A rock sample from the moon includes a mineral that contains small amounts of the radioactive isotope Potassium-40 and its daughter element Argon-40 (half-life of 1.3 billion years). This mineral would not form with any Argon-40. Consider a crystal with 7 atoms of Argon-40 for every 1 atom of Potassium-40. How many atoms of Potassium-40 were present when the crystal formed for each atom of Potassium-40 that exists today

The half-life of a radioactive material is the time taken for half the original material to decay or the time required for a quantity of the radioactive substance to reduce to half of its initial value.

If the original material formed without any Argon-40, it means that the atoms originally present were Potassium-40 atoms.

Presently, there are 7 Argon-40 atoms for every 1 of Potassium-40, we can deduce the number of half-lifes the Potassium-40 has undergone as follows :

After one half-life, (1/2) there will be one Potassium-40 atom for every Argon-40 atom.

After a second half life, 1/2 × 1/2 = 1/4: there will be one Potassium-40 atom for every three atoms of Argon-40.

After a third half-life, 1/4 × 1/2 = 1/8: there will be one Potassium-40 atom for every 7 atoms of Argon-40.

Since there are 1/8 atoms of Potassium-40 presently, there were originally 8 atoms of Potassium-40.