How often are vulgar fractions used in mathematical writing? In (American) primary school, they first teach you how to reckon with simple fractions in numerator-over-denominator "vulgar" form, often using the image of a pie as an illustration. They teach you how to reduce fractions. They teach you about fractions like $3/2$ where the numerator is higher than the denominator, but call these "improper" and require that they should be reduced to to a mixed number such as $1½$. They then forget about all of the above and teach you decimal fractions, which you then use more or less exclusively for the remainder of your primary and secondary education.
How often are vulgar rather than decimal fractions used in modern mathematical writing? Obviously they're often used to represent mathematical expressions involving variables—$\frac{x}{2}$ seems preferred to $0.5x$—but how often will you see simple rational constants like one-half represented in the form of a numerator over a denominator? Are mixed numbers ever used?
As far as I can tell, there are three ways to render a vulgar fraction: $\frac{1}{2}$, $½$, and $1/2$. I don't know what these are properly called, so I'm going to call them "vertical form", "slanted form", and "solidus form". Is there any dominant preference between these forms, and if so, how strong of a preference?
I personally like vulgar fractions because I feel they more strongly suggest precise ratios where a decimal fraction hints that a quantity may have been rounded or truncated. I don't mind the solidus form, but to me it suggests integer division, possibly because that's what it often means in various programming languages. Are these impressions at all in line with actual mathematical usage?
If I submitted an article to a mathematical journal that used vulgar fractions (or, for that matter, mixed numbers), would it be rejected on those grounds? Would the form in which they were typeset matter? Would it be accepted, but edited to better match convention?
 A: In brief:

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*Mixed numbers are never used. I suppose there may be some random journal article that happens to use one, but I've never run across any serious math book, class, or paper that uses them. The only virtue mixed numbers offer is knowing the nearest integer to the number given. If that's an issue, just write (as is often done, e.g., when bounds of this sort come up in combinatorics or number theory) something like "$C = 67890/12345 = 5.4994\dots$"

*Decimal expansions are generally only used for approximations, or when a number is simply a constant that has no significance or would be too cumbersome to write out as a fraction. In this example you give, $\frac{1}{2}x$ or $x/2$ would be strongly preferred to $0.5x$. As you mention in the third paragraph, the latter looks more like what you mean is actually "$Cx$ for $C \approx  0.5000$." (For an example of the latter, I had a number of students in a statistics class I was TAing in grad school mention on a homework problem that the normal cdf function $\Phi$ had $\Phi(0) = 0.5000$. I did make a comment there that it's $\frac{1}{2}$ rather than $0.5000$, because I suspect that they got that value from the table of approximate $z$-values to $4$ decimal places at the back of the textbook. The value is exactly $\frac{1}{2}$ by symmetry.)

*As for the exact text formatting of fractions, math is not a programming language; anything readable works, and the most readable thing is preferable. Some journals may be particular about particular points of style, but it generally doesn't matter.

*Mixed numbers would be weird. A journal wouldn't reject your paper for using fractions instead of decimals or vice versa, but you may get notes back suggesting or demanding that you change it before publication. If it's something exceptionally egregious, or appears in conjunction with a lot of other formatting issues, it may be rejected for general quality issues (or simply because the reviewers don't want to slog through it).

