Two vessels A and B contain spirit and water mixed in the ratio 5:2 and 7:6 respectively. Find the ratio in which these mixture be mixed to obtain a new mixture in vessel C containing spirit and water in the ratio 8:5?

Options

A. a. 4:8
B. b. 3:9
C. c. 7:9
D. d. 5:6

Answer

To solve the problem, we need to find the ratio in which the mixtures from vessels A and B should be mixed to obtain a new mixture in vessel C containing spirit and water in the ratio of 8:5.

### Step-by-Step Solution:

1. **Identify the Ratios in Vessels A and B:**
- Vessel A contains spirit and water in the ratio of 5:2.
- Vessel B contains spirit and water in the ratio of 7:6.

2. **Calculate the Fraction of Spirit in Each Vessel:**
- For Vessel A:
- Total parts = 5 (spirit) + 2 (water) = 7
- Fraction of spirit in A = \( \frac{5}{7} \)
- For Vessel B:
- Total parts = 7 (spirit) + 6 (water) = 13
- Fraction of spirit in B = \( \frac{7}{13} \)

3. **Calculate the Fraction of Spirit in Vessel C:**
- Vessel C contains spirit and water in the ratio of 8:5.
- Total parts = 8 (spirit) + 5 (water) = 13
- Fraction of spirit in C = \( \frac{8}{13} \)

4. **Set Up the Equation Using Alligation:**
- Let the quantity of mixture from A be \( x \) and from B be \( y \).
- According to the alligation rule:
\[ \frac{\text{Spirit in A} - \text{Spirit in C}}{\text{Spirit in B} - \text{Spirit in C}} = \frac{y}{x} \]
- Substitute the values:
\[ \frac{\frac{5}{7} - \frac{8}{13}}{\frac{7}{13} - \frac{8}{13}} = \frac{y}{x} \]

5. **Calculate the Numerator and Denominator:**
- Find a common denominator to subtract the fractions:
- For \( \frac{5}{7} - \frac{8}{13} \):
- Common denominator = 91
- Convert \( \frac{5}{7} = \frac{65}{91} \)
- Convert \( \frac{8}{13} = \frac{56}{91} \)
- So, \( \frac{5}{7} - \frac{8}{13} = \frac{65 - 56}{91} = \frac{9}{91} \)

- For \( \frac{7}{13} - \frac{8}{13} \):
- \( \frac{7 - 8}{13} = \frac{-1}{13} \)

6. **Set Up the Ratio:**
- Now we have:
\[ \frac{\frac{9}{91}}{\frac{-1}{13}} = \frac{y}{x} \]
- This simplifies to:
\[ \frac{9}{91} \times \frac{13}{-1} = \frac{y}{x} \]
- Which gives:
\[ \frac{-117}{91} = \frac{y}{x} \]

7. **Final Ratio:**
- The ratio of \( x:y \) (vessel A to vessel B) is:
\[ x:y = 91:117 \]
- Simplifying this gives:
\[ x:y = 7:9 \]

### Final Answer:
The ratio in which the mixtures from vessels A and B should be mixed to obtain the new mixture in vessel C is **7:9**.