The “Dangling Participle” rule is simple: Write participles and prepositions before their objects, never after.
Most editors, and even many English teachers, are happy to let them slide these days. But I think you should correct danglers, because they go “clank!” in your reader’s inner ear. The trick is how to fix them without resorting to old-fashioned or pompous language.
Prepositions like ‘with’, ‘at’, ‘by’ and ‘on’ are often left swinging in the breeze. Here’s an example from an ‘FAQ’ page on the web. I’ve highlighted the dangling preposition in purple, and its object in green.
How many computers can I install my single-user licence on?
Danglers are a syntax error. The preposition appears after its object, instead of before it, which is standard in English. For the old-fashioned fix, just move the preposition ahead of its object:
On how many computers can I install my single-user licence? (10 words)
While grammatically correct, this does sound a bit stuffy. But put the phrase after the clause, and we have the same thing in more modern syntax:
I can install my single-user licence on how many computers? (10 words)
Only now we’ve lost what was best about the original sentence. The question is ‘How many?’, which is strongest at the head of the sentence. To keep this emphasis, while losing the dangler, make ‘computers’ the subject of the sentence, like this: (Subject in blue, verb in red.)
How many computers are covered by my single-user licence? (9 words)
Oops — this requires a passive verb. But with ‘licence’ as the subject, we can make it active, like this:
How many computers does my single-user licence cover? (8 words)
The video shows this in animated form, with a bit more detail.