Let \( S \) be the total number of seats in the classroom. According to the information given, 18 students occupy \( \frac{3}{5} \) of the seats. We can write the equation:
\[ \frac{3}{5} S = 18 \]
To find the total number of seats \( S \), we can solve for \( S \):
\[ S = 18 \times \frac{5}{3} = 30 \]
Now we know that there are 30 seats in the classroom. We need to find out how many students would occupy \( \frac{2}{3} \) of the seats. First, let's calculate \( \frac{2}{3} \) of the total seats:
\[ \frac{2}{3} S = \frac{2}{3} \times 30 = 20 \]
Next, we determine how many students occupy these 20 seats. Since 18 students occupy \( \frac{3}{5} \) of the seats, we can find the ratio of students to seats:
\[ \text{Number of students per seat} = \frac{18 \text{ students}}{\frac{3}{5} \times 30 \text{ seats}} = \frac{18}{18} = 1 \text{ student per seat} \]
Therefore, if \( 20 \) seats are occupied, the number of students would be:
\[ \text{Number of students for } \frac{2}{3} \text{ of the seats} = 20 \text{ students} \]
Thus, the number of students that would occupy \( \frac{2}{3} \) of the seats in the classroom is \( \boxed{20} \).