There is a new pub, Jenny's Pub. The pub sells both domestic and import beers at a contribution margin of $$1\mathrm{.}50$ each. The owner of the pub is considering adding sandwiches and hamburgers to its menu as more and more customers are asking for food in addition to drink. This will also help the owner to meet competition from other bars nearby, some of which provide extensive menus.

The pub needs to hire a part$-$time cook for its new menu and the monthly wage is $$3,000.$ The additional fixed cost for the new menu will be $$800$ per month. Each sandwich will cost $$0\mathrm{.}30,$ $$1\mathrm{.}50,$ and $$0\mathrm{.}70$ for bread, meat, and other items, respectively. Each hamburger will cost $$0\mathrm{.}50,$ $$1\mathrm{.}90,$ and $$0\mathrm{.}80$ for buns, meat, and other items, respectively. The sandwiches will be sold for $$4\mathrm{.}00$ each while the hamburger will be $$4\mathrm{.}20$ each. We assume that there are $30$ days in a month.

$(1)$ If the numbers of sandwiches and hamburgers can be sold are projected to be in the ratio of $1$ to $2,$ what is the monthly break$-$even point in terms of sandwiches and hamburgers?

$(2)$ If the numbers of sandwiches and hamburgers can be sold are projected to be in the ratio of $2$ to $1,$ what is the monthly break$-$even point in terms of dollar amount?

$(3)$ At the end of $3$ months, the owner finds the pub has sold $5,000$ sandwiches and $10,000$ hamburgers. What is the operating profit per month on the extended menu?

$(4)$ The pub owner believes at least $50$ extra beers are sold daily because of the extended menu. How does this affect the monthly operating income?