Growth of an Account

Question

Russ McClelland, who is self-employed, wants to invest $60,000 in a pension plan.  One investment offers 5% compounded quarterly. Another offers 4.75% compounded continuously. If Russ chooses the plan with continuous compounding, how long will it take for his $60,000 to grow to $80,000?

Answer 1

The model equation for periodic interest is A = P(1 + r/n)^nt

 

Thus,  80,000 = 60,000(1 + 0.05/4)^4t, since we are solving for the exponent t we muse use logarithms.

 

      80,000/60,000 = (1.0125)^4t

 

                4/3 = (1.0125)^4t

 

           log (4/3) = log (1.0125)^4t

 

           log (4/3) = 4t(log 1.0125)

 

         log (4/3) / log 1.0125 = 4t

 

         [log (4/3) / log 1.0125] / 4 = t

 

                            t = 5.79 years (correct to the nearest hundredth0

 

For continuous compounding the model equation is A = Pe^rt

 

Thus,       80,000 = 60,000e^(0.0475t)

 

                   4/3 = e^(0.0475t)

 

                ln (4/3) = ln e^(0.0475t)

 

                ln (4/3) = 0.0475t (ln e) and note that ln e = 1 

 

                ln (4/3) = 0.0475t

 

             ln (4/3) / 0.0475 = t

 

                         t = 6.06 years (correct to the nearest hundredth)