Question

In a bank principal increases at the rate of 5% per year. In how many year Rs. 1000 double itself.

Answer 1

Let P be the principal at any time t 

Then, dp/dt = 5 % of p = 5p/100

dp/dt = p/20

(1/p).dp = (1/20)dt

Integrating, we get

∫(1/p) dp = (1/20)∫dt

log p = (1/20)t + log c (log c is arbitrary constant)

log(p/c) = (1/20)t

p = ce^t/20

when t = 0, p = 1000 ....(1)

By (1), we get c = 1000

∴ p = 1000 e^t/20

Let T years be the require time to double the principal.

i.e., t = T, p = 2000

So, By (1), 2000 = 1000 e^T/20

e^T/20 = 2

T/20 = log e²

T = 20, log 2

Hence, principal double in 20 log 2 years.