Should we have bet the$$ kids' college fund at the dog$$ track? Let's look at one specific case of a college professor$($let$\text{'}$s call him Prof.$$ ME$)$ with two young children. years$$ ago, Prof. ME invested $$$$170,000$ hoping to have $$$$410,000$ available $14$ years later when his first child started college.$$ However, the$$ account's balance is now only $$$$150,000.$Let$\text{'}$s figure out what is needed to get Prof.$$ ME's college savings plan back on track.

a$.$ What was the original annual rate of return needed to reach Prof.$$ ME's goal when he started the fund years$$ ago?

b$.$ Now with only $$$$150,000$ in the fund and $12$ years remaining until his first child starts$$ college, what APR would the fund have to earn to reach Prof.$$ ME's $$$$410,000$ goal if he adds nothing to the$$ account?

c$.$ Shocked by his experience of the past $2$years$,$ Prof. ME feels the college mutual fund has invested too much in stocks. He wants a$$ low$-$risk fund in order to ensure he has the necessary $$$$410,000$ in $12$years$,$ and he is willing to make$$ end$-$of$-$the$-$month deposits to the fund as well. He later finds a fund that promises to pay a guaranteed APR of $5\mathrm{.}5$ percent compounded monthly. Prof. ME decides to transfer the $$$$150,000$ to this new fund and make the necessary monthly deposits. How large of a monthly deposit must Prof. ME make into this new fund to meet his $$$$410,000$goal$?$

d$.$ Now Prof. ME gets sticker shock from the necessary monthly deposit he has to make into the guaranteed fund in the preceding question. He decides to invest the $$$$150,000$ today and $$$$300$ at the end of each month for the next $12$ years into a fund consisting of $50$ percent stock and $50$ percent$$ bonds, and hope for the best. What APR would the fund have to earn for Prof. ME to reach his $$$$410,000$goal$?$

Question content area bottom

Part $1$

a$.$ If Prof. ME invested $$$ into a fund years ago and hoped to have $$$ available years later when his first child started$$ college, what was the original APR needed to reach his$$ goal?

$$

enter your response here$\%($Round to two decimal$$ places.$)$

Part $2$

b$.$ Now with only $$$ in the fund and years remaining until his first child starts$$ college, what APR would the fund have to earn to reach Prof.$$ ME's $$$ goal if he adds nothing to the$$ account?

$$

enter your response here$\%($Round to two decimal$$ places.$)$

Part $3$

c$.$ If Prof. ME decides to transfer the $$$ to a new fund that promises to pay a guaranteed APR of percent compounded monthly and makes the necessary$$ end$-$of$-$the$-$month deposits, how large of a monthly deposit must he make into this new fund to meet his $$$ goal in $$years$?$

$$$

$$

enter your response here$($Round to the nearest$$ cent.$)$

Part $4$

d$.$ Now Prof. ME decides to invest the $$$ today and $$$ at the end of each month for the next years into a fund consisting of $50$ percent stock and $50$ percent$$ bonds, and hope for the best. What APR would the fund have to earn for Prof. ME to reach his $$$ $$goal$?$

$$

enter your response here$\%($Round to two decimal$$ places.$)$