Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though). I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?

my high school math would tell me 12. Could you explain why it wouldn't be?

No, I can't explain why it wouldn't be, but almost everyone I asked said 3 (you resolve the 3x first for some reason). They couldn't give me a good reason for it though.

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though). I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?

Poorly written, and it appears to be another one of these "write the expression in an ambiguous manner so people can argue about order of operations and feel superior to each other" questions.

That said, typically you would read 3x as being a single "unit" that is equal to 6, so that the entire expression is equal to 3. If it had been written 18/3x it would probably be worse, and that's why mathematicians tend to take up lots of space to write fractions so you can clearly see what's in the numerator and what's in the denominator and didn't we argue about one of these in this thread about ten pages ago oh my god my head hurts.

I'm not trying to start an argument, I legitimately want to make sure that there's not some rule I'm somehow unaware of that says you do "smash up multiplication" before everything else. And I want to be very sure that I'm correct, because the book I was working from consistently resolved the 3x first, and I had to try to explain to a student why they were doing that, even though it seems very wrong to me. I really think the book is wrong, but I want to be 100% sure of that before I tell my boss that the book has a major error in it that needs to be fixed.

Of course I agree that you should always write it in a less ambiguous way, but I have to be able to tell students what to do if they see it written like that.

The argument for 12 is that multiplication and division have equal priority, so you go from left to right. 18 divided by 3 is 6, 6x is 12. The only possible argument for 3 that I could think of is that 3x is somehow fundamentally different from 3*x (with a symbol between them), since I think we could all agree 18÷3*x is 12. Then I'd have to ask about 18÷3(x), and 18÷3(2). But I would think 3x is exactly the same as 3*x, and I wasn't aware of any special rule that distinguishes them.