$($a$)$ In $2022$ Widget Company, a calendar year and cash method taxpayer, sold $$50,000$ of

widgets on credit to one of its customers. In $2023,$ Widget Company failed to collect the

account receivable after pursuing every possible means of collection. In $2023,$ Widget

Company has $$300,000$ of net operational income. How much of a bad debt deduction

can Widget take in $2023?$

$($b$)$ In $2022$ Sally loaned Jane $$5,500,$ as a personal loan. In $2023($when the outstanding loan

is still $$5,500),$ Sally informs Jane that she will not be able to repay the loan. In $2023,$

Jane has $$600$ short$-$term capital gain and $$44,000$ wage income. In $2024,$ Sally repays

the $$5,500.$ How much, if any, of that does Jane have to include in her income for $2024?$