Proportion of gases in Ideal Gas Mixture?
Your interpretation of specific gravity is incorrect: 


  Specific gravity (SG) refers to the ratio between a given density and that of a reference.  Usually it is defined for liquids with respect to water and gasses with respect to air so in this case
  
  
    $0.6=\frac{n_{CH_4}\rho_{CH_4}+n_{C_2H_6}\rho_{C_2H_6}}{\text{air}}$ 
  
  
  where $n_i$ are proportions by mole factor. This is the incorrect assumption 
  
  
    $0.6=\frac{\rho_{CH_4}}{\rho_{C_2H_6}}$.
  
  
  Gasses are always a little dicier since the reference is much more important with changes in temperature and pressure affecting the resulting density.  


Example


  The density form of the ideal gas laws goes like: 
  
  
    $\rho = \text{mass / volume = molecular weight * pressure / constant / temperature}$.  
  
  
  Therefore for constant temperature and pressure, the specific gravity
  is equal to the ratio of molecular weights.  This follows because
  $\frac{\rho_1}{\rho_2} = \frac{MW_1}{MW_2}$ at constant conditions.


Equations and Solution


  
    $M_{\text{air}} = 28.97$ (I got $28.84$ using the simple $0.21*32+0.79*28$ and this agrees with  this)
    
    $\text{SG}=0.6=\frac{n_{CH_4}\rho_{CH_4}+n_{C_2H_6}\rho_{C_2H_6}}{\text{air}}$ 
    
    $1=n_{CH_4}+n_{C_2H_6}$
  
  
  so by the first equation and the second equation the density of the mixture is $0.6 * 28.97 = 17.38$. The third equation holds only if $0.9*16 + 0.1*30 = 17.4$ so
  
  
    methane is 90% and ethane is 10%.