The ratios of acid and water in vessels A and B are 4 : 5 and 7:5, respectively. In what ratio should the contents of A and B be mixed to get a solution containing 50% acid?

Options

A. A)`4:3`
B. B)`2:3`
C. C)`3:2`
D. D)`3:4`

Answer

To solve the problem of mixing the contents of vessels A and B to achieve a solution containing 50% acid, we can follow these steps:

### Step 1: Determine the Composition of Each Vessel
- **Vessel A** has a ratio of acid to water of 4:5.
- This means there are 4 parts acid and 5 parts water.
- Total parts = 4 + 5 = 9 parts.
- Acid in A = \( \frac{4}{9} \) of the total mixture.

- **Vessel B** has a ratio of acid to water of 7:5.
- This means there are 7 parts acid and 5 parts water.
- Total parts = 7 + 5 = 12 parts.
- Acid in B = \( \frac{7}{12} \) of the total mixture.

### Step 2: Set Up the Allegation Method
To find the ratio in which the two vessels should be mixed, we can use the allegation method.

1. **Calculate the percentage of acid in each vessel:**
- For Vessel A: \( \frac{4}{9} \) (approximately 0.444 or 44.44%)
- For Vessel B: \( \frac{7}{12} \) (approximately 0.583 or 58.33%)

2. **Desired percentage of acid in the mixture:**
- We want a mixture that has 50% acid.

### Step 3: Apply the Allegation Formula
Using the allegation method, we can set up the following:

- **Difference between the percentage of acid in A and the desired percentage:**
\[ 50\% - 44.44\% = 5.56\% \]

- **Difference between the percentage of acid in B and the desired percentage:**
\[ 58.33\% - 50\% = 8.33\% \]

### Step 4: Find the Ratio
Now, we can find the ratio of the quantities of A and B to be mixed:

- The ratio of A to B is given by the differences calculated:
\[ \text{Ratio of A to B} = \frac{8.33}{5.56} \]

To simplify this, we can convert the percentages into fractions:
- \( \frac{8.33}{5.56} \) can be approximated as \( \frac{5}{3} \) after simplifying.

### Step 5: Final Ratio
Thus, the ratio in which the contents of A and B should be mixed to obtain a solution containing 50% acid is:
\[ \text{Ratio of A to B} = 3:2 \]

### Conclusion
Therefore, the final answer is that the contents of vessels A and B should be mixed in the ratio of **3:2**.

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