The ratio of milk and water in two vessels are ` 3 : 4 ` and ` 5 : 6`. The capacity of both vessels are in the ratio `1 : 2`. If the whole quantity of first vessel and half quantity of second vessel are poured into a third vessel, then find the ratio of milk and water in third vessel.

Answer

To solve the problem step by step, we will follow the outlined process to find the ratio of milk and water in the third vessel.

### Step 1: Understand the Ratios in Each Vessel
- For the first vessel, the ratio of milk to water is \(3:4\).
- For the second vessel, the ratio of milk to water is \(5:6\).

### Step 2: Calculate Total Parts in Each Vessel
- In the first vessel, the total parts = \(3 + 4 = 7\).
- In the second vessel, the total parts = \(5 + 6 = 11\).

### Step 3: Equalize the Ratios
To compare the two vessels, we need to express them in terms of a common total. We can use the least common multiple (LCM) of the total parts (7 and 11).
- LCM of 7 and 11 is \(77\).

### Step 4: Scale the Ratios
- For the first vessel, multiply each part by \(11\):
- Milk: \(3 \times 11 = 33\)
- Water: \(4 \times 11 = 44\)

- For the second vessel, multiply each part by \(7\):
- Milk: \(5 \times 7 = 35\)
- Water: \(6 \times 7 = 42\)

### Step 5: Consider the Capacities of the Vessels
The capacity of the first vessel is in the ratio \(1:2\) with the second vessel. Therefore, if we denote the capacity of the first vessel as \(x\), the second vessel will have a capacity of \(2x\).

### Step 6: Calculate the Amounts Used in the Third Vessel
- We pour the whole quantity of the first vessel into the third vessel:
- Total milk from the first vessel = \(33\)
- Total water from the first vessel = \(44\)

- We pour half the quantity of the second vessel into the third vessel:
- Total milk from half of the second vessel = \(35 / 2 = 17.5\)
- Total water from half of the second vessel = \(42 / 2 = 21\)

### Step 7: Combine the Quantities in the Third Vessel
- Total milk in the third vessel:
\[ 33 + 17.5 = 50.5 \]

- Total water in the third vessel:
\[ 44 + 21 = 65 \]

### Step 8: Find the Ratio of Milk to Water in the Third Vessel
The ratio of milk to water in the third vessel is:
\[ \text{Ratio} = \frac{50.5}{65} \]

To simplify this ratio:
- Multiply both parts by \(2\) to eliminate the decimal:
\[ \text{Ratio} = \frac{101}{130} \]

### Step 9: Final Simplification
The ratio of milk to water in the third vessel is \(101:130\).

### Conclusion
Thus, the ratio of milk and water in the third vessel is \(101:130\).

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