The equation for determining the half life of a radioactive species based on the decay rate is as follows:
t(1/2) = (t*ln2)/(ln(N(0)/N(t))
Where t(1/2) is the half life, N(0) is the initial quantity of the substance, and N(t) is the amount of substance remaining after t amount of time. So we can input our specific values into the equation and solve for t(1/2).
t(1/2) = (30 min)(0.693)/(ln(160/20)) = 20.79/2.079 = 10 min
So the half life is 10 minutes.
The half-life of a radioactive element is the time needed for half of the material to decay.
Thus , let half life period is t minutes.
decay from 160 to 80 it needed t minutes.
decay from 80 to 40 it needed t minutes.
decay from 40 to 20 ,it neded t minutes.
decay from 160 to 20 ,it needed three half life periods ie. 3t minutes.
But we have given that from 160 to 20 ,it took time 30 minutes.
So half life period is 10 minutes for this particular substance.