Most styles of any art will have a prescribed syllabus which lays out what you need to know and/or achieve for each ranking, and they will also usually contain a suggested time period to be spent at each rank level.
Note the word suggested - an instructor has the right to advance students or invite them to attend gradings as he sees fit. It is quite common for some students to skip through the lower ranks if they learn fast or have previous experience. Some instructors may also allow you to double-grade at lower ranks, i.e. you skip the yellow tip and go straight to yellow belt.
In summary, it's at the instructor's discretion. Provided that they apply the rules evenly and fairly then there is no problem when some students get advanced faster than others.
If I keep working about that hard, how long should it take to get a black belt?
We can't answer that. But you should remeberremember that a black belt is just another rank, you still have plenty of learning to do after that. By all means treat it as a target, but don't treat it as the end of the journey.