Question

The population of town A is double that of town B but it is decreasing at the rate of 5% per year. The population of town B is increasing at the rate of 5% per year. In how many years will the population of town B be double that of town A

Answer 1

B0*(1.05)^n = 2A0*(.95)^n
or B0*(1.05)^n = 2*2*B0*(.95)^n
or (1.05/.95)^n = 4
or n = ln(4) / ln(1.05/.95)
or n = 13.851383446 years

Answer 2

Another Solution :
Solve for t such that B(t) = 2 * A(t)
B0 * 1.05^t = 2 * (A0 * 0.95^t)
Substitute to eliminate A0:
B0 * 1.05^t = 2 * (2 * B0 * 0.95^t)
B0 * 1.05^t = 4 * B0 * 0.95^t
1.05^t = 4 * 0.95^t
log(1.05^t) = log(4 * 0.95^t)
log(1.05^t) = log(4) + log(0.95^t)
t * log(1.05) = log(4) + t * log(0.95)
t * log(1.05) - t * log(0.95) = log(4)
t * (log(1.05) - log(0.95)) = log(4)
t = log(4) / (log(1.05) - log(0.95))

Answer 3

@stefffi

Answer 4

@hba thanks a bunch :) especially for providing multiple solutions

Answer 5

@stefffi You Are Welcome :)