Problem 67 - Contemporary Precalculus

Question

The minimum monthly payment for a certain bank credit card is the larger of \(1 / 25\) of the outstanding balance or \(\$ 5\). If the balance is less than \(\$ 5,\) the entire balance is due. If you make only the minimum payment each month, how long will it take to pay off a balance of \(\$ 200\) (excluding any interest that might be due)?

Step-by-Step Solution

Answer

Answer: It will take 36 months to pay off the initial balance of $200 without any accumulated interests.

Step 1: Determine the Payment Plan

According to the given conditions, for a balance of $200, the monthly payment would be 1/25 of the balance.

Step 2: Calculate the Monthly Payments

Each month, calculate the minimum payment which is her 1/25 of the remaining balance, as long as the balance is over $200.

Step 3: Adjust the Payment Plan If Needed

If, at some point, the remaining balance falls below \(200, adjust the monthly payment to \)5 (as long as the balance is over \(5). Finally, if the remaining balance falls below \)5, then the full balance is due.

Step 4: Track the Number of Months Until the Balance Is Paid Off

Start with the initial balance of $200, and for each month, subtract the appropriate monthly payment amount. At the same time, keep track of the number of months it takes to reach a balance of 0.

Step 5: Calculate the specific monthly balance

The monthly balance for the first 8 months would be \(200-(200/25) = \)192,\( \)192-(192/25) = \(184.32,\) \(184.32-(184.32/25) = \)176.97,\( etc. It would continue being 1/25 of the remaining balance until the balance drops below \)200.

Step 6: Adjustments According to Payment Plan

In the 9th month, the remaining balance would be slightly below \(200, thus the minimum monthly payment would be \)5. This would continue until the remaining balance falls below $5.

Step 7: Final Calculation

Following the conditions and calculations outlined above, the balance will continue to decrease each month. It will take 36 months to pay off the balance without any accumulated interests.

Key Concepts

Minimum Monthly Payment

Understanding the concept of the minimum monthly payment is crucial when managing credit card debts or loans. It refers to the lowest amount you can pay by the due date on your account each month without facing penalties. This value is usually set by the credit card company or lender and is often a small percentage of your total outstanding balance or a fixed value, whichever is greater. For example, if you have an outstanding credit card balance of \(200 and the terms require the larger of 1/25th of the balance or \)5, your minimum payment would be \(8 (since 1/25 of \)200 is \(8, and it's larger than \)5). However, only paying the minimum can result in long repayment periods and high interest costs. Therefore, understanding how to calculate this amount and the impact it has on your overall debt is essential for effective financial planning.

Payment Plan Calculation

Creating a payment plan for your debts involves more than just knowing the minimum payment; it requires a systematic approach to determine how long it will take to pay off the entire amount. To calculate a payment plan, one must consider the outstanding balance, the minimum payment percentage, and any applicable fixed minimum payment amounts. In our example with a \(200 balance, a payment of 1/25 per month is calculated until the balance drops below \)200. If the remaining balance is less than \(200 but more than \)5, the payment switches to \(5 per month. Should the balance dip below \)5, the full amount is due. It's important to track these payments over time and adjust the plan as needed to accurately forecast when the debt will be cleared.

Linear Equations in Finance

Linear equations are at the heart of many financial calculations, including the determination of payment schedules. They provide a way to model financial situations, such as loan repayments or investment growth, by setting up relationships between different variables. In the context of our monthly payment problem, the equation starts as a straightforward linear relationship: each month, the payment is calculated to be a fixed proportion (1/25) of the current balance. As payments are made, the balance decreases in a predictable linear fashion, which is interrupted only by the fixed payment thresholds as dictated by the policy (\(5 and the full balance if under \)5). Thus, having a firm grasp on how linear equations apply to financial scenarios can empower students to manage their personal finances more effectively.