A butler stole wine from a butt of sherry containing `50%` of spirit, then he replenished 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. 44256
B. 44257
C. 44228
D. 44287

Answer

To solve the problem, we will use the method of alligation. Here's a step-by-step breakdown of the solution:

### Step 1: Identify the strengths of the wines involved
- The original wine has a spirit strength of 50%.
- The wine used to replenish has a spirit strength of 20%.
- The resulting mixture has a spirit strength of 30%.

### Step 2: Set up the alligation
We will set up the alligation to find the ratio of the quantities of the two wines.

- The formula for alligation is:
\[ \text{Difference between the strengths} = \text{Strength of one wine} - \text{Strength of mixture} \]
\[ \text{Difference between the strengths} = \text{Strength of mixture} - \text{Strength of other wine} \]

### Step 3: Calculate the differences
- For the 50% wine:
\[ 50\% - 30\% = 20\% \]
- For the 20% wine:
\[ 30\% - 20\% = 10\% \]

### Step 4: Set up the ratio
Now we can set up the ratio of the quantities of the two wines:
\[ \text{Ratio} = \frac{20}{10} = 2:1 \]
This means that for every 2 parts of the original wine (50% spirit), there is 1 part of the replenishing wine (20% spirit).

### Step 5: Determine the total parts
The total parts in the mixture is:
\[ 2 + 1 = 3 \text{ parts} \]

### Step 6: Calculate the proportion of the original wine
The proportion of the original wine in the mixture is:
\[ \frac{2 \text{ parts}}{3 \text{ parts}} = \frac{2}{3} \]

### Step 7: Conclusion
Thus, the butler stole \(\frac{2}{3}\) of the original wine.

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