You have a balance of $7500 on your credit card and you make no further charges. If
your APR is 12.5% and each month you make only the minimum payment of 5% of
your balance, then find a formula for the balance after
t
monthly payments.
A) 7500(0.959896)
t
B) 7500(0.940104)
t
C) 7500(1.03906)
t
D) 7500(1.06094)
t
i = .125/12 = .01041666..
balance after 1 month before payment
= 7500(1.01041666.)
payment = .05 of balance
balance at end of month 1 = 7500(.95)(1.0104166)
balance after month 2 before payment
= 7500(.95)(1.0104166)^2
payment = .05 of that balance
balance at end of month 2
= 7500(.95)^2 (1.0104166)^2
= 7500(.9598958..)^2
balance at end of month t
= 7500(.9598958..)^t
the closest would be A) if it had been typed correctly
To calculate the balance after t monthly payments, you need to understand the concept of compound interest. Here's how you can calculate it step by step.
1. Calculate the monthly interest rate. Convert the annual percentage rate (APR) to a monthly rate by dividing it by 12. In this case, the monthly interest rate is 12.5% / 12 = 0.0104167.
2. Calculate the minimum payment for the first month. Multiply the balance by 5% to find the minimum payment. The minimum payment for the first month is 7500 * 0.05 = 375.
3. Calculate the interest charged for the first month. Multiply the balance by the monthly interest rate. The interest charged for the first month is 7500 * 0.0104167 = 78.125.
4. Subtract the minimum payment from the balance. The new balance after the first month is 7500 - 375 = 7125.
5. Add the interest charged to the new balance. The balance after the first month with interest added is 7125 + 78.125 = 7203.125.
6. Repeat steps 2-5 for each subsequent month until you reach t months.
Based on the steps above, the formula to calculate the balance after t monthly payments is:
Balance after t months = (Initial balance - (Minimum payment * t)) + (Monthly interest rate * (Initial balance - (Minimum payment * t)))
Now, let's simplify the answer choices to find the correct formula:
A) 7500 * (0.959896)^t
B) 7500 * (0.940104)^t
C) 7500 * (1.03906)^t
D) 7500 * (1.06094)^t
Based on the formula we derived, none of the answer choices match exactly. However, we can see that answer choice A) is very close to the formula we derived. So, the closest formula for the balance after t monthly payments is:
A) 7500 * (0.959896)^t