Answer
It takes $~~t = 5.75~ns~~$ for 25% of the sample to decay.
Work Step by Step
Let $P$ be the probability that an excited atom will emit a photon.
We can find the lifetime of the excited state:
$P = \lambda~\Delta t$
$P = (\frac{1}{\tau})~(\Delta t)$
$\tau = \frac{\Delta t}{P}$
$\tau = \frac{0.20~ns}{0.010}$
$\tau = 20~ns$
We can find the time it takes for 25% of the sample to decay:
$N = N_0~e^{-t/\tau}$
$0.75~N_0 = N_0~e^{-t/\tau}$
$0.75 = e^{-t/\tau}$
$ln(0.75) = -\frac{t}{\tau}$
$t = -ln(0.75)~(\tau)$
$t = -ln(0.75)~(20~ns)$
$t = 5.75~ns$
It takes $~~t = 5.75~ns~~$ for 25% of the sample to decay.
