Answer
We can rank the systems in decreasing order of the frequency of oscillations:
$a = c = e \gt b \gt d$
Work Step by Step
We can write an expression for frequency:
$f = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$
We can find the frequency for each mass-spring system:
(a) $f_a = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$
(b) $f_b = \frac{1}{2\pi}\sqrt{\frac{k}{2m}} = \frac{\sqrt{2}}{2}\times \frac{1}{2\pi}\sqrt{\frac{k}{m}}$
(c) $f_c = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$
(d) $f_d = \frac{1}{2\pi}\sqrt{\frac{k/2}{2m}} = \frac{1}{2}\times \frac{1}{2\pi}\sqrt{\frac{k}{m}}$
(e) $f_e = \frac{1}{2\pi}\sqrt{\frac{2k}{2m}} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$
We can rank the systems in decreasing order of the frequency of oscillations:
$a = c = e \gt b \gt d$
