n what proportion can three varieties of milk solutions, containing 40%, 72% and 90% milk respectively, should be mixed so that the concentration of milk in the final solution is 60%?

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 a given price must be mixed to produce a mixture of a desired price.

Here are the steps:

1. First, we need to identify the mean value. The mean value is the concentration of milk in the final solution, which is 60%.

2. Next, we need to find the differences between the mean value and the concentrations of milk in each of the three solutions.

   • For the 40% milk solution, the difference is 60% - 40% = 20%.
   • For the 72% milk solution, the difference is 72% - 60% = 12%.
   • For the 90% milk solution, the difference is 90% - 60% = 30%.

3. Now, we can use these differences to find the ratios in which the three solutions should be mixed.

   • The ratio for the 40% solution is 12% (the difference for the 72% solution) : 20% (the difference for the 40% solution) = 3 : 5.
   • The ratio for the 72% solution is 20% (the difference for the 40% solution) : 12% (the difference for the 72% solution) = 5 : 3.
   • The ratio for the 90% solution is not possible to mix with 60% because its concentration is higher than the mean value.

So, the three varieties of milk solutions should be mixed in the proportion of 3 : 5 : 0 (40% : 72% : 90%) to get a final solution with 60% milk concentration.