Let's try this:
Suppose I tell you that I have a secret number. When I increase that number by 10%, the result is 550. You want to find my number.
Following your method, you would subtract 10% of 550, which is 55, and tell me that my number was 550 - 55 = 495.
But if you check that, you find that 495, increased by 10%, is 495 + 49.5 = 544.5, so that can't be my secret number!
Instead, you can recognize that by adding 10% to my number, I am adding 100% and 10%, so that the result is 110% of my number, that is, 1.1 times my number. Then you can find my number by undoing that multiplication: 550/1.1 = 500.
And that checks out: If you increase 500 by 10%, you get 500 + 50 = 550.
Do you see that this way works, and the other doesn't?
The reason is that the 10% that was added is not 10% of the 550 I got after the increase, but 10% of the original number, 500. So subtracting 55 was wrong.
In the sales tax case, similarly, the 10% tax is 10% of the before-tax amount, not 10% of the 660. Your method subtracts too much.