To solve the problem, let's break it down step by step:

Information given:

• Three vessels, each with a 10-liter capacity.
• The ratio of milk to water in the three vessels is:
  1. First vessel: $2:1$
  2. Second vessel: $3:1$
  3. Third vessel: $3:2$

We need to find the proportion of milk and water when all three vessels are emptied into a single large vessel.

Step 1: Determine the amount of milk and water in each vessel.

1. First vessel (Ratio 2:1):

   • Total liquid: 10 liters.
   • Ratio of milk to water = 2:1. This means for every 3 parts (2 parts milk + 1 part water), 2 parts are milk and 1 part is water.
   • Amount of milk = $\frac{2}{3} \times 10 = 6.67 \, \text{liters}$
   • Amount of water = $\frac{1}{3} \times 10 = 3.33 \, \text{liters}$

2. Second vessel (Ratio 3:1):

   • Total liquid: 10 liters.
   • Ratio of milk to water = 3:1. This means for every 4 parts (3 parts milk + 1 part water), 3 parts are milk and 1 part is water.
   • Amount of milk = $\frac{3}{4} \times 10 = 7.5 \, \text{liters}$
   • Amount of water = $\frac{1}{4} \times 10 = 2.5 \, \text{liters}$

3. Third vessel (Ratio 3:2):

   • Total liquid: 10 liters.
   • Ratio of milk to water = 3:2. This means for every 5 parts (3 parts milk + 2 parts water), 3 parts are milk and 2 parts are water.
   • Amount of milk = $\frac{3}{5} \times 10 = 6 \, \text{liters}$
   • Amount of water = $\frac{2}{5} \times 10 = 4 \, \text{liters}$

Step 2: Add the quantities of milk and water from all vessels.

• Total milk = $6.67 + 7.5 + 6 = 20.17 \, \text{liters}$
• Total water = $3.33 + 2.5 + 4 = 9.83 \, \text{liters}$

Step 3: Find the ratio of milk to water in the final mixture.

• The ratio of milk to water = $\frac{20.17}{9.83}$.
• Simplifying the ratio: $20.17 \div 9.83 = 2.05$.

Thus, the proportion of milk to water in the final mixture is approximately 2.05:1.

Would you like further details or have any questions?

Related questions:

1. How would the answer change if the vessels had different capacities?
2. How is the ratio affected if one vessel had a different mixture of milk and water?
3. What is the percentage of milk in the final mixture?
4. Can the proportion of milk to water ever be equal given these initial ratios?
5. How would you calculate the total amount of liquid in the large vessel?

Tip:

When working with ratios, remember that they represent parts of a whole. It's important to first break down the ratio into individual quantities before combining them.