Correct Option (3)
The problem involves a process of successive dilution and replacement. The initial volume of liquid A is 20 litres. The total volume of the mixture remains constant at 20 litres throughout the process.
The quantity of liquid A remaining after 'n' operations, where a quantity 'x' is removed and replaced from a total volume 'V', can be calculated using the formula for successive dilution:
Final Quantity of A = Initial Quantity of A * (1 - (x/V))^n
In this case:
- Initial Quantity of A = 20 litres
- Quantity removed and replaced (x) = 4 litres
- Total Volume (V) = 20 litres
- Number of operations (n) = 2
Calculating the quantity of liquid A after two operations:
Quantity of A = 20 * (1 - (4/20))^2
= 20 * (1 - 1/5)^2
= 20 * (4/5)^2
= 20 * (16/25)
= (4 * 16) / 5
= 64/5 litres
Since the total volume of the liquid in the bottle remains 20 litres, the quantity of liquid B in the final mixture is:
Quantity of B = Total Volume - Quantity of A
= 20 - 64/5
= (100 - 64) / 5
= 36/5 litres
The ratio of the quantity of liquid A to that of liquid B in the final mixture is:
Ratio A : B = (64/5) : (36/5)
= 64 : 36
Dividing both sides by 4:
= 16 : 9
Thus, the final ratio of liquid A to liquid B is 16 : 9.
Incorrect Options:
Options 1 (4 : 1), 2 (5 : 1), and 4 (17 : 8) are incorrect. These ratios do not accurately reflect the final composition of the mixture after two successive removal and replacement operations, as calculated by the principles of mixtures and dilutions. They may result from errors in calculation or incomplete application of the dilution process.