Mixtures and Alligation

thank you so much!

Finding ratio of water added to milk:

52.5/3.5 is 15/1. This means for every 15 litres of milk, he adds 1 litre water. This in turn means, for every 16 litre milk he sells, he saves 1 litre of cost. Therefore, profit is 1/15.

Why not 1/16? His cost here is 15 litres and he's selling it at price of 16. Thus, profit = margin/cost which is 1/15.

Now for 496 litres of adulterated milk, there will be 465 litres of milk and 31 litres of water. His profit here will be 31/465 which is nothing but 1/15.

To make the profit half, ie 1/30, we can use ratio proportion but on water. Profit here is 1/30, which means for every 30 litres of milk, he's adding 1 litre water. Question has asked to add ONLY milk, thus quantity of water will be same which is 31 litres.

If there's 1 litre water for every 30 litre milk, then there will be 31 litres water for 930 litres milk.

Amount to be added will be 930-465 which is 465.