What is the most common process used in amending the constitution

The most common method used to add amendments to the Constitution is the first method.  An Amendment may be proposed by two- thirds vote in each House of Congress and ratified by three- fourths of State Legislators.

Article 5 gives the process of amending the US Constitution.  Both the Senate and the House of Representatives, in the US Congress, approve by a 2/3 majority voting.  This is a joint resolution that amends the Constitution.  By this the President's signature is not required.

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