Problem:
A mixture of 40% liter of milk and water contains 10% of water. how much is to be added to the mixture so that the water may be 20% in the new mixture?

Solution:

To solve this problem, we need to use the concept of mixtures and alligations. We have to find out how much water needs to be added to the mixture to make it 20% in the new mixture.

Step 1:
Let's assume that we have 1 liter of the mixture.

Step 2:
As per the problem, the mixture contains 40% milk and water and 10% water.

This means that:
- 40% of 1 liter mixture = 0.4 liters of milk
- 10% of 1 liter mixture = 0.1 liters of water

Step 3:
We need to find out how much water needs to be added to the mixture to make it 20% water in the new mixture.

Let's assume that x liters of water need to be added.

Step 4:
After adding x liters of water, the total volume of the mixture will be (1 + x) liters.

Step 5:
In the new mixture, the percentage of water should be 20%.

This means that:
- 80% of (1 + x) liters = 0.8(1 + x) liters of milk
- 20% of (1 + x) liters = 0.2(1 + x) liters of water

Step 6:
Now, we can use the concept of alligations to find out the value of x.

Initial mixture: Final mixture:
Milk Water Milk Water
40% 10% 80% 20%

The ratio of milk and water in the initial mixture is:
= 40 : 10
= 4 : 1

The ratio of milk and water in the final mixture is:
= 80 : 20
= 4 : 1

So, the ratio of milk and water in the mixture after adding x liters of water is:
= 4 : (1 + x)

To get a ratio of 4 : 1, we need to add 3 liters of water.

Therefore, 3 liters of water need to be added to the mixture to make it 20% water in the new mixture.