The ratios of spirit and water in vessels A and B are 4 : 5 and 7: 11, respectively. The contents of A and B are mixed in the ratio 2 : 5. What is the ratio of spirit and water in the resulting solution?

Options

A. `22:75`
B. `53:75`
C. `17:25`
D. `13:20`

Answer

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

### Step 1: Determine the individual ratios of spirit and water in vessels A and B.
- For vessel A, the ratio of spirit to water is 4:5.
- For vessel B, the ratio of spirit to water is 7:11.

### Step 2: Calculate the total parts of spirit and water in each vessel.
- In vessel A:
- Total parts = 4 (spirit) + 5 (water) = 9 parts.
- Spirit in A = \( \frac{4}{9} \) of the total volume.
- Water in A = \( \frac{5}{9} \) of the total volume.

- In vessel B:
- Total parts = 7 (spirit) + 11 (water) = 18 parts.
- Spirit in B = \( \frac{7}{18} \) of the total volume.
- Water in B = \( \frac{11}{18} \) of the total volume.

### Step 3: Determine the mixing ratio of the two vessels.
- The contents of A and B are mixed in the ratio 2:5.

### Step 4: Assign a variable for the total volume of the mixture.
Let the total volume of the mixture be \( V \). Then:
- Volume from A = \( \frac{2}{7} V \) (since 2 parts out of 7 total parts).
- Volume from B = \( \frac{5}{7} V \) (since 5 parts out of 7 total parts).

### Step 5: Calculate the amount of spirit and water from each vessel in the mixture.
- Spirit from A:
\[ \text{Spirit from A} = \frac{2}{7} V \times \frac{4}{9} = \frac{8}{63} V \]
- Water from A:
\[ \text{Water from A} = \frac{2}{7} V \times \frac{5}{9} = \frac{10}{63} V \]

- Spirit from B:
\[ \text{Spirit from B} = \frac{5}{7} V \times \frac{7}{18} = \frac{35}{126} V = \frac{15}{54} V = \frac{35}{126} V \]
- Water from B:
\[ \text{Water from B} = \frac{5}{7} V \times \frac{11}{18} = \frac{55}{126} V \]

### Step 6: Combine the amounts of spirit and water from both vessels.
- Total spirit in the mixture:
\[ \text{Total Spirit} = \frac{8}{63} V + \frac{35}{126} V \]
To add these, we need a common denominator:
\[ \frac{8}{63} V = \frac{16}{126} V \]
So,
\[ \text{Total Spirit} = \frac{16}{126} V + \frac{35}{126} V = \frac{51}{126} V \]

- Total water in the mixture:
\[ \text{Total Water} = \frac{10}{63} V + \frac{55}{126} V \]
Again, converting \( \frac{10}{63} V \):
\[ \frac{10}{63} V = \frac{20}{126} V \]
Thus,
\[ \text{Total Water} = \frac{20}{126} V + \frac{55}{126} V = \frac{75}{126} V \]

### Step 7: Find the ratio of spirit to water in the resulting mixture.
The ratio of spirit to water is:
\[ \text{Ratio} = \frac{\text{Total Spirit}}{\text{Total Water}} = \frac{\frac{51}{126} V}{\frac{75}{126} V} = \frac{51}{75} \]

### Step 8: Simplify the ratio.
\[ \frac{51}{75} = \frac{17}{25} \]

Thus, the ratio of spirit to water in the resulting solution is **17:25**.

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