If the results of a poll with two options is 20% - 80%, we can be certain that at least 5 people have voted, because the 20:80 ratio can't be reached with fewer than 5 people.
Question
Is there any way to calculate how many votes are required to reach a given result? E.g if a vote ends
29,6% - 70,4%, what is the minimum amount of votes that could produce this result?Is there any way to calculate this for polls with more than 2 options?
As with Batting Averages, it gets tricky when there is rounding (which there usually is). For example if $2$ out of $7$ vote for a certain thing, and the other $5$ vote against it, then you'd have $28.571429\dots \%$ and $71.428571\dots\%$ which you would probably round to something like $28.57,\,71.43$. But if you used those numbers you'd look at the fraction $\frac {2857}{10000}$ and, as that is in least terms, you'd deduce that you needed $10000$ voters.
One way to handle this is to use Continued Fractions. For example, the Continued Fractions expansion for $\frac {2857}{10000}$ has the terms $[0;3,1,1,1428]$ and of course $\frac 27$ is one of the convergents.
To illustrate using your values: starting from $.296$ we'd look at the convergents to the continued fraction for $\frac {296}{1000}=\frac {37}{125}$. They are: $$\{0, \frac 13, \frac 27, \frac 3{10}, \frac {5}{17}, \frac {8}{27}, \frac {37}{125}\}$$
Now...$\frac 5{17} \approx .29411\dots$ so that's unlikely. However $\frac 8{27}=0.296296296296\dots $ so if you are rounding to the tenths place in the percent then this would work, so I'd argue that the answer here could well be $\boxed {27}$.