Answer: $\frac{27}{25}$

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