4 L are drawn from a container full of milk and then is filled with water.This operation is performed three more times. The ratio of the quantity of milk in the container and that of water is `16:65`.How much milk did the container hold initially?

Options

A. 24 L
B. 12 L
C. 15 L
D. 25 L

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript and apply the relevant mathematical principles.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a container initially filled with milk. We draw 4 liters of milk and replace it with water. This operation is repeated three more times (a total of four times). At the end, the ratio of milk to water in the container is given as 16:65.

2. **Setting Up the Variables**:
Let the initial volume of milk in the container be \( x \) liters. After each operation, the amount of milk left can be calculated using the formula for successive dilution.

3. **Calculating the Amount of Milk Left After Each Operation**:
The formula for the amount of milk left after drawing \( a \) liters from \( x \) liters of milk is:
\[ \text{Milk left} = x \left(1 - \frac{a}{x}\right)^n \]
where \( n \) is the number of times the operation is performed.

Here, \( a = 4 \) liters and \( n = 4 \).

4. **Substituting Values**:
After 4 operations, the amount of milk left can be expressed as:
\[ \text{Milk left} = x \left(1 - \frac{4}{x}\right)^4 \]

5. **Using the Given Ratio**:
The final ratio of milk to the total liquid (milk + water) is given as \( 16:65 \). This means:
\[ \frac{\text{Milk}}{\text{Total}} = \frac{16}{81} \]
(since \( 16 + 65 = 81 \)).

Therefore, we can set up the equation:
\[ \frac{x \left(1 - \frac{4}{x}\right)^4}{x} = \frac{16}{81} \]

6. **Simplifying the Equation**:
This simplifies to:
\[ \left(1 - \frac{4}{x}\right)^4 = \frac{16}{81} \]

7. **Taking the Fourth Root**:
Taking the fourth root of both sides gives:
\[ 1 - \frac{4}{x} = \frac{2}{3} \]

8. **Solving for \( x \)**:
Rearranging gives:
\[ \frac{4}{x} = 1 - \frac{2}{3} = \frac{1}{3} \]
Cross-multiplying results in:
\[ 4 = \frac{x}{3} \implies x = 4 \times 3 = 12 \]

9. **Conclusion**:
Thus, the initial volume of milk in the container was \( 12 \) liters.

### Final Answer:
The container held **12 liters** of milk initially.

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