Paying Off Credit Card Debt

Each month, a credit card statement will come with the option for you to pay a minimum amount of your charge, usually 2% of the balance due. However, the credit card company earns money by charging interest on the balance that you don't pay. So even if you pay credit card payments on time, interest is still accruing on the outstanding balance.
Say you've made a $5,000 purchase on a credit card with an 18% annual interest rate and a 2% minimum monthly payment rate. If you only pay the minimum monthly amount for a year, how much is the remaining balance?
You can think about this in the following way.
At the beginning of month 0 (when the credit card statement arrives), assume you owe an amount we will call      (b for balance; subscript 0 to indicate this is the balance at month 0).
Any payment you make during that month is deducted from the balance. Let's call the payment you make in month 0,     . Thus, your unpaid balance for month 0,     , is equal to         .
            
 
At the beginning of month 1, the credit card company will charge you interest on your unpaid balance. So if your annual interest rate is  , then at the beginning of month 1, your new balance is your previous unpaid balance     , plus the interest on this unpaid balance for the month. In algebra, this new balance would be
                 
 
In month 1, we will make another payment,     . That payment has to cover some of the interest costs, so it does not completely go towards paying off the original charge. The balance at the beginning of month 2,     , can be calculated by first calculating the unpaid balance after paying     , then by adding the interest accrued:

To calculate the remaining balance after making minimum monthly payments for a year, follow these steps:

Step 1: Convert the annual interest rate to a monthly interest rate. Divide the annual interest rate by 12. In this case, the annual interest rate is 18%, so the monthly interest rate would be 18% / 12 = 1.5%.

Step 2: Calculate the monthly minimum payment. Multiply the balance by the minimum monthly payment rate (2%) to find the minimum payment amount for each month. In this case, the balance is $5000, so the minimum monthly payment would be $5000 * 2% = $100.

Step 3: Calculate the remaining balance after each monthly payment and interest charge.

Month 0:
- Starting Balance (b0): $5000
- Minimum Monthly Payment: $100
- Unpaid balance after payment (ub0): b0 - minimum monthly payment = $5000 - $100 = $4900
- Balance at the beginning of month 1 (b1): ub0 + monthly interest accrued = $4900 + ($4900 * 1.5%) = $4973.50

Month 1:
- Starting Balance (b1): $4973.50
- Minimum Monthly Payment: $100
- Unpaid balance after payment (ub1): b1 - minimum monthly payment = $4973.50 - $100 = $4873.50
- Balance at the beginning of month 2 (b2): ub1 + monthly interest accrued = $4873.50 + ($4873.50 * 1.5%) = $4943.31

Repeat these calculations for each subsequent month until you reach month 12:

Month 2:
- Starting Balance: $4943.31
- Minimum Monthly Payment: $100
- Unpaid balance after payment: $4843.31
- Balance at the beginning of month 3: $4843.31 + ($4843.31 * 1.5%)

...

Month 12:
- Starting Balance: $XXX
- Minimum Monthly Payment: $XXX
- Unpaid balance after payment: $XXX
- Remaining Balance at the end of month 12: $XXX

By performing these calculations for each month, you will be able to determine the remaining balance after making minimum monthly payments for a year on a credit card with a given balance and interest rate.