The u.s. senate failed to pass the balanced budget amendment in 1996 and 1997. if the effort had been successful, what steps would have been necessary for this proposal to become a permanent amendment to the constitution?

The correct answer is that: Two-thirds of both houses have to vote to propose an amendment and three-fourths of the states need to ratify the proposal.
For a proposal to become a permanent amendment to the constitution it has to be voted by two thirds of Congress and ratified by three-fourths of the states.

  First, if it was passed on the senate with two-thirds majority votes it goes to the House of Representatives where it must be passed with the same amount of votes.

  When Congress successfully passed the amendment it proposes in the form of a joint resolution and forwarded directly to the National Archives and Records Administration (NARA).

  Then the NARA submit the joint resolution to the governors of all the States for their consideration. Once they receive it they formally submit the amendment to the state legislatures for ratification or negation.

  Finally, if three-fourths of the States (38 of 50) ratified the amendment it became part of the Constitution.

  I hope this answer helps you

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