In three vessels each of 10 litres capacity, mixture of milk and water is filled. The ratios of milk and water are `2 : 1`, `3 : 1` and `3 : 2` in the three respective vessels. If all the three vessels are emptied into a single large vessel, find the proportion of milk and and water in the mixture.

Options

A. `181:49`
B. `101:49`
C. `121:59`
D. `131:69`

Answer

To solve the problem of finding the proportion of milk and water when three vessels are combined, we will follow these steps:

### Step 1: Determine the amount of milk and water in each vessel.

1. **Vessel 1**: The ratio of milk to water is 2:1.
- Total parts = 2 + 1 = 3
- Amount of milk = (2/3) * 10 liters = 20/3 liters
- Amount of water = (1/3) * 10 liters = 10/3 liters

2. **Vessel 2**: The ratio of milk to water is 3:1.
- Total parts = 3 + 1 = 4
- Amount of milk = (3/4) * 10 liters = 30/4 = 7.5 liters
- Amount of water = (1/4) * 10 liters = 10/4 = 2.5 liters

3. **Vessel 3**: The ratio of milk to water is 3:2.
- Total parts = 3 + 2 = 5
- Amount of milk = (3/5) * 10 liters = 30/5 = 6 liters
- Amount of water = (2/5) * 10 liters = 20/5 = 4 liters

### Step 2: Calculate the total amount of milk and water when all vessels are combined.

- **Total Milk**:
- From Vessel 1: 20/3 liters
- From Vessel 2: 7.5 liters = 30/4 liters
- From Vessel 3: 6 liters
- Total Milk = (20/3) + (30/4) + 6

To add these fractions, we need a common denominator. The least common multiple of 3, 4, and 1 (the denominator of 6) is 12.

- Convert each term:
- (20/3) = (20 * 4)/(3 * 4) = 80/12
- (30/4) = (30 * 3)/(4 * 3) = 90/12
- 6 = (6 * 12)/12 = 72/12

Now, add them:
Total Milk = (80/12) + (90/12) + (72/12) = (80 + 90 + 72)/12 = 242/12 liters

- **Total Water**:
- From Vessel 1: 10/3 liters
- From Vessel 2: 2.5 liters = 10/4 liters
- From Vessel 3: 4 liters
- Total Water = (10/3) + (10/4) + 4

Using the same common denominator of 12:
- Convert each term:
- (10/3) = (10 * 4)/(3 * 4) = 40/12
- (10/4) = (10 * 3)/(4 * 3) = 30/12
- 4 = (4 * 12)/12 = 48/12

Now, add them:
Total Water = (40/12) + (30/12) + (48/12) = (40 + 30 + 48)/12 = 118/12 liters

### Step 3: Find the ratio of milk to water.

Now that we have the total amounts:
- Total Milk = 242/12 liters
- Total Water = 118/12 liters

The ratio of milk to water is:
Ratio = Total Milk : Total Water = (242/12) : (118/12)

This simplifies to:
Ratio = 242 : 118

To simplify further, divide both numbers by their greatest common divisor (GCD). The GCD of 242 and 118 is 2.
- Simplified Ratio = (242/2) : (118/2) = 121 : 59

### Final Answer:
The proportion of milk to water in the mixture is **121 : 59**.

---