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Ruby, 127 bytes

b=r if(r=(q*d-=1).round.to_r/d).round(p).to_f.to_s==n while d>1

Defines a function f which returns the result as a Rational. E.g. if you append this code

p f["1.7"]
p f["0."]
p f["0.001"]
p f["3.1416"]

You get


The loop is over denominators. I'm starting with the full fraction, e.g. 31416/10000 for the last example. Then I'm decrementing the denominator, proportionally decrement the numerator (and round it). If the resulting rational rounds to the same as the input number, I'm remembering a new best fraction.