$m$ arithmetic means are inserted between 1 and 31, so that the ratio of the 7th and $(m-1)$ ^th means is $5:9$. Then the value of m is

$d=\frac{b-a}{n+1}=\frac{31-1}{m+1}=\frac{30}{m+1}$

Now

The ratio of 7^th and $m-1^{th}$means $=5:9$

$\Rightarrow \frac{A_7}{A_{m-1}}=\frac{5}{9}$

$\Rightarrow \frac{a+7d}{a+(m-1)d}=\frac{5}{9}$

$\Rightarrow \frac{1+7\left(\frac{30}{\mathrm{m}+1}\right)}{1+(\mathrm{m}-1)\left(\frac{30}{\mathrm{m}+1}\right)}=\frac{5}{9}$

$\Rightarrow \frac{m+1+210}{m+1+30m-30}=\frac{5}{9}$

$\Rightarrow \frac{m+211}{31m-29}=\frac{5}{9}$

$\Rightarrow 155m-145=9m+1899$

$\Rightarrow 146m=2044$

$\Rightarrow m=\frac{2044}{146}$

$\Rightarrow m=14$