Indigo Holding's accounting records show the following balances on December $31,2022$:

$13\%$ Preferred stock $-$ $$100$ par value, outstanding $18,0000$ shares $-$ $$1,800,000$

Common stock $-$ $$1$ par value, outstanding $50,000$ shares $-$ $$50,000$

Retained Earnings $-1,400,000$

Assuming that the directors decide to declare total dividends in the amount of $$500,000,$ determine how much each class of stock should receive under each of the conditions stated below. One year's dividends are in arrears on the preferred stock.

$($a$)$ Preferred stock is cumulative and fully participating.

$($b$)$ Preferred stock is cumulative and participating up to $10\%.$

$($c$)$ The preferred stock is noncumulative and is participating in distribution in excess of a $13\%$dividend rate on the common stock.

Please show work for all three problems.