Question

Credit card A has an APR of 19.3% and an annual fee of $84, while credit card B has an APR of 24.6% and no annual fee. All else being equal, which of these equations can be used to solve for the principal P for which the cards offer the same deal over the course of a year? (Assume all interest is compounded monthly.)

Answer 1

The equations are missing but we can solve the problem from the given information

The equation
`P(1+(0.193)/(12))^(12)+84=P(1+(0.246)/(12))^(12)` can be used to solve for the principal P for which the cards offer the same deal over the course of a year

Explanation:

The formula of the compounded interest is
`A=P(1 + (r)/(n))^(nt)` , where

• A is the future value of the investment/loan, including interest
• P is the principal investment amount
• r is the annual interest rate (decimal)
• n is the number of times that interest is compounded per unit t
• t is the time the money is invested or borrowed for

Card A:

∵ Credit card A has an APR of 19.3% and an annual fee of $84

∴ r = 19.3% = 19.3 ÷ 100 = 0.193

∴ Annual fee = 84

∵ Interest is compounded monthly for a year

∴ n = 12 and t = 1

- Substitute the values of r, n and t in the formula above

∴
`A=P(1 + (0.193)/(12))^(12(1))`

- Add the value of the annual fee

∴
`A=P(1 + (0.193)/(12))^(12)+84`

Card B:

∵ Credit card B has an APR of 24.6% and no annual fee

∴ r = 24.6% = 24.6 ÷ 100 = 0.246

∴ Annual fee = 0

∵ Interest is compounded monthly for a year

∴ n = 12 and t = 1

- Substitute the values of r, n and t in the formula above

∴
`A=P(1 + (0.246)/(12))^(12(1))`

∴
`A=P(1 + (0.246)/(12))^(12)`

- Equate The equations of cards A and B

∴
`P(1+(0.193)/(12))^(12)+84=P(1+(0.246)/(12))^(12)`

The equation
`P(1+(0.193)/(12))^(12)+84=P(1+(0.246)/(12))^(12)` can be used to solve for the principal P for which the cards offer the same deal over the course of a year