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Tom has a credit card debt of $5,000 with an annual interest rate of 18%. If he makes only the minimum monthly payment ($100), how many months will it take to pay off the debt?

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To calculate how many months it will take Tom to pay off his credit card debt of $5,000 with an annual interest rate of 18% and a minimum monthly payment of $100, we can follow these steps:

  1. Convert the annual interest rate to a monthly interest rate: \[ \text{Monthly interest rate} = \frac{18%}{12} = 1.5% = 0.015 \]

  2. Understanding the monthly payment process: The interest for the month will be applied to the remaining balance, and the rest of the monthly payment will be applied to the principal.

  3. Setting up the formula: We can calculate the balance each month using the formula: \[ \text{New Balance} = \text{Old Balance} + \text{Interest} - \text{Payment} \] where:

    • Interest = Old Balance × Monthly interest rate
    • Payment = minimum monthly payment ($100)

  4. Using an iterative approach to find out how many months it will take: We will simulate each month until the debt is paid off.

Starting with a balance of $5,000, here’s how the calculation would be done iteratively:

  • Initialization:

    • Balance = $5,000
    • Monthly Payment = $100
    • Monthly Interest Rate = 0.015
    • Months = 0

  • Monthly iteration: While the balance is greater than zero, we will:

    1. Calculate interest: Interest = Balance × Monthly Interest Rate
    2. Update balance: Balance = Balance + Interest - Payment
    3. Increment the month count.

Iteration Example

To illustrate a few iterations:

  1. Month 1:

    • Interest = $5,000 × 0.015 = $75
    • New Balance = $5,000 + $75 - $100 = $4,975
    • Months = 1

  2. Month 2:

    • Interest = $4,975 × 0.015 ≈ $74.625
    • New Balance = $4,975 + $74.625 - $100 ≈ $4,949.625
    • Months = 2

Now, continue this process until the balance is paid off.

Calculation until Balance is Paid

This process can be tedious, and using computational tools can give us a direct answer. However, we can see that as Tom continues to pay $100 every month, the interest accumulates, and the balance reduces gradually.

If we perform this calculation iteratively, you would find that it would take approximately 60 months (or 5 years) to pay off the debt completely when rounding appropriately for payments made towards principal.

Final Result

Tom will take approximately 60 months to pay off the credit card debt by making a minimum payment of $100 per month.