What quantity of water is required to be mixed with a mixture having alcohol & water in ratio 4:1 such that final mixture contains equal quantity of both? (final mixture quantity is 50 lit)

Options

A. A)25 lit
B. B)20.75 lit
C. C)18.75 lít
D. D)22.75 lit

Answer

To solve the problem of how much water is required to be mixed with a mixture of alcohol and water in a ratio of 4:1 to achieve a final mixture containing equal quantities of both, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Mixture**:
The initial mixture has alcohol and water in the ratio of 4:1. This means for every 4 parts of alcohol, there is 1 part of water.

2. **Define the Total Parts**:
The total parts in the initial mixture = 4 (alcohol) + 1 (water) = 5 parts.

3. **Determine the Quantity of Alcohol and Water in the Initial Mixture**:
Let’s denote the total quantity of the initial mixture as \( x \) liters.
- Alcohol = \(\frac{4}{5}x\)
- Water = \(\frac{1}{5}x\)

4. **Final Mixture Requirement**:
We need to add water such that the final mixture has equal quantities of alcohol and water. The final mixture quantity is given as 50 liters.

5. **Set Up the Equation for Final Mixture**:
Let \( y \) be the quantity of water added. After adding \( y \) liters of water, the new quantities will be:
- Alcohol = \(\frac{4}{5}x\)
- Water = \(\frac{1}{5}x + y\)

For the final mixture to have equal quantities of alcohol and water, we set:
\[ \frac{4}{5}x = \frac{1}{5}x + y \]

6. **Express Total Mixture**:
The total mixture after adding water is:
\[ x + y = 50 \]

7. **Substituting for \( y \)**:
From the equation \( y = \frac{4}{5}x - \frac{1}{5}x \):
\[ y = \frac{3}{5}x \]

8. **Substituting into Total Mixture Equation**:
Substitute \( y \) in the total mixture equation:
\[ x + \frac{3}{5}x = 50 \]
\[ \frac{8}{5}x = 50 \]

9. **Solving for \( x \)**:
Multiply both sides by \( \frac{5}{8} \):
\[ x = 50 \times \frac{5}{8} = 31.25 \text{ liters} \]

10. **Finding \( y \)**:
Now substitute \( x \) back to find \( y \):
\[ y = \frac{3}{5} \times 31.25 = 18.75 \text{ liters} \]

### Final Answer:
The quantity of water required to be mixed is **18.75 liters**.