Alan writes the equation A=500(1.25)t to figure out how much it will cost him for a one-year loan of $500 with an interest rate of 25% compounded only once. He wants to determine the annual percentage rate (APR) for a loan that would cost the same amount overall, if it is compounded monthly, instead of only once. Which equation should he use, and what is the APR?

A≈500(1.25)12tAPR≈1.88%

A≈500(1.0188)12tAPR≈22.56%

A≈500(1.0188)12tAPR≈1.88%

A≈500(1.2512)112tAPR≈14.55%

, APR ≅ 22.56% ⇒ 2nd answer

Step-by-step explanation:

The formula of the compounded interest is , where

∵ The equation of one-year loan of $500 with an interest rate

   of 25% compounded only once is

∴ t = 1

∴ A = 500(1.25) = 625

∵ A loan that would cost the same amount overall, if it is

    compounded monthly, instead of only once

∴ n = 12 ⇒ compounded monthly

∵ t = 1 and n = 12

∵ P = 500

- Use the formula of the compounded interest above

∴

- Equate A by 625 (the value of money of the 1st equation)

∵

- Divide both sides by 500

∴

- Reverse the exponent and take it the the other side

∴

- Subtract 1 from both sides

∴

∴

- Multiply both sides by 12

∴ r ≅ 0.2256

- Multiply it by 100%

∴ r = 22.56%

The annual percentage rate (APR) is 22.56%

Substitute the value of r in the equation above

∴

∵

∴

He should use the equation

898.1666667

step-by-step explanation:

5389/6 = 898.1666667