From a container, 6 litres milk was drawn out and was replaced by water. Again 6 litres of mixture was drawn out & was replaced by the water. Thus the quantity of milk and water in the container after these two operations is 9 : 16. The quantity of mixture is :

Options

A. 15
B. 16
C. 25
D. 31

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript and apply the necessary mathematical principles.

### Step 1: Understand the Problem
We start with a container filled with milk. We remove 6 liters of milk and replace it with water. We then remove 6 liters of the mixture (which now contains both milk and water) and replace it with more water. After these operations, the ratio of milk to water in the container is given as 9:16.

### Step 2: Set Up the Ratios
Let the total quantity of the mixture in the container be \( x \) liters. After the operations, the ratio of milk to the total mixture is given as:
\[ \frac{\text{Milk}}{\text{Milk} + \text{Water}} = \frac{9}{25} \]
This means that the quantity of milk is \( \frac{9}{25}x \) and the quantity of water is \( \frac{16}{25}x \).

### Step 3: Apply the Formula for Replacement
We know that when a certain amount of liquid is removed and replaced, the quantity of the remaining liquid can be calculated using the formula:
\[ \text{Remaining liquid} = x \left(1 - \frac{y}{x}\right)^n \]
Where:
- \( y \) is the quantity removed (6 liters),
- \( n \) is the number of operations (2 in this case).

### Step 4: Substitute Values into the Formula
After the first operation, the quantity of milk left can be expressed as:
\[ \text{Milk after 1st operation} = x \left(1 - \frac{6}{x}\right) \]
After the second operation, the quantity of milk left becomes:
\[ \text{Milk after 2nd operation} = x \left(1 - \frac{6}{x}\right)^2 \]

### Step 5: Set Up the Equation
We know that after two operations, the remaining milk is \( \frac{9}{25}x \). Therefore, we can set up the equation:
\[ x \left(1 - \frac{6}{x}\right)^2 = \frac{9}{25}x \]
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ \left(1 - \frac{6}{x}\right)^2 = \frac{9}{25} \]

### Step 6: Solve the Equation
Taking the square root of both sides:
\[ 1 - \frac{6}{x} = \frac{3}{5} \quad \text{or} \quad 1 - \frac{6}{x} = -\frac{3}{5} \]
We will only consider the first equation since the second would not yield a valid positive quantity for \( x \):
\[ 1 - \frac{6}{x} = \frac{3}{5} \]
Rearranging gives:
\[ \frac{6}{x} = 1 - \frac{3}{5} = \frac{2}{5} \]
Thus,
\[ x = \frac{6 \cdot 5}{2} = 15 \text{ liters} \]

### Final Answer
The quantity of the mixture in the container is **15 liters**.