Average, Mixture and Alligation Question 2 - CAT 2024
A certain amount of water was poured into a 300-litre container and the remaining portion of the container was filled with milk. Then an amount of this solution was taken out from the container which was twice the volume of water that was earlier poured into it, and water was poured to refill the container again. If the resulting solution contains 72% milk, then the amount of water, in litres, that was initially poured into the container was:
Step-by-Step Explanation
Let the initial volume of water poured into the container be W litres. Since the total capacity of the container is 300 litres, the remaining portion filled with milk is 300 − W litres.
- Analyze the Removal Step:
A volume equal to 2W is removed from the 300-litre mixture.
Fraction of solution remaining = (300 − 2W) / 300 = 1 − W / 150
- Track the Milk Component:
Refilling the container with water changes only the water volume; the absolute quantity of milk remains unchanged after the refill.
Final Milk Volume = Initial Milk Volume × Fraction Remaining
Final Milk Volume = (300 − W) × (1 − W / 150)
We are given that the final solution contains 72% milk, which means:
Final Milk Volume = 72% of 300 = 216 litres
- Solve the Quadratic Equation:
(300 − W) × (150 − W) / 150 = 216
(300 − W)(150 − W) = 216 × 150
45000 − 450W + W² = 32400
W² − 450W + 12600 = 0
(W − 30)(W − 420) = 0
Since the total capacity of the container is only 300 litres, the initial volume of water W cannot be 420 litres. Therefore, W = 30.
The amount of water initially poured was 30 litres.
Correct Answer: 30
💡 FundaMakers Tip:
In replacement problems, always focus on the component that is not being used to refill the container. Since water is added to refill the tank, track the concentration changes of milk. Using the standard replacement formula Final = Initial × (1 − Out / Total) simplifies the algebraic setup and prevents double-counting.