Initial Solution Analysis

We need to determine the final ratio of milk and water after mixing two different solutions.

• Solution 1: Quantity = 40 litres, Milk:Water Ratio = 3:2.
• Solution 2: Quantity = 30 litres, Milk:Water Ratio = 2:1.

Calculating Components of Solution 1

The total parts in the ratio for Solution 1 are $3 + 2 = 5$.

Calculate the amount of milk in Solution 1:

$ \text{Milk}_1 = \left( \frac{3}{3+2} \right) \times 40 \, \text{litres} = \frac{3}{5} \times 40 \, \text{litres} = 24 \, \text{litres} $

Calculate the amount of water in Solution 1:

$ \text{Water}_1 = \left( \frac{2}{3+2} \right) \times 40 \, \text{litres} = \frac{2}{5} \times 40 \, \text{litres} = 16 \, \text{litres} $

Calculating Components of Solution 2

The total parts in the ratio for Solution 2 are $2 + 1 = 3$.

Calculate the amount of milk in Solution 2:

$ \text{Milk}_2 = \left( \frac{2}{2+1} \right) \times 30 \, \text{litres} = \frac{2}{3} \times 30 \, \text{litres} = 20 \, \text{litres} $

Calculate the amount of water in Solution 2:

$ \text{Water}_2 = \left( \frac{1}{2+1} \right) \times 30 \, \text{litres} = \frac{1}{3} \times 30 \, \text{litres} = 10 \, \text{litres} $

Mixing Solutions for Resultant Ratio

Combine the amounts of milk and water from both solutions to find the total quantities in the mixture.

Total milk in the resultant solution:

$ \text{Total Milk} = \text{Milk}_1 + \text{Milk}_2 = 24 \, \text{litres} + 20 \, \text{litres} = 44 \, \text{litres} $

Total water in the resultant solution:

$ \text{Total Water} = \text{Water}_1 + \text{Water}_2 = 16 \, \text{litres} + 10 \, \text{litres} = 26 \, \text{litres} $

Determining the Final Ratio

The ratio of milk to water in the resultant solution is the ratio of the total milk to the total water.

$ \text{Resultant Ratio} = \text{Total Milk} : \text{Total Water} = 44 : 26 $

Simplify the ratio by dividing both quantities by their greatest common divisor, which is 2.

$ \text{Simplified Ratio} = \frac{44}{2} : \frac{26}{2} = 22 : 13 $