Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Two years ago, Paul borrowed $$\$ 10,000$$ from his sister Gerri to start a business. Paul agreed to pay Gerri interest for the loan at the rate of \(6 \% /\) year, compounded continuously. Paul will now begin repaying the amount he owes by amortizing the loan (plus the interest that has accrued over the past 2 yr) through monthly payments over the next 5 yr at an interest rate of \(5 \% /\) year compounded monthly. Find the size of the monthly payments Paul will be required to make.

Step-by-Step Solution

Answer

The size of the monthly payments that Paul will be required to make is approximately $$\$189.85.$$

Step 1: Calculate the accrued interest over 2 years compounded continuously

To find the amount of interest accrued over 2 years, use the formula for continuously compounded interest: \(A = P \cdot e^{rt}\), where A is the final amount, P is the principal borrowed, r is the annual interest rate, and t is the time in years. In this case, \(P = \$10,000\), \(r = 6\% = 0.06\), and \(t = 2\) years. \(A = 10000 \cdot e^{0.06 \cdot 2}\).

Step 2: Calculate the total amount owed after interest

Evaluate the expression from Step 1 to find the total amount owed, including principal and interest accrued: \(A = 10000 \cdot e^{0.12} \approx $ \$11,275.83$$.

Step 3: Calculate the monthly interest rate

We know that the annual interest rate is 5%. Next, we need to find the equivalent monthly interest rate, r. With the formula: \(1 + r_{annual} = (1 + r_{monthly})^{12}\). Solving for r_{monthly}: \(r_{monthly} = (1 + 0.05)^{\frac{1}{12}} - 1 \approx 0.4074\%\).

Step 4: Calculate the number of monthly payments

We are given that the loan is to be paid off in 5 years, and the payments are to be made monthly. Therefore, the total number of monthly payments, n, is: \(n = 5 \text{ years} \cdot 12 \text{ payments/year} = 60 \text{ payments}\).

Step 5: Calculate the monthly payment amount

We will now use the annuity formula to find the monthly payment, P. The annuity formula is: \(P = \dfrac{A \cdot r_{monthly}}{1 - (1 + r_{monthly})^{-n}}\), where A is the amount owed, r_{monthly} is the monthly interest rate, and n is the number of payments. Plugging in the values from steps 2, 3, and 4, we have: \(P = \dfrac{11275.83 \cdot 0.004074}{1 - (1 + 0.004074)^{-60}}\).

Step 6: Evaluate the expression for the monthly payment amount

Evaluate the expression from Step 5 to find the monthly payment, P: \(P \approx $ \$189.85$$. The size of the monthly payments that Paul will be required to make is approximately $$\$189.85.$$

Key Concepts

Understanding Continuously Compounded Interest

When interest is compounded continuously, it grows not just at the set intervals (like annually or monthly) but at every possible moment. Imagine watching your money grow by a tiny bit every second!
This is different from, say, annual compounding where interest is added once a year.
Here’s the magic formula for continuously compounded interest:

• \( A = P \cdot e^{rt} \)
• \( A \) is the amount after time \( t \)
• \( P \) is the initial principal (\(\\)10,000$ in our example)
• \( r \) is the annual interest rate (expressed as a decimal, so \(6\%\) becomes \(0.06\))
• \( t \) is the time in years (2 years here)

This formula lets us find out how much Paul will have to pay back after two years before he starts monthly payments.

Monthly Interest Rate Calculation Made Easy

When dealing with loans or savings, interest rates are often expressed annually, but payments might be monthly. To align this, we convert from an annual rate to a monthly one.
This involves a neat formula that looks like this:

• \(1 + r_{annual} = (1 + r_{monthly})^{12}\)
• We solve for \(r_{monthly}\) using the above relationship
• In this case, \(r_{annual}\) is \(5\% (=0.05)\)

By using the formula, you find that \(r_{monthly} \approx 0.4074\%\).
This will be the interest rate used for calculating the monthly payments.

Deciphering the Annuity Formula

An annuity formula helps compute regular payments needed to pay off a loan over time. For Paul, he pays off the loan in 60 monthly installments, making this formula essential.The formula used is:

• \(P = \frac{A \cdot r_{monthly}}{1 - (1 + r_{monthly})^{-n}}\)
• \(P\) is the monthly payment we want to find
• \(A\) is the total amount owed, \(\\(11,275.83\)
• \(r_{monthly}\) is the monthly interest rate found earlier
• \(n\) is the total number of payments (60 in this case)

The equation helps balance between what is owed and the length of the loan.
After applying the numbers, Paul's monthly payment is about \)\\(189.85\).

Mathematics Education: Bringing Concepts to Life

Understanding these financial concepts has real-world applications.
Mathematics education isn't just about numbers; it's about interpreting these numbers so they support decision-making.

• Hands-on learning allows students to simulate real-life situations, like applying for loans.
• Seeing how interest accumulates over time can encourage better financial planning.
• Using formulas enhances problem-solving skills and instills confidence.

By mastering these concepts, you can approach financial decisions with a critical eye. This empowerment through education makes handling such calculations less daunting and more intuitive.