There are three lumps of a given active substance. Their activities are in the ratio of $1 : 2 : 3$  now. What will be the ratio of their activities at any further date?

Suppose the activities of the lumps at present are  ${\mathrm{A}}_1,{\mathrm{A}}_{2 }\mathrm{and} {\mathrm{A}}_3.$
${\mathrm{A}}_1:{\mathrm{A}}_2:{\mathrm{A}}_3=\mathrm{\lambda }{\mathrm{N}}_1:\mathrm{\lambda }{\mathrm{N}}_2:\mathrm{\lambda }{\mathrm{N}}_3=1:2:3.$
  After time t, Let their activities be  ${\mathrm{A}}_1^1,{\mathrm{A}}_2^1\mathrm{and} {\mathrm{A}}_3^1$
$\begin{array}{l}\therefore {\mathrm{A}}_1^1:{\mathrm{A}}_2^1:{\mathrm{A}}_3^1={\mathrm{\lambda N}}_1^1:{\mathrm{\lambda N}}_2^1:{\mathrm{\lambda N}}_3^1=\mathrm{N}{}_1{}^{ }\mathrm{e}^{-\mathrm{\lambda t}}:\mathrm{N}{}_2{}^{ }\mathrm{e}^{-\mathrm{\lambda t}}:\mathrm{N}{}_3{}^{ }\mathrm{e}^{-\mathrm{\lambda t}}\\ =1:2:3.\end{array}$