Ethel and Jack each separately apply for and receive a loan worth \( \$ 7,725 \) apiece. Ethel has a relatively average credit rating, so her loan has an APR of \( 9.14 \% \), compounded monthly. Jack's credit rating is excellent, so his loan has an APR of \( 6.88 \% \), compounded monthly. If they both pay off their respective loans by making six years of identical monthly payments, how much more will Ethel pay than Jack? (Round all dollar values to the nearest cent.) a. \( \$ 1,047.51 \) b. \( \$ 1,681.17 \) c. \( \$ 687.48 \) d. \( \$ 613.44 \)

To find out how much more Ethel will pay than Jack for their loans, we need to calculate the monthly payments for both loans and then find the total amount paid by each. Finally, we will subtract Jack's total payment from Ethel's total payment. ### Step 1: Calculate Monthly Payment Formula The formula for the monthly payment \( M \) on a loan is given by: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] Where: - \( P \) = principal amount (loan amount) - \( r \) = monthly interest rate (annual rate / 12) - \( n \) = total number of payments (loan term in months) ### Step 2: Calculate Ethel's Monthly Payment - Principal \( P = 7725 \) - Annual interest rate \( = 9.14\% \) or \( 0.0914 \) - Monthly interest rate \( r = \frac{0.0914}{12} \) - Loan term \( = 6 \) years \( = 6 \times 12 = 72 \) months Now, we will calculate \( r \) and then \( M \) for Ethel. ### Step 3: Calculate Jack's Monthly Payment - Principal \( P = 7725 \) - Annual interest rate \( = 6.88\% \) or \( 0.0688 \) - Monthly interest rate \( r = \frac{0.0688}{12} \) - Loan term \( = 6 \) years \( = 72 \) months Now, we will calculate \( r \) and then \( M \) for Jack. ### Step 4: Calculate Total Payments Total payment for each will be \( M \times n \). ### Step 5: Calculate the Difference Finally, we will find the difference between Ethel's total payment and Jack's total payment. Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(\frac{7725\left(\frac{0.0914}{12}\right)\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}}{\left(\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}-1\right)}\) - step1: Remove the parentheses: \(\frac{7725\left(\frac{0.0914}{12}\right)\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}}{\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}-1}\) - step2: Divide the terms: \(\frac{7725\left(\frac{0.0914}{12}\right)\left(1+\frac{457}{60000}\right)^{6\times 12}}{\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}-1}\) - step3: Add the numbers: \(\frac{7725\left(\frac{0.0914}{12}\right)\left(\frac{60457}{60000}\right)^{6\times 12}}{\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}-1}\) - step4: Divide the terms: \(\frac{7725\left(\frac{0.0914}{12}\right)\left(\frac{60457}{60000}\right)^{6\times 12}}{\left(1+\frac{457}{60000}\right)^{6\times 12}-1}\) - step5: Add the numbers: \(\frac{7725\left(\frac{0.0914}{12}\right)\left(\frac{60457}{60000}\right)^{6\times 12}}{\left(\frac{60457}{60000}\right)^{6\times 12}-1}\) - step6: Divide the terms: \(\frac{7725\times \frac{457}{60000}\left(\frac{60457}{60000}\right)^{6\times 12}}{\left(\frac{60457}{60000}\right)^{6\times 12}-1}\) - step7: Multiply the numbers: \(\frac{7725\times \frac{457}{60000}\left(\frac{60457}{60000}\right)^{72}}{\left(\frac{60457}{60000}\right)^{6\times 