A young couple buying their first home borrow $$90,000$ for $30$ years at $7\mathrm{.}8\%,$ compounded monthly, and make payments of

$$647\mathrm{.}88.$ After $2$ years, they are able to make a one$-$time payment of $$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 $$647\mathrm{.}88$ 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.$)$

$

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.$)$