What was the interest rate as a percent?

Question

Suppose $80,000 is invested into an account where interest is compounded monthly. After 30 years, the balance is $637,803.
What was the interest rate as a percent?

Answer 1

Let A(t) = amount in t years

        P = principal (initial investment)

        r = annual interest rate as a decimal

       n = number of times that interest is compounded per year

 

Then, A(t) = P(1 + r/n)^nt

 

Substituting in the given values, we have 

 

      637803 = 80000(1 + r/12)³⁶⁰

 

      7.9725375 = (1 + r/12)³⁶⁰

 

     ln(7.9725375) = ln(1 + r/12)³⁶⁰

 

     ln(7.9725375) = 360ln(1 + r/12)

 

     .00576667 = ln(1 + r/12)

 

      1 + r/12 = e^0.00576667

 

      1 + r/12 = 1.005783334

 

           r/12 = 0.005783334

 

              r = 0.0694 = 6.94%