Let's say you deposited $$$$150$ comma $000$ in a $529$ plan$($a tax advantaged college savings$$ plan$)$ hoping to have $$$$420$ comma $000$ available $12$ years later when your first child starts college. $$ However, you$$ didn't invest very$$ well, and $3$ years later the account balance dropped to $$$$130$ comma $000.$ Let's look at what you need to do to get the college savings plan back on track.

a$.$What was the original annual rate of return needed to reach your goal when you started the fund $3$ years$$ ago?

b$.$ With only $$$$130$ comma $000$ in the fund and $9$ years remaining until your first child starts$$ college, what annual rate of return would the fund have to make to reach your $$$$420$ comma $000$ goal if you add nothing to the$$ account?

c$.$Shocked by your experience of the past $3$years$,$ you feel the college fund has invested too much in$$ stocks, and you want a$$ low$-$risk fund in order to ensure you have the necessary $$$$420$ comma $000$ in $9$ years. You are willing to make$$ end$-$of$-$the$-$month deposits to the fund as well. You find you can get a fund that promises to pay a guaranteed annual return of $5$ percent which is compounded monthly. You decide to transfer the $$$$130$ comma $000$ to this new fund and make the necessary monthly deposits. How large of a monthly deposit must you make into this new$$ fund?

d$.$After seeing how large the monthly deposit would be$($in part c of this$$ problem$),$ you decide to invest the $$$$130$ comma $000$ today and $$$$450$ at the end of each month for the next $9$ years into a fund consisting of $50$ percent stock and $50$ percent bonds and hope for the best. What APR would the fund have to earn in order to reach your $$$$420$ comma $000$goal$?$