From a cask of wine, containing 64 litres, 8 litres are drawn out and the cask is filled up with water. If the same process is repeated a second, then a third time, what will be the number of litres of wine left in the cask ?

Options

A. 42 1/8 kg
B. 42 3/8 kg
C. 48 7/8 kg
D. 42 7/8 kg

Answer

To solve the problem of how much wine is left in the cask after drawing out 8 liters and replacing it with water three times, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- The cask initially contains 64 liters of wine.
- Each time, 8 liters of wine is drawn out.

2. **Calculate the Fraction of Wine Remaining After Each Draw**:
- After drawing out 8 liters, the remaining wine in the cask is \(64 - 8 = 56\) liters.
- The fraction of wine remaining after drawing out 8 liters is:
\[ \frac{56}{64} = \frac{7}{8} \]

3. **Apply the Process Repeatedly**:
- This process is repeated three times. The amount of wine left after each process can be calculated using the formula:
\[ \text{Remaining wine} = \text{Initial wine} \times \left(\frac{7}{8}\right)^n \]
where \(n\) is the number of times the process is repeated.

4. **Substitute the Values**:
- Here, the initial amount of wine is 64 liters, and \(n = 3\):
\[ \text{Remaining wine} = 64 \times \left(\frac{7}{8}\right)^3 \]

5. **Calculate \(\left(\frac{7}{8}\right)^3\)**:
- First, calculate \(\left(\frac{7}{8}\right)^3\):
\[ \left(\frac{7}{8}\right)^3 = \frac{7^3}{8^3} = \frac{343}{512} \]

6. **Multiply by the Initial Amount**:
- Now, multiply this fraction by the initial volume of wine:
\[ \text{Remaining wine} = 64 \times \frac{343}{512} \]

7. **Simplify the Calculation**:
- To simplify:
\[ 64 = \frac{64 \times 343}{512} = \frac{21952}{512} \]
- Now, divide:
\[ \frac{21952}{512} = 42.5 \]

8. **Final Result**:
- Therefore, the amount of wine left in the cask after three repetitions of the process is:
\[ 42.5 \text{ liters} \]

### Summary:
The final answer is that there will be **42.5 liters** of wine left in the cask after the process is repeated three times.