A total of $n$ people were asked a question in a survey. Exactly $76 \%$ of the $n$ people responded "yes" and exactly $24 \%$ of the $n$ people responded "no". What is the smallest possible value of $n$ ?
Let $Y$ denote the number of people who answered "yes" and let $N$ denote the number of people who answered "no".

The number of people is $n$, so $n=Y+N$. As well, since $76 \%$ of respondents said "yes", we have that $\frac{76}{100}=\frac{Y}{n}=\frac{Y}{Y+N}$.

The equation $\frac{76}{100}=\frac{Y}{Y+N}$ implies $76 Y+76 N=100 Y$ which can be rearranged to $76 N=24 Y$. Dividing both sides by 4 gives $19 N=6 Y$.

Since $Y$ and $N$ are numbers of people, they are non-negative integers. Furthermore, since a non-zero percentage of people responded with each of "yes" and "no", they must be positive integers.

Therefore, the equation $19 N=6 Y$ implies that $Y$ is a positive multiple of 19 since 6 and 19 do not have any positive factors in common other than 1.

The smallest that $Y$ can be is 19 , and if $Y=19$, then $N=\frac{(6)(19)}{19}=6$.

If $Y=19$ and $N=6$, then $n=25$, which means $\frac{Y}{n}=\frac{19}{25}=0.76$ and $\frac{N}{n}=\frac{6}{25}=0.24$.

Therefore, $Y=19$ and $N=6$ satisfy the conditions, and we have already argued that $Y$ cannot be any smaller.

If $Y$ is greater than 19, then $n$ is greater than $2(19)=38$ since $Y$ must be a positive multiple of 19. Therefore, the answer is 25 .

ANSWER: 25