Quotes in my city run from £25 to £70 an hour — does the price track the teaching at all?

I have paid at three points on that scale over about a decade. The cheapest was a lovely person teaching children, and everything I asked outside the exam syllabus got a vague answer. The middle was the best value by miles: an active performer who taught adults deliberately and had actual opinions about how to practise. The most expensive was a genuinely brilliant player who was not a very good teacher, which is a combination that exists more often than people admit. Price correlates with playing ability far more than with teaching ability, and only one of those is the thing you are buying.

Bear in mind rates vary enormously by country and city, so the numbers you get in a thread like this are not really comparable. What is comparable is the ratio: locally, the difference between the bottom and top of the range is usually about a factor of two and a half, and the teaching quality does not vary in the same order.

I paid at the top of my local range for two years and it was worth every penny, but for a specific reason: I brought a recording to every lesson and he took the practice process apart rather than the piece. If you are going to turn up and be taught the piece, you can get that cheaper. If you want someone to change how you work between lessons, that is what the money buys.

Online lessons broke the local price ceiling for me. I pay about a third less than the in-person rate here and my teacher is in another country entirely, which is only possible because I already had the basics and did not need someone physically moving my hand. Latency is a nuisance for anything played together, so it does not suit everyone. For technique and interpretation it has been completely fine.

Ask a different question of each of them: how do you teach an adult who wants to play well but will never take an exam. The cheap children's-grades teacher will struggle with it, and the audition teacher may not be interested. The answer tells you far more than the rate does. Also ask whether they teach any adults currently, and how many. Anyone with a real adult roster has already solved the problems you are about to have.

Fortnightly with the good teacher beat weekly with the cheap one, at the same monthly spend. Same money, twice the progress. That was the entire lesson for me.