You have a balance of $12,000 for your tuition on your credit card and you make no further charges. If your APR is 16.7% and each month you make only the minimum payment of 2% of your balance, then find a formula for the balance after t monthly payments.
A. 12,000(0.966362)t
B. 12,000(0.993638)t
C. 12,000(1.0342)t
D. 12,000(1.00581)t
To find the formula for the balance after t monthly payments, we need to understand how the balance changes each month.
The minimum payment is 2% of the balance. So, each month, you pay 0.02 times the balance towards the tuition. This means the remaining balance after each payment is (1 - 0.02) times the previous balance.
The APR is 16.7%, which is equivalent to an annual interest rate. To find the monthly interest rate, we divide the annual rate by 12 (the number of months in a year). So, the monthly interest rate is 16.7% / 12.
Now, let's go through the steps to find the formula:
1. Start with the initial balance of $12,000.
2. After the first payment, the remaining balance is (1 - 0.02) times $12,000.
3. After the second payment, the remaining balance is (1 - 0.02) times the balance from step 2, and so on.
4. Each month, the balance decreases by (1 - 0.02) due to the payment, but increases by the monthly interest rate.
Therefore, the formula for the balance after t monthly payments would be:
Balance = $12,000 times (1 - 0.02)t
Simplifying the equation, we have:
Balance = $12,000 times (0.98)t
To match one of the answer choices, we can rewrite 0.98 as a decimal:
0.98 = 0.993638 (approximately)
Hence, the formula for the balance after t monthly payments would be:
Balance = $12,000(0.993638)t
Therefore, option B. 12,000(0.993638)t is the correct answer.