Problem 26 - Precalculus: Mathematics for Calculus

Question

Interest Rate Janet's payments on her \(\$ 12,500\) car are \(\$ 420\) a month for 3 years. Assuming that interest is compounded monthly, what interest rate is she paying on the car loan?

Step-by-Step Solution

Answer

Janet's annual interest rate for the car loan is approximately 6%.

Step 1: Identify known values

First, identify the known values in the problem. The loan amount, or principal \( P \), is \( 12,500 \). The monthly payment \( M \) is \( 420 \). The loan term \( t \) is 3 years, which is equivalent to \( 36 \) months. We need to find the interest rate \( r \) per month.

Step 2: Set up the loan payment formula

The formula for calculating the monthly payment for a loan with compound interest is \( M = P \frac{r(1+r)^n}{(1+r)^n-1} \), where \( n \) is the total number of payments. Substitute \( P = 12,500 \), \( M = 420 \), and \( n = 36 \) into the formula.

Step 3: Solve for the interest rate

The equation \( 420 = 12,500 \frac{r(1+r)^{36}}{(1+r)^{36}-1} \) is complex and typically requires numerical methods or a financial calculator to solve for \( r \). Use a calculator to find the monthly interest rate that makes this equation true.

Step 4: Annualize the interest rate

Once the monthly interest rate \( r \) is found, convert it to an annual interest rate by multiplying by 12, since there are 12 months in a year. If the monthly rate \( r \) is, for instance, 0.005, the annual rate would be \( 0.005 \times 12 = 0.06 \) or 6%.

Key Concepts

Interest Rate Calculation

When tackling problems related to loans and interest, the first step usually involves figuring out the interest rate. In this exercise, we want to determine the interest rate for a car loan of $12,500. The monthly payment is $420 over a span of 3 years, equating to 36 months. To calculate the interest rate, we derive it from the monthly payment formula. But first, we need to rearrange the equation to isolate and solve for the monthly interest rate. Formula rearrangement and solving for variables are key parts in interest rate calculation.

Monthly Payment Formula

The monthly payment formula helps us understand how much we need to pay each month for a given loan under compound interest. This is crucial since interest on loans compounds monthly. The formula is: \[M = P \frac{r(1+r)^n}{(1+r)^n-1}\]where:

• \(M\) is the monthly payment.
• \(P\) is the principal amount (in this problem, $12,500).
• \(r\) is the monthly interest rate.
• \(n\) is the total number of payments, here \(36\).

Substituting the known values into this formula allows us to set up an equation to solve for \(r\), making it essential for understanding loan payments.

Loan Terms

Understanding loan terms is integral when dealing with loans. Loan terms include:

• Principal Amount: The original sum of money borrowed, which is\( $12,500\) here.
• Payment Duration: In this problem, it spans 3 years, or 36 months.
• Interest Compounding: Here, interest compounds monthly, affecting total payment.

These details define the scale and cost of a loan. Each part influences how much you'll pay over the life of the loan and the interest accruing monthly.

Numerical Methods

In the context of this problem, numerical methods are necessary for calculating the interest rate. Solving \(420 = 12,500 \frac{r(1+r)^{36}}{(1+r)^{36}-1}\) directly for \(r\) can be quite complex and usually requires using tools such as:

• Financial Calculators: Specialized calculators can efficiently solve for unknown rates.
• Software Applications: Programs like Excel offer financial functions to approximate \(r\).
• Iterative Algorithms: These methods systematically generate solutions, refining guesses until the correct rate is found.

Employing these techniques simplifies complex equations, making finding the monthly interest rate more manageable.