Answer
The important thing to note is that these abbreviations are much more common in handwritten correspondence than they are online. c/o is often used when addressing post to someone via a third party, and w/ and w/o are common written shorthand for with and without.
It was quite common in older written texts to abbreviate words using some identifying letters and a line, for example:
- w— for with
- D—r for Doctor, which eventually became just Dr.
- Rev—d for Reverend, which became Revd or Rev.
These are part of a long history of written abbreviations that are just a bit awkward in print. I would guess that printing a slash in these abbreviations arose simply as a way of representing the shorthand that people had already been using in writing for ages. Using a diagonal slash-like mark (c/o) is just narrower and faster to write than something like c—o—, which looks awkward.
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