Two vessels A and B contain solution of acid and water. In A and B the ratio of acid and water are 7 : 3 and 4 : 1, respectively. They are mixed in the ratio 1 : 2. What is the ratio of acid and water in the resulting solution?

Options

A. ` 15 :8`
B. `16 :7`
C. `23 :7`
D. `25 :9`

Answer

To find the ratio of acid and water in the resulting solution after mixing the solutions from vessels A and B, we can follow these steps:

### Step 1: Determine the composition of vessel A
In vessel A, the ratio of acid to water is given as 7:3.
- Total parts = 7 (acid) + 3 (water) = 10 parts.
- Therefore, in 1 unit of vessel A:
- Quantity of acid = \( \frac{7}{10} \) units
- Quantity of water = \( \frac{3}{10} \) units

### Step 2: Determine the composition of vessel B
In vessel B, the ratio of acid to water is given as 4:1.
- Total parts = 4 (acid) + 1 (water) = 5 parts.
- Therefore, in 2 units of vessel B (since A and B are mixed in the ratio 1:2):
- Quantity of acid = \( \frac{4}{5} \times 2 = \frac{8}{5} \) units
- Quantity of water = \( \frac{1}{5} \times 2 = \frac{2}{5} \) units

### Step 3: Combine the quantities of acid and water from both vessels
Now, we combine the quantities of acid and water from both vessels:
- Total acid = Acid from A + Acid from B
\[ = \frac{7}{10} + \frac{8}{5} \]
To add these, we need a common denominator (which is 10):
\[ = \frac{7}{10} + \frac{16}{10} = \frac{23}{10} \]

- Total water = Water from A + Water from B
\[ = \frac{3}{10} + \frac{2}{5} \]
Again, we need a common denominator (which is 10):
\[ = \frac{3}{10} + \frac{4}{10} = \frac{7}{10} \]

### Step 4: Calculate the final ratio of acid to water
Now, we can find the ratio of acid to water in the final mixture:
\[ \text{Ratio of acid to water} = \frac{\text{Total acid}}{\text{Total water}} = \frac{\frac{23}{10}}{\frac{7}{10}} = \frac{23}{7} \]

### Final Answer
The ratio of acid to water in the resulting solution is **23:7**.

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