jmc

The problem tells us that $45.7\%$ of the people who think rats carry diseases equates to $21$ people total. If the total number of people who think that rats carry diseases is $x$, then we solve for $x$ as follows: $0.457x=21$, so $x=45.95$. However, we know that $x$ must be an integer, so rounding up, we get $x=46.$ Now, these $46$ people represent $86.8\%$ of the total number of people surveyed. So, if we let $y$ be the total number of people surveyed, we can solve for $y$ as follows: $0.868y=46$, so $y=52.995$. Rounding up, we get that John surveyed $\boxed{53}$ total people.