A butler stole wine from a butt of sherry containing 50% of spirit, then he replaced it by different wine containing 20% spirit. Thus there was only 30% strength (spirit) in the new mixture. How much of the original wine did he steal?

Options

A. `1/3`
B. `2/3`
C. `1/2`
D. `1/4`

Answer

To solve the problem step by step, we can use the method of alligation. Let's break down the solution:

### Step 1: Understand the Problem
The butler stole some wine from a butt of sherry that contained 50% spirit and replaced it with wine that contained 20% spirit. The resulting mixture has a spirit strength of 30%. We need to find out how much of the original wine was stolen.

### Step 2: Set Up the Alligation
We will use the alligation method to find the ratio of the quantities of the two wines involved.

- **Strength of original wine (A)**: 50% spirit
- **Strength of replaced wine (B)**: 20% spirit
- **Strength of the mixture (C)**: 30% spirit

### Step 3: Apply Alligation Formula
Using the alligation formula, we can find the ratio of the quantities of the original wine and the replaced wine.

- Difference between the strength of original wine and the mixture: \( A - C = 50 - 30 = 20 \)
- Difference between the strength of replaced wine and the mixture: \( C - B = 30 - 20 = 10 \)

### Step 4: Set Up the Ratio
The ratio of the quantities of the original wine to the replaced wine is given by the differences calculated:

\[ \text{Ratio} = \frac{A - C}{C - B} = \frac{20}{10} = 2:1 \]

This means for every 2 parts of the original wine, there is 1 part of the replaced wine.

### Step 5: Determine the Total Parts
Let’s denote the total parts of the mixture as \( x \). According to the ratio we found:
- Original wine = 2 parts
- Replaced wine = 1 part

Thus, the total parts = \( 2 + 1 = 3 \) parts.

### Step 6: Calculate the Fraction of Wine Stolen
Since the butler stole the original wine, we need to find out how much of the original wine he stole in terms of the total mixture.

The fraction of the original wine stolen is:
\[ \text{Fraction stolen} = \frac{\text{Original wine}}{\text{Total mixture}} = \frac{2}{3} \]

### Conclusion
The butler stole \( \frac{2}{3} \) of the original wine.

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