Vessel A contains milk and water in the ratio $4 : 5$. Vessel B contains milk and water in the ratio $2 : 1$. If $x$ litres of mixture from Vessel A is mixed with $y$ litres of mixture from Vessel B, the resulting mixture has milk to water ratio $8 : 5$. Find the ratio $x : y$.

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**Solution:**

1. Let the quantity of mixture taken from Vessel A be $x$ litres and from Vessel B be $y$ litres.

2. Milk and water ratio in Vessel A is $4 : 5$, so:
   - Milk in $x$ litres of A = $\frac{4}{9}x$
   - Water in $x$ litres of A = $\frac{5}{9}x$

3. Milk and water ratio in Vessel B is $2 : 1$, so:
   - Milk in $y$ litres of B = $\frac{2}{3}y$
   - Water in $y$ litres of B = $\frac{1}{3}y$

4. After mixing, the ratio of milk to water is $8 : 5$. Therefore,
   $$\frac{	ext{Milk from A} + 	ext{Milk from B}}{	ext{Water from A} + 	ext{Water from B}} = \frac{8}{5}$$

Substituting values:
   $$\frac{\frac{4}{9}x + \frac{2}{3}y}{\frac{5}{9}x + \frac{1}{3}y} = \frac{8}{5}$$

5. Cross-multiply to solve for $x$ and $y$:
   $$5\left(\frac{4}{9}x + \frac{2}{3}yight) = 8\left(\frac{5}{9}x + \frac{1}{3}yight)$$

6. Multiply out:
   $$\frac{20}{9}x + \frac{10}{3}y = \frac{40}{9}x + \frac{8}{3}y$$

7. Multiply entire equation by 9 to clear denominators:
   $$20x + 30y = 40x + 24y$$

8. Rearranging terms:
   $$30y - 24y = 40x - 20x$$
   $$6y = 20x$$

9. Simplify:
   $$y = \frac{20}{6}x = \frac{10}{3}x$$

10. Therefore, the ratio $x : y$ is:
    $$x : y = 3 : 10$$

**Answer:** $3 : 10$

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