two different buyers purchased a home $10$ years ago. The first buyer took a $30-$year $150,000$ fix interest rate mortgage with an APR $4\mathrm{.}5\%.$ The second buyer took a $30-$year $$150,000$ adjustable rate mortgage $($ARM$)$ with starting interest rate of $3\%.$ after $4$ years the interest rate rose to $5\mathrm{.}5\%($it had an interest rate of $3\%$ for years $1,2,3,$ and $4$ and the interest rate changed to $5\mathrm{.}5\%$ in year $5$ for the remaining balance and remaining time$).$ both homeowners sold their house after $10$ years. which homeowner paid the lease for the $10$ years they paid on their loan? $($pay attention to time on the ARM$).$