You just bought a new car for $X. To pay for it, you took out a loan that requires regular monthly payments of $1,850 for 7 months and a special payment of $13,000 in 3 months. The interest rate on the loan is 1.27 percent per month and the first regular payment will be made in 1 month. What is X?

To find the value of $ X $, which represents the present value (price of the car) of the loan, we need to account for both the regular monthly payments and the special payment of $13,000 in 3 months, while also applying the monthly interest rate of 1.27%.

### Key Variables:
- Monthly payment = $1,850
- Number of regular payments = 7
- Special payment in 3 months = $13,000
- Monthly interest rate = 1.27% = 0.0127

### Formula:
The value $ X $ is the present value of all future cash flows (monthly payments and the special payment). We discount each payment using the monthly interest rate.

The formula for the present value $ X $ is:

\[
X = \text{PV of regular payments} + \text{PV of special payment}
\]

#### 1. Present value of the regular payments
The first regular payment happens in 1 month, and subsequent payments continue for 7 months. The present value of a series of regular payments is given by:

\[
\text{PV of regular payments} = \sum_{t=1}^{7} \frac{1,850}{(1 + 0.0127)^t}
\]

#### 2. Present value of the special payment
The special payment of $13,000 is made in 3 months. To calculate its present value:

\[
\text{PV of special payment} = \frac{13,000}{(1 + 0.0127)^3}
\]

Now, let's calculate both components to find $ X $.The present value of the loan, which corresponds to the price of the car $ X $, is approximately $24,833.45.

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

Here are 5 related questions to expand your understanding:
1. How would changing the monthly interest rate affect the present value?
2. What if the special payment occurred after 5 months instead of 3? How would that change $ X $?
3. How is the present value formula applied in mortgage calculations?
4. How would the result change if there were no special payment at all?
5. How would a lower monthly payment, but over a longer period, impact the total present value?

**Tip:** When dealing with loans, always consider how interest rates and timing of payments affect the overall cost.

Learn how to calculate the present value of a car loan with monthly payments of $1,850 for 7 months and a special payment of $13,000 in 3 months, using a 1.27% monthly interest rate. This solution covers key financial concepts like loan amortization and discounting cash flows.

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