Average, Mixture and Alligation Question 1 - CAT 2023
A mixture P is formed by removing a certain amount of coffee from a coffee jar and replacing the same amount with cocoa powder. The same amount is again removed from mixture P and replaced with the same amount of cocoa powder to form a new mixture Q. If the ratio of coffee and cocoa in mixture Q is 16 : 9, then the ratio of cocoa in mixture P to that in mixture Q is:
Step-by-Step Explanation
Let the total capacity of the coffee jar be 25 units.
- Analyze the Final State (Mixture Q):
The ratio of coffee to cocoa in mixture Q is 16 : 9. This means the total volume is 16 + 9 = 25 units, out of which 16 units is coffee.
- Find the Removal Factor:
Since the replacement process is executed twice, we use the standard successive dilution relationship for the remaining component (coffee):
Final Coffee Volume = Initial Coffee Volume × (1 − x / V)²
16 = 25 × (1 − x / V)²
⟹ (1 − x / V)² = 16 / 25
Taking the square root on both sides:
1 − x / V = 4 / 5 = 0.80
- Track Cocoa in Mixture P and Mixture Q:
In Mixture P (after 1 operation):
The remaining coffee fraction is 4 / 5, so coffee volume = 25 × 4 / 5 = 20 units.
The remaining volume is filled with cocoa:
Cocoa in P = 25 − 20 = 5 units
In Mixture Q (after 2 operations):
We are already given the final state ratio, where cocoa constitutes 9 units.
Cocoa in Q = 9 units
The ratio of cocoa in mixture P to mixture Q is 5 : 9.
Correct Option: C
💡 FundaMakers Tip:
When tracking multiple operations, remember that the remaining native liquid follows an exponential factor, but the added tracking component (cocoa) does not. Find the structural multiplier (4 / 5) first from the final remaining coffee pool, map out the intermediate state volumes step-by-step, and avoid calculating complex variable weights.