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Suzanne has purchased a car with a list price of $\$23,860$. She traded in her previous car, which was a Dodge in good condition, and financed the rest of the cost for five years at a rate of $11.62\%$, compounded monthly. The dealer gave her $85\%$ of the listed trade-in price for her car. She was also responsible for $8.11\%$ sales tax, a $\$1,695$ vehicle registration fee, and a $\$228$ documentation fee. If Suzanne makes a monthly payment of $\$455.96$, which of the following was her original car?




\multicolumn{6}{|c|}{Dodge Cars in Good Condition}

Model/Year		20042005200620072008

Viper		\$7,068\$7,225\$7,626\$7,901\$8,116

Neon		\$6,591\$6,777\$6,822\$7,191\$7,440

Intrepid		\$8,285\$8,579\$8,699\$9,030\$9,121

Dakota		\$7,578\$7,763\$7,945\$8,313\$8,581





A2004 Intrepid
B2008 Neon
C2005 Viper
D2007 Dakota

Verified Answer :

  • Calculate the total cost of the new car, including tax and fees, which is \(\$27,718.05\).
  • Determine the financed amount using the present value of an annuity formula, resulting in \(\$20,675.74\).
  • Subtract the financed amount from the total cost to find the trade-in value, which is \(\$7,042.31\).
  • Calculate the original listed price as \(\frac{7,042.31}{0.85} \approx \$8,285\), which matches the 2004 Intrepid in the provided table.
  • Final answer: \(\boxed{\text{a. 2004 Intrepid}}\)

Explanation

  1. Calculate the Total Purchase Cost

To identify Suzanne's original car, we need to calculate its trade-in value based on the financing terms and the total purchase cost of the new car.
First, the total cost for the new car, excluding the trade-in, is:
\[\text{Total Cost} = \text{List Price} + \text{Sales Tax} + \text{Registration Fee} + \text{Documentation Fee}\]
The sales tax is \(\$23,860 \times 0.0811 = \$1,935.046\).
Adding the fees and the list price gives the total cost before the trade-in:
\[\text{Total Cost} = \$23,860 + \$1,935.046 + \$1,695 + \$228 = \$27,718.046.\]

  1. Calculate the Financed Amount

Next, we calculate the amount Suzanne financed using the present value of an annuity formula:
\[PV = M \times \frac{1-(1+r)^{-n}}{r}\]
Where:

  • Monthly payment \(M = \$455.96\)
  • Annual interest rate = \(11.62\%\), so the monthly rate \(r = \frac{0.1162}{12} \approx 0.00968333\)
  • Number of payments \(n = 5 \times 12 = 60\)

Substituting these values:
\[PV = 455.96 \times \frac{1-(1+0.00968333)^{-60}}{0.00968333} \approx \$20,675.74\]
This is the amount Suzanne financed.

  1. Determine the Original Car Model

Now, we determine the trade-in value of Suzanne's original car by subtracting the financed amount from the total cost:
\[\text{Trade-in Value} = \text{Total Cost} - \text{Financed Amount} = \$27,718.046 - \$20,675.74 \approx \$7,042.31\]
The dealer provided \(85\%\) of the car's listed trade-in price. Let \(LP\) be the listed trade-in price:
\[0.85 \times LP = \$7,042.31 \implies LP = \frac{\$7,042.31}{0.85} \approx \$8,285.07\]
Comparing this result to the table provided, the value \(\$8,285\) corresponds to the 2004 Intrepid. Therefore, Suzanne's original car was a 2004 Intrepid.

Examples

Understanding the calculation of financed amounts and trade-in values is essential for making informed financial decisions when purchasing expensive items like cars. By breaking down the costs, interest rates, and resale values, one can evaluate whether a deal is fair and align financial commitments with their budget. This process of isolating a specific variable—like the original list price—is a common technique used in personal finance to track the value of assets over time.

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