In mixture P the ratio between milk & water is 3:2 and in another mixture Q the ratio between milk & water is9:1.In what ratio these two mixtures must be mixed so that the final mixture will contain milk and water in the ratio 4:1 ?

To solve this problem, we can use the method of allegation which is a rule of mixture to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a desired price.

Step 1: Find the concentration of milk in both mixtures.

In mixture P, the ratio of milk to water is 3:2. So, the concentration of milk in mixture P is 3/(3+2) = 3/5 = 0.6 or 60%.

In mixture Q, the ratio of milk to water is 9:1. So, the concentration of milk in mixture Q is 9/(9+1) = 9/10 = 0.9 or 90%.

Step 2: Find the concentration of milk in the final mixture.

The final mixture should have milk and water in the ratio 4:1. So, the concentration of milk in the final mixture is 4/(4+1) = 4/5 = 0.8 or 80%.

Step 3: Use the method of allegation.

The method of allegation states that the difference in the value of a cheaper and costlier quantity and the mean value gives the ratio in which the two quantities are mixed.

So, the difference between the concentration of milk in mixture Q and the final mixture is 0.9 - 0.8 = 0.1.

And the difference between the concentration of milk in the final mixture and mixture P is 0.8 - 0.6 = 0.2.

Therefore, the two mixtures must be mixed in the ratio 0.1:0.2 or 1:2 to get a final mixture that contains milk and water in the ratio 4:1.