You need a $25-$year, fixed$-$rate mortgage to buy a new home for $$240,000.$ Your mortgage bank will lend you the money at a $8\mathrm{.}6$ percent APR for this $300-$month loan. However, you can afford monthly payments of only $$850,$ so you offer to pay off any remaining loan balance at the end of the loan in the form of a single balloon payment. How large will this balloon payment have to be for you to keep your monthly payments at $$850?$

You need a $25-$year, fixed$-$rate mortgage to buy a new home for $$240,000.$ Your mortgage bank will lend you the money at a $7\mathrm{.}6$ percent APR for this $300-$month loan. However, you can afford monthly payments of only $$850,$ so you offer to pay off any remaining loan balance at the end of the loan in the form of a single balloon payment.

How large will this balloon payment have to be for you to keep your monthly payments at $$850?$