The ratio of milk and water in two vessels having same capacity are 2 : 3 and 5 : 4. Both are poured into a third vessel, then find the ratio of milk and water in third vessel.

Answer

To solve the problem, we need to find the ratio of milk and water in the third vessel after mixing the contents of the first two vessels. Let's break it down step by step.

### Step 1: Understand the Ratios in Each Vessel
- **Vessel 1** has a milk to water ratio of **2:3**.
- **Vessel 2** has a milk to water ratio of **5:4**.

### Step 2: Determine the Total Parts in Each Vessel
- For **Vessel 1**:
- Total parts = 2 (milk) + 3 (water) = **5 parts**.

- For **Vessel 2**:
- Total parts = 5 (milk) + 4 (water) = **9 parts**.

### Step 3: Equalize the Capacities of the Vessels
Since both vessels have the same capacity, we need to find a common multiple of the total parts:
- The least common multiple (LCM) of 5 and 9 is **45**.

### Step 4: Calculate the Amount of Milk and Water in Each Vessel
- For **Vessel 1** (scaled to a total of 45):
- Milk: \( \frac{2}{5} \times 45 = 18 \) units
- Water: \( \frac{3}{5} \times 45 = 27 \) units

- For **Vessel 2** (scaled to a total of 45):
- Milk: \( \frac{5}{9} \times 45 = 25 \) units
- Water: \( \frac{4}{9} \times 45 = 20 \) units

### Step 5: Combine the Amounts in the Third Vessel
Now, we add the amounts of milk and water from both vessels into the third vessel:
- Total Milk in Vessel 3: \( 18 + 25 = 43 \) units
- Total Water in Vessel 3: \( 27 + 20 = 47 \) units

### Step 6: Find the Ratio of Milk to Water in the Third Vessel
The ratio of milk to water in the third vessel is:
- Ratio = Milk : Water = \( 43 : 47 \)

### Final Answer
The ratio of milk and water in the third vessel is **43:47**.

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