12}-1}\) - step8: Multiply the numbers: \(\frac{7725\times \frac{457}{60000}\left(\frac{60457}{60000}\right)^{72}}{\left(\frac{60457}{60000}\right)^{72}-1}\) - step9: Multiply: \(\frac{\frac{47071\times 60457^{72}}{800\times 60000^{72}}}{\left(\frac{60457}{60000}\right)^{72}-1}\) - step10: Subtract the numbers: \(\frac{\frac{47071\times 60457^{72}}{800\times 60000^{72}}}{\frac{60457^{72}-60000^{72}}{60000^{72}}}\) - step11: Multiply by the reciprocal: \(\frac{47071\times 60457^{72}}{800\times 60000^{72}}\times \frac{60000^{72}}{60457^{72}-60000^{72}}\) - step12: Rewrite the expression: \(\frac{47071\times 60457^{72}}{800\times 60000^{72}}\times \frac{800^{72}\times 75^{72}}{60457^{72}-60000^{72}}\) - step13: Reduce the numbers: \(\frac{47071\times 60457^{72}}{60000^{72}}\times \frac{800^{71}\times 75^{72}}{60457^{72}-60000^{72}}\) - step14: Rewrite the expression: \(\frac{47071\times 60457^{72}}{800^{72}\times 75^{72}}\times \frac{800^{71}\times 75^{72}}{60457^{72}-60000^{72}}\) - step15: Reduce the numbers: \(\frac{47071\times 60457^{72}}{800\times 75^{72}}\times \frac{75^{72}}{60457^{72}-60000^{72}}\) - step16: Rewrite the expression: \(\frac{47071\times 60457^{72}}{800\times 75^{72}}\times \frac{25^{72}\times 3^{72}}{60457^{72}-60000^{72}}\) - step17: Rewrite the expression: \(\frac{47071\times 60457^{72}}{25\times 32\times 75^{72}}\times \frac{25^{72}\times 3^{72}}{60457^{72}-60000^{72}}\) - step18: Reduce the numbers: \(\frac{47071\times 60457^{72}}{32\times 75^{72}}\times \frac{25^{71}\times 3^{72}}{60457^{72}-60000^{72}}\) - step19: Rewrite the expression: \(\frac{47071\times 60457^{72}}{32\times 25^{72}\times 3^{72}}\times \frac{25^{71}\times 3^{72}}{60457^{72}-60000^{72}}\) - step20: Reduce the numbers: \(\frac{47071\times 60457^{72}}{32\times 25}\times \frac{1}{60457^{72}-60000^{72}}\) - step21: Multiply the fractions: \(\frac{47071\times 60457^{72}}{800\left(60457^{72}-60000^{72}\right)}\) - step22: Multiply: \(\frac{47071\times 60457^{72}}{800\times 60457^{72}-800\times 60000^{72}}\) Calculate or simplify the expression \( 7725 * (0.0688/12) * (1 + (0.0688/12))^(6*12) / ((1 + (0.0688/12))^(6*12) - 1) \). Calculate the value by following steps: - step0: Calculate: \(\frac{7725\left(\frac{0.0688}{12}\right)\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}}{\left(\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}-1\right)}\) - step1: Remove the parentheses: \(\frac{7725\left(\frac{0.0688}{12}\right)\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}}{\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}-1}\) - step2: Divide the terms: \(\frac{7725\left(\frac{0.0688}{12}\right)\left(1+\frac{43}{7500}\right)^{6\times 12}}{\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}-1}\) - step3: Add the numbers: \(\frac{7725\left(\frac{0.0688}{12}\right)\left(\frac{7543}{7500}\right)^{6\times 12}}{\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}-1}\) - step4: Divide the terms: \(\frac{7725\left(\frac{0.0688}{12}\right)\left(\frac{7543}{7500}\right)^{6\times 12}}{\left(1+\frac{43}{7500}\right)^{6\times 12}-1}\) - step5: Add the numbers: \(\frac{7725\left(\frac{0.0688}{12}\right)\left(\frac{7543}{7500}\right)^{6\times 12}}{\left(\frac{7543}{7500}\right)^{6\times 12}-1}\) - step6: Divide the terms: \(\frac{7725\times \frac{43}{7500}\left(\frac{7543}{7500}\right)^{6\times 12}}{\left(\frac{7543}{7500}\right)^{6\times 12}-1}\) - step7: Multiply the numbers: \(\frac{7725\times \frac{43}{7500}\left(\frac{7543}{7500}\right)^{72}}{\left(\frac{7543}{7500}\right)^{6\times 12}-1}\) - step8: Multiply the numbers: \(\frac{7725\times \frac{43}{7500}\left(\frac{7543}{7500}\right)^{72}}{\left(\frac{7543}{7500}\right)^{72}-1}\) - step9: Multiply: \(\frac{\frac{4429\times 7543^{72}}{100\times 7500^{72}}}{\left(\frac{7543}{7500}\right)^{72}-1}\) - step10: Subtract the numbers: \(\frac{\frac{4429\times 7543^{72}}{100\times 7500^{72}}}{\frac{7543^{72}-7500^{72}}{7500^{72}}}\) - step11: Multiply by the reciprocal: \(\frac{4429\times 7543^{72}}{100\times 7500^{72}}\times \frac{7500^{72}}{7543^{72}-7500^{72}}\) - step12: Rewrite the expression: \(\frac{4429\times 7543^{72}}{100\times 7500^{72}}\times \frac{100^{72}\times 75^{72}}{7543^{72}-7500^{72}}\) - step13: Reduce the numbers: \(\frac{4429\times 7543^{72}}{7500^{72}}\times \frac{100^{71}\times 75^{72}}{7543^{72}-7500^{72}}\) - step14: Rewrite the expression: \(\frac{4429\times 7543^{72}}{100^{72}\times 75^{72}}\times \frac{100^{71}\times 75^{72}}{7543^{72}-7500^{72}}\) - step15: Reduce the numbers: \(\frac{4429\times 7543^{72}}{100\times 75^{72}}\times \frac{75^{72}}{7543^{72}-7500^{72}}\) - step16: Rewrite the expression: \(\frac{4429\times 7543^{72}}{100\times 75^{72}}\times \frac{25^{72}\times 3^{72}}{7543^{72}-7500^{72}}\) - step17: Rewrite the expression: \(\frac{4429\times 7543^{72}}{25\times 4\times 75^{72}}\times \frac{25^{72}\times 3^{72}}{7543^{72}-7500^{72}}\) - step18: Reduce the numbers: \(\frac{4429\times 7543^{72}}{4\times 75^{72}}\times \frac{25^{71}\times 3^{72}}{7543^{72}-7500^{72}}\) - step19: Rewrite the expression: \(\frac{4429\times 7543^{72}}{4\times 25^{72}\times 3^{72}}\times \frac{25^{71}\times 3^{72}}{7543^{72}-7500^{72}}\) - step20: Reduce the numbers: \(\frac{4429\times 7543^{72}}{4\times 25}\times \frac{1}{7543^{72}-7500^{72}}\) - step21: Multiply the fractions: \(\frac{4429\times 7543^{72}}{100\left(7543^{72}-7500^{72}\right)}\) - step22: Multiply: \(\frac{4429\times 7543^{72}}{100\times 7543^{72}-100\times 7500^{72}}\) Calculate or simplify the expression \( 139.784638 * 72 \). Calculate the value by following steps: - step0: Calculate: \(139.784638\times 72\) - step1: Multiply the numbers: \(10064.493936\) Calculate or simplify the expression \( 131.258895 * 72 \). Calculate the value by following steps: - step0: Calculate: \(131.258895\times 72\) - step1: Multiply the numbers: \(9450.64044\) Calculate or simplify the expression \( 10064.493936 - 9450.64044 \). Calculate the value by following steps: - step0: Calculate: \(10064.493936-9450.64044\) - step1: Subtract the numbers: \(613.853496\) ### Summary of Calculations 1. **Ethel's Monthly Payment**: - Monthly payment \( M_E \approx 139.78 \) - Total payment over 72 months: \[ 139.78 \times 72 \approx 10064.49 \] 2. **Jack's Monthly Payment**: - Monthly payment \( M_J \approx 131.26 \) - Total payment over 72 months: \[ 131.26 \times 72 \approx 9450.64 \] 3. **Difference in Total Payments**: \[ 10064.49 - 9450.64 \approx 613.85 \] ### Final Answer Ethel will pay approximately \( \$ 613.85 \) more than Jack. Thus, the correct answer is: **d. \( \$ 613.44 \)** (rounded to the nearest cent).