You just bought a new car for $X. To pay for it, you took out a loan that requires regular monthly payments for $1,930 for 12 months and a special payment of $38,000 in 11 months. The interest rate on the loan  is 0.67 percent per month and the first regular payment will be made today. What is X? Round tothe nearest dollar

To find the value of $ X $, the present value of the car, we need to calculate the present value (PV) of the monthly payments and the special payment. This is a time-value-of-money problem, where we discount future payments to the present using the given interest rate.

### Given:
- Regular monthly payment = $1,930
- Special payment in month 11 = $38,000
- Interest rate = 0.67% per month = 0.0067
- Total number of payments = 12
- First payment is made today, so it is an annuity due (payments start immediately).

### Steps:
1. **Present value of regular monthly payments:**
   The first payment is made today, so we need to calculate the present value of 12 monthly payments, with the first payment at $ t = 0 $. For an annuity due, the formula for present value is:

   \[
   PV_{\text{annuity due}} = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
   \]
   where:
   - $ P = 1930 $ (monthly payment)
   - $ r = 0.0067 $ (monthly interest rate)
   - $ n = 12 $ (number of payments)

2. **Present value of the special payment:**
   The special payment of $38,000 is made in month 11, so we discount it back to the present:

   \[
   PV_{\text{special}} = \frac{38000}{(1 + r)^{11}}
   \]

3. **Total present value $ X $:**
   Finally, the total present value $ X $ is the sum of the present values of the regular payments and the special payment.

\[
X = PV_{\text{annuity due}} + PV_{\text{special}}
\]

Let's compute this now.The total present value $ X $, which represents the price of the car, is approximately **$57,640**.

Do you need further details or explanations? Here are 5 related questions for deeper understanding:

1. How would the result change if the interest rate were 1% per month?
2. How does the concept of present value apply to other financial decisions?
3. What if the first payment was made after one month instead of today?
4. How would the calculation change if there were no special payment?
5. Can you use this method for investments and returns?

**Tip:** When payments start immediately, always adjust the annuity formula to account for the first payment happening right away (annuity due).

You just bought a new car for $X. To pay for it, you took out a loan that requires regular monthly payments for $1,930 for 12 months and a special payment of $38,000 in 11 months. The interest rate on the loan is 0.67 percent per month and the first regular payment will be made today. What is X? Round to the nearest dollar.

Learn how to calculate the present value of a car loan where monthly payments of $1,930 are made for 12 months, and a special payment of $38,000 is due in month 11. The loan carries a 0.67% monthly interest rate. This detailed solution covers concepts such as annuity due and present value.

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