Vessels A and B contain 60 litres of alcohol and 60 litres of water, respectively. A certain volume is taken out from A and poured into B. After stirring, the same volume is taken out from B and poured into A. If the resultant ratio of alcohol and water in A is 15 : 4, then the volume, in litres, initially taken out from A is

The correct answer is 16.

Here is the step-by-step logical derivation and solution to the problem:

Step 1: Initial state of the vessels
Vessel A contains 60 litres of pure alcohol.
Vessel B contains 60 litres of pure water.

Step 2: Transferring a volume of liquid from Vessel A to Vessel B
Let the volume taken out from Vessel A be $x$ litres. Since Vessel A contains only alcohol, we are transferring $x$ litres of pure alcohol into Vessel B.
After this transfer, the state of both vessels becomes:

• Vessel A: Contains $(60-x)$ litres of alcohol.
• Vessel B: Contains 60 litres of water and $x$ litres of alcohol. The total volume in Vessel B is now $(60+x)$ litres.

Step 3: Calculating the concentrations in Vessel B
After stirring Vessel B thoroughly, the mixture is homogeneous. The concentration fraction of water and alcohol in Vessel B are:
Fraction of alcohol in B = $\frac{x}{60+x}$

Fraction of water in B = $\frac{60}{60+x}$

Step 4: Transferring the same volume back from Vessel B to Vessel A
Now, the same volume $x$ litres of the mixture is taken from Vessel B and poured back into Vessel A. The quantities of alcohol and water present in this $x$ litres are:
Alcohol transferred = $x\times \frac{x}{60+x}=\frac{x^2}{60+x}$ litres

Water transferred = $x\times \frac{60}{60+x}=\frac{60x}{60+x}$ litres

Step 5: Determining the final quantities in Vessel A
After this second transfer, the final quantities in Vessel A are:
Total Alcohol in A = $(60-x)+\frac{x^2}{60+x}$
To simplify this expression:
Total Alcohol in A = $\frac{(60-x)(60+x)+x^2}{60+x}=\frac{3600-x^2+x^2}{60+x}=\frac{3600}{60+x}$ litres

Total Water in A = $\frac{60x}{60+x}$ litres

Step 6: Setting up the ratio equation
We are given that the final ratio of alcohol to water in Vessel A is 15 : 4. Therefore:
$\frac{\text{Alcohol in A}}{\text{Water in A}}=\frac{15}{4}$

Substituting the simplified expressions:
$\frac{\left(\frac{3600}{60+x}\right)}{\left(\frac{60x}{60+x}\right)}=\frac{15}{4}$

Simplifying the left-hand side:
$\frac{3600}{60x}=\frac{15}{4}$

$\frac{60}{x}=\frac{15}{4}$

Step 7: Solving for $x$
Cross-multiplying gives:
$15x=60\times 4$

$15x=240$

$x=\frac{240}{15}=16$

Thus, the volume initially taken out from Vessel A is 16 litres.