From a vessel filled with alcohol. 1/5 of its contents is removed and the vessel is than filled up with water. If this be done 5 times in succession, what proportion of the alcohol originally contained in the vessel will have been removed from it ?

Options

A. (1024)/(3125)
B. (2101)/(3125)
C. (1024)/(2101)
D. None of these

Answer

To solve the problem, we need to determine the proportion of alcohol that remains in the vessel after removing 1/5 of its contents and replacing it with water, repeated 5 times.

### Step-by-Step Solution:

1. **Understanding the Removal Process**:
When we remove 1/5 of the contents, we are left with 4/5 of the original contents of alcohol in the vessel.

2. **Setting Up the Formula**:
The formula to calculate the amount of substance left after repeated removals is:
\[ \text{Amount left} = \left(1 - \frac{B}{N}\right)^N \]
where:
- \( B = \frac{1}{5} \) (the fraction removed each time),
- \( N = 5 \) (the number of times the process is repeated).

3. **Substituting Values**:
In our case, we can substitute \( B \) and \( N \) into the formula:
\[ \text{Amount left} = \left(1 - \frac{1}{5}\right)^5 = \left(\frac{4}{5}\right)^5 \]

4. **Calculating \( \left(\frac{4}{5}\right)^5 \)**:
We need to calculate \( \left(\frac{4}{5}\right)^5 \):
\[ \left(\frac{4}{5}\right)^5 = \frac{4^5}{5^5} = \frac{1024}{3125} \]

5. **Finding the Proportion of Alcohol Removed**:
To find the proportion of alcohol that has been removed, we subtract the amount left from the original amount (which we consider as 1):
\[ \text{Proportion removed} = 1 - \text{Amount left} = 1 - \frac{1024}{3125} \]
\[ = \frac{3125 - 1024}{3125} = \frac{2101}{3125} \]

6. **Final Result**:
The proportion of alcohol originally contained in the vessel that has been removed is:
\[ \frac{2101}{3125} \]