Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The price of a new car is $$\$ 16,000$$. Assume that an individual makes a down payment of \(25 \%\) toward the purchase of the car and secures financing for the balance at the rate of \(10 \% /\) year compounded monthly. a. What monthly payment will she be required to make if the car is financed over a period of 36 mo? Over a period of \(48 \mathrm{mo}\) ? b. What will the interest charges be if she elects the 36 -mo plan? The 48-mo plan?

Step-by-Step Solution

Answer

In conclusion, the monthly payments and interest charges are as follows: - 36-month plan: Monthly payments of approximately $387.97 with interest charges of approximately $1,967.02. - 48-month plan: Monthly payments of approximately $304.35 with interest charges of approximately $2,609.48.

Step 1: 1. Calculate Loan Amount

To find the loan amount, we first need to calculate the amount of the down payment, which is 25% of the car's price: Down payment = 0.25 × \(16,000 = \)4,000 Now subtract the down payment from the car's price to get the loan amount: Loan amount = \(16,000 - \)4,000 = $12,000

Step 2: 2. Convert Annual Interest Rate to Monthly Rate

The annual interest rate is 10%, but since we are asked to calculate the required monthly payments, we must convert the annual rate to a monthly rate. To do so, divide the annual interest rate by 12 months: Monthly interest rate (r) = 0.10 / 12 = 0.008333

Step 3: 3. Calculate Monthly Payments for 36-Month Plan

Now, we will use the monthly payment formula with the loan amount, the monthly interest rate, and 36 months as the number of payments: M = $12,000 × \(\frac{0.008333(1 + 0.008333)^{36}}{(1 + 0.008333)^{36} - 1}\) M ≈ $387.97 Therefore, the required monthly payment for the 36-month plan is approximately $387.97.

Step 4: 4. Calculate Monthly Payments for 48-Month Plan

Using the same formula, we'll calculate the monthly payments for the 48-month plan: M = $12,000 × \(\frac{0.008333(1 + 0.008333)^{48}}{(1 + 0.008333)^{48} - 1}\) M ≈ $304.35 Therefore, the required monthly payment for the 48-month plan is approximately $304.35.

Step 5: 5. Calculate Interest Charges for 36-Month Plan

To find the interest charges for the 36-month plan, first find the total amount paid: Total amount paid = Monthly payment × Number of payments Total amount paid ≈ \(387.97 × 36 = \)13,967.02 Now subtract the loan amount from the total amount paid to get the interest charges: Interest charges ≈ \(13,967.02 - \)12,000 = $1,967.02 Therefore, the interest charges for the 36-month plan are approximately $1,967.02.

Step 6: 6. Calculate Interest Charges for 48-Month Plan

Similarly, find the interest charges for the 48-month plan: Total amount paid = Monthly payment × Number of payments Total amount paid ≈ \(304.35 × 48 = \)14,609.48 Interest charges ≈ \(14,609.48 - \)12,000 = $2,609.48 Therefore, the interest charges for the 48-month plan are approximately $2,609.48. In conclusion, the monthly payments and interest charges are as follows: - 36-month plan: Monthly payments of approximately \(387.97 with interest charges of approximately \)1,967.02. - 48-month plan: Monthly payments of approximately \(304.35 with interest charges of approximately \)2,609.48.

Key Concepts

Compound Interest

Compound interest is a valuable concept in finance math, impacting how money grows over time. Unlike simple interest, which is calculated on the principal alone, compound interest is calculated on the principal plus the accumulated interest from previous periods. This means that, as time progresses, interest accumulates on a growing amount, potentially leading to larger returns.

When applied to loans or savings, compound interest can significantly affect the amount to be paid or received. In our car loan example, the interest rate is compounded monthly. This means interest is added every month, and the next month's interest calculation includes the interest from the previous month as part of the principal. Understanding how compound interest works can help you make better financial decisions.

• Principal is the original amount of the loan or investment.
• Compounding frequency affects how often interest is added to the principal.
• The effective interest rate can be higher than the stated rate due to compounding.

Monthly Payments

Monthly payments are a constant financial commitment made regularly to pay off a loan over its term. Determining the correct monthly payment involves considering several factors, including the principal (initial loan amount), the interest rate, and the loan duration. A clear understanding of these payments can help you manage your budget and financial planning.

In our scenario, the car loan requires monthly payments determined using the compound interest formula. By using this formula, the borrower ensures that the debt is fully repaid by the end of the loan period, including interest. Factors such as loan duration significantly affect the amount of each payment.

• Shorter terms typically result in higher monthly payments but less total interest paid.
• Longer terms usually lower the monthly payment but increase the total interest paid.
• Calculating monthly payments involves finding a balance between what you can afford and minimizing total interest.

Interest Rates

Interest rates are a crucial element in finance math, acting as the cost to borrow money or the reward for investing money. An interest rate is expressed as a percentage of the principal and can be annual, monthly, or for another specified duration. When dealing with loans, the interest rate tells you how much extra you will pay on top of the amount borrowed.

In the example given, we have an annual interest rate of 10%, which needs to be converted to a monthly rate to calculate monthly payments accurately. This conversion is necessary because payments are made monthly, not annually. The monthly interest rate is found by dividing the annual rate by 12. Understanding interest rates and how they are applied can help you evaluate the true cost of a loan.

• Higher interest rates mean higher borrowing costs.
• Compound interest rates can make understanding the true cost of a loan more complex.
• Always convert the interest rate to match the payment frequency.

Loan Calculation

Loan calculations involve determining how much you need to repay over time, considering principal, interest rates, and the loan term. Calculating the terms of a loan can be complex because of the various factors involved, particularly interest rates and compounding. By using the right formulas, you can calculate both the monthly payments and the total cost of borrowing.

The scenario describes calculating the car loan's monthly payments and interest charges using the monthly payment formula, which accounts for compound interest. First, you identify the principal (after down payment), then calculate the monthly payment for the given duration. Finally, total payments help determine the total interest paid.

• Calculating the loan amount involves subtracting any down payments from the purchase price.
• The monthly payment formula uses the principal, monthly interest rate, and loan term.
• Total interest paid is the difference between total payments made and the principal amount.