Two containers A and B contain 15 litres of milk and 15 litres of water respectively. A can of volume V is filled with the milk in A. This is emptied into B. The can is then filled with the contents of B and emptied into A. The ratio of milk and water in A at this stage is 5 : 1. What is the value of V (in litres)?
Question
Two containers A and B contain 15 litres of milk and 15 litres of water respectively. A can of volume V is filled with the milk in A. This is emptied into B. The can is then filled with the contents of B and emptied into A. The ratio of milk and water in A at this stage is 5 : 1. What is the value of V (in litres)?
Solution
Let's solve this step by step:
Step 1: Let's denote the volume of milk in container A after the first operation as M1 and the volume of water in container B as W1. After the first operation (transferring V litres of milk from A to B), we have:
M1 = 15 - V (since V litres of milk was removed from A) W1 = 15 + V (since V litres of milk was added to B)
Step 2: In the second operation, V litres of the mixture in B (which contains both milk and water) is transferred to A. Let's denote the volume of milk in this V litres as M2 and the volume of water as W2. Since the mixture in B is a uniform mixture, the ratio of milk to water in the V litres will be the same as the ratio in the entire container B. Therefore, we have:
M2 = V * (15 / (15 + V)) (since the ratio of milk to total volume in B is 15 / (15 + V)) W2 = V * (V / (15 + V)) (since the ratio of water to total volume in B is V / (15 + V))
Step 3: After the second operation, the volume of milk in A is M1 + M2 and the volume of water is W2. According to the problem, the ratio of milk to water in A is 5 : 1. Therefore, we have:
(M1 + M2) / W2 = 5 / 1
Substitute M1, M2, and W2 from steps 1 and 2 into this equation, we get:
((15 - V) + V * (15 / (15 + V))) / (V * (V / (15 + V))) = 5
Solving this equation, we get V = 3 litres. Therefore, the volume of the can is 3 litres.
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