What are the three ways the constitution can be amended?

1. proposed by three-fourths of both houses and ratified by two-thirds of the states
2. proposed by two-thirds of both houses and ratified by three-fourths of the states
3. ratified by special conventions in three-fourths of the states
4. ratified by special conventions in one-half of the states
5. proposed by special convention called by two-thirds of state legislatures and ratified by three-fourths of state legislatures

*The constitution can be amended when the amendments are proposed by two-thirds of both houses and ratified by three-fourths of the states.

*ratified by special conventions in three-fourths of the states

*proposed by special convention called by two-thirds of state legislatures and ratified by three-fourths of state legislatures.

Explanation:

A convention to propose amendments to the United States Constitution, also called an Article V Convention or amendments convention. This convention is called for by two-thirds of the state legislatures and it is one of two processes authorized by Article Five of the United States Constitution whereby the United States Constitution may be altered. Amendments may also be proposed by the Congress with a two-thirds vote in both the House of Representatives and the Senate.  

To become part of the Constitution, an amendment must be ratified by either the legislatures of three-fourths  of the states, or state ratifying conventions in three-fourths of the states.