The answer is 6×10⁹ years.
K-Ar dating is radioactive dating method used to measure a time elapsed since the rock is cool enough to trap Ar after a radioactive decay of 40K to 40Ar. It can be expressed as:
[tex]t= \frac{ t_{ \frac{1}{2} } }{ln(2)} ln( \frac{ K_{f} + \frac{ Ar_{f} }{0.109} }{ K_{f} } )[/tex]
where:
[tex]t[/tex] - elapsed time
[tex] t_{ \frac{1}{2} } [/tex] - half-life of K40
[tex] K_{f} [/tex] - amount of K40
[tex] Ar_{f} [/tex] - amount of Ar40
We know that a half-life of K40 is 1.251×10⁹ years:
[tex]t_{ \frac{1}{2} } =1.251* 10^{9} [/tex]
We do not know absolute value of amount of K40 and Ar40, but we know percentage and can express them as following:
[tex] K_{f} =25%=0.25[/tex]
[tex] Ar_{f} =75%=0.75[/tex]
So:
[tex]t= \frac{1.251* 10^{9} }{ln(2)} ln( \frac{0.25+ \frac{0.75}{0.109} }{0.25} )[/tex]
⇒ t = 6×10⁹
Thus, the rock is old 6×10⁹ years.
