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Charles is going to purchase a new car that has a list price of [tex]\$ 21,450[/tex]. He is planning on trading in his good-condition 2004 Dodge Neon and financing the rest of the cost over three years, paying monthly. His finance plan has an interest rate of [tex]12.28\%[/tex], compounded monthly. Charles will also be responsible for [tex]6.88\%[/tex] sales tax, a [tex]\$ 1,089[/tex] vehicle registration fee, and a [tex]\$ 124[/tex] documentation fee. If the dealer gives Charles [tex]80\%[/tex] of the listed trade-in price on his car, once the financing is paid off, what percent of the total amount paid will the interest be? (Consider the trade-in to be a reduction in the amount paid.)

\begin{tabular}{|l|c|c|c|c|c|}
\hline \multicolumn{6}{|c|}{ Dodge Cars in Good Condition } \\
\hline Model/Year & 2004 & 2005 & 2006 & 2007 & 2008 \\
\hline Viper & [tex]\$ 7,068[/tex] & [tex]\[tex]$ 7,225[/tex] & [tex]\$[/tex] 7,626[/tex] & [tex]\$ 7,901[/tex] & [tex]\$ 8,116[/tex] \\
\hline Neon & [tex]\$ 6,591[/tex] & [tex]\$ 6,777[/tex] & [tex]\[tex]$ 6,822[/tex] & [tex]\$[/tex] 7,191[/tex] & [tex]\$ 7,440[/tex] \\
\hline Intrepid & [tex]\$ 8,285[/tex] & [tex]\$ 8,579[/tex] & [tex]\$ 8,699[/tex] & [tex]\[tex]$ 9,030[/tex] & [tex]\$[/tex] 9,121[/tex] \\
\hline Dakota & [tex]\$ 7,578[/tex] & [tex]\$ 7,763[/tex] & [tex]\$ 7,945[/tex] & [tex]\$ 8,313[/tex] & [tex]\$ 8,581[/tex] \\
\hline \hline
\end{tabular}

a. [tex]17.64\%[/tex]
b. [tex]15.67\%[/tex]
c. [tex]16.70\%[/tex]
d. [tex]12.86\%[/tex]



Answer :

GinnyAnswer
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]

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