Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Dan is contemplating trading in his car for a new one. He can afford a monthly payment of at most $$\$ 400 .$$ If the prevailing interest rate is \(7.2 \% /\) year compounded monthly for a 48 -mo loan, what is the most expensive car that Dan can afford, assuming that he will receive $$\$ 8000$$ for the trade-in?

Step-by-Step Solution

Answer

The most expensive car Dan can afford is approximately \$23,921.15.

Step 1: Convert the given interest rate to a monthly interest rate

The given interest rate is 7.2% per year compounded monthly. To find the monthly interest rate, divide the annual interest rate by 12. \(Monthly \, interest \, rate = \frac{7.2}{12}\) \(Monthly \, interest \, rate = 0.6 \%\)

Step 2: Calculate the monthly interest rate in decimal form

Convert the monthly percentage interest rate to a decimal by dividing by 100. \(Decimal \, monthly \, interest \, rate = \frac{0.6}{100}\) \(Decimal \, monthly \, interest \, rate = 0.006\)

Step 3: Calculate the maximum loan amount based on the monthly payment and interest rate

We use the loan payment formula, where \(P\) is the loan payment, \(r\) is the decimal monthly interest rate, \(L\) is the loan amount, and \(n\) is the number of payments (months): \(P = L\frac{r(1 + r)^n}{(1 + r)^n - 1}\) Now, rearrange the formula to solve for the maximum loan amount, \(L\): \(L = P \frac{(1 + r)^n - 1}{r(1 + r)^n}\) Plug in the values for \(P\), \(r\), and \(n\) (400, 0.006, and 48, respectively) and solve for \(L\): \(L = 400 \frac{(1 + 0.006)^{48} - 1}{0.006(1 + 0.006)^{48}}\) \(L \approx\$ 15921.15\)

Step 4: Add the trade-in value to find the maximum car price

Now, add the trade-in value of Dan's current car ($8000) to the maximum loan amount to find the most expensive car he can afford: \(Car \, price = Loan \, amount + Trade \, in \, value\) \(Car \, price = \$ 15921.15 + \$ 8000\) \(Car \, price = \$ 23921.15\) So, the most expensive car Dan can afford is approximately $23921.15.

Key Concepts

Interest Rate Conversion

When you're dealing with loans, understanding how interest rates work is crucial. Typically, interest rates are given annually, but when payments are made more frequently, like monthly, you need to convert this annual rate into a monthly rate. This is important because loan calculations usually depend on the frequency of compounding.

To convert an annual interest rate to a monthly rate, you divide by 12, since there are 12 months in a year. In Dan's case, the annual interest rate is 7.2%. So, the monthly rate becomes:

• Annual Rate: 7.2%
• Monthly Rate: \( \frac{7.2}{12} = 0.6\% \)

Keep in mind that percentages must be converted to decimals for most calculations by dividing by 100, which gives:

• Decimal Monthly Rate: \( \frac{0.6}{100} = 0.006 \)

Monthly Payment Calculation

Loan payments made on a monthly basis require careful calculation to ensure you can afford them. The formula for calculating your monthly payment revolves around the loan amount, the term of the loan, and the interest rate.

The formula in question here is:

• \( P = L \frac{r(1 + r)^n}{(1 + r)^n - 1} \)

Where:

• \( P \) is the monthly payment
• \( L \) is the loan amount
• \( r \) is the monthly interest rate in decimal form
• \( n \) is the total number of payments

Understanding this formula will help you to determine what you can realistically afford based on the monthly payments.

Loan Amount Calculation

Calculating the maximum loan amount is often necessary when deciding how much you can borrow based on a fixed monthly payment. Utilizing the monthly payment formula, you can rearrange it to solve for the loan amount, \( L \), using this equation:

• \( L = P \frac{(1 + r)^n - 1}{r(1 + r)^n} \)

For Dan, this translates to:

• \( P = 400 \)
• \( r = 0.006 \)
• \( n = 48 \)

Substituting these values into the formula provides a loan amount of about $15,921.15. This is the foundation upon which Dan's purchasing decisions for the car will be made.

Trade-in Value

The trade-in value of your current car plays an essential role in determining the overall budget for your new vehicle purchase. This is because the trade-in value effectively reduces the amount you need to borrow or pay out of pocket.

In Dan's scenario, he will receive \(8,000 for his old car. Therefore, you add this trade-in value to the maximum loan amount you calculated earlier. This forms the total spending limit for purchasing a new car.

• Maximum Loan: \)15,921.15
• Trade-in Value: \(8,000
• Total Possible Budget: \( 15,921.15 + 8,000 = 23,921.15 \)

Thus, the total maximum price for a new car that Dan can afford is around \)23,921.15. Always remember to consider the trade-in value as it can significantly alter your spending power.