Equal quantitles 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 and milk in the combined mixture of three different mixtures, we will follow these steps:

### Step 1: Understand the Ratios
We have three mixtures with the following ratios:
1. Mixture 1: Milk : Water = 1 : 2
2. Mixture 2: Milk : Water = 2 : 3
3. Mixture 3: Milk : Water = 3 : 4

### Step 2: Calculate the Total Parts in Each Mixture
For each mixture, we will calculate the total parts:
1. Mixture 1: Total parts = 1 + 2 = 3
2. Mixture 2: Total parts = 2 + 3 = 5
3. Mixture 3: Total parts = 3 + 4 = 7

### Step 3: Determine the Proportions of Milk and Water
Now we will find the proportion of milk and water in each mixture:
1. Mixture 1:
- Milk = 1 part
- Water = 2 parts

2. Mixture 2:
- Milk = 2 parts
- Water = 3 parts

3. Mixture 3:
- Milk = 3 parts
- Water = 4 parts

### Step 4: Equal Quantities of Each Mixture
Since we are mixing equal quantities of each mixture, we can assume we take 1 liter from each mixture for simplicity. Thus, we will calculate the total milk and water from each mixture.

### Step 5: Calculate the Amount of Milk and Water
1. From Mixture 1 (1 liter):
- Milk = (1/3) liters
- Water = (2/3) liters

2. From Mixture 2 (1 liter):
- Milk = (2/5) liters
- Water = (3/5) liters

3. From Mixture 3 (1 liter):
- Milk = (3/7) liters
- Water = (4/7) liters

### Step 6: Total Milk and Water
Now we will add the amounts of milk and water from all three mixtures:
- Total Milk = (1/3) + (2/5) + (3/7)
- Total Water = (2/3) + (3/5) + (4/7)

### Step 7: Finding a Common Denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of 3, 5, and 7 is 105.

### Step 8: Convert Each Fraction
Convert each fraction to have a denominator of 105:
- Total Milk:
- (1/3) = 35/105
- (2/5) = 42/105
- (3/7) = 45/105
- Total Milk = 35/105 + 42/105 + 45/105 = 122/105

- Total Water:
- (2/3) = 70/105
- (3/5) = 63/105
- (4/7) = 60/105
- Total Water = 70/105 + 63/105 + 60/105 = 193/105

### Step 9: Ratio of Water to Milk
Now we can find the ratio of water to milk:
- Ratio = Total Water : Total Milk = (193/105) : (122/105)

### Step 10: Simplifying the Ratio
To simplify, we can multiply both sides by 105:
- Ratio = 193 : 122

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

---