You take out a car loan of $$10,000$ at $12\%$ with monthly compounding. The loan is to be paid off

in monthly installments over a $50-$month period.

$($a$)$ Construct the amortization schedule for the loan and provide the monthly payment, interest

payment, principal payment and outstanding balance for the $15$th payment period the $25$th

payment period, and the $45$th payment period $($outstanding balance for the end of the

periods$).$

$($b$)$ After $15$ months you decide to pay off the entire loan. What is the balance that needs to be

paid off?

$($c$)$ How much interest is saved by paying off the loan after $15$ months?

$($d$)$ Suppose instead of paying off the loan in $15$months, you increase your payments by a fixed

amount, $$100$ per month $($beginning at the $15$th period$)$ for the remaining period of the loan.

When is the loan paid off $($what payment period$)?$

$($e$)$ How much interest is saved in this scenario?

$($f$)$ Would it be more suitable to pay off the loan at the end of the $15$th month, or increase the

payments by $$100$ beginning at the end of the $15$th month?