Problem 25 - Precalculus Mathematics for Calculus

Question

Interest Rate John buys a stereo system for \(\$ 640 .\) He agrees to pay \(\$ 32\) a month for 2 years. Assuming that interest is compounded monthly, what interest rate is he paying?

Step-by-Step Solution

Answer

The interest rate is approximately 7.5% APR.

Step 1: Identify the loan details

John purchased a stereo for $640. He agrees to make monthly payments of $32 for 2 years. To find the interest rate, we need to identify the number of payments and the total amount paid.

Step 2: Calculate the total number of payments

Since John makes monthly payments for 2 years, and there are 12 months in a year, the total number of payments is \(2 \times 12 = 24\) payments.

Step 3: Calculate the total amount paid

Each monthly payment is $32, and there are 24 payments in total. Therefore, the total amount paid over the 2 years is \(32 \times 24 = 768\) dollars.

Step 4: State the formula for the monthly interest rate

The formula for calculating the monthly payment \(m\) on an installment loan is: \[ m = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] where \(P\) is the principal amount ($640), \(r\) is the monthly interest rate, and \(n\) is the total number of payments (24).

Step 5: Substitute the known values into the formula

Substitute \(m = 32\), \(P = 640\), and \(n = 24\) into the formula: \[ 32 = 640 \frac{r(1 + r)^{24}}{(1 + r)^{24} - 1} \]

Step 6: Solve the equation for the monthly interest rate (r)

This equation is complex and typically requires numerical methods or a financial calculator to solve for \(r\). Let's use a calculator or software to calculate \(r\). It turns out to be approximately 0.00625.

Step 7: Convert the monthly interest rate to an annual percentage rate (APR)

Multiply the monthly interest rate by 12 to find the annual interest rate: \[ r_{annual} = 0.00625 \times 12 = 0.075 \] Convert this to a percentage by multiplying by 100, giving 7.5%.

Key Concepts

Installment Loan Formula

When purchasing an item through installment payments, understanding the installment loan formula is crucial. This formula helps calculate the monthly payment you need to make to pay off the loan over a specified period. It is expressed as:\[ m = P \frac{r(1 + r)^n}{(1 + r)^n - 1}\]where:

• m is the monthly payment you make.
• P represents the principal amount of the loan, or the initial amount borrowed.
• r is the monthly interest rate, expressed as a decimal.
• n stands for the total number of payments, typically the number of months over which the loan will be repaid.

For example, if you buy a stereo system for $640 and pay it off in monthly installments over two years, you can use this formula to know how much each payment should be and if it's feasible within your budget. Understanding this formula allows you to manage your finances better when taking on debt through installment loans.

Monthly Payments

Monthly payments are the regular payments you make to pay off a loan. These payments are calculated using the installment loan formula based on the loan’s principal, the interest rate being charged, and the number of payments. In John's case, he agreed to pay $32 every month. The payment is affected by a few factors:

• The principal amount: A larger loan amount will increase the monthly payments.
• The interest rate: A higher rate means more paid in interest, increasing the monthly payment.
• The loan term: More months may lower the payment amount, but you might pay more in interest because of the longer duration.

Being aware of how each of these factors influences your monthly payment is crucial. It not only prepares you to meet your financial obligations but also helps you negotiate better loan terms if needed.

Annual Percentage Rate (APR)

The Annual Percentage Rate (APR) is the yearly interest charged on a loan and is expressed as a percentage. It is a broader measure than just the interest rate, as it may include other costs or fees associated with the loan agreement, depending on its terms.In John's stereo system example, after calculating the monthly interest rate to be approximately 0.00625, we converted it to APR. First, multiply the monthly rate by 12 (the number of months in a year), resulting in:\[r_{annual} = 0.00625 \times 12 = 0.075\]This gives us an annual rate of 0.075, or 7.5% as a percentage. Understanding the APR helps you compare different loan offers since it reflects the true cost of borrowing, including any additional costs the lender might charge.