$T=2\pi \sqrt{\frac{M}{k}}$ when mass is increased by $m$ then ...(i)

$T=2\pi \sqrt{\frac{M+m}{k}}$

$\Rightarrow$ $\frac{5T}{3}=2\pi \sqrt{\frac{M+m}{k}}$ ...(ii)

Dividing Eq. (i) by Eq. (ii), we get

$\frac{3}{5}=\sqrt{\frac{M}{M+m}}$

$\frac{9}{25}=\frac{M}{M+m}$

$\Rightarrow$ $9M+9m=25M$

$\Rightarrow$ $16M=9m$

$\frac{m}{M}=\frac{16}{9}$