Three bottles of equal capacity have mixture of milk and water in ratio `5 : 7, 7 : 9 and 2 : 1` respectively. These three bottles are emptied into a large bottle. What is the percentage of milk in the new mixture?

Options

A. `49.6`
B. `52.3`
C. `51.2`
D. `50.7`

Answer

To solve the problem of finding the percentage of milk in the new mixture from three bottles with different ratios of milk and water, we can follow these steps:

### Step 1: Identify the Ratios
The ratios of milk to water in the three bottles are:
- Bottle 1: 5:7
- Bottle 2: 7:9
- Bottle 3: 2:1

### Step 2: Determine the Total Parts in Each Ratio
For each bottle, calculate the total parts of the mixture:
- Bottle 1: 5 + 7 = 12 parts
- Bottle 2: 7 + 9 = 16 parts
- Bottle 3: 2 + 1 = 3 parts

### Step 3: Equalize the Capacities
To compare the mixtures, we can assume each bottle has a capacity that can be scaled to a common multiple. Let's use 48 as a common capacity:
- For Bottle 1 (12 parts): Each part = 48 / 12 = 4
- Milk = 5 parts × 4 = 20
- Water = 7 parts × 4 = 28
- For Bottle 2 (16 parts): Each part = 48 / 16 = 3
- Milk = 7 parts × 3 = 21
- Water = 9 parts × 3 = 27
- For Bottle 3 (3 parts): Each part = 48 / 3 = 16
- Milk = 2 parts × 16 = 32
- Water = 1 part × 16 = 16

### Step 4: Calculate Total Milk and Water
Now, we can add the amounts of milk and water from all bottles:
- Total Milk = 20 (from Bottle 1) + 21 (from Bottle 2) + 32 (from Bottle 3) = 73
- Total Water = 28 (from Bottle 1) + 27 (from Bottle 2) + 16 (from Bottle 3) = 71

### Step 5: Calculate the Total Mixture
The total mixture is the sum of total milk and total water:
- Total Mixture = Total Milk + Total Water = 73 + 71 = 144

### Step 6: Calculate the Percentage of Milk
To find the percentage of milk in the mixture:
\[ \text{Percentage of Milk} = \left( \frac{\text{Total Milk}}{\text{Total Mixture}} \right) \times 100 = \left( \frac{73}{144} \right) \times 100 \]
Calculating this gives:
\[ \text{Percentage of Milk} \approx 50.7\% \]

### Final Answer
The percentage of milk in the new mixture is approximately **50.7%**.

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