Rates above $50 might seem like a lot. I would argue that they are not if this is your primary business and you are treating it like one. I use to charge less but found that it wasn't worth my time and I could easily make more money teaching. So the solution, at least for me, was to charge a rate that would be sustainable. Meaning that I could support myself by earning equivalent income even with the added expenses of being self employed. I get that in different parts of the world that amount would be different.
In my opinion, here are the biggest things to consider as a private tutor:
Time: So after transportation my one hour session was at least 1.5 hours, sometimes 2 hours if there's prep invovled. There's also billing and bookkeeping that takes time.
Expenses: Mileage .55/mi, space $3/coffee, paper, calculators etc $100/year, website.... I also needed to think about health insurance now that I don't have a W2 job
Taxes: business taxes ($200), income (25%)
Limited SupplyThere's only a few prime tutoring hours in the day 3:30-6pm in which you can meet 2 students, maybe more with a home school student. So there's a bit of supply and demand in effect, you have limited time and everyone wants the same time slots.
Variability: Clients cancel, go on vacation, summer happens...