How much water must be added to 60 litres of milk at 3/2 litres for Rs. 20 so as to have a mixture worth Rs. 32/3 a litres? Options : 12 L 15 L 18 L 20 L

The problem is about creating a mixture of milk and water that has a certain cost per litre. The cost of the milk is given as Rs. 20 for 3/2 litres, and the desired cost of the mixture is Rs. 32/3 per litre. The amount of milk is fixed at 60 litres, and we need to find out how much water (which is free) needs to be added to achieve the desired cost.

Step 1: Calculate the cost per litre of the milk The cost of the milk is Rs. 20 for 3/2 litres, so the cost per litre is Rs. 20 / (3/2) = Rs. 40/3.

Step 2: Calculate the difference in cost per litre between the milk and the desired mixture The desired cost of the mixture is Rs. 32/3 per litre, so the difference in cost per litre is Rs. 40/3 - 32/3 = Rs. 8/3.

Step 3: Calculate the total cost of the milk The total cost of the milk is the cost per litre times the number of litres, or Rs. 40/3 * 60 = Rs. 800.

Step 4: Calculate the total cost of the mixture The total cost of the mixture is the same as the total cost of the milk, because we're only adding water, which is free. So the total cost of the mixture is also Rs. 800.

Step 5: Calculate the total volume of the mixture The total cost of the mixture is Rs. 800, and the cost per litre is Rs. 32/3, so the total volume of the mixture is Rs. 800 / (32/3) = 75 litres.

Step 6: Calculate the volume of water to be added The total volume of the mixture is 75 litres, and we started with 60 litres of milk, so the volume of water to be added is 75 - 60 = 15 litres.

So, the answer is 15 L.

To solve this problem, we need to use the concept of mixtures and allegations. Here are the steps:

1. First, we need to find the cost price of 1 litre of milk. Given that 3/2 litres of milk cost Rs. 20, the cost price of 1 litre of milk is Rs. 20 / (3/2) = Rs. 40/3.

2. The cost price of 1 litre of water is Rs. 0 (since water is being added for free).

3. The mean price of the mixture is given as Rs. 32/3 per litre.

4. Now, we use the rule of allegation. The rule of allegation states that the ratio of quantities of cheaper to dearer (more expensive) ingredients is inversely proportional to the difference in their prices to the mean price.

5. The difference in price of milk and the mean price is Rs. 40/3 - Rs. 32/3 = Rs. 8/3.

6. The difference in price of water and the mean price is Rs. 32/3 - Rs. 0 = Rs. 32/3.

7. Therefore, the ratio of quantity of water to quantity of milk is (8/3) : (32/3) = 1 : 4.

8. We know that the quantity of milk is 60 litres. Therefore, the quantity of water should be 1/4 of this, which is 60/4 = 15 litres.

So, the correct answer is 15 litres of water must be added to the milk.