smc

In order, the points from left to right are $A,P,Q,B$. Let the lengths between successive points be $x,2,y$, respectively. Since $\frac{AP}{PB}=\frac{2}{3}$, we have $\frac{x}{2+y}=\frac{2}{3}$. Since $\frac{AQ}{QB}=\frac{3}{4}$, we have $\frac{x+2}{y}=\frac{3}{4}$. The first equation gives $x=\frac{2}{3}(2+y)$. Plugging that in to the second equation gives: $\frac{\frac{2}{3}(2+y)+2}{y}=\frac{3}{4}$. $\frac{2}{3}(2+y)+2=\frac{3}{4}y$ $\frac{4}{3}+\frac{2}{3}y+2=\frac{3}{4}y$ $\frac{10}{3}=\frac{1}{12}y$ $y=40$ Since $x=\frac{2}{3}(2+y)$, we have $x=28$. The length of the entire segment is $x+2+y$, which is $70$, or option $\boxed{70}$.