Equal quantities of three mixtures of milk and water are mixed in the ratio of `1:2, 2:3` and `3:4`. The ratio of water and milk in the mixture is

Options

A. `193:122`
B. `122:193`
C. `61:97`
D. `137:178`

Answer

To solve the problem of finding the ratio of water to milk when equal quantities of three mixtures of milk and water are mixed in the ratios of 1:2, 2:3, and 3:4, we can follow these steps:

### Step 1: Understand the Ratios
We have three mixtures:
1. Mixture A: Milk:Water = 1:2
2. Mixture B: Milk:Water = 2:3
3. Mixture C: Milk:Water = 3:4

### Step 2: Convert Ratios to Parts
Convert each ratio into parts:
- Mixture A: 1 part milk + 2 parts water = 3 parts total
- Mixture B: 2 parts milk + 3 parts water = 5 parts total
- Mixture C: 3 parts milk + 4 parts water = 7 parts total

### Step 3: Find a Common Quantity
Since we are mixing equal quantities of each mixture, we can assume we take a common quantity that is a multiple of the total parts. Let's take 105 units (the least common multiple of 3, 5, and 7).

### Step 4: Calculate Milk and Water in Each Mixture
Now, we can calculate the amount of milk and water in each mixture based on the total parts:
- For Mixture A (3 parts):
- Total = 105 units
- Milk = (1/3) * 105 = 35 units
- Water = (2/3) * 105 = 70 units

- For Mixture B (5 parts):
- Total = 105 units
- Milk = (2/5) * 105 = 42 units
- Water = (3/5) * 105 = 63 units

- For Mixture C (7 parts):
- Total = 105 units
- Milk = (3/7) * 105 = 45 units
- Water = (4/7) * 105 = 60 units

### Step 5: Sum Up Milk and Water
Now, we sum up the quantities of milk and water from all three mixtures:
- Total Milk = 35 + 42 + 45 = 122 units
- Total Water = 70 + 63 + 60 = 193 units

### Step 6: Find the Ratio of Water to Milk
Finally, the ratio of water to milk is:
Water:Milk = 193:122

### Final Answer
Thus, the ratio of water to milk in the mixture is **193:122**.

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