Three vessels of equal volumes contain water and syrup in the ratio 4: 1. 5: 2 and 7: 3 respectively. When they are thoroughly mixed together in a large vessel, find the resulting ratio of water and syrup in the mixture. (Assume that in the mixture total volume remains unaltered).

Options

A. `11:30`
B. `19:11`
C. `31:11`
D. `11:35`

Answer

To solve the problem of finding the resulting ratio of water and syrup when mixing three vessels with different ratios of water and syrup, we can follow these steps:

### Step 1: Determine the total volume of each vessel
Each vessel has water and syrup in different ratios:
- Vessel 1: Water : Syrup = 4 : 1
- Vessel 2: Water : Syrup = 5 : 2
- Vessel 3: Water : Syrup = 7 : 3

### Step 2: Calculate the total volume for each vessel
For each vessel, the total volume can be calculated as follows:
- Vessel 1: Total volume = 4 + 1 = 5 units
- Vessel 2: Total volume = 5 + 2 = 7 units
- Vessel 3: Total volume = 7 + 3 = 10 units

### Step 3: Find the least common multiple (LCM) of the total volumes
To mix the vessels, we need to find a common volume. The LCM of 5, 7, and 10 is 70.

### Step 4: Scale the ratios to the common volume
Now we will scale each vessel's ratio to the common volume of 70:
- For Vessel 1 (5 units):
\[ \text{Water} = 4 \times 14 = 56 \quad \text{Syrup} = 1 \times 14 = 14 \]
- For Vessel 2 (7 units):
\[ \text{Water} = 5 \times 10 = 50 \quad \text{Syrup} = 2 \times 10 = 20 \]
- For Vessel 3 (10 units):
\[ \text{Water} = 7 \times 7 = 49 \quad \text{Syrup} = 3 \times 7 = 21 \]

### Step 5: Calculate the total amount of water and syrup
Now we will add the amounts of water and syrup from all three vessels:
- Total Water:
\[ 56 + 50 + 49 = 155 \]
- Total Syrup:
\[ 14 + 20 + 21 = 55 \]

### Step 6: Determine the resulting ratio of water to syrup
The resulting ratio of water to syrup is:
\[ \text{Ratio} = \frac{155}{55} \]

### Step 7: Simplify the ratio
To simplify the ratio, we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 5:
\[ \text{Simplified Ratio} = \