To find the ratio of water to milk in the mixture, we can follow these steps:

Step 1: Define the quantities and ratios

Let's assume that the quantities of the three samples taken are equal. Since we are mixing equal quantities, we can assume a common quantity for each sample. For simplicity, let's assume each sample has a total volume of $x$ liters.

Step 2: Determine the milk and water content in each sample

Given the ratios:

1. For the first sample, the ratio of milk to water is $2:1$.

   • Milk content: $\frac{2}{2+1} = \frac{2}{3}$ of $x$ liters
   • Water content: $\frac{1}{2+1} = \frac{1}{3}$ of $x$ liters

2. For the second sample, the ratio of milk to water is $3:2$.

   • Milk content: $\frac{3}{3+2} = \frac{3}{5}$ of $x$ liters
   • Water content: $\frac{2}{3+2} = \frac{2}{5}$ of $x$ liters

3. For the third sample, the ratio of milk to water is $5:3$.

   • Milk content: $\frac{5}{5+3} = \frac{5}{8}$ of $x$ liters
   • Water content: $\frac{3}{5+3} = \frac{3}{8}$ of $x$ liters

Step 3: Calculate total milk and water in the mixture

Since each sample has an equal quantity of $x$ liters, the total quantity of the mixture is $3x$ liters.

1. Total milk in the mixture:

   $\text{Milk from first sample} = \frac{2x}{3}$ $\text{Milk from second sample} = \frac{3x}{5}$ $\text{Milk from third sample} = \frac{5x}{8}$

   Adding them together:

   $\text{Total milk} = \frac{2x}{3} + \frac{3x}{5} + \frac{5x}{8}$

   To add these fractions, find a common denominator, which is 120:

   $\text{Total milk} = \frac{2x \cdot 40}{120} + \frac{3x \cdot 24}{120} + \frac{5x \cdot 15}{120}$ $= \frac{80x}{120} + \frac{72x}{120} + \frac{75x}{120}$ $= \frac{227x}{120}$

2. Total water in the mixture:

   $\text{Water from first sample} = \frac{x}{3}$ $\text{Water from second sample} = \frac{2x}{5}$ $\text{Water from third sample} = \frac{3x}{8}$

   Adding them together:

   $\text{Total water} = \frac{x}{3} + \frac{2x}{5} + \frac{3x}{8}$

   Find a common denominator, which is 120:

   $\text{Total water} = \frac{x \cdot 40}{120} + \frac{2x \cdot 24}{120} + \frac{3x \cdot 15}{120}$ $= \frac{40x}{120} + \frac{48x}{120} + \frac{45x}{120}$ $= \frac{133x}{120}$

Step 4: Calculate the ratio of water to milk in the mixture

The ratio of water to milk is:

$\text{Ratio} = \frac{\text{Total water}}{\text{Total milk}} = \frac{\frac{133x}{120}}{\frac{227x}{120}}$

Simplifying this:

$= \frac{133}{227}$

Final Answer

The ratio of water to milk in the mixture is $\frac{133}{227}$.

Would you like further details or have any questions? Here are some related questions:

1. What happens if the quantities of the three samples are different?
2. How would the ratio change if the initial ratios of milk to water were different?
3. What if we mixed different proportions of the samples instead of equal quantities?
4. Can you generalize a method to find the mixture ratio for any number of samples?
5. How does the choice of total volume $x$ affect the final ratio?

Tip: When dealing with ratios, always try to simplify calculations by assuming convenient values for quantities (like $x$), which can often help in obtaining a clearer answer.