An isotope of potassium \( \mathrm{K}^{40} \) undergoes two parallel types of decay, one by elec...

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An isotope of potassium \( \mathrm{K}^{40} \) undergoes two parallel types of decay, one by electron capture and other by \( \beta \) decay with half lives as \( 1.3 \times 10^{9} \) years and \( \frac{1.3}{9} \times 10^{9} \) years respectively. If in a sample of a mineral, mass ratio of \( \mathrm{K}: \mathrm{Ar}: \mathrm{Ca} \) is \( 5,1.5 \) and 16 then calculate age of the mineral if it is known that all \( \mathrm{Ca}^{40} \) in the mineral is not from the radioactive decay of potassium.
[Express your answer in terms of \( 10^{7} \) years]
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