Answer:
About 3 years...longer if you're younger.
It takes as long as it takes. You can't put a date on earning a black belt, because everyone is different. In most traditional Okinawan styles of karate it takes around 4 to 5 years.
If a place guarantees you a black belt in three years, run, don't walk, away!
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I began learning TKD when I was 10 years old and I achieved my blackbelt in four years. However all the other blackbelts I know, who are all older than me, have taken longer to get their blackbelts. It depends how focused and determined you are, not to mention how seriously blackbelt is your goal and where it stands on a list of your life priorities. As a young person it was top on my list for the entire four years, as I had no job or important exams to worry about yet at that time.
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This is almost impossible to answer as it depends on how often the student trains, how hard they train and how quickly they learn. Students who are more dedicated and train hard will obviously achieve their black belt before other less devoted students but to put a time span on it is difficult. I would say that even the most dedicated of students should train at least three years before being allowed to Dan Grade
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I've been taking karate for 13 years. I started when I was 2 and I got my black belt when I was 11. I'm not saying it's gonna take you 9 years, but it is going to take a while. It also depends on the DoJo, not all DoJo's are going to give 11 year olds black belts and say they can teach, it takes time. Also, it's going to take a long time and a lot of tests to get Sensei, there's a lot of Que's and it takes time to get through them all.