Problem 74 - Algebra and Trigonometry

Question

Paying off a Debt Margarita borrows \(\$ 10,000\) from her uncle and agrees to repay it in monthly installments of \(\$ 200 .\) Her uncle charges 0.5\(\%\) interest per month on the balance. (a) Show that her balance \(A_{n}\) in the nth month is given recursively by \(A_{0}=10,000\) and $$A_{n}=1.005 A_{n-1}-200$$ (b) Find her balance after six months.

Step-by-Step Solution

Answer

After six months, the balance is $9,088.67.

Step 1: Understanding the Problem Formulation

Margarita's initial debt is $10,000, and she will repay $200 each month. Her uncle adds an interest of 0.5% per month. We need to express the balance in terms of a recursive formula and calculate the balance after six months.

Step 2: Proving the Recursive Formula - Initial Condition

Start with the initial condition: Margarita's initial debt is \[ A_0 = 10,000. \] This sets the starting point for the recursive relationship.

Step 3: Understanding the Monthly Balance Calculation

Each month, the balance incurs a 0.5% interest, and then \(200 is deducted as repayment. If the balance from the previous month is \( A_{n-1} \), the balance after adding interest is \[ 1.005 \times A_{n-1}. \] After deducting \)200 for repayment, the new balance is \[ A_{n} = 1.005 \times A_{n-1} - 200. \] This confirms the recursive formula \( A_{n} = 1.005A_{n-1} - 200 \).

Step 4: Calculating the Balance After Six Months - Month by Month Approach

Using the recursive formula, we'll calculate the balance for each month up to the sixth month:1. \( A_0 = 10,000 \)2. \( A_1 = 1.005 \times 10,000 - 200 = 10,050 - 200 = 9,850 \)3. \( A_2 = 1.005 \times 9,850 - 200 = 9,899.25 - 200 = 9,699.25 \)4. \( A_3 = 1.005 \times 9,699.25 - 200 = 9,747.74625 - 200 = 9,547.74625 \)5. \( A_4 = 1.005 \times 9,547.74625 - 200 = 9,595.48498125 - 200 = 9,395.48498125 \)6. \( A_5 = 1.005 \times 9,395.48498125 - 200 = 9,442.46240515625 - 200 = 9,242.46240515625 \)7. \( A_6 = 1.005 \times 9,242.46240515625 - 200 = 9,288.6747171828125 - 200 = 9,088.6747171828125 \)

Step 5: Conclusion - Balance After Six Months

Based on our calculations, the balance after six months is \( 9,088.67 \). This is the amount still owed by Margarita to her uncle after six months, taking into account the interest and repayments.

Key Concepts

Debt Repayment

Debt repayment is a process where borrowed money is paid back over time. Margarita, in our example, borrowed $10,000 from her uncle, and she plans to repay it in monthly installments. This repayment plan is structured so that Margarita pays a fixed amount each month, which is $200 in this case. This approach allows borrowers to manage their debt efficiently by spreading the repayment over a longer period.

Debt repayment typically involves both principal and interest. The principal is the original amount borrowed, and interest is a charge for borrowing the money. Regular repayment schedules ensure that borrowers consistently reduce their debt, thereby avoiding large, unexpected expenditures at the end of their repayment period.

If you are planning to repay a debt, consider the following steps:

• Determine the monthly installment you can afford.
• Understand the total cost of borrowing, including interest.
• Adhere to a repayment schedule to avoid late fees or additional interest charges.

Interest Calculation

Interest calculation is crucial in understanding how debt grows or reduces over time. Interest is the additional amount charged on the debt itself and is usually expressed as a percentage of the principal balance. For Margarita, her uncle charges 0.5% interest per month on the outstanding balance.

To calculate the new balance each month, the interest rate is applied to the remaining balance from the previous month before any repayment is made. The formula used is:

• Multiply the previous balance by the interest rate to get the interest amount.
• Add this interest to the previous balance to update the balance.

The interest calculation for Margarita's debt looks like this: 1. Start with the previous balance, say for the first month, it's $10,000. 2. Compute the interest: 0.5% of $10,000, which is $50. 3. Add this $50 to the original amount to get $10,050. This gives the new balance before repayment. 4. Subtract the $200 repayment to arrive at the balance for the next month, which is $9,850.

Mathematical Modeling

Mathematical modeling involves using equations and formulas to represent real-world situations, like debt repayment. In this example, Margarita's debt repayment structure is expressed using a recursive sequence. A recursive sequence is one where the next term is defined in terms of the previous one, using a specific rule.

The rule for Margarita's debt repayment is represented by the recursive formula: \( A_n = 1.005 A_{n-1} - 200 \), where:

• \( A_n \) is the balance after month \( n \).
• \( A_{n-1} \) is the balance from the previous month.
• 1.005 is the factor representing a 0.5% interest rate.
• 200 is the monthly repayment amount.

Through this formula, we see how the principle of recursive sequences allows us to compute each subsequent month's balance using the previous month's balance. This method is handy for projecting future balances and repayments efficiently without needing to redo calculations from scratch for each period.