In the following figure, triangle $ABC$ is similar to triangle $DEF$. What is the value of $\tan{E}$?

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Because the triangles are similar, the measure of the corresponding angles of both triangles will be congruent.

Therefore, since $m\angle{E}=m\angle{B},   \tan{E}=\tan{B}$.

$$\tan{B}=\frac{\text{opposite}}{\text{adjacent}}$$ $$= \boxed{\frac{5}{12}}$$