How to prove that $y = 0,273273273...,$ is a rational number?

Question

How to prove that $$y = 0.273273273...$$ is a rational number?

I don't have any experience with proofs... Can I get your help and your advice?

Answer 1

Since $y$ has a repeating cycle of $3$ digits, just write: $\;1000y=273+y$, solve for $y$, and simplify.

This procedure, though intuitively clear, is not rigourously justified, but it can easily be justified with the interpretation of the positional notation for numbers via a geometric series.

Answer 2

$1000y = 273.273273...$. Therefore, $1000y-y = 273$. Solving for $y$, $y = \frac{273}{999}$.