A sample of uranium is a mixture of three isotopes ₉₂U²³⁴ , ₉₂U²³⁵ and ₉₂U²³⁸ present in the ratio 0.006%, 0.71% and 99.284% respectively. The half lives of then isotopes are 2.5×10⁵ years, 7.1×10⁸ years and 4.5×10⁹ years respectively.  The contribution to activity (in %) of each isotope in the sample respectively

Let m is the total mass of the uranium mixture. The masses of the isotopes  ₉₂U²³⁴ , ₉₂U²³⁵ and ₉₂U²³⁸  in the mixture are ${\mathrm{m}}_1=\frac{0.006}{100}\mathrm{m};$ ${\mathrm{m}}_2=\frac{0.71}{100}\mathrm{m},\text{ and }{\mathrm{m}}_3=\frac{99.284}{100}\mathrm{m}$.

If N_A is the Avogadro number, then number of atoms of three isotopes are ${\mathrm{N}}_1=\frac{{\mathrm{m}}_1{\mathrm{N}}_{\mathrm{A}}}{{\mathrm{M}}_1};$${\mathrm{N}}_2=\frac{{\mathrm{m}}_2{\mathrm{N}}_{\mathrm{A}}}{{\mathrm{M}}_2},\text{ and }{\mathrm{N}}_3=\frac{{\mathrm{m}}_3{\mathrm{N}}_{\mathrm{A}}}{{\mathrm{M}}_3}$.

Activity of radioactive sample A = λN.

$\mathrm{As} \mathrm{\lambda } =\frac{0.693}{{\mathrm{t}}_{1/2}}, \therefore \mathrm{A}=\frac{0.693}{{\mathrm{t}}_{1/2}}\mathrm{N}$

$\text{ If }{\mathrm{t}}_1,{\mathrm{t}}_2\text{ and }{\mathrm{t}}_3\text{ be the half lives, then }$

$\begin{array}{l}{\mathrm{A}}_1:{\mathrm{A}}_2:{\mathrm{A}}_3=\frac{{\mathrm{N}}_1}{{\mathrm{t}}_1}:\frac{{\mathrm{N}}_2}{{\mathrm{t}}_2}:\frac{{\mathrm{N}}_3}{{\mathrm{t}}_3}\\ \text{ or }{\mathrm{A}}_1:{\mathrm{A}}_2:{\mathrm{A}}_3=\frac{{\mathrm{m}}_1}{{\mathrm{M}}_1{\mathrm{t}}_1}:\frac{{\mathrm{m}}_2}{{\mathrm{M}}_2{\mathrm{t}}_2}:\frac{{\mathrm{m}}_3}{{\mathrm{M}}_3{\mathrm{t}}_3}\end{array}$

$\begin{array}{l}=\frac{0.006}{234\left(2.5\times {10}^5\right)}:\frac{0.71}{235\left(7.5\times {10}^8\right)}:\frac{99.284}{238\left(4.5\times {10}^9\right)}\\ =51.41\mathrm{\%}:2.13\mathrm{\%}:46.46\mathrm{\%}\end{array}$