X works twice as fast as Y . Y alone can finish the work in nine days . X and Y together can finish the work in _____ days.

Y alone can finish the work in 9 days.

⇒ Y’s 1-day work $= \dfrac{1}{9}$

X works twice as fast as Y.

⇒ X’s 1-day work $= 2 \times \dfrac{1}{9} = \dfrac{2}{9}$

Together, their 1-day work $= \dfrac{2}{9} + \dfrac{1}{9} = \dfrac{3}{9} = \dfrac{1}{3}$

Time taken to complete the whole work $= \dfrac{1}{(1/3)} = 3 \text{ days}$

Answer: $\boxed{3\text{ days}}$ ✅