Questions: You have a balance of 9700 for your tuition on your American Express credit card. Assume that you make no more charges on the card. Also assume that American Express charges 13% APR and that each month you make only the minimum payment of 3% of the balance. How long will it take to get the balance below 50?

Solution Steps

To solve this problem, we need to simulate the monthly balance reduction on the credit card. Each month, the balance is reduced by the minimum payment (5% of the balance), but it also increases due to the interest (13% APR, which is approximately 1.0833% per month). We will iterate month by month, updating the balance until it falls below $50.

We start with a balance of \( B_0 = 9700 \) and an annual percentage rate (APR) of \( 13\% \). The monthly interest rate is calculated as: \[ r = \frac{0.13}{12} \approx 0.0108333 \] The minimum payment rate is \( 5\% \), so the minimum payment each month is: \[ P = 0.05 \times B \]

Each month, the balance is updated by adding the interest and subtracting the minimum payment: \[ B_{n+1} = B_n + (B_n \times r) - (B_n \times 0.05) \] This simplifies to: \[ B_{n+1} = B_n \times (1 + r - 0.05) \] Substituting \( r \): \[ B_{n+1} = B_n \times (1 + 0.0108333 - 0.05) \approx B_n \times 0.9608333 \]

We continue this process until the balance \( B_n \) falls below \( 50 \). After \( 132 \) months, the balance is approximately: \[ B_{132} \approx 49.6954 \]

Final Answer