Meeting

First of all, we know that 7 women have a woman on their right hand side so the simplest scenario is:

ww m ww m ww m ww m ww m ww m ww m

where 'w' is a woman and 'm' is a man. However, we are then told that 12 women have a man on their right hand side. At the moment only 7 women have a man on their right hand side so we need to add in 5 more women with a man on their right hand side without affecting the number of women with a woman on their right hand side. We can do this by adding in 5 lots of 'w m', each one just after an existing 'm'. The list now looks like this:

ww m w m ww m w m ww m w m ww m w m ww m w m ww m ww m

We are then given that 75% of men have a woman on their right side. At the moment all 12 men have a woman on their right side so we need to add in four more men, each one to the right of a man so that 12 out of the 16 men have a woman on their right side but 4 have a man. This can be done as follows:

ww m w mm ww m w mm ww m w mm ww m w mm ww m w m ww m ww

(This is not the only way of ordering the people, it is just an example.)

Therefore there are $\boxed{35}$ people sitting at the round table.

The solution is: $m w w w w w w w w - (1)$ $w$ $m$ $w$ $m$ $w$ $m$ $w$ $m$ $w$ $m$ $w$ $m$ $w$ $m$ $w$ $m$ $w$ $m$ $w$ $m$ $w$ $m m m m m - (1)$