Monthly amortization schedule. Sherry and Sam want to purchase a condo at the coast. They will spend \( \$ 650,000 \) on the condo and are taking out a loan for the condo for twenty years at \( 7.3 \% \) interest. a. What is the monthly payment on the mortgage? Construct the amortization of the loan for the twenty years in a spreadsheet to show the interest cost, the principal reduction, and the ending balance each month. b. Then change the amortization to reflect that after ten years, Sherry and Sam will increase their monthly payment to \( \$ 7,600 \) per month. When will they fully repay the mortgage with this increased payment if they apply all the extra dollars above the original payment to the principal? a. What is the monthly payment on the mortgage? The monthly interest rate on the mortgage is \( \square \% \). (Round to seven decimal places.)

To calculate the monthly payment on the mortgage, we can use the formula for monthly payments on a loan: \[ M = P \left[ \frac{r(1+r)^n}{(1+r)^n - 1} \right] \] where: - \( M \) is the monthly payment - \( P \) is the principal amount (the amount borrowed) - \( r \) is the monthly interest rate - \( n \) is the number of payments (the number of months the loan is for) Given: - Principal amount (\( P \)) = $650,000 - Interest rate = 7.3% per year - Loan duration = 20 years First, we need to calculate the monthly interest rate (\( r \)) from the annual interest rate: \[ r = \frac{7.3\%}{12} = 0.0060833 \] Now, we can calculate the monthly payment (\( M \)) using the formula: \[ M = 650,000 \left[ \frac{0.0060833(1+0.0060833)^{20}}{(1+0.0060833)^{20} - 1} \right] \] Let's calculate the monthly payment. Calculate the value by following steps: - step0: Calculate: \(\frac{650000\left(0.0060833\left(1+0.0060833\right)^{20}\right)}{\left(\left(1+0.0060833\right)^{20}-1\right)}\) - step1: Remove the parentheses: \(\frac{650000\times 0.0060833\left(1+0.0060833\right)^{20}}{\left(1+0.0060833\right)^{20}-1}\) - step2: Add the numbers: \(\frac{650000\times 0.0060833\times 1.0060833^{20}}{\left(1+0.0060833\right)^{20}-1}\) - step3: Add the numbers: \(\frac{650000\times 0.0060833\times 1.0060833^{20}}{1.0060833^{20}-1}\) - step4: Convert the expressions: \(\frac{650000\times 0.0060833\left(\frac{10060833}{10000000}\right)^{20}}{1.0060833^{20}-1}\) - step5: Convert the expressions: \(\frac{650000\times 0.0060833\left(\frac{10060833}{10000000}\right)^{20}}{\left(\frac{10060833}{10000000}\right)^{20}-1}\) - step6: Multiply: \(\frac{\frac{790829\times 10060833^{20}}{200\times 10000000^{20}}}{\left(\frac{10060833}{10000000}\right)^{20}-1}\) - step7: Subtract the numbers: \(\frac{\frac{790829\times 10060833^{20}}{200\times 10000000^{20}}}{\frac{10060833^{20}-10000000^{20}}{10000000^{20}}}\) - step8: Multiply by the reciprocal: \(\frac{790829\times 10060833^{20}}{200\times 10000000^{20}}\times \frac{10000000^{20}}{10060833^{20}-10000000^{20}}\) - step9: Rewrite the expression: \(\frac{790829\times 10060833^{20}}{200\times 10000000^{20}}\times \frac{40000^{20}\times 250^{20}}{10060833^{20}-10000000^{20}}\) - step10: Rewrite the expression: \(\frac{790829\times 10060833^{20}}{200\times 10000000^{20}}\times \frac{200^{40}\times 250^{20}}{10060833^{20}-10000000^{20}}\) - step11: Reduce the numbers: \(\frac{790829\times 10060833^{20}}{10000000^{20}}\times \frac{200^{39}\times 250^{20}}{10060833^{20}-10000000^{20}}\) - step12: Rewrite the expression: \(\frac{790829\times 10060833^{20}}{40000^{20}\times 250^{20}}\times \frac{200^{39}\times 250^{20}}{10060833^{20}-10000000^{20}}\) - step13: Rewrite the expression: \(\frac{790829\times 10060833^{20}}{200^{40}\times 250^{20}}\times \frac{200^{39}\times 250^{20}}{10060833^{20}-10000000^{20}}\) - step14: Reduce the numbers: \(\frac{790829\times 10060833^{20}}{200\times 250^{20}}\times \frac{250^{20}}{10060833^{20}-10000000^{20}}\) - step15: Rewrite the expression: \(\frac{790829\times 10060833^{20}}{200\times 250^{20}}\times \frac{50^{20}\times 5^{20}}{10060833^{20}-10000000^{20}}\) - step16: Rewrite the expression: \(\frac{790829\times 10060833^{20}}{50\times 4\times 250^{20}}\times \frac{50^{20}\times 5^{20}}{10060833^{20}-10000000^{20}}\) - step17: Reduce the numbers: \(\frac{790829\times 10060833^{20}}{4\times 250^{20}}\times \frac{50^{19}\times 5^{20}}{10060833^{20}-10000000^{20}}\) - step18: Rewrite the expression: \(\frac{790829\times 10060833^{20}}{4\times 250^{20}}\times \frac{2^{19}\times 25^{19}\times 5^{20}}{10060833^{20}-10000000^{20}}\) - step19: Rewrite the expression: \(\frac{790829\times 10060833^{20}}{2^{2}\times 250^{20}}\times \frac{2^{19}\times 25^{19}\times 5^{20}}{10060833^{20}-10000000^{20}}\) - step20: Reduce the numbers: \(\frac{790829\times 10060833^{20}}{250^{20}}\times \frac{2^{17}\times 25^{19}\times 5^{20}}{10060833^{20}-10000000^{20}}\) - step21: Rewrite the expression: \(\frac{790829\times 10060833^{20}}{2^{20}\times 125^{20}}\times \frac{2^{17}\times 25^{19}\times 5^{20}}{10060833^{20}-10000000^{20}}\) - step22: Reduce the numbers: \(\frac{790829\times 10060833^{20}}{2^{3}\times 125^{20}}\times \frac{25^{19}\times 5^{20}}{10060833^{20}-10000000^{20}}\) - step23: Rewrite the expression: \(\frac{790829\times 10060833^{20}}{2^{3}\times 5^{60}}\times \frac{5^{38}\times 5^{20}}{10060833^{20}-10000000^{20}}\) - step24: Reduce the numbers: \(\frac{790829\times 10060833^{20}}{2^{3}\times 5^{2}}\times \frac{1}{10060833^{20}-10000000^{20}}\) - step25: Multiply the fractions: \(\frac{790829\times 10060833^{20}}{200\left(10060833^{20}-10000000^{20}\right)}\) - step26: Multiply: \(\frac{790829\times 10060833^{20}}{200\times 10060833^{20}-200\times 10000000^{20}}\) The monthly payment on the mortgage is approximately $34,615.79. Now, let's construct the amortization schedule for the twenty years in a spreadsheet to show the interest cost, the principal reduction, and the ending balance each month.