# Volume of a Part of a Cube

Geometry Level 5

Cube $$ABCDEFGH$$, labeled as shown above, has edge length $$1$$ and is cut by a plane passing through vertex $$D$$ and the midpoints $$M$$ and $$N$$ of $$\overline{AB}$$ and $$\overline{CG}$$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $$\frac{p}{q}$$, where $$p$$ and $$q$$ are relatively prime positive integers. Find $$p+q$$.

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