Two vessels of equal capacity contain juice and water in the ratio of 5:1 and 5: 7 respec tively. The mixtures of both the vessels are mixed and trans Serred into a bigger vessel. What is the ratio of juice and water in the new mixture?

Options

A. `3:2`
B. `5:3`
C. `5:3`
D. `1:2`

Answer

To solve the problem step by step, we will analyze the contents of each vessel and then find the overall ratio of juice to water after mixing the two vessels.

### Step 1: Understand the contents of each vessel
- **Vessel 1** has juice and water in the ratio of 5:1.
- **Vessel 2** has juice and water in the ratio of 5:7.

### Step 2: Define the total capacity of each vessel
Let the capacity of each vessel be \( X \).

### Step 3: Calculate the amounts of juice and water in Vessel 1
In Vessel 1:
- Total parts = 5 (juice) + 1 (water) = 6 parts.
- Amount of juice = \( \frac{5}{6} \times X = \frac{5X}{6} \)
- Amount of water = \( \frac{1}{6} \times X = \frac{X}{6} \)

### Step 4: Calculate the amounts of juice and water in Vessel 2
In Vessel 2:
- Total parts = 5 (juice) + 7 (water) = 12 parts.
- Amount of juice = \( \frac{5}{12} \times X = \frac{5X}{12} \)
- Amount of water = \( \frac{7}{12} \times X = \frac{7X}{12} \)

### Step 5: Combine the contents of both vessels
Now, we will add the amounts of juice and water from both vessels:
- Total juice = Amount of juice from Vessel 1 + Amount of juice from Vessel 2
\[ = \frac{5X}{6} + \frac{5X}{12} \]
To add these fractions, we need a common denominator, which is 12:
\[ = \frac{10X}{12} + \frac{5X}{12} = \frac{15X}{12} \]

- Total water = Amount of water from Vessel 1 + Amount of water from Vessel 2
\[ = \frac{X}{6} + \frac{7X}{12} \]
Again, we need a common denominator of 12:
\[ = \frac{2X}{12} + \frac{7X}{12} = \frac{9X}{12} \]

### Step 6: Find the ratio of juice to water in the new mixture
Now, we have:
- Total juice = \( \frac{15X}{12} \)
- Total water = \( \frac{9X}{12} \)

The ratio of juice to water is:
\[ \text{Ratio} = \frac{\text{Total Juice}}{\text{Total Water}} = \frac{\frac{15X}{12}}{\frac{9X}{12}} = \frac{15}{9} \]

### Step 7: Simplify the ratio
\[ \frac{15}{9} = \frac{5}{3} \]

Thus, the final ratio of juice to water in the new mixture is **5:3**.