Suppose that you wanted to buy a house in Launceston and

you borrowed $$500,000$ from the Commonwealth Bank. The bank offered an interest rate

of j$12=8\%$ p$.$a$.$ You agreed to repay this loan with monthly payments within $30$ years,

the first payment occurring one month after the loan was taken out.

$($a$)$ Determine the monthly payment.

$($b$)$ After making $25$ payments you lost your job and missed the $26$th$,27$th$,28$th$,$ and

$29$th payments. By the time of the $30$th$,$ the payments were resumed. How big

does the $30$th payment have to be to return to the original repayment schedule?

$($c$)$ After making $240$ payments, the interest rate decrease to j$12=6\%.$ Determine the new size of the monthly payment so that you can still kill off the debt after $30$ years as you agreed with the bank.

$($d$)$ Construct an amortization table showing the last three payments $($i$.$e two full pay$-$

ments and a partial payment$),$ and describe and perform a sanity check on the final

outstanding principal.