Two vessels contain milk water in the ratio `5:9 and 7: 11`. If both vessels are mixed in ratio 4:3. Find the ratio of milk and water in new mixture?

Options

A. `141:131`
B. `107:89`
C. `109:185`
D. `114:175`

Answer

To find the ratio of milk and water in the new mixture formed by mixing the contents of two vessels, we can follow these steps:

### Step 1: Determine the composition of each vessel.
- **Vessel 1** has milk and water in the ratio of \(5:9\).
- The total parts in Vessel 1 = \(5 + 9 = 14\).
- Milk in Vessel 1 = \(\frac{5}{14}\) of the total volume.
- Water in Vessel 1 = \(\frac{9}{14}\) of the total volume.

- **Vessel 2** has milk and water in the ratio of \(7:11\).
- The total parts in Vessel 2 = \(7 + 11 = 18\).
- Milk in Vessel 2 = \(\frac{7}{18}\) of the total volume.
- Water in Vessel 2 = \(\frac{11}{18}\) of the total volume.

### Step 2: Define the volumes of the vessels.
Assume the total volume of Vessel 1 is \(4x\) and the total volume of Vessel 2 is \(3y\), where \(x\) and \(y\) are constants representing the volumes.

### Step 3: Calculate the quantities of milk and water in each vessel.
- For Vessel 1 (volume = \(4x\)):
- Milk = \(\frac{5}{14} \times 4x = \frac{20x}{14} = \frac{10x}{7}\)
- Water = \(\frac{9}{14} \times 4x = \frac{36x}{14} = \frac{18x}{7}\)

- For Vessel 2 (volume = \(3y\)):
- Milk = \(\frac{7}{18} \times 3y = \frac{21y}{18} = \frac{7y}{6}\)
- Water = \(\frac{11}{18} \times 3y = \frac{33y}{18} = \frac{11y}{6}\)

### Step 4: Combine the quantities from both vessels.
Now we need to find the total quantities of milk and water when both vessels are mixed together.

- Total Milk = Milk from Vessel 1 + Milk from Vessel 2
\[ = \frac{10x}{7} + \frac{7y}{6} \]

- Total Water = Water from Vessel 1 + Water from Vessel 2
\[ = \frac{18x}{7} + \frac{11y}{6} \]

### Step 5: Find a common denominator and simplify.
To add the fractions, we need a common denominator. The least common multiple of 7 and 6 is 42.

- For Milk:
\[ = \frac{10x \times 6}{42} + \frac{7y \times 7}{42} = \frac{60x + 49y}{42} \]

- For Water:
\[ = \frac{18x \times 6}{42} + \frac{11y \times 7}{42} = \frac{108x + 77y}{42} \]

### Step 6: Set the ratio of milk to water.
Now we can express the ratio of milk to water:
\[ \text{Ratio of Milk to Water} = \frac{60x + 49y}{108x + 77y} \]

### Step 7: Substitute values for x and y to find a numerical ratio.
Assuming \(x = 1\) and \(y = 1\) for simplicity:
- Milk = \(60(1) + 49(1) = 109\)
- Water = \(108(1) + 77(1) = 185\)

Thus, the ratio of milk to water in the new mixture is:
\[ \text{Ratio} = 109:185 \]

### Conclusion
The final ratio of milk to water in the new mixture is \(109:185\).