If a mixture contains milk and water in the ratio of 4: 1. If 20% of mixture is taken out and same amount of milk and water are added into mixture, then difference between milk and water in final mixture becomes 72 liters. Find the initial amount of mixture.

Correct Answer/Option: 150 liter

Let's solve the problem step-by-step.

Step 1: Understand the Initial Mixture
Let the initial volume of the mixture be $T$ liters.
The ratio of milk to water in the mixture is given as 4 : 1.
Therefore, the initial quantities of milk and water are:
$\text{Initial Milk}=\frac{4}{5}T=0.8T$
$\text{Initial Water}=\frac{1}{5}T=0.2T$

Step 2: Calculate Remaining Quantities after removing 20% of the Mixture
When 20% of the mixture is taken out, the remaining volume of the mixture is 80% of $T$ (i.e., $0.8T$).
Since the ratio of milk to water remains the same (4 : 1) in the left-out mixture, the remaining quantities are:
$\text{Remaining Milk}=0.8\times 0.8T=0.64T$
$\text{Remaining Water}=0.8\times 0.2T=0.16T$

Step 3: Analyze the Addition of Milk and Water
According to the question, the same amount of mixture that was taken out (which is 20% of $T$, or $0.2T$) is restored by adding equal amounts of milk and water.
Let the volume of milk added be $x$ and the volume of water added be $x$.
The total added volume is:
$x+x=0.2T$
$2x=0.2T\Rightarrow x=0.1T$
So, $0.1T$ liters of milk and $0.1T$ liters of water are added back to the mixture.

Step 4: Determine the Final Quantities
The final quantities of milk and water in the mixture are:
$\text{Final Milk}=0.64T+0.1T=0.74T$
$\text{Final Water}=0.16T+0.1T=0.26T$

Step 5: Set Up the Equation to Find the Initial Mixture Volume
We are given that the difference between milk and water in the final mixture is 72 liters:
$\text{Final Milk}-\text{Final Water}=72$
$0.74T-0.26T=72$
$0.48T=72$
$T=\frac{72}{0.48}=150$
Thus, the initial amount of the mixture is 150 liters.