During the first eleven days, 700 people responded to the survey question. Each of them chose exactly one of the three offered options. The frequency ratio of the individual responses was $4: 7: 14$. On the twelfth day, several more people participated in the survey, changing the frequency ratio of the responses to $6: 9: 16$. How many people at minimum must have responded to the survey on the twelfth day?

(L. Šimünek)
700 people are to be divided in the ratio $4: 7: 14$ (a total of 28 parts, with 25 people per part), which in the given ratio is 112 : 196 : 392. The new ratio $6: 9: 16$ has 31 parts, and the new number must be divisible by 31. It holds that $700: 31=22$ (remainder 18). The nearest multiple of 31 greater than 700 is therefore $31 \cdot 23=713$.

Check: 713 divided in the ratio $6: 9: 16$ results in $138: 207: 368$. According to this result, the number of respondents for the third question has decreased, which is not possible. Therefore, the number 713 does not work. Similarly, we process and test the next multiples of 31: the number 744 does not work for the same reason, the number 775 works. If we divide this number in the given ratio, it results in $150: 225: 400$. Now, the number has not decreased for any question. On the twelfth day, at least 75 people must have responded to the survey.

The problem can also be solved using other methods.