A light string passing over a smooth light fixed pulley connects two blocks of masses $$m_1$$ and $$m_2$$. If the acceleration of the system is $$\frac{g}{8}$$, then the ratio of masses is

Lets start writing equations for both the blocks

For $$m_1$$ : -

$$m_1g-T=m_1a$$

For $$m_2$$: - 

$$T-m_2g=m_2a$$

Given that $$a=\frac{g}{8}$$

On substituting a in both equations and adding them we get , 

$$m_1g-\ m_2g\ =\ m_1\left(\frac{g}{8}\right)+m_2\left(\frac{g}{8}\right)$$

$$\left(m_1-\ m_2\right)g\ =\ \left(m_1+m_2\right)\left(\frac{g}{8}\right)$$

$$\frac{\left(m_1-\ m_2\right)}{m_1+m_2}=\frac{1}{8}$$

On dividing Numerator and denominator with $$m_2$$

$$\frac{\left(\frac{m_1}{m_2}-1\right)}{\left(\frac{m_1}{m_2}+1\right)}=\frac{1}{8}$$

consider $$\frac{m_1}{m_2}$$ as x and solve for x

$$\frac{\left(x-1\right)}{\left(x+1\right)}=\frac{1}{8}$$

$$\left(x-1\right)=\frac{1}{8}\left(x+1\right)$$

$$x-\frac{1}{8}x=\frac{1}{8}+1$$

$$\frac{7}{8}x=\frac{9}{8}$$

$$x=\frac{9}{7}$$