You just bought a new car for $X. To pay for it, you took out a loan that requires regular monthly payments of $2,060 for 8 months and a special payment of $21,000 in 7 months. The interest rate on the loan is 1.03 percent per month and the first regular payment will be made today. What is X

To find the value of $ X $ (the price of the car), we need to compute the present value of all the payments, taking into account the 1.03% monthly interest rate. This is because the loan payments represent a future cash flow, and we need to discount them back to the present to find the total amount that the car costs today.

The given information is:

- Regular monthly payments: $2,060 for 8 months
- Special payment in 7 months: $21,000
- Monthly interest rate: 1.03% = 0.0103
- First payment is made today (so no discount for the first payment)

We'll break this down into two parts:

1. **Regular Monthly Payments**: The first regular payment is made today, and the remaining 7 payments are made at the end of each month. We need to discount each of these future payments back to today.

2. **Special Payment**: A special payment of $21,000 is made in 7 months, and we also need to discount this back to today.

### Step 1: Discount the Regular Payments
The present value of the regular payments can be computed as the sum of discounted payments. The first payment is made today, so it is not discounted, but the remaining 7 payments must be discounted.

The formula for the present value of a single payment is:
\[
PV = \frac{C}{(1 + r)^n}
\]
Where:
- $ C $ is the payment ($2,060 for the regular payments),
- $ r $ is the interest rate per period (0.0103),
- $ n $ is the number of periods (month).

So, we calculate the present value of each future payment (starting from month 1 through 7).

### Step 2: Discount the Special Payment
The special payment of $21,000 is made at the end of 7 months, so we need to discount this payment back to the present.

The present value of this special payment is:
\[
PV = \frac{21,000}{(1 + 0.0103)^7}
\]

### Total Present Value of the Loan (i.e., $ X $):
The total present value, which represents $ X $, will be the sum of the present value of all regular payments and the present value of the special payment.

Let me compute this.The total present value of the loan, which represents the price of the car $ X $, is approximately **$35,450.19**.

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

Here are 5 related questions to consider:
1. How would the value of $ X $ change if the interest rate increased to 2% per month?
2. What if the first regular payment was made after one month instead of today?
3. How would the calculations change if the special payment was due in 9 months instead of 7?
4. How does the length of the loan (number of months) affect the present value of the car?
5. What is the impact of making a lump-sum payment in the middle of the loan period?

**Tip**: Always ensure that you account for the timing of payments when calculating present value, especially with non-uniform payments.

You just bought a new car for $X. To pay for it, you took out a loan that requires regular monthly payments of $2,060 for 8 months and a special payment of $21,000 in 7 months. The interest rate on the loan is 1.03 percent per month and the first regular payment will be made today. What is X?

Learn how to calculate the present value of a car loan with regular monthly payments of $2,060 for 8 months and a special payment of $21,000 in 7 months. The problem involves applying the present value formula with an interest rate of 1.03% per month, suitable for finance students.

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