TITLE: Bad Fraction Reduction That Actually Works
QUESTION [45 upvotes]: $$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$
This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there are other such examples.  Is there a systematic way to generate examples of this kind of bad fraction reduction?

REPLY [4 votes]: T.J. Osler calls these "lucky fractions" in his article; in particular, he shows how other lucky fractions can be constructed from known lucky fractions through their "repetition properties".
To use Isaac's example as the demonstration of the two repetition properties:
$$\frac{16}{64}=\frac{166}{664}=\frac{1666}{6664}=\cdots$$
$$\frac{16}{64}=\frac{1616}{6464}=\frac{161616}{646464}=\cdots$$
Osler also gives a pile of lucky fractions found through a computer search. See also this article by Ralph Boas and this article on the more general concept of a "Smarandache Lucky Method".