PROBLEM SET 4, QUESTION 1:

Twenty years elapse from 1995 to 2015. Since the half-life is 20 years, the number of nuclei that decay in that period is half of the number that existed at the start. Hence 1,000,000 nuclei decay, and 1,000,000 nuclei remain in 2015.

(This reduction by half can be applied to numbers of nuclei, or equally well to total weight.)

The time from 2015 to 2035 is also 20 years. Hence, during that period the number of nuclei that decay is half of what there was at the beginning of that period , namely half of 1,000,000. So 500,000 nuclei decay in this 20-year period; 500,000 are left at the end, in 2035. Note that what happens between 2015 and 2035 is not calculated on the basis of what we have in 1995. It is calculated only on the basis of what we have in 2015.

In the same way, between 2035 and 2055, half of the initial number of nuclei decay. This initial number is 500,000. So 250,000 decay, and 250,000 remain. You can now see the progression. After another 20 years, the number is reduced again by half, to 125,000.