A bottle contains 20 litres of liquid A. 4 litres of liquid A is taken out of it and replaced by same quantity of liquid B. Again 4 litres of the mixture is taken out and replaced by same quantity of liquid B. What is the ratio of quantity of liquid A to that of liquid B in the final mixture?

The correct option is 16 : 9.

Let us solve this step-by-step using the standard replacement formula for mixtures.

Step 1: Identify the initial parameters
Initial volume of liquid A in the bottle, $V=20$ litres.
Volume of liquid drawn out and replaced in each operation, $x=4$ litres.
Number of times the operation is repeated, $n=2$.

Step 2: Formula for remaining quantity of liquid A
When a container contains $V$ units of a liquid, and $x$ units are taken out and replaced by another liquid, and this process is repeated $n$ times, the remaining quantity of the original liquid (liquid A) is given by:

$\text{Remaining quantity of Liquid A}=V\×{\left(1-\frac{x}{V}\right)}^n$

Step 3: Calculate the remaining quantity of liquid A
Substitute the values into the formula:

$\text{Remaining Liquid A}=20\×{\left(1-\frac{4}{20}\right)}^2$

$\text{Remaining Liquid A}=20\×{\left(1-\frac{1}{5}\right)}^2$

$\text{Remaining Liquid A}=20\×{\left(\frac{4}{5}\right)}^2$

$\text{Remaining Liquid A}=20\×\frac{16}{25}=\frac{64}{5}=12.8\text{litres}$

Step 4: Calculate the quantity of liquid B in the final mixture
Since the total volume of the liquid in the bottle remains constant at 20 litres throughout the process:

$\text{Quantity of Liquid B}=\text{Total Volume}-\text{Remaining Liquid A}$

$\text{Quantity of Liquid B}=20-12.8=7.2\text{litres}$

Step 5: Find the ratio of Liquid A to Liquid B
Now, find the ratio of the final quantity of liquid A to liquid B:

$\text{Ratio (A : B)}=\frac{12.8}{7.2}=\frac{128}{72}$

Dividing both the numerator and the denominator by 8:

$\text{Ratio (A : B)}=\frac{16}{9}$

Thus, the ratio of quantity of liquid A to that of liquid B in the final mixture is 16 : 9.