You just bought a new car for $X. To pay for it, you took out a loan that requires regular monthly payments of $1,970 for 33 months and a special payment of $33,200 in 10 months. The interest rate on the loan is 0.61 percent per month and the first regular payment will be made in 1 month. What is X?

We are tasked with calculating the present value of a loan that involves regular monthly payments of $1,970, a special payment of $33,200 in 10 months, and a monthly interest rate of 0.61%. Let's break it down step by step.

### Step 1: Determine the Present Value of Regular Monthly Payments

You have regular monthly payments of $1,970 for 33 months, with the first payment starting in 1 month. We can calculate the present value of these payments using the formula for the present value of an ordinary annuity:

\[
PV_{\text{annuity}} = P \times \frac{1 - (1 + r)^{-n}}{r}
\]

Where:
- $ P = 1,970 $ is the monthly payment,
- $ r = 0.0061 $ is the monthly interest rate (0.61%),
- $ n = 33 $ is the number of payments.

### Step 2: Determine the Present Value of the Special Payment

The special payment of $33,200 is made in 10 months. To find the present value of this payment, we discount it by the interest rate for 10 months using the formula for present value of a single future payment:

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

Where:
- $ F = 33,200 $ is the future payment,
- $ r = 0.0061 $ is the monthly interest rate,
- $ t = 10 $ is the number of months until the payment is made.

### Step 3: Calculate the Total Present Value (X)

The total present value $ X $, which is the amount of the loan (or the cost of the car), is the sum of the present value of the regular payments and the present value of the special payment:

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

Let's calculate these values.The present value of the loan, which corresponds to the cost of the car, is approximately **$89,964.16**.

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How would the present value change if the interest rate increased?
2. What happens to the present value if the special payment is delayed by a few months?
3. How would the calculation differ if the loan had no special payment?
4. How can we calculate the total interest paid over the life of this loan?
5. What is the impact of paying a higher monthly amount over a shorter loan term?

**Tip:** Present value helps you understand the real cost of future payments in today’s money, considering the time value of money.

Learn how to calculate the present value of a loan that includes regular monthly payments of $1,970 over 33 months and a special payment of $33,200 in 10 months with an interest rate of 0.61% per month. This detailed solution covers key financial concepts such as present value and annuities.

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