In Congress, each state gets two Senators in the Senate and Representatives in the House of Representatives based on population. Why would this aspect of the USA government make the North and South cautious of a new territory becoming a state?

They were worried about overrepresentation.

Explanation:

The North and the South were both anxious about adding new states on either side in worry that this would disrupt the equal balance of Northern and Southern Senators. Somewhat of a modern example would be the Republican worry of D.C. statehood overrepresenting Democrats.

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