Two equal vessels are filled with the mixtures of water and milk in the ratio of 3:4 and 5:3 respectively. If the mixtures are poured into a third vessel, the ratio of water and milk in the third vessel will be

Options

A. `15:12`
B. `53:59`
C. `20:9`
D. `59:53`

Answer

To solve the problem, we need to find the ratio of water and milk when two mixtures are combined in a third vessel. Let's break down the solution step by step.

### Step 1: Understand the Ratios in Each Vessel
- The first vessel has a mixture of water and milk in the ratio of 3:4. This means:
- Total parts = 3 + 4 = 7 parts
- Water in the first vessel = (3/7) of the total volume
- Milk in the first vessel = (4/7) of the total volume

- The second vessel has a mixture of water and milk in the ratio of 5:3. This means:
- Total parts = 5 + 3 = 8 parts
- Water in the second vessel = (5/8) of the total volume
- Milk in the second vessel = (3/8) of the total volume

### Step 2: Assume the Volume of Each Vessel
Let's assume the volume of each vessel is 1 liter. Therefore:
- For the first vessel:
- Water = (3/7) liters
- Milk = (4/7) liters

- For the second vessel:
- Water = (5/8) liters
- Milk = (3/8) liters

### Step 3: Calculate Total Water and Milk in the Third Vessel
Now, we need to combine the contents of both vessels into the third vessel.

- Total Water:
\[ \text{Total Water} = \left(\frac{3}{7} + \frac{5}{8}\right) \]

To add these fractions, we need a common denominator. The least common multiple of 7 and 8 is 56.

Converting each fraction:
- \(\frac{3}{7} = \frac{3 \times 8}{7 \times 8} = \frac{24}{56}\)
- \(\frac{5}{8} = \frac{5 \times 7}{8 \times 7} = \frac{35}{56}\)

Now, add them:
\[ \text{Total Water} = \frac{24}{56} + \frac{35}{56} = \frac{59}{56} \text{ liters} \]

- Total Milk:
\[ \text{Total Milk} = \left(\frac{4}{7} + \frac{3}{8}\right) \]

Using the same common denominator of 56:
- \(\frac{4}{7} = \frac{4 \times 8}{7 \times 8} = \frac{32}{56}\)
- \(\frac{3}{8} = \frac{3 \times 7}{8 \times 7} = \frac{21}{56}\)

Now, add them:
\[ \text{Total Milk} = \frac{32}{56} + \frac{21}{56} = \frac{53}{56} \text{ liters} \]

### Step 4: Find the Ratio of Water to Milk
Now that we have the total amounts of water and milk in the third vessel, we can find the ratio:
\[ \text{Ratio of Water to Milk} = \frac{\text{Total Water}}{\text{Total Milk}} = \frac{\frac{59}{56}}{\frac{53}{56}} = \frac{59}{53} \]

### Final Answer
The ratio of water to milk in the third vessel is **59:53**.

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