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Articles often claim percentages such as "46.1%" for some statistic. In this case, one can tell that they definitely had more than two measurement points, otherwise the statistic would have been either 0%, 50% or 100%.

Using brute force, I made a tool that will figure out the answer. In this particular case, there is only one result that will lead to 46.0512%, which is 414/899 (for just 46.1%, there are multiple smaller solutions).

Is there a method, more efficient than brute force, to find possible inputs that lead to this result? Or find the lower bound for either the numerator or denominator?

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Do you think claimed percentages that you can read in articles are exact, or that they are most of the time rounded to one or two decimal places ?

That being said, even if that was not the case, there is one single irreducible fraction that leads to the result, but you also have an infinite number of non-irreducible fractions so that bounding the numerator or denominator is not really informative for getting an upper bound on the pool size. To see this, think that if you double the number of people taken into the survey but the number of people who verify some predicate also doubles, your number does not change.

It might only indicate lower bounds (I repeat that this is only true in the case where this value is the exact one).

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  • $\begingroup$ Thanks for the follow-up. No, I don't think the reported percentages are exact, and I know that an upper bound cannot be given (I realised the same thing as you mentioned about any multiple of the figures). Figuring out the number of samples from a reported result is also just one application of this, it is also generally interesting if there exists a solution. $\endgroup$ – Luc Aug 23 '18 at 16:08
  • $\begingroup$ Yes I understand what you would like to get, but I don't think you can get anything meaningful. Look at how you can approximate $\pi$ for example with many rational forms 22/7, 201/64, 355/113 and observe how close these numbers are. $\endgroup$ – ippiki-ookami Aug 24 '18 at 7:24

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