The technique known as potassiumargon dating is used to date volcanic rock and ash, and thus establish dates for nearby fossils, like this 1.8 -millionyear-old hominid skull. The potassium isotope \({ }^{40} \mathrm{~K}\) decays with a 1.28 -billionyear half-life and is naturally present at very low levels. The most common decay mode is beta-minus decay into the stable isotope \({ }^{40} \mathrm{Ca}\), but \(10.9 \%\) of decays result in the stable isotope \({ }^{40} \mathrm{Ar}\). The high temperatures in volcanoes drive argon out of solidifying rock and ash, so there is no argon in newly formed material. After formation, argon produced in the decay of \({ }^{40} \mathrm{~K}\) is trapped, so \({ }^{40} \mathrm{Ar}\) builds up steadily over time. Accurate dating is possible by measuring the ratio of the number of atoms of \({ }^{40} \mathrm{Ar}\) and \({ }^{40} \mathrm{~K}\). 1.8 million years after its formation,
a. What fraction of the \({ }^{40} \mathrm{~K}\) initially present in a sample has decayed?
b. What is the \({ }^{40} \mathrm{Ar} /{ }^{40} \mathrm{~K}\) ratio of the sample?