Three vessels having volumes in the ratio of 2:3:5 are full of a mixture of water and milk. In the first vessel ratio of water and milk is 1:3, in second 2:3 and in third vessel 2:5. If all the three vessels were poured out in a large container what is the resulting ratio of milk and water?

Options

A. `43:96`
B. `438:962`
C. `348:962`
D. `962:438`

Answer

To find the resulting ratio of milk and water when the contents of three vessels are poured into a large container, we will follow these steps:

### Step 1: Determine the volumes of the vessels
The volumes of the three vessels are in the ratio 2:3:5. Let’s denote the volumes as:
- Volume of Vessel 1 = 2x
- Volume of Vessel 2 = 3x
- Volume of Vessel 3 = 5x

### Step 2: Determine the ratios of water and milk in each vessel
1. **Vessel 1**: The ratio of water to milk is 1:3.
- Total parts = 1 + 3 = 4
- Water in Vessel 1 = (1/4) * (2x) = (2x/4) = (x/2)
- Milk in Vessel 1 = (3/4) * (2x) = (6x/4) = (3x/2)

2. **Vessel 2**: The ratio of water to milk is 2:3.
- Total parts = 2 + 3 = 5
- Water in Vessel 2 = (2/5) * (3x) = (6x/5)
- Milk in Vessel 2 = (3/5) * (3x) = (9x/5)

3. **Vessel 3**: The ratio of water to milk is 2:5.
- Total parts = 2 + 5 = 7
- Water in Vessel 3 = (2/7) * (5x) = (10x/7)
- Milk in Vessel 3 = (5/7) * (5x) = (25x/7)

### Step 3: Calculate the total amounts of water and milk
Now, we will sum the amounts of water and milk from all three vessels.

**Total Water**:
- Water from Vessel 1 = \( \frac{x}{2} \)
- Water from Vessel 2 = \( \frac{6x}{5} \)
- Water from Vessel 3 = \( \frac{10x}{7} \)

To add these, we need a common denominator. The least common multiple of 2, 5, and 7 is 70.

- Convert each fraction:
- \( \frac{x}{2} = \frac{35x}{70} \)
- \( \frac{6x}{5} = \frac{84x}{70} \)
- \( \frac{10x}{7} = \frac{100x}{70} \)

Now, add them together:
\[ \text{Total Water} = \frac{35x + 84x + 100x}{70} = \frac{219x}{70} \]

**Total Milk**:
- Milk from Vessel 1 = \( \frac{3x}{2} \)
- Milk from Vessel 2 = \( \frac{9x}{5} \)
- Milk from Vessel 3 = \( \frac{25x}{7} \)

Again, we will convert these to have a common denominator of 70.

- Convert each fraction:
- \( \frac{3x}{2} = \frac{105x}{70} \)
- \( \frac{9x}{5} = \frac{126x}{70} \)
- \( \frac{25x}{7} = \frac{250x}{70} \)

Now, add them together:
\[ \text{Total Milk} = \frac{105x + 126x + 250x}{70} = \frac{481x}{70} \]

### Step 4: Calculate the resulting ratio of milk to water
Now, we have:
- Total Water = \( \frac{219x}{70} \)
- Total Milk = \( \frac{481x}{70} \)

The ratio of Milk to Water is:
\[ \text{Ratio of Milk to Water} = \frac{481x/70}{219x/70} = \frac{481}{219} \]

### Step 5: Simplify the ratio
To simplify \( \frac{481}{219} \), we can check for common factors. The GCD of 481 and 219 is 1, so the ratio is already in its simplest form.

Thus, the resulting ratio of milk to water when all three vessels are poured into a large container is:
\[ \text{Milk : Water} = 481 : 219 \]