A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table?
1. Let \( W \) be the number of women and \( M \) be the number of men at the table.
2. According to the problem, 7 women have a woman on their right side, and 12 women have a man on their right side. Therefore, the total number of women is:
   \[
   W = 7 + 12 = 19
   \]
3. It is also given that 75% of the men have a woman on their right side. Let \( x \) be the number of men who have a woman on their right side. Then:
   \[
   x = 0.75M
   \]
4. Since the table is round, each person must have exactly one person on their right side. Therefore, the total number of people at the table is equal to the total number of people who have someone on their right side. This means:
   \[
   W + M = 19 + M
   \]
5. The number of men who have a man on their right side is:
   \[
   M - x = M - 0.75M = 0.25M
   \]
6. Since the total number of people at the table must be equal to the total number of people who have someone on their right side, we can write:
   \[
   19 + 0.25M = W + M
   \]
7. Substituting \( W = 19 \) into the equation, we get:
   \[
   19 + 0.25M = 19 + M
   \]
8. Simplifying the equation:
   \[
   0.25M = M
   \]
9. This equation is not possible unless \( M = 0 \), which contradicts the given information that there are men at the table. Therefore, we need to re-evaluate our approach.

10. Let's consider the total number of people at the table as \( N \). Since each person has exactly one person on their right side, we have:
    \[
    N = W + M
    \]
11. From the given information, 75% of the men have a woman on their right side, so:
    \[
    0.75M = 12
    \]
12. Solving for \( M \):
    \[
    M = \frac{12}{0.75} = 16
    \]
13. Now, substituting \( M = 16 \) and \( W = 19 \) into the equation for the total number of people:
    \[
    N = W + M = 19 + 16 = 35
    \]

The final answer is \(\boxed{35}\).