A mixture contains wine and water in the ratio 3 : 2 and another mixture contains them in the ratio 4: 5. How many litres of the later must be mixed with 3 litres of the former so that the resulting mixture may contain equal quantities of wine and water?
एक मिश्रण में शराब व पानी का अनुपात 3:2 है और किसी दूसरे मिश्रण में यह 4:5 है, तो दूसरे मिश्रण की कितनी मात्रा पहले मिश्रण की 3 लीटर मात्रा में मिलाई जाए कि एक मिश्रण में शराब व पानी 1:1 के अनुपात में हो जाए ?

Options

A. A. `5(2)/5` litres
B. B. `5(2)/(3)` litres
C. C. `4(1)/(2)` litres
D. D. `3(3)/(4)` litres

Explanation

To solve the problem, we need to find out how many liters of the second mixture (which has a wine to water ratio of 4:5) must be mixed with 3 liters of the first mixture (which has a wine to water ratio of 3:2) in order to achieve a final mixture where the quantities of wine and water are equal.

### Step-by-Step Solution:

1. **Determine the quantities of wine and water in the first mixture (3 liters)**:
- The ratio of wine to water in the first mixture is 3:2.
- Total parts = 3 + 2 = 5 parts.
- Quantity of wine in the first mixture = \(\frac{3}{5} \times 3 = 1.8\) liters.
- Quantity of water in the first mixture = \(\frac{2}{5} \times 3 = 1.2\) liters.

2. **Define the quantities of wine and water in the second mixture**:
- Let the quantity of the second mixture to be mixed be \(x\) liters.
- The ratio of wine to water in the second mixture is 4:5.
- Total parts = 4 + 5 = 9 parts.
- Quantity of wine in the second mixture = \(\frac{4}{9} \times x\) liters.
- Quantity of water in the second mixture = \(\frac{5}{9} \times x\) liters.

3. **Set up the equation for equal quantities of wine and water in the final mixture**:
- Total quantity of wine in the final mixture = Wine from the first mixture + Wine from the second mixture:
\[ 1.8 + \frac{4}{9}x \]
- Total quantity of water in the final mixture = Water from the first mixture + Water from the second mixture:
\[ 1.2 + \frac{5}{9}x \]
- Set these two expressions equal to each other:
\[ 1.8 + \frac{4}{9}x = 1.2 + \frac{5}{9}x \]

4. **Solve the equation**:
- Rearranging gives:
\[ 1.8 - 1.2 = \frac{5}{9}x - \frac{4}{9}x \]
\[ 0.6 = \frac{1}{9}x \]
- Multiply both sides by 9:
\[ x = 0.6 \times 9 = 5.4 \text{ liters} \]

5. **Convert the answer to a mixed number**:
- \(5.4\) liters can be expressed as \(5 \frac{2}{5}\) liters.

### Final Answer:
The amount of the second mixture that must be mixed with 3 liters of the first mixture is **5 liters and 2/5 liters**.