jmc

We use the line-segment ratios to infer area ratios and height ratios. Areas: $AD:DC=1:2⟹AD:AC=1:3⟹[ABD]=\frac{[ABC]}{3}=120$. $BE:BD=1:2\text{ (midpoint)}⟹[ABE]=\frac{[ABD]}{2}=\frac{120}{2}=60$. Heights: Let $h_A$ = height (of altitude) from $\overline{BC}$ to $A$. $AD:DC=1:2⟹CD:CA=2:3⟹\text{height }{h}_D$ from $\overline{BC}$ to $D$ is $\frac{2}{3}{h}_A$. $BE:BD=1:2\text{ (midpoint)}⟹\text{height }{h}_E$ from $\overline{BC}$ to $E$ is $\frac{1}{2}{h}_D=\frac{1}{2}(\frac{2}{3}{h}_A)=\frac{1}{3}{h}_A$. Conclusion: $\frac{[EBF]}{[ABF]}=\frac{[EBF]}{[EBF]+[ABE]}=\frac{[EBF]}{[EBF]+60}$, and also $\frac{[EBF]}{[ABF]}=\frac{h_E}{h_A}=\frac{1}{3}$. So, $\frac{[EBF]}{[EBF]+60}=\frac{1}{3}$, and thus, $[EBF]=\boxed{30}$.