CEMC Galois, 2010

Geometry Difficulty 2.0 Junior Prove it Canada

In the diagram, points $X$ , $Y$ and $Z$ are on the sides of $\triangle UVW$ , as shown. Line segments $UY$ , $VZ$ and $WX$ intersect at $P$ . Point $Y$ is on $VW$ such that $VY : YW = 4 : 3$ . If $\triangle PYW$ has an area of $30$ and $\triangle PZW$ has an area of $35$ , determine the area of $\triangle UXP$ .

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