Final answer:
To calculate how long it will take Phoebe to repay her $6,400 credit card debt with a 16% APR and monthly payments of $225, we use the amortization formula for loans, adjusted to solve for the number of payments using logarithms.
Step-by-step explanation:
Phoebe is dealing with a credit card debt of $6,400 and wants to pay it off by making monthly payments of $225. The credit card has an Annual Percentage Rate (APR) of 16%, compounded monthly. To determine how long it will take her to pay off the debt, we can use the formula for the amortization of a loan, which takes into account the charges due to the interest rate.
Let P be the principal amount ($6,400), r be the monthly interest rate (16% APR divided by 12 months), and n be the total number of payments. The formula for the monthly payment M on an amortizing loan is:
M = P * (r / (1 - (1 + r)^(-n)))
However, since we know the monthly payments (M) and need to find the number of payments (n), we rearrange the formula to solve for n. This involves using logarithms, and the rearranged formula is:
n = -(log(1 - (P*r)/M)) / log(1+r)
By substituting the values into the formula, we calculate the number of months Phoebe will need to completely pay off her credit card debt. This calculation would typically be done using a calculator or spreadsheet software capable of handling such financial functions.