You just bought a new car for $X. To pay for it, you took out a loan that requires regular monthly payments of $1,920 for 29 months and a special payment of $38,800 in 9 months. The interest rate on the loan is 1.29 percent per month and the first regular payment will be made in 1 month. What is X?



This is a loan valuation problem where we need to find the present value of the loan (denoted as $ X $), given monthly payments, a special payment, and the interest rate. The total cost of the car is equivalent to the present value of the future payments, discounted at the given interest rate.

### Problem Summary:

- Regular monthly payments: $ 1,920 $ for 29 months.
- Special payment: $ 38,800 $ in 9 months.
- Monthly interest rate: $ 1.29\% $ (or 0.0129 as a decimal).
- The first payment occurs in 1 month.

### Steps:

1. **Present value of the regular monthly payments:**
   The present value of an annuity formula will be used to calculate the present value of the 29 monthly payments:
   \[
   PV_{\text{regular}} = P \times \left[ 1 - (1 + r)^{-n} \right] / r
   \]
   where:
   - $ P = 1,920 $ (monthly payment),
   - $ r = 0.0129 $ (monthly interest rate),
   - $ n = 29 $ (number of monthly payments).

2. **Present value of the special payment:**
   The special payment of $ 38,800 $ occurs 9 months from now. The present value of this lump sum payment is calculated using the formula:
   \[
   PV_{\text{special}} = \frac{F}{(1 + r)^t}
   \]
   where:
   - $ F = 38,800 $ (future value),
   - $ r = 0.0129 $ (monthly interest rate),
   - $ t = 9 $ (time in months until payment).

3. **Total present value (X):**
   The total cost $ X $ is the sum of the present values of the regular payments and the special payment:
   \[
   X = PV_{\text{regular}} + PV_{\text{special}}
   \]

Let's calculate these values.The total present value of the car, or $ X $, is approximately \$80,778.29.

Would you like more details on any part of the calculation or have any other questions?

Here are 5 related questions to explore:
1. How would the present value change if the interest rate were 1.5% per month?
2. What if the special payment was made after 12 months instead of 9 months?
3. How would the present value be affected if the regular monthly payment amount increased?
4. What would be the impact if the loan period was extended to 35 months instead of 29?
5. How is the formula for the present value of an annuity derived?

**Tip:** When dealing with loans, understanding how payments are spread out and discounted over time helps in evaluating the real cost of borrowing.

You just bought a new car for $X. To pay for it, you took out a loan that requires regular monthly payments of $1,920 for 29 months and a special payment of $38,800 in 9 months. The interest rate on the loan is 1.29 percent per month and the first regular payment will be made in 1 month. What is X?

Learn how to calculate the present value of a car loan with monthly payments of $1,920 over 29 months and a special payment of $38,800 in 9 months at a 1.29% monthly interest rate. This problem covers key concepts in loan valuation, present value calculations, and the time value of money. Suitable for advanced high school or college-level students.

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