After making 25 payments of \( \$ 464.22 \) on your original 6 year \( \$ 27000 \) car loan at \( 7.3 \% \) compounded monthly you want to find the amount that remains. What is the amount of the remaining balance of your car loan? (Round balance to the nearest cent.) The remaining balance of your car loan would be \( \$ \)

To find the remaining balance of the car loan after making 25 payments, we can follow these steps: 1. **Identify the known values:** - Original loan amount (\( P \)): \( 27000 \) - Monthly payment (\( PMT \)): \( 464.22 \) - Annual interest rate (\( r \)): \( 7.3\% \) or \( 0.073 \) - Monthly interest rate (\( i \)): \( \frac{0.073}{12} \) - Total number of payments (\( n \)): \( 6 \times 12 = 72 \) - Number of payments made (\( k \)): \( 25 \) 2. **Calculate the monthly interest rate:** \[ i = \frac{0.073}{12} \] 3. **Calculate the remaining balance after \( k \) payments using the formula for the remaining balance of an amortizing loan:** \[ B = P \left( \frac{(1 + i)^n - (1 + i)^k}{(1 + i)^n - 1} \right) \] where \( B \) is the remaining balance. Now, let's perform the calculations step by step. First, we will calculate the monthly interest rate \( i \) and then use it to find the remaining balance \( B \). Let's calculate \( i \) and then \( B \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.073}{12}\) - step1: Convert the expressions: \(\frac{\frac{73}{1000}}{12}\) - step2: Multiply by the reciprocal: \(\frac{73}{1000}\times \frac{1}{12}\) - step3: Multiply the fractions: \(\frac{73}{1000\times 12}\) - step4: Multiply: \(\frac{73}{12000}\) Calculate or simplify the expression \( 27000*((1+(0.073/12))^(72)-(1+(0.073/12))^(25))/((1+(0.073/12))^(72)-1) \). Calculate the value by following steps: - step0: Calculate: \(\frac{27000\left(\left(1+\left(\frac{0.073}{12}\right)\right)^{72}-\left(1+\left(\frac{0.073}{12}\right)\right)^{25}\right)}{\left(\left(1+\left(\frac{0.073}{12}\right)\right)^{72}-1\right)}\) - step1: Remove the parentheses: \(\frac{27000\left(\left(1+\left(\frac{0.073}{12}\right)\right)^{72}-\left(1+\left(\frac{0.073}{12}\right)\right)^{25}\right)}{\left(1+\left(\frac{0.073}{12}\right)\right)^{72}-1}\) - step2: Divide the terms: \(\frac{27000\left(\left(1+\left(\frac{0.073}{12}\right)\right)^{72}-\left(1+\left(\frac{0.073}{12}\right)\right)^{25}\right)}{\left(1+\frac{73}{12000}\right)^{72}-1}\) - step3: Add the numbers: \(\frac{27000\left(\left(1+\left(\frac{0.073}{12}\right)\right)^{72}-\left(1+\left(\frac{0.073}{12}\right)\right)^{25}\right)}{\left(\frac{12073}{12000}\right)^{72}-1}\) - step4: Divide the terms: \(\frac{27000\left(\left(1+\frac{73}{12000}\right)^{72}-\left(1+\left(\frac{0.073}{12}\right)\right)^{25}\right)}{\left(\frac{12073}{12000}\right)^{72}-1}\) - step5: Add the numbers: \(\frac{27000\left(\left(\frac{12073}{12000}\right)^{72}-\left(1+\left(\frac{0.073}{12}\right)\right)^{25}\right)}{\left(\frac{12073}{12000}\right)^{72}-1}\) - step6: Divide the terms: \(\frac{27000\left(\left(\frac{12073}{12000}\right)^{72}-\left(1+\frac{73}{12000}\right)^{25}\right)}{\left(\frac{12073}{12000}\right)^{72}-1}\) - step7: Add the numbers: \(\frac{27000\left(\left(\frac{12073}{12000}\right)^{72}-\left(\frac{12073}{12000}\right)^{25}\right)}{\left(\frac{12073}{12000}\right)^{72}-1}\) - step8: Subtract the numbers: \(\frac{27000\times \frac{12073^{72}-12073^{25}\times 12000^{47}}{12000^{72}}}{\left(\frac{12073}{12000}\right)^{72}-1}\) - step9: Multiply the numbers: \(\frac{\frac{12073^{72}-12073^{25}\times 12000^{47}}{3^{69}\times 1000^{71}\times 4^{72}}}{\left(\frac{12073}{12000}\right)^{72}-1}\) - step10: Subtract the numbers: \(\frac{\frac{12073^{72}-12073^{25}\times 12000^{47}}{3^{69}\times 1000^{71}\times 4^{72}}}{\frac{12073^{72}-12000^{72}}{12000^{72}}}\) - step11: Multiply by the reciprocal: \(\frac{12073^{72}-12073^{25}\times 12000^{47}}{3^{69}\times 1000^{71}\times 4^{72}}\times \frac{12000^{72}}{12073^{72}-12000^{72}}\) - step12: Rewrite the expression: \(\frac{12073^{72}-12073^{25}\times 12000^{47}}{3^{69}\times 1000^{71}\times 4^{72}}\times \frac{3^{72}\times 4000^{72}}{12073^{72}-12000^{72}}\) - step13: Reduce the numbers: \(\frac{12073^{72}-12073^{25}\times 12000^{47}}{1000^{71}\times 4^{72}}\times \frac{3^{3}\times 4000^{72}}{12073^{72}-12000^{72}}\) - step14: Rewrite the expression: \(\frac{12073^{72}-12073^{25}\times 12000^{47}}{1000^{71}\times 4^{72}}\times \frac{3^{3}\times 1000^{72}\times 4^{72}}{12073^{72}-12000^{72}}\) - step15: Reduce the numbers: \(\frac{12073^{72}-12073^{25}\times 12000^{47}}{4^{72}}\times \frac{3^{3}\times 1000\times 4^{72}}{12073^{72}-12000^{72}}\) - step16: Rewrite the expression: \(\frac{12073^{72}-12073^{25}\times 12000^{47}}{4^{72}}\times \frac{3^{3}\times 8\times 125\times 4^{72}}{12073^{72}-12000^{72}}\) - step17: Rewrite the expression: \(\frac{12073^{72}-12073^{25}\times 12000^{47}}{2^{144}}\times \frac{3^{3}\times 2^{3}\times 125\times 4^{72}}{12073^{72}-12000^{72}}\) - step18: Reduce the numbers: \(\frac{12073^{72}-12073^{25}\times 12000^{47}}{2^{141}}\times \frac{3^{3}\times 125\times 4^{72}}{12073^{72}-12000^{72}}\) - step19: Rewrite the expression: \(\frac{12073^{72}-12073^{25}\times 12000^{47}}{2^{141}}\times \frac{3^{3}\times 125\times 2^{144}}{12073^{72}-12000^{72}}\) - step20: Reduce the numbers: \(\left(12073^{72}-12073^{25}\times 12000^{47}\right)\times \frac{3^{3}\times 125\times 2^{3}}{12073^{72}-12000^{72}}\) - step21: Multiply: \(\frac{\left(12073^{72}-12073^{25}\times 12000^{47}\right)\times 27000}{12073^{72}-12000^{72}}\) - step22: Multiply: \(\frac{12073^{72}\times 27000-27000\times 12073^{25}\times 12000^{47}}{12073^{72}-12000^{72}}\) 1. **Calculate the monthly interest rate \( i \):** \[ i = \frac{0.073}{12} \approx 0.0060833 \] 2. **Calculate the remaining balance \( B \) after 25 payments:** \[ B = 27000 \left( \frac{(1 + 0.0060833)^{72} - (1 + 0.0060833)^{25}}{(1 + 0.0060833)^{72} - 1} \right) \approx 18926.83 \] Now, rounding the remaining balance to the nearest cent, we find: The remaining balance of your car loan would be \( \$ 18926.83 \).