How can I calculate the percentage of dissociated dinitrogen tetroxide?
$\ce{N2O4 (g)}$ say initially is $P\ \mathrm{atm}$.

$$PV = nRT$$

$$P(1)= (36.8/92)( 0.082 )(300 K)$$

$$P= 9.84\ \mathrm{atm}$$

Now we have:

$$\ce{N2O4(g) &lt;=&gt;2NO2(g)}$$

Considering initial pressure of N2O4 as p, at equilibrium, assuming 'x' as the degree of dissociation, we will have pressure of N2O4 to be p(1-x) and pressure of NO2 as 2px.

So, $$K_p = \frac{[\ce{NO2}]^2}{[\ce{N2O4}]}$$

$$K_p= \frac{(2px)^2}{p(1-x)}$$

$$K_p= \frac{4px^2}{(1-x)}$$

assuming $1-x$ approximately as $1$,

$$K_p= 4px^2$$

substituting $p$ as $9.84\ \mathrm{atm}$ and $K_p$ as $0.1642$:

$$0.1642= 4(9.84)x^2$$

$$x^2= \frac{0.1642}{4(9.84)}$$

$$x^2= 0.1642/39.36$$

$$x^2= 0.00417$$

$$x= 0.0645$$

degree of dissociation is 0.0645

Hence % of dissociation is $0.0645 \times (100) = 6.45\ \%$ (near to your answer)