$[0\mathrm{.}36/2\mathrm{.}17$ Points$]$

A young couple buying their first home borrow $$85,000$ for $30$ years at $7\mathrm{.}9\%,$ compounded monthly, and make payments of $$617\mathrm{.}78.$ After $2$ years, they are able to make a one$-$time payment of $$2,000$ along with their $24$ th payment.

$($a$)$ Find the unpaid balance immediately after they pay the extra $$2,000$ and their $24$ th payment. $($Round your answer to the nearest cent.$)$

$($b$)$ How many regular payments of $$617\mathrm{.}78$ will amortize the unpaid balance from part $($a$)?$ Give the answer to one decimal point.

payments

$($c$)$ How much will the remaining debt be after the number of full payment periods in part $($b$)$ is made? $($Round your answer to the nearest cent.$)$

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How much extra must be included with the last full payment to pay off the debt? $($Round your answer to the nearest cent.$)$ $

$($d$)$ How much will the couple pay over the life of the loan by paying the extra $$2,000?($Round your answer to the nearest cent.$)$ $

$($e$)$ How much will the couple save over the life of the loan by paying the extra $$2,000?($Use your answer from part $($b$).$ Round your answer to the nearest cent.$)$

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