Answer :
To solve the problem, let's go through each step systematically.
1. Determine the trade-in value Charles will receive for his 2004 Dodge Neon:
- The listed trade-in value for a 2004 Dodge Neon is \[tex]$6,591. - Charles receives 80% of this value: \[ \text{Trade-in value received} = 0.80 \times 6591 = 5272.8 \, \text{USD} \] 2. Calculate the total additional fees: - The fees include a registration fee of \$[/tex]1,089 and a documentation fee of \[tex]$124: \[ \text{Total additional fees} = 1089 + 124 = 1213 \, \text{USD} \] 3. Calculate the sales tax: - The sales tax rate is 6.88%. - The list price of the car is \$[/tex]21,450:
[tex]\[ \text{Sales tax} = 0.0688 \times 21450 = 1475.76 \, \text{USD} \][/tex]
4. Calculate the total price of the car after trade-in and including fees and taxes:
[tex]\[ \text{Total price} = \text{List price} + \text{Total fees} + \text{Sales tax} - \text{Trade-in value received} \][/tex]
[tex]\[ = 21450 + 1213 + 1475.76 - 5272.8 = 18865.96 \, \text{USD} \][/tex]
5. Determine the details of the financing plan:
- Annual interest rate: 12.28%
- Monthly interest rate:
[tex]\[ \text{Monthly interest rate} = \frac{12.28\%}{12} = 0.01023333333 \, \text{(decimal form)} \][/tex]
- Financing period: 3 years with monthly payments:
[tex]\[ \text{Number of payments} = 3 \times 12 = 36 \, \text{payments} \][/tex]
6. Calculate the monthly payment using the loan payment formula:
[tex]\[ \text{Monthly payment} = \frac{P \times r}{1 - (1 + r)^{-n}} \][/tex]
where [tex]\( P = 18865.96 \)[/tex], [tex]\( r = 0.01023333333 \)[/tex], and [tex]\( n = 36 \)[/tex]:
[tex]\[ \text{Monthly payment} = 629.1458454614684 \, \text{USD} \][/tex]
7. Calculate the total amount paid over the financing period:
[tex]\[ \text{Total paid} = \text{Monthly payment} \times \text{Number of payments} \][/tex]
[tex]\[ = 629.1458454614684 \times 36 = 22649.250436612863 \, \text{USD} \][/tex]
8. Calculate the total interest paid:
[tex]\[ \text{Interest paid} = \text{Total paid} - \text{Total price} \][/tex]
[tex]\[ = 22649.250436612863 - 18865.96 = 3783.290436612864 \, \text{USD} \][/tex]
9. Determine the percentage of the total amount paid that is interest:
[tex]\[ \text{Interest percentage} = \left( \frac{\text{Interest paid}}{\text{Total paid}} \right) \times 100 \][/tex]
[tex]\[ = \left( \frac{3783.290436612864}{22649.250436612863} \right) \times 100 = 16.70382182050986\% \][/tex]
Thus, the answer is:
c. [tex]\( 16.70 \% \)[/tex]
1. Determine the trade-in value Charles will receive for his 2004 Dodge Neon:
- The listed trade-in value for a 2004 Dodge Neon is \[tex]$6,591. - Charles receives 80% of this value: \[ \text{Trade-in value received} = 0.80 \times 6591 = 5272.8 \, \text{USD} \] 2. Calculate the total additional fees: - The fees include a registration fee of \$[/tex]1,089 and a documentation fee of \[tex]$124: \[ \text{Total additional fees} = 1089 + 124 = 1213 \, \text{USD} \] 3. Calculate the sales tax: - The sales tax rate is 6.88%. - The list price of the car is \$[/tex]21,450:
[tex]\[ \text{Sales tax} = 0.0688 \times 21450 = 1475.76 \, \text{USD} \][/tex]
4. Calculate the total price of the car after trade-in and including fees and taxes:
[tex]\[ \text{Total price} = \text{List price} + \text{Total fees} + \text{Sales tax} - \text{Trade-in value received} \][/tex]
[tex]\[ = 21450 + 1213 + 1475.76 - 5272.8 = 18865.96 \, \text{USD} \][/tex]
5. Determine the details of the financing plan:
- Annual interest rate: 12.28%
- Monthly interest rate:
[tex]\[ \text{Monthly interest rate} = \frac{12.28\%}{12} = 0.01023333333 \, \text{(decimal form)} \][/tex]
- Financing period: 3 years with monthly payments:
[tex]\[ \text{Number of payments} = 3 \times 12 = 36 \, \text{payments} \][/tex]
6. Calculate the monthly payment using the loan payment formula:
[tex]\[ \text{Monthly payment} = \frac{P \times r}{1 - (1 + r)^{-n}} \][/tex]
where [tex]\( P = 18865.96 \)[/tex], [tex]\( r = 0.01023333333 \)[/tex], and [tex]\( n = 36 \)[/tex]:
[tex]\[ \text{Monthly payment} = 629.1458454614684 \, \text{USD} \][/tex]
7. Calculate the total amount paid over the financing period:
[tex]\[ \text{Total paid} = \text{Monthly payment} \times \text{Number of payments} \][/tex]
[tex]\[ = 629.1458454614684 \times 36 = 22649.250436612863 \, \text{USD} \][/tex]
8. Calculate the total interest paid:
[tex]\[ \text{Interest paid} = \text{Total paid} - \text{Total price} \][/tex]
[tex]\[ = 22649.250436612863 - 18865.96 = 3783.290436612864 \, \text{USD} \][/tex]
9. Determine the percentage of the total amount paid that is interest:
[tex]\[ \text{Interest percentage} = \left( \frac{\text{Interest paid}}{\text{Total paid}} \right) \times 100 \][/tex]
[tex]\[ = \left( \frac{3783.290436612864}{22649.250436612863} \right) \times 100 = 16.70382182050986\% \][/tex]
Thus, the answer is:
c. [tex]\( 16.70 \% \)[/tex]
