It's not so much the time, as what you do with it which matters, and everyone has a different make-up and will find different aspects of the skill mix you need to acquire easier or harder to manage, so you really can't put a timescale on it at all. Even the guesstimate that most will do a grade a year is totally unhelpful - and doesn't say at all much about how levels are being achieved. You can do a grade by learning your three or four pieces, and the scales, and doing aural tests/sightreading for each level on a very superficial level, scrape through, and embark on the next - gets the bits of paper, but isn't really a benchmark for where you're at...or you can explore the repertoire at each level, do some duetting and ensemble playing and expand your musical education in all sorts of ways. It depends what you want out of music too.
My music is something I do for pleasure and for my sanity - it gets me release from the humdrum side of life, and I practise intensively but often in very short bursts, sometimes only ten minutes in one go, sometimes an hour and occasionally even more than that. I might work on two bars, I might play something related to what I'm working on, or perhaps do a bit of playing through with my husband accompanying. I rarely get more than four or five hours' worth in a week, but it has been enough to sustain a good level of progress, even now, when I have done my grade eight and a couple of exams after that as well..
I am glad though that you're going to get a teacher - for me this has made the difference between playing notes and making real music

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