In each of the following independent cases, the corporation has outstanding $20,000,$ $$0\mathrm{.}80,$ preferred shares, with a carrying value of $$200,000,$ and $80,000$ common shares, with a carrying value of $$800,000.$ No dividends have been declared for $2017$ or $2018.$

$($a$)$ At December $31,2019,$ the board of directors wants to distribute $$125,000$ in dividends. How much will the preferred shareholders and common shareholders receive if their shares are cumulative and nonparticipating?

$($b$)$ On December $31,2019,$ the preferred shareholders received an $$80,000$ dividend on their shares, which are cumulative and fully participating. How much money was distributed to common shareholders and how much in total for dividends?

$($c$)$ At December $31,2019,$ the board of directors wants to distribute $$200,000$ in dividends. How much will the preferred shareholders and common shareholders receive if their shares are cumulative and fully participating?