Starting with the continuous compound interest formula, we have A = Pe^rt where P is the principal amount to be invested, r is the rate of annual interest to be compounded continuously, t is the period of time in years for which the amount P invested, A is the total amount in the account at the end of time t.
Given that a family paid $99,000 cash for a house and it was sold for $195,000 after fifteen years. Now we have to find the annual nominal rate of interest did the original $99,000 investment earn if interest is compounded continuously. Substituting the values we have P=$99,000, A=$195,000, t =15 years.
Hence, the equation above becomes 195,000 = 99,000e^r(15) => e^r(15) = 195,000/99,000 => e^r15 = 65/33.
ð Ln(e^r15) = ln (65/33) => 15rln€ = 0.67788
ð R = 0.67788 /15 = 0.045192 = 4.52%
The annual nominal rate of interest is 4.52%.
