Jack and Diane are considering purchasing a new home. They have identified a suitable option, a two bedroom townhouse that will cost $$212,400$ to purchase. They are now trying to select the best method of financing and have identified a couple of suitable options. As their most trusted friend, they have asked for your help in selecting the best option for their needs. They plan to live in this townhouse for only $10$ years, after which they anticipate needing a larger home, with a backyard, to accommodate their growing family.

Option A: Conventional $30$ year mortgage with an interest rate of $4\mathrm{.}375\%$ APR with monthly payments. If Jack and Diane choose this option they would need to make a $20\%$ down payment and would owe an additional $$5250$ in closing costs and fees. They will make the down payment from their savings, however the lender will allow them to include the closing costs and fees in the loan finance amount $($i$.$e$.$ the amount borrowed$).$

Option B: A special $25$ year mortgage with an interest rate of $4\mathrm{.}76\%$ compounded monthly with monthly payments. This loan begins with a $5$ year period in which Jack and Diane would only pay interest on the loan. If they choose this option they will make an $18\%$ down payment and would owe an additional $$4750$ in closing costs and fees. Once again closing costs and fees can be included into the loan amount. After the interest only period is over, Jack and Diane's monthly payment would be applied to principal and interest.

Option C: A special $15$ year fixed rate mortgage, with bi$-$m payments $($i$.$e$.$ every $2$ weeks or two times per month$).$ The loan's interest rate of $3\mathrm{.}94\%$ compounded monthly. Because of the short term $($i$.$e$.$ length$)$ of this loan the bank will only require a $10\%$ down payment, which will be paid from their savings. Jack and Diane would owe an additional $$3200$ in closing costs and fees, which can once again be included in the loan finance amount.

a$)$ Determine the monthly payments that Jack and Diane will make for each period of each loan in the first ten years of ownership.

b$)$ How much will Jack and Diane still owe $($i$.$e$.$ what would the remaining balance be$)$ at the end of the ${10}^{th}$ year for each loan?

c$)$ Which of these option should Jack and Diane want to choose, if their objective is to pay the least for their home over the first ten years of purchase. Support your decision by comparing the total interest paid over this period of time.