Three utensils contain equal quanlity of mixtures of milk and water in the ratio 6:1,5: 2 and 3:1 respectively. If all the solutions are mixed together, the ratio of milk and water in the final mixture is

Options

A. `65 : 28`
B. `65 : 19`
C. `19 : 65`
D. `19 : 28`

Answer

To solve the problem, we need to find the final ratio of milk and water when three mixtures are combined. Let's break down the solution step by step.

### Step 1: Determine the quantities of milk and water in each mixture.

1. **First Mixture (Ratio 6:1)**:
- Total parts = 6 + 1 = 7 parts
- Milk = (6/7) of the total quantity
- Water = (1/7) of the total quantity

2. **Second Mixture (Ratio 5:2)**:
- Total parts = 5 + 2 = 7 parts
- Milk = (5/7) of the total quantity
- Water = (2/7) of the total quantity

3. **Third Mixture (Ratio 3:1)**:
- Total parts = 3 + 1 = 4 parts
- Milk = (3/4) of the total quantity
- Water = (1/4) of the total quantity

### Step 2: Assume a common quantity for each mixture.

Let’s assume each mixture has a total quantity of 1 liter for simplicity. Therefore, we have:

- **First Mixture**:
- Milk = (6/7) liters
- Water = (1/7) liters

- **Second Mixture**:
- Milk = (5/7) liters
- Water = (2/7) liters

- **Third Mixture**:
- Milk = (3/4) liters
- Water = (1/4) liters

### Step 3: Calculate the total quantities of milk and water after mixing.

1. **Total Milk**:
- From the first mixture: (6/7) liters
- From the second mixture: (5/7) liters
- From the third mixture: (3/4) liters

To add these, we need a common denominator. The least common multiple of 7 and 4 is 28.

- Convert each quantity:
- (6/7) = (6 * 4) / (7 * 4) = 24/28
- (5/7) = (5 * 4) / (7 * 4) = 20/28
- (3/4) = (3 * 7) / (4 * 7) = 21/28

Now, add them:
\[ \text{Total Milk} = \frac{24}{28} + \frac{20}{28} + \frac{21}{28} = \frac{65}{28} \text{ liters} \]

2. **Total Water**:
- From the first mixture: (1/7) liters
- From the second mixture: (2/7) liters
- From the third mixture: (1/4) liters

Again, using the common denominator of 28:
- Convert each quantity:
- (1/7) = (1 * 4) / (7 * 4) = 4/28
- (2/7) = (2 * 4) / (7 * 4) = 8/28
- (1/4) = (1 * 7) / (4 * 7) = 7/28

Now, add them:
\[ \text{Total Water} = \frac{4}{28} + \frac{8}{28} + \frac{7}{28} = \frac{19}{28} \text{ liters} \]

### Step 4: Find the final ratio of milk to water.

The ratio of milk to water is:
\[ \text{Ratio} = \frac{\text{Total Milk}}{\text{Total Water}} = \frac{65/28}{19/28} = \frac{65}{19} \]

### Final Answer:
The ratio of milk to water in the final mixture is **65:19**.

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