A container has a capacity of 20 gallons and is full of spirit. 4 gallons of spirit is drawn out and the container is again filled with water. This process is repeated 5 times. Find out how much spirit is left in the resulting mixture finally?

Options

A. `6 (257/525)` gallons
B. `6(346)/625` gallons
C. 6.5 gallons
D. 6.25 gallons

Answer

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

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
The container has a capacity of 20 gallons and is full of spirit. Therefore, the initial amount of spirit in the container is 20 gallons.

**Hint**: Always start by identifying the total capacity and the initial contents of the container.

2. **Calculating the Amount Removed**:
Each time we draw out 4 gallons of spirit. After the first draw, the amount of spirit left in the container is:
\[ 20 - 4 = 16 \text{ gallons} \]

**Hint**: When removing a certain amount, subtract it from the total to find the remaining quantity.

3. **Filling with Water**:
After removing 4 gallons of spirit, we fill the container back up to 20 gallons with water. Thus, the total volume remains 20 gallons, but now it contains 16 gallons of spirit and 4 gallons of water.

**Hint**: Remember that the total volume of the container stays constant even after adding water.

4. **Finding the Fraction of Spirit Remaining**:
The fraction of spirit remaining after the first draw is:
\[ \frac{16}{20} = \frac{4}{5} \]

**Hint**: To find the fraction remaining, divide the remaining amount by the total capacity.

5. **Repeating the Process**:
This process is repeated 5 times. Each time, the fraction of spirit remaining is multiplied by the fraction of spirit left after each draw. Therefore, after 5 repetitions, the amount of spirit left is given by:
\[ \text{Remaining spirit} = 20 \times \left(\frac{4}{5}\right)^5 \]

**Hint**: Use exponentiation to simplify repeated multiplication of the same fraction.

6. **Calculating the Final Amount**:
Now, we calculate \(\left(\frac{4}{5}\right)^5\):
\[ \left(\frac{4}{5}\right)^5 = \frac{4^5}{5^5} = \frac{1024}{3125} \]
Therefore, the amount of spirit left is:
\[ 20 \times \frac{1024}{3125} = \frac{20480}{3125} \]

**Hint**: Break down the powers and simplify fractions step by step for easier calculations.

7. **Final Calculation**:
To get the final value, we can perform the division:
\[ \frac{20480}{3125} \approx 6.5536 \text{ gallons} \]

**Hint**: When dividing large numbers, ensure you simplify or convert to decimal for easier interpretation.

### Conclusion:
After repeating the process 5 times, the amount of spirit left in the container is approximately 6.5536 gallons.