Problem 21 - Precalculus: Mathematics for Calculus

Question

Financing a Car Jane agrees to buy a car for a down payment of \(\$ 2000\) and payments of \(\$ 220\) per month for 3 years. If the interest rate is \(8 \%\) per year, compounded monthly, what is the actual purchase price of her car?

Step-by-Step Solution

Answer

The total purchase price of the car is approximately $8948.26.

Step 1: Understanding the Loan Terms

Jane made a down payment of $2000 and will pay $220 per month for 3 years. The interest rate is 8% per year, compounded monthly.

Step 2: Calculate Total Payment Period

Since Jane pays monthly for 3 years, we calculate the number of total payments: \[ 3 \text{ years} \times 12 \text{ months/year} = 36 \text{ payments} \]

Step 3: Find Monthly Interest Rate

Convert the annual interest rate to a monthly interest rate. The annual rate is 8%, so the monthly interest rate is: \[ \frac{8\%}{12} = \frac{0.08}{12} \approx 0.00667 \text{ per month} \]

Step 4: Use the Present Value of Annuity Formula

Use the present value of an ordinary annuity formula to calculate the loan's present value: \[ PV = P \times \frac{1 - (1 + r)^{-n}}{r} \]where \(P = 220\), \(r = 0.00667\), and \(n = 36\).

Step 5: Calculate Present Value of Annuity

Substitute the values into the formula: \[ PV = 220 \times \frac{1 - (1 + 0.00667)^{-36}}{0.00667} \]Calculate \( (1 + 0.00667)^{-36} \), then the full expression to find \(PV\).

Step 6: Compute Present Value

Calculate the expression from the previous step to find the present value of the annuity: \[ PV \approx 220 \times 31.583 \approx 6948.26 \]

Step 7: Determine Total Purchase Price

Sum the present value and the down payment to find the total purchase price: \[ \text{Total Purchase Price} = 6948.26 + 2000 = 8948.26 \]

Key Concepts

Compound Interest

Compound interest is a key concept to understand when dealing with loans or investments over any period of time. It's the process where the interest earned or paid is added to the principal balance, so that new interest is calculated on the accumulated amount in the next period. This means you're earning or paying interest not just on your initial amount, but also on any interest that is added to it.
Let's consider the scenario of a car loan with an annual interest rate of 8%, compounded monthly. Since the interest is compounded monthly, it means that the interest for each month is calculated using the total amount of the remaining debt from the previous month.
What makes compound interest powerful or intimidating is how it accelerates the growth of the balance over time. For borrowers, this means you end up paying interest on the interest that was added during each previous period, making the total repayment higher than if the interest were not compounded. Understanding compound interest helps us better anticipate the true cost of financing a purchase.

Monthly Interest Rate

When dealing with loans or investments that involve compound interest, it's crucial to understand how to convert an annual interest rate into a monthly interest rate. This is particularly important when payments or compositions are made on a monthly basis, as in many car loans or mortgage agreements.
The annual interest rate given in a problem must be divided by 12 to convert it into a monthly rate. For example, if the annual rate is 8%, the monthly interest rate can be calculated as \( \frac{8\%}{12} \), which is roughly 0.00667 or 0.667% per month.
This conversion is essential because it accurately reflects how often the interest is being applied, and it determines the exact amount of interest accrued every single month. Knowing the monthly interest rate helps in calculating monthly payments and understanding how each month's interest contributes to the total cost over the life of the loan.

Total Purchase Price Calculation

Calculating the total purchase price of an item when financing involves a few steps to account for all costs accurately. The total purchase price is the sum of any initial payments (down payment) and the present value of all future annuity payments (regular installments).
In Jane's case, for buying a car, we start by recognizing her down payment, which is \(2000. Then, we add the present value of her monthly payments over the 3-year loan period. This present value is found using the Present Value of Annuity formula:

• Given: Monthly Payment \(P = 220\), Monthly Interest Rate \(r = 0.00667\), Number of Payments \(n = 36\).
• Formula: \( PV = P \times \frac{1 - (1 + r)^{-n}}{r} \).
• Calculation: \( PV = 220 \times 31.583 \approx 6948.26 \).

Summing the present value of the annuity with the down payment, the total purchase price computed is approximately \)8948.26. This figure represents the amount Jane effectively agrees to pay for the car, including all future payments adjusted for present value. Understanding this total purchase price ensures you have a clear picture of the financial commitment being undertaken in any financed purchase.