Last year a local restaurant realized sales of $$250,000$ with fixed costs of $$120,000$ and total variable costs of $$70,000.$

a$)$ What was the restaurants contribution rate $($as a decimal$)$ last year?

b$)$ If the restaurant has the same contribution rate this year, what net income can be expected this year from a revenue of $$180,000?$ Use the rounded answer from part $($a$)$ to solve this.

c$)$ If the restaurant has the same fixed and variable costs this year, what sales this year will result in a profit of $$48,000?$ Use the rounded answer from part $($a$)$ to solve this.

d$)$ Suppose that the fixed costs this year rise to $$140,000$ and the variable costs remain the same. What is break$-$even revenue? Use the rounded answer from part $($a$)$ to solve this.

e$)$ Suppose that the fixed costs remain at $$120,000$ this year but the variable costs fall to $$57,000.$ What is the break$-$even revenue? Calulate a new contribution rate to solve this.