Binary strings such that the sum of 0's is not equal to twice the sum of 1's
Distinguish $n_0(w) &lt; 2n_1(w)$ and $n_0(w) &gt; 2n_1(w)$. That is, in your grammar for $n_0(w) = 2n_1(w)$ either generate extra $0$ or (in a separate half of the grammar) extra $1$.