https://file.aiquickdraw.com/mathbot/file/1730667264249af56yk51.jpg

To calculate the **purchase price** of the car given the lease terms, we need to account for the monthly payments, interest rate, term, residual value, and down payment.

The formula for calculating the **present value of lease payments** (PV of an annuity) plus the present value of the residual value is as follows:

### Formula:
1. **Monthly Rate (r)** = Annual Rate / 12 = 8.25% / 12 = 0.6875% per month (convert to decimal: 0.006875).
2. **Present Value of Monthly Payments (PV\_annuity)**:
   \[
   PV\_annuity = PMT \times \frac{1 - (1 + r)^{-n}}{r}
   \]
   where:
   - $ PMT = 334.08 $
   - $ r = 0.006875 $
   - $ n = 48 $

3. **Present Value of Residual Value (PV\_residual)**:
   \[
   PV\_residual = FV \times (1 + r)^{-n}
   \]
   where:
   - $ FV = 2545.66 $
   - $ r = 0.006875 $
   - $ n = 48 $

4. **Total Purchase Price**:
   \[
   \text{Purchase Price} = PV\_annuity + PV\_residual + \text{Down Payment}
   \]

Let's go ahead and calculate each component:

1. **Calculate PV\_annuity** (present value of monthly payments),
2. **Calculate PV\_residual** (present value of residual value),
3. **Add down payment** of $4500 to the sum of PV values to get the purchase price.

I’ll compute these values.The **purchase price** of the car, based on the provided lease terms, is approximately **$19,951.20**.

Would you like a more detailed breakdown of each calculation step? Let me know if you have any further questions.

Here are some related questions you might find helpful:
1. How would changes in the interest rate affect the purchase price?
2. What if the residual value was different—how would that impact the calculations?
3. How does increasing the down payment alter the overall cost?
4. What would the calculations look like for a different term, say 36 months instead of 48?
5. How would the calculation differ if payments were made quarterly instead of monthly?

**Tip:** For lease or loan calculations, ensure you understand how to convert annual interest rates to monthly rates by dividing by 12, as this is essential for accuracy in monthly payment scenarios.

Patricia is looking at leasing a car. Compute the purchase price given the lease information in the table below.

Learn how to calculate the purchase price of a car lease given a down payment, monthly payment, annual interest rate, term, and residual value. This guide covers the use of present value calculations for annuities and residual values. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation