You left one important piece of the pie out of your posited scenario. That is the book distributor or wholesaler.
Here is a quote from Brett Sampson, CEO of Outskirts Press, “Industry standards for retail margins are difficult to define because, ultimately, it comes down to negotiation between all parties involved. Publishers have the power to negotiate with distributors, who have the power to negotiate with retailers, who have the ability to negotiate with the reader, but the typical trade discount is around 55%, which allows for a typical retail margin of 40%.” You can read the whole article at http://www.outskirtspress.com/articles/retailmargin.php
So if we take his average figure with your example, if the cover price of the book (the retail price) was $10 (Get me that book! These days that is cheap!), the wholesaler would have gotten a 55% discount. He allows the retailer a 40% margin. So, publisher gets 45% or $4.50, wholesaler gets 15% or $1.50 and retailer gets 40% or $4.00. The author’s cut comes out of the publisher’s piece and lis22 explained it well.
Whip me, beat me, make me do math! I’m a word person!