The ratio of milk and water in four vessels are `1 : 2, 3 : 2, 7 : 3` and `1 : 1`. The ratio of capacity of all vessels are in the ratio `1 : 2 : 3 : 4`. All are poured into a fifth vessel, then find the ratio of milk and water in a fifth vessel, then find the ratio of milk and water in fifth vessel.

Options

A. 169:141
B. 237:129
C. 156:57
D. 105:63

Answer

To solve the problem step by step, we will first determine the amount of milk and water in each vessel based on their ratios and capacities, then combine these amounts to find the overall ratio in the fifth vessel.

### Step 1: Determine the capacities of the vessels
The capacities of the vessels are in the ratio 1:2:3:4. Let's assume the capacities are:
- Vessel 1: 1 liter
- Vessel 2: 2 liters
- Vessel 3: 3 liters
- Vessel 4: 4 liters

### Step 2: Calculate the quantity of milk and water in each vessel

#### Vessel 1 (Ratio of milk to water = 1:2)
- Total parts = 1 + 2 = 3
- Milk = (1/3) * 1 = 1/3 liter
- Water = (2/3) * 1 = 2/3 liter

#### Vessel 2 (Ratio of milk to water = 3:2)
- Total parts = 3 + 2 = 5
- Milk = (3/5) * 2 = 6/5 liters
- Water = (2/5) * 2 = 4/5 liters

#### Vessel 3 (Ratio of milk to water = 7:3)
- Total parts = 7 + 3 = 10
- Milk = (7/10) * 3 = 21/10 liters
- Water = (3/10) * 3 = 9/10 liters

#### Vessel 4 (Ratio of milk to water = 1:1)
- Total parts = 1 + 1 = 2
- Milk = (1/2) * 4 = 2 liters
- Water = (1/2) * 4 = 2 liters

### Step 3: Sum the quantities of milk and water from all vessels

#### Total Milk:
- From Vessel 1: 1/3
- From Vessel 2: 6/5
- From Vessel 3: 21/10
- From Vessel 4: 2

To add these, we need a common denominator. The least common multiple of 3, 5, 10, and 1 is 30.

Converting each fraction:
- 1/3 = 10/30
- 6/5 = 36/30
- 21/10 = 63/30
- 2 = 60/30

Now, sum them:
Total Milk = 10/30 + 36/30 + 63/30 + 60/30 = 169/30 liters

#### Total Water:
- From Vessel 1: 2/3
- From Vessel 2: 4/5
- From Vessel 3: 9/10
- From Vessel 4: 2

Again, we convert to a common denominator of 30:
- 2/3 = 20/30
- 4/5 = 24/30
- 9/10 = 27/30
- 2 = 60/30

Now, sum them:
Total Water = 20/30 + 24/30 + 27/30 + 60/30 = 131/30 liters

### Step 4: Find the ratio of milk to water in the fifth vessel
The ratio of milk to water is given by:
\[ \text{Ratio} = \frac{\text{Total Milk}}{\text{Total Water}} = \frac{169/30}{131/30} = \frac{169}{131} \]

### Final Answer
The ratio of milk to water in the fifth vessel is **169 : 131**.

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