A 1 . 5 × 10 ^ 9 years old rock contains U 238 which disintegrates...

Let ${N}_{0}$ be the initial number of uranium nuclei. After time $t$, let ${N}_{U}$ be the number of uranium nuclei and ${N}_{Pb}$ be the number of lead nuclei.

The number of half-lives passed is

$n = \dfrac{t}{{T}_{half}} = \dfrac{1 . 5 \times {10}^{9}}{4 . 5 \times {10}^{9}} = \dfrac{1}{3}$

${N}_{U} = {N}_{0}{\left(\dfrac{1}{2}\right)}^{n} = {N}_{0}\left(\dfrac{1}{2}\right)\dfrac{1}{3}$

${N}_{Pb} = {N}_{0} - {N}_{0}\left(\dfrac{1}{2}\right)\dfrac{1}{3}$
$\dfrac{{N}_{Pb}}{{N}_{U}} = 2\dfrac{1}{3} - 1 = 0 . 26$