Please explain the solution to this problem to me conceptually as well as mathematically. I'd like to fully understand the transactions and the logic behind them. Thank you.

In year 1, Rim Corporation purchases 1,000 shares of treasury stock for $10 per share. In year 2, Rim reissues 100 shares of treasury stock for $12 per share. In year 3, Rim reissues 500 shares of its treasury stock for $9 per share. The journal entry to record the reissuance of treasury stock in year 3 will include which of the following entries?

Select all that may apply.

Credit paid-in-capital treasury shares for $500

Debit paid-in-capital treasury shares for $500

Debit paid-in-capital treasury shares for $200

Debit retained earnings for $300

Here is the solution:

Account names and explanations	Dr.	Cr.
Cash (500 shares x $9)	$4,500
Paid-in-capital treasury shares (100 x 2)	$200
Retained earnings ((10-9)x500-200)	$300
Treasury stock (500 x $10)		$5,000

First off, what are the relationships between treasury stock, retained earnings, and paid-in-capital treasury?

Secondly, how was the paid-in-capital and retained earnings journal entries calculated? What is the logic behind it?

Please explain it in a simple way.

Thanks!