In three vessels, the ratio of water and milk is 6:7,5:9 and 8: 7 respectively. If the mixtures of the three vessels are mixed together, then what will be the ratio of water and milk?

Answer

To find the ratio of water and milk when the mixtures from three vessels are combined, we can follow these steps:

### Step 1: Understand the Ratios
We have three vessels with the following ratios of water to milk:
- Vessel 1: Water : Milk = 6 : 7
- Vessel 2: Water : Milk = 5 : 9
- Vessel 3: Water : Milk = 8 : 7

### Step 2: Calculate the Total Parts
For each vessel, we calculate the total parts of the mixture:
- Vessel 1: 6 + 7 = 13 parts
- Vessel 2: 5 + 9 = 14 parts
- Vessel 3: 8 + 7 = 15 parts

### Step 3: Calculate the Proportion of Water and Milk
Now, we can find the proportion of water and milk in each vessel:
- Water in Vessel 1 = 6/13
- Milk in Vessel 1 = 7/13
- Water in Vessel 2 = 5/14
- Milk in Vessel 2 = 9/14
- Water in Vessel 3 = 8/15
- Milk in Vessel 3 = 7/15

### Step 4: Find the Total Water and Milk
To find the total amount of water and milk, we will assume a common quantity for each vessel. Let's assume we take 1 liter from each vessel.

- Total Water:
\[ \text{Water from Vessel 1} = 1 \times \frac{6}{13} = \frac{6}{13} \]
\[ \text{Water from Vessel 2} = 1 \times \frac{5}{14} = \frac{5}{14} \]
\[ \text{Water from Vessel 3} = 1 \times \frac{8}{15} = \frac{8}{15} \]

- Total Milk:
\[ \text{Milk from Vessel 1} = 1 \times \frac{7}{13} = \frac{7}{13} \]
\[ \text{Milk from Vessel 2} = 1 \times \frac{9}{14} = \frac{9}{14} \]
\[ \text{Milk from Vessel 3} = 1 \times \frac{7}{15} = \frac{7}{15} \]

### Step 5: Find the LCM for Denominators
To add these fractions, we need to find the least common multiple (LCM) of the denominators (13, 14, 15). The LCM of 13, 14, and 15 is 2730.

### Step 6: Convert Fractions to a Common Denominator
Now we convert each fraction to have a denominator of 2730:
- Water:
\[ \frac{6}{13} = \frac{6 \times 210}{2730} = \frac{1260}{2730} \]
\[ \frac{5}{14} = \frac{5 \times 195}{2730} = \frac{975}{2730} \]
\[ \frac{8}{15} = \frac{8 \times 182}{2730} = \frac{1456}{2730} \]

- Total Water:
\[ \text{Total Water} = \frac{1260 + 975 + 1456}{2730} = \frac{3691}{2730} \]

- Milk:
\[ \frac{7}{13} = \frac{7 \times 210}{2730} = \frac{1470}{2730} \]
\[ \frac{9}{14} = \frac{9 \times 195}{2730} = \frac{1755}{2730} \]
\[ \frac{7}{15} = \frac{7 \times 182}{2730} = \frac{1274}{2730} \]

- Total Milk:
\[ \text{Total Milk} = \frac{1470 + 1755 + 1274}{2730} = \frac{4499}{2730} \]

### Step 7: Form the Final Ratio
Now, we can form the ratio of total water to total milk:
\[ \text{Ratio of Water to Milk} = \frac{3691}{4499} \]

### Conclusion
The final ratio of water to milk when the mixtures from the three vessels are combined is:
\[ \text{Water : Milk} = 3691 : 4499 \]