To find the total number of residents surveyed (r), we know that two-fifths of the residents said they like spending their vacation at the beach, and that number is equal to 80 residents. We can represent this situation with the equation:
\[ \frac{2}{5} r = 80 \]
To isolate \( r \), we can multiply both sides of the equation by the reciprocal of \(\frac{2}{5}\), which is \(\frac{5}{2}\):
\[ r = 80 \times \frac{5}{2} \]
Calculating that gives:
\[ r = 80 \times 2.5 = 200 \]
So, the equation that represents the number of residents surveyed is:
\[ \frac{2}{5} r = 80 \]
And the total number of residents surveyed is:
200 residents were surveyed.
Therefore, the correct response is:
\(\frac{2}{5} r = 80 ; 200 residents were surveyed.\)