Jina borrowed money from her credit union to invest in the stock market. She took out a personal, amortized loan for \( \$ 25,000 \), at an interest rate of \( 7.65 \% \), with monthly payments for a term of 1 year. For each part, do not round any intermediate computations and round your final answers to the nearest cent. If necessary, refer to the list of financial formulas. (a) Find Jina's monthly payment. \( \$ \square \) (b) If Jina pays the monthly payment each month for the full term, find her total amount to repay the loan. \( \$ \square \) (c) If Jina pays the monthly payment each month for the full term, find the total amount of interest she will pay. \( \$ \square \)

To solve Jina's loan problem, we will use the formula for the monthly payment on an amortized loan, which is given by: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] where: - \( M \) is the monthly payment, - \( P \) is the principal amount (the loan amount), - \( r \) is the monthly interest rate (annual interest rate divided by 12), - \( n \) is the number of payments (loan term in months). ### Given Data: - \( P = 25000 \) - Annual interest rate = \( 7.65\% \) - Monthly interest rate \( r = \frac{7.65}{100} \div 12 \) - Loan term = 1 year = 12 months, so \( n = 12 \) Let's calculate \( r \) first: \[ r = \frac{7.65}{100} \div 12 \] Now, we can calculate \( M \) using the formula. ### Step 1: Calculate Monthly Interest Rate \[ r = \frac{7.65}{100} \div 12 = \frac{0.0765}{12} \] ### Step 2: Calculate Monthly Payment \( M \) Now we can substitute \( P \), \( r \), and \( n \) into the monthly payment formula. \[ M = 25000 \frac{r(1 + r)^{12}}{(1 + r)^{12} - 1} \] Let's perform these calculations. ### Step 3: Perform Calculations I'll calculate \( r \) and then \( M \). 1. Calculate \( r \). 2. Calculate \( M \). Let's proceed with these calculations. Calculate the value by following steps: - step0: Calculate: \(\frac{\frac{7.65}{100}}{12}\) - step1: Divide the terms: \(\frac{\frac{153}{2000}}{12}\) - step2: Multiply by the reciprocal: \(\frac{153}{2000}\times \frac{1}{12}\) - step3: Reduce the numbers: \(\frac{51}{2000}\times \frac{1}{4}\) - step4: Multiply the fractions: \(\frac{51}{2000\times 4}\) - step5: Multiply: \(\frac{51}{8000}\) Calculate or simplify the expression \( 25000*(7.65/100/12)*(1+(7.65/100/12))^12/((1+(7.65/100/12))^12-1) \). Calculate the value by following steps: - step0: Calculate: \(\frac{25000\left(\frac{\frac{7.65}{100}}{12}\right)\left(1+\left(\frac{\frac{7.65}{100}}{12}\right)\right)^{12}}{\left(\left(1+\left(\frac{\frac{7.65}{100}}{12}\right)\right)^{12}-1\right)}\) - step1: Remove the parentheses: \(\frac{25000\left(\frac{\frac{7.65}{100}}{12}\right)\left(1+\left(\frac{\frac{7.65}{100}}{12}\right)\right)^{12}}{\left(1+\left(\frac{\frac{7.65}{100}}{12}\right)\right)^{12}-1}\) - step2: Divide the terms: \(\frac{25000\left(\frac{\frac{7.65}{100}}{12}\right)\left(1+\left(\frac{\frac{153}{2000}}{12}\right)\right)^{12}}{\left(1+\left(\frac{\frac{7.65}{100}}{12}\right)\right)^{12}-1}\) - step3: Divide the terms: \(\frac{25000\left(\frac{\frac{7.65}{100}}{12}\right)\left(1+\frac{51}{8000}\right)^{12}}{\left(1+\left(\frac{\frac{7.65}{100}}{12}\right)\right)^{12}-1}\) - step4: Add the numbers: \(\frac{25000\left(\frac{\frac{7.65}{100}}{12}\right)\left(\frac{8051}{8000}\right)^{12}}{\left(1+\left(\frac{\frac{7.65}{100}}{12}\right)\right)^{12}-1}\) - step5: Divide the terms: \(\frac{25000\left(\frac{\frac{7.65}{100}}{12}\right)\left(\frac{8051}{8000}\right)^{12}}{\left(1+\left(\frac{\frac{153}{2000}}{12}\right)\right)^{12}-1}\) - step6: Divide the terms: \(\frac{25000\left(\frac{\frac{7.65}{100}}{12}\right)\left(\frac{8051}{8000}\right)^{12}}{\left(1+\frac{51}{8000}\right)^{12}-1}\) - step7: Add the numbers: \(\frac{25000\left(\frac{\frac{7.65}{100}}{12}\right)\left(\frac{8051}{8000}\right)^{12}}{\left(\frac{8051}{8000}\right)^{12}-1}\) - step8: Divide the terms: \(\frac{25000\left(\frac{\frac{153}{2000}}{12}\right)\left(\frac{8051}{8000}\right)^{12}}{\left(\frac{8051}{8000}\right)^{12}-1}\) - step9: Divide the terms: \(\frac{25000\times \frac{51}{8000}\left(\frac{8051}{8000}\right)^{12}}{\left(\frac{8051}{8000}\right)^{12}-1}\) - step10: Multiply: \(\frac{\frac{51\times 8051^{12}}{2^{75}\times 5^{34}}}{\left(\frac{8051}{8000}\right)^{12}-1}\) - step11: Subtract the numbers: \(\frac{\frac{51\times 8051^{12}}{2^{75}\times 5^{34}}}{\frac{8051^{12}-8000^{12}}{8000^{12}}}\) - step12: Multiply by the reciprocal: \(\frac{51\times 8051^{12}}{2^{75}\times 5^{34}}\times \frac{8000^{12}}{8051^{12}-8000^{12}}\) - step13: Rewrite the expression: \(\frac{51\times 8051^{12}}{2^{75}\times 5^{34}}\times \frac{64^{12}\times 125^{12}}{8051^{12}-8000^{12}}\) - step14: Rewrite the expression: \(\frac{51\times 8051^{12}}{2^{75}\times 5^{34}}\times \frac{2^{72}\times 125^{12}}{8051^{12}-8000^{12}}\) - step15: Reduce the numbers: \(\frac{51\times 8051^{12}}{2^{3}\times 5^{34}}\times \frac{125^{12}}{8051^{12}-8000^{12}}\) - step16: Rewrite the expression: \(\frac{51\times 8051^{12}}{2^{3}\times 5^{34}}\times \frac{5^{36}}{8051^{12}-8000^{12}}\) - step17: Reduce the numbers: \(\frac{51\times 8051^{12}}{2^{3}}\times \frac{5^{2}}{8051^{12}-8000^{12}}\) - step18: Multiply the fractions: \(\frac{51\times 8051^{12}\times 5^{2}}{2^{3}\left(8051^{12}-8000^{12}\right)}\) - step19: Multiply: \(\frac{1275\times 8051^{12}}{8\times 8051^{12}-8\times 8000^{12}}\) ### Step 1: Calculate Monthly Interest Rate The monthly interest rate \( r \) is calculated as follows: \[ r = \frac{7.65}{100} \div 12 \approx 0.006375 \] ### Step 2: Calculate Monthly Payment \( M \) Now substituting \( P = 25000 \), \( r \approx 0.006375 \), and \( n = 12 \) into the monthly payment formula: \[ M = 25000 \frac{0.006375(1 + 0.006375)^{12}}{(1 + 0.006375)^{12} - 1} \] Calculating \( M \): \[ M \approx 25000 \frac{0.006375(1.079646)}{0.079646} \] Calculating the values: \[ M \approx 25000 \frac{0.006875}{0.079646} \approx 25000 \times 0.086307 \approx 2157.675 \] ### Final Monthly Payment Rounding to the nearest cent, Jina's monthly payment is: \[ M \approx 2157.68 \] ### Step 3: Total Amount to Repay the Loan To find the total amount to repay the loan, we multiply the monthly payment by the number of payments: \[ \text{Total Repayment} = M \times n = 2157.68 \times 12 \] Calculating this: \[ \text{Total Repayment} \approx 25892.16 \] ### Step 4: Total Amount of Interest Paid To find the total interest paid, we subtract the principal from the total repayment: \[ \text{Total Interest} = \text{Total Repayment} - P = 25892.16 - 25000 \] Calculating this: \[ \text{Total Interest} \approx 892.16 \] ### Summary of Results (a) Jina's monthly payment: \( \$ 2157.68 \) (b) Total amount to repay the loan: \( \$ 25892.16 \) (c) Total amount of interest she will pay: \( \$ 892.16 \)