$($Future value of an annuity$)$ Let's say you deposited $$160,000$ in a $529$ plan $($a tax advantaged college savings plan$)$ hoping to have $$450,000$ available $10$ years later when your first child starts college. However, you didn't invest very well, and $2$ years later the account balance dropped to $$140,000.$ 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 $2$ years ago? the account?

c$.$ Shocked by your experience of the past $2$ 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 $$450,000$ in $8$ years. You are willing to make end$-$of$-$the$-$month deposits to the fund as well. You find you can get a fund that promises to pay a guaranteed annual return of $6$ percent which is compounded monthly. You decide to transfer the $$140,000$ 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 $$140,000$ today and $$500$ at the end of each month for the next $8$ 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 $$450,000$ goal?