If $1/(2+1/(3+1/x))=7/16$, then $x$ equals:

Turn both sides upside down: $2+1/(3+1/x)=16/7$, so $1/(3+1/x)=16/7-2=2/7$. Flipping again, $3+1/x=7/2$, hence $1/x=7/2-3=1/2$ and $x=2$. Check with $x=2$: 3 + 1/2 = 7/2, then 2 + 2/7 = 16/7, and its reciprocal is 7/16.