited mandate, the States still retain a large number of rights.
The Great Compromise, or the Connecticut Compromise, pleased both the small states and the large states because it gave them both the representation they wanted. In the lower house (the House of Representatives), the small states got the proportional representation they wanted. In the upper house (the Senate), the large states got the equal representation they wanted with every state getting two senators.