jmc

The Angle Bisector Theorem tells us that $$\frac{BX}{AX}=\frac{BC}{AC}=\frac{27}{30}=\frac{9}{10}.$$Since $△BCX$ and $△ACX$ share the same height, the ratio of their areas is simply the ratio of their bases, so our answer is $$\frac{BX}{AX}=\boxed{\frac{9}{10}}.$$