CASE STUDY $-2$ BORROWING TO START A BUSINESS Jane, a budding entrepreneur, was excited that her presentation at her college annual Entrepreneurial competition was voted the best by every judge, An angel investor in the audience was very impressed by her business plan and offered to help her start her designer clothing store by offering her a loan at $6\%$ compounded semi annually After two years, her business had savings of $$80,654$ and she used the entire amount to completely pay off her outstanding debt with the investor. a$)$ What was the loan amount provided to her by the angel investor and what was the accumulated interest over the two years period? b$)$ What rate, compounded monthly, would have resulted in the same accumulated debt? c$)$ How long $($rounded up to the next month$)$ would it take for her debt to reach $$100,000$ if she does not repay any amount throughout the term? Assume the same interest rate of $6\%$ compounded semi$-$annually throughout this extended period. d$)$ If she had obtained the same loan amount from a local bank, it would have accumulated to $$80,654$ in $18$ months instead of two years. What is the interest rate compounded semi$-$annually charged by the local bank? e$)$ What would her savings be if the loan had been issued to her at an annually compounding frequency? f$)$ If her contract with the investor required that she settles all her dues in two years, how much could she have borrowed initially if she was sure that she could repay $$25,000$ in one year and $$40,000$ at the end of two years? g$)$ What was the size of the loan provided by the investor if she was charged $6\%$ compounded semi$-$annually for the first year, and $8\%$ compounded quarterly for the second year and it accumulated to $$80,654$ in two years?