In $\triangle ABC$ points $D$ and $E$ lie on $BC$ and $AC$, respectively. If $AD$ and $BE$ intersect at $T$ so that $AT/DT=2$ and $BT/ET=3$, what is $CD/BD$?

Using mass points: put masses $a,b,c$ at $A,B,C$, so that $D$ (on $BC$) has mass $b+c$ and $E$ (on $AC$) has mass $a+c$. Since $AT/DT=2$, mass$(D)=b+c=2a$. Since $BT/ET=3$, mass$(E)=a+c=3b$.
From $b+c=2a$: $c=2a-b$. Substituting into $a+c=3b$: $a+2a-b=3b\Rightarrow 3a=4b\Rightarrow a=\frac{4b}3$.
$c=2\left(\frac{4b}3\right)-b=\frac{8b}3-b=\frac{5b}3$.
Since $D$ is the balance point of $B$ and $C$, $CD/BD=b/c=b/\frac{5b}3=\boxed{\frac35}$.