Requests 1-173 were asked and answered back when there were only 25 books.
Also, some of the early questions were unintentionally truncated and cannot be restored. However, the answers are shown in their totality.
How frequent is it that a Free Woman is beheaded rather than letting herself become enslaved? What is the population of Free Women to slaves?
Answer
The beheading of a Free Woman is certainly not outside the realm of possibility. This is proven when Samos is going to cut the head off of Lady Rowena. There are other examples but here is that quote.
"What are you going to do!" she screamed.
"Behead you," said Samos.
"Why!" she cried.
"There is no place in my holding for a free woman," he said.
"Enslave me!" she cried.
"I cannot believe my ears," he said, skeptically.
"Enslave me!" she cried. "Enslave me!"
Players of Gor Book 20 Page 15
However none of those beheadings ever took place.
The case of Lord Yamada beheading 10 women does not apply to your question since the women were not offered slavery.
Lord Yamada chose ten of these by lot, and had them beheaded.
Rebels of Gor Book 33 Page 250
Therefore, to answer your question, the beheading of a Free Woman rather than letting herself become enslaved is never shown to happen.
As for the ratio of Free Women to slaves, this page is already at ~The Gorean Cave~.