Two radioactive substances A and B have decay constants 5λ and λ respectively. At t=0, they have the same number of nuclei. The ratio of number of nuclei of A to those of B will be (1/e²) after a time:

Given :
(No)A = (No)B = No
Using : Kt = ln(N/No)
where, K is the decay constant
⇒ Kt = lnN - lnNo
For substance A :
λt = lnNA - ln(No)A
⇒ λt = lnNA - lnNo (1)
For substance B :
5λt = lnNB - ln(No)B
⇒ 5λt = lnNB - lnNo (2)
Subtracting (2) from (1) we get:
λt - 5λt = ln(NA/NB)
-4λt = ln(1/e²)
-4λt = -2;
⇒ t = 1/2λ