Two vessels A and B contains acid and water in the ratio 4:3 and 5:3 respectively. Then the ratio in which these mixtures to be mixed to obtain a new mixture in vessel C containing acid and water in the ratio 3:2 is

Options

A. `5:8`
B. `7:8`
C. `7:5`
D. `4:7`

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 that has a specific ratio of acid to water. Let's break it down step by step.

### Step 1: Understand the Ratios in Vessels A and B
- Vessel A contains acid and water in the ratio of 4:3.
- Vessel B contains acid and water in the ratio of 5:3.

From these ratios, we can determine the fraction of acid in each vessel:
- In vessel A, the total parts = 4 + 3 = 7. The fraction of acid = \( \frac{4}{7} \).
- In vessel B, the total parts = 5 + 3 = 8. The fraction of acid = \( \frac{5}{8} \).

### Step 2: Understand the Desired Ratio in Vessel C
- Vessel C should contain acid and water in the ratio of 3:2.
- The total parts = 3 + 2 = 5. The fraction of acid = \( \frac{3}{5} \).

### Step 3: Set Up the Alligation Method
We will use the alligation method to find the ratio in which the two mixtures should be combined.

1. **Calculate the difference between the acid fractions:**
- For vessel A: \( \frac{4}{7} \) (acid in A) - \( \frac{3}{5} \) (acid in C)
- For vessel B: \( \frac{5}{8} \) (acid in B) - \( \frac{3}{5} \) (acid in C)

2. **Finding a common denominator for calculations:**
- The common denominator for 7, 8, and 5 is 280.

### Step 4: Calculate the Differences
- Convert \( \frac{4}{7} \) to a fraction with a denominator of 280:
\[ \frac{4}{7} = \frac{4 \times 40}{7 \times 40} = \frac{160}{280} \]

- Convert \( \frac{5}{8} \) to a fraction with a denominator of 280:
\[ \frac{5}{8} = \frac{5 \times 35}{8 \times 35} = \frac{175}{280} \]

- Convert \( \frac{3}{5} \) to a fraction with a denominator of 280:
\[ \frac{3}{5} = \frac{3 \times 56}{5 \times 56} = \frac{168}{280} \]

### Step 5: Calculate the Differences
- For vessel A:
\[ \frac{4}{7} - \frac{3}{5} = \frac{160}{280} - \frac{168}{280} = -\frac{8}{280} = -\frac{1}{35} \]

- For vessel B:
\[ \frac{5}{8} - \frac{3}{5} = \frac{175}{280} - \frac{168}{280} = \frac{7}{280} = \frac{1}{40} \]

### Step 6: Set Up the Ratio
Now, we take the absolute values of the differences to find the ratio:
- The ratio of the amounts of A to B is given by:
\[ \text{Ratio} = \frac{\frac{1}{40}}{\frac{1}{35}} = \frac{35}{40} = \frac{7}{8} \]

### Final Answer
The ratio in which the mixtures from vessels A and B should be mixed to obtain the desired mixture in vessel C is **7:8**.

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