Stuck converting recurring decimals with a non-repeating start into fractions

I’m working on converting recurring decimals into fractions, and I think I understand the basic idea, but I keep getting tangled when there’s a non-repeating part at the start.

For a simple one like x = 0.\overline{3}, I write 10x = 3.\overline{3}, so 10x − x = 3. That seems clean because the repeating tails line up and cancel. So far, so good.

Where I get stuck is something like x = 0.1\overline{8} (so 0.18888…). I tried:
– 10x = 1.\overline{8}
– 100x = 18.\overline{8}
If I do 100x − 10x, the decimals look like they cancel and I get 90x = 17. But I’m not 100% sure why 90 is the correct coefficient here (and not 99 or 9), or whether subtracting 100x − x would be wrong. I also wondered if I should split it first as 0.1 + 0.0\overline{8}, but that feels like I’m moving the repeating block to the wrong place. What’s the reliable way to set this up?

Another example giving me trouble is y = 2.\overline{054}. Here’s what I wrote:
– 1000y = 2054.\overline{054}
– y = 2.\overline{054}
Subtracting should give 999y = 2052 (or is it 2052.\overline{0}? I’m second-guessing the alignment). I also tried mixing different multiples (like 1000y − 2y) after thinking about shifting the non-repeating part, but then I get a different integer on the right. Clearly I’m not lining things up consistently.

For z = 0.\overline{36}, I wrote 100z = 36.\overline{36}, so 99z = 36, which feels straightforward. But what if I accidentally pick a longer block than necessary? For example, if I multiply by 1000 and also by 10, I get 1000z − 10z = 990z, and the subtraction 363.\overline{6} − 3.\overline{6} looks like it gives 360. That seems to still work, just with bigger numbers. Is this always guaranteed to produce the same final fraction, even if I choose a longer repeating block than needed? How do I know the repeating block I chose is actually minimal?

Could someone please spell out the precise, step-by-step rule for which powers of 10 to use before subtracting when there’s a non-repeating start followed by a repeating block? I’d really like a principle I can apply mechanically so the tails always cancel cleanly without off-by-one mistakes.

Follow-up: Is there a general formula in terms of m = length of the non-repeating part and n = length of the repeating block that I can use to check my setup? And once I get a fraction, is there a quick way to predict the length of the repeating part when converting back to a decimal (or at least verify that the repeating block I picked was minimal)?