From a container of wine, a thief has stolen 15 litres of wine and replaced it with same quantity of water. He again repeated the same process. Thus in three attempts the ratio of wine and water became 343:169. The initial amount of wine in the container was :

Options

A. 75 litres
B. 100 litres
C. 150 litres
D. 120 litres

Answer

To solve the problem step by step, we need to understand the process of successive replacement and how it affects the ratio of wine to water in the container.

### Step 1: Understand the Problem
The thief steals 15 liters of wine and replaces it with 15 liters of water. This process is repeated three times, and after these three attempts, the ratio of wine to water is given as 343:169.

### Step 2: Calculate the Total Mixture
The total mixture after the three attempts can be calculated by adding the parts of wine and water in the ratio:
- Wine = 343 parts
- Water = 169 parts
- Total mixture = 343 + 169 = 512 parts

### Step 3: Determine the Ratio of Wine to Mixture
The ratio of wine to the total mixture after three attempts is:
- Ratio of wine = 343/512
- Ratio of water = 169/512

### Step 4: Set Up the Equation
Let the initial amount of wine in the container be \( x \) liters. After each replacement, the amount of wine left in the container can be calculated using the formula:
\[ \text{Remaining Wine} = x \left(1 - \frac{15}{x}\right)^n \]
where \( n \) is the number of times the process is repeated.

After three attempts:
\[ \text{Remaining Wine} = x \left(1 - \frac{15}{x}\right)^3 \]

### Step 5: Set Up the Ratio Equation
We know that after three attempts, the remaining wine to the total mixture is:
\[ \frac{x \left(1 - \frac{15}{x}\right)^3}{512} = \frac{343}{512} \]

### Step 6: Simplify the Equation
By cross-multiplying, we get:
\[ x \left(1 - \frac{15}{x}\right)^3 = 343 \]

### Step 7: Solve for \( x \)
To solve for \( x \), we need to simplify:
1. Substitute \( y = 1 - \frac{15}{x} \), then \( \frac{15}{x} = 1 - y \) which gives \( x = \frac{15}{1 - y} \).
2. Substitute this back into the equation and solve for \( y \).

However, we can also directly solve:
\[ \left(1 - \frac{15}{x}\right)^3 = \frac{343}{x} \]

### Step 8: Find \( x \)
We know that \( 343 = 7^3 \), so we can assume:
\[ 1 - \frac{15}{x} = \frac{7}{8} \]
This gives us:
\[ \frac{15}{x} = \frac{1}{8} \]
Thus:
\[ x = 15 \times 8 = 120 \]

### Conclusion
The initial amount of wine in the container was **120 liters**.