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$$ \frac{2}{3} \times \frac{5}{6} \times X = \frac{3}{4} \times
\frac{4}{5} \times Y$$ $$ \frac{5X}{9} = \frac{3Y}{5}$$ $$ 25X =
27Y$$ $$ \frac{X}{Y} = \frac{27}{25}$$

*This problem is taken from the UKMT Mathematical Challenges.*