You just bought a new car for $X. To pay for it, you took out a loan that requires regular monthly payments of $1,150 for 32 months and a special payment of $39,900 in 8 months. The interest rate on the loan is 1.20 percent per month and the first regular payment will be made in 1 month. What is X? Round your answer to the nearest dollar.

To find $ X $, the value of the car, we need to calculate the present value of all payments made under the loan agreement. These payments include:

1. **Regular payments**: Monthly payments of $1,150 for 32 months.
2. **Special payment**: A one-time payment of $39,900 in 8 months.
3. **Interest rate**: 1.20% per month.

The formula for the present value of regular monthly payments (an annuity) and the present value of a lump sum (the special payment) is:

### 1. Present Value of Regular Payments (Annuity)
The formula for the present value of an ordinary annuity is:

\[
PV_{\text{annuity}} = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]

Where:
- $ P $ = $1,150 (monthly payment),
- $ r $ = 0.012 (monthly interest rate),
- $ n $ = 32 months (number of payments).

### 2. Present Value of the Special Payment
The present value of a lump sum is calculated using:

\[
PV_{\text{lump sum}} = F \times (1 + r)^{-t}
\]

Where:
- $ F $ = $39,900 (special payment),
- $ r $ = 0.012 (monthly interest rate),
- $ t $ = 8 months (when the payment will be made).

Now, let's calculate $ X $, the total present value (sum of both parts). I'll compute this.The value of the car, $ X $, is approximately \$66,677.

Would you like any further details on the calculation, or do you have any other questions?

Here are some related questions to explore:

1. How would the present value change if the interest rate were higher?
2. What happens to the present value if the special payment occurs after 12 months instead of 8?
3. How do different payment amounts affect the present value of a loan?
4. What is the effect of making a larger down payment upfront?
5. How would increasing the loan term from 32 months to 48 months affect the regular payments?

**Tip**: When dealing with loan payments, understanding how different terms (like interest rate and payment schedule) affect the total cost can help you make informed financial decisions.

Learn how to calculate the present value of a car loan with monthly payments of $1,150 for 32 months and a special payment of $39,900 in 8 months. This problem involves applying the formulas for the present value of annuities and lump sums, with an interest rate of 1.20% per month. Suitable for college-level finance or advanced high school math students.

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