Problem 72 - Algebra and Trigonometry

Question

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 \(n\) th 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

The balance after six months is \$9,100.80.

Step 1: Understand the Recursive Formula

The formula given in the problem for the balance is a recursive relation. It tells us how to find the balance after each month based on the previous month's balance. The formula is:\[ A_{n} = 1.005 A_{n-1} - 200 \]Where \( A_0 = 10,000 \). The term \(1.005 A_{n-1}\) accounts for adding 0.5% interest to the previous month's balance, and \(-200\) accounts for the repayment.

Step 2: Verify the Initial Condition

First, confirm the initial balance:\[ A_0 = 10,000 \]This means Margarita starts with \$10,000 owing, as stated in the problem.

Step 3: Calculate Each Monthly Balance

Apply the recursive formula to determine the balance for each month up to the sixth month:- **Month 1**: \[ A_1 = 1.005 imes 10000 - 200 = 10050 - 200 = 9,850 \]- **Month 2**: \[ A_2 = 1.005 imes 9850 - 200 = 9899.25 - 200 = 9,699.25 \]- **Month 3**: \[ A_3 = 1.005 imes 9699.25 - 200 = 9748.74625 - 200 = 9548.74625 \]- **Month 4**: \[ A_4 = 1.005 imes 9548.74625 - 200 = 9607.4899812 - 200 = 9407.4899812 \]- **Month 5**: \[ A_5 = 1.005 imes 9407.4899812 - 200 = 9454.526430106 - 200 = 9254.526430106 \]- **Month 6**: \[ A_6 = 1.005 imes 9254.526430106 - 200 = 9300.79996285653 - 200 = 9100.79996285653 \]

Step 4: Conclusion - Balance After Six Months

After calculating each month's balance using the recursive formula, the balance after six months is approximately:\[ A_6 = 9,100.80 \] rounded to two decimal places.

Key Concepts

Interest Calculation

Understanding interest calculation is crucial when dealing with any loan. Margarita's loan from her uncle involves a monthly interest rate of 0.5%. This interest is applied to the remaining balance after each monthly payment is made. Here's a breakdown of how it works: - Interest is calculated by multiplying the current balance by the interest rate. In Margarita's case, the interest rate is 0.5%, which translates to multiplying her balance by 1.005. - This calculation increases the total amount Margarita still owes before her monthly payment. - It's important to note that the interest compounds on each remaining balance, which means the interest you pay also depends on how much is left unpaid. In simpler terms, every time there's a remaining balance, interest is calculated and added to it before subtracting her fixed monthly payment of $200. This makes it a key component of the recursive formula used to determine her monthly debt balance.

Loan Repayment

Loan repayment strategies must account for both principal reduction and interest payment. Margarita's method involves paying $200 each month towards reducing her loan. The recursive formula given in the problem illustrates how both the interest and principal payment interplay: - **Principal Payment**: This is the part of Margarita's $200 payment that reduces the actual borrowed amount. As interest grows with the remaining balance, reducing the principal as much as possible each month can decrease the overall interest paid. - **Total Monthly Payment**: The $200 monthly payment gets divided between the interest and principal. The interest is cleared first, and whatever remains goes towards reducing the principal balance. Margarita's repayment strategy is straightforward, but due to the interest, the decrease in her principal loan is not equal to $200 every month. Monthly calculations must account for both interest and repayment, emphasizing why understanding these two concepts can help in effectively managing and repaying loans.

Algebra Problem Solving

When faced with situations like Margarita's loan repayment, algebra problem solving becomes essential. The recursive formula, \( A_{n} = 1.005 A_{n-1} - 200 \), is a great tool for tracking how her debt changes monthly.This type of formula allows you to:- Establish a sequence of values, with each value depending on the preceding one. In Margarita's case, each month relies on the previous month's balance.- Solve real-world problems by methodically calculating following steps. For example, her initial balance is known (\( A_0 = 10,000 \)), and each subsequent balance can be determined step-by-step using the recursive formula.- Grasp complex scenarios by breaking them down into easier parts. Rather than calculating complex interest accumulations manually, the formula simplifies it into manageable monthly bits.Through step-by-step computation, the recursive formula is a powerful algebraic method that simplifies large problems into smaller, visible progress -- much like Margarita's monthly repayments. By calculating each balance successively, algebra serves as a systematic approach to keep the problem organized and solvable.