A butler stole wine from a butt of sherry which contained 32% of spirit and then replaced what he stole, by wine containing only 18% spirit. The butt was then of 24% strength only. How much of the butt had he stolen?

Options

A. `3/8`
B. `5/7`
C. `4/7`
D. `7/11`

Answer

To solve the problem step by step, we will use the concept of alligation.

### Step 1: Identify the strengths of the wines
We have two types of wine:
- Wine A (the original butt of sherry): 32% spirit
- Wine B (the wine used to replace what was stolen): 18% spirit
- Final mixture strength: 24% spirit

### Step 2: Set up the alligation
We will use the alligation method to find the ratio in which the two wines are mixed.

1. Calculate the difference between the strength of Wine A and the final mixture:
\[ 32\% - 24\% = 8\% \]
2. Calculate the difference between the strength of Wine B and the final mixture:
\[ 24\% - 18\% = 6\% \]

### Step 3: Write the alligation formula
Now we can set up the alligation:
- The difference from Wine A (32% to 24%) is 8%.
- The difference from Wine B (18% to 24%) is 6%.

### Step 4: Find the ratio
The ratio of the quantities of Wine A to Wine B is given by the differences calculated:
\[ \text{Ratio} = \frac{6}{8} = \frac{3}{4} \]

### Step 5: Calculate the total parts
The total parts in the mixture is:
\[ 3 + 4 = 7 \]

### Step 6: Determine the amount stolen
Let \( x \) be the amount of wine stolen. According to the ratio, the part of Wine B (the wine with 18% spirit) is 4 parts out of a total of 7 parts. Therefore, the amount of wine stolen can be calculated as:
\[ \text{Amount stolen} = \frac{4}{7} \text{ of the total volume of the butt} \]

### Conclusion
Thus, the butler stole \(\frac{4}{7}\) of the total volume of the butt.

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