Using Dalton's law of partial pressure to find percentages of composition
First find the mole fraction of each gas in the original mixture using the formula, P(gas) = X * P(tot). For He:

$$X = \frac{368}{716} = 0.514$$

For Ne: 

$$X = 1 - X(\ce{He}) = 0.486$$

Say you have 1 mol of mixture then you have 0.514 mol He and 0.486 mol Ne.
Converting these to weights using periodic table: 

$$\rm 0.514~mol~He = 2.06~g~He \\\
0.486~mol~Ne = 9.81~g~Ne$$

The total mass of your mixture is the sum of these two masses, 11.9 g. Divide the individual masses by the total mass to get mass percent.

$$\rm\% w. He = 2.06/11.9 = 17.3 \%$$

Subtracting this from one for Ne (or you could apply the formula again):

$$\rm\% w. Ne = 1 - 17.3\% = 82.7\%$$ 

Note: I didn't carefully carry sig figs but you should!