The volume of the pyramid $ABCD$ is 1. Points $K, L$, and $M$ are taken on the edges $AD, BD, CD$ respectively, such that $2AK = KD, BL = 2LD$, and $2CM = 3MD$. Find the volume of the polyhedron $ABCKLM$.

#
$$
V_{K L M D}=\frac{D K}{D A} \cdot \frac{D L}{D B} \cdot \frac{D M}{D C} \cdot V_{A B C D}=\frac{2}{3} \cdot \frac{1}{3} \cdot \frac{2}{5} \cdot 1=\frac{4}{45}
$$

Therefore,

$$
V_{A B C K L M}=V_{A B C D}-V_{K L M D}=1-\frac{4}{45}=\frac{41}{45}
$$

## Answer

$\frac{41}{45}$