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?

This problem can be approached using the concept of the present value of an annuity for the regular monthly payments and a lump sum payment for the special payment in 3 months. Here's how to solve for $ X $, the price of the car.

### Step 1: Present Value of Regular Monthly Payments
The regular monthly payments form an annuity, and we can calculate the present value of this 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 = 1850 $ is the regular monthly payment,
- $ r = 0.0127 $ is the monthly interest rate (1.27%),
- $ n = 7 $ is the number of payments (7 months).

### Step 2: Present Value of the Special Payment
The special payment is a lump sum made in 3 months, and we can discount it back to the present using the following formula for the present value of a single payment:

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

Where:
- $ F = 13000 $ is the special payment,
- $ r = 0.0127 $ is the monthly interest rate,
- $ t = 3 $ is the time in months until the payment.

### Step 3: Total Present Value (Price of the Car)
The total present value, $ X $, is the sum of the present values of the regular payments and the lump sum payment:

\[
X = PV_{\text{annuity}} + PV_{\text{lump sum}}
\]

Now, let's calculate it.The total price of the car, $ X $, is approximately $24,833.45.

Would you like more details on this calculation or have any questions?

Here are 5 related questions to consider:
1. How would the price change if the interest rate were higher?
2. What happens to the monthly payments if you extend the loan term?
3. How is the calculation different for a loan with compounding interest?
4. How does a down payment affect the total loan amount?
5. How can you reduce the total interest paid on a loan?

**Tip:** Lowering your monthly payments often means extending the loan term, but this can increase the total interest you pay over time.

Learn how to calculate the price of a car using the present value of regular monthly payments of $1,850 for 7 months and a special payment of $13,000 in 3 months with an interest rate of 1.27% per month. This problem covers key concepts in present value calculations, annuities, and lump sum payments.

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