Assume that you have a balance of $5000 on your Visa credit card and that you make no more charges. If your APR is 22% and each month you make only the minimum payment of 3% of your balance, then find a formula for the balance after t monthly payments.
A. 5000(0.987783)t
B. 5000(1.048883)t
C. 5000(1.011117)t
D. 5000(0.952217)t
To find a formula for the balance after t monthly payments, we need to first understand how the balance decreases each month.
The minimum payment is 3% of the balance, so each month you will pay 0.03 times the balance. This means the remaining balance after making the minimum payment will be 97% (or 0.97) of the original balance.
Next, since the APR is 22%, we can calculate the monthly interest rate by dividing the APR by 12 (the number of months in a year). So, the monthly interest rate is 22% / 12 = 1.83%.
We can now construct the formula for the balance after t monthly payments:
Balance after t months = Initial balance * (1 - Minimum payment percentage)^(t)
Balance after t months = $5000 * (0.97)^(t)
Now we can compare this formula to the given options:
A. 5000(0.987783)^t: This is not the correct formula. The exponent should be t, not t multiplied by 0.987783.
B. 5000(1.048883)^t: This is not the correct formula. The value inside the parentheses should be less than 1 to reflect the decreasing balance due to the minimum payment.
C. 5000(1.011117)^t: This is not the correct formula. The value inside the parentheses should be less than 1 to reflect the decreasing balance due to the minimum payment.
D. 5000(0.952217)^t: This is the correct formula. The value inside the parentheses is the correct reduction factor for each month's balance due to the minimum payment.
Therefore, the correct formula for the balance after t monthly payments is 5000(0.952217)^t, which corresponds to option D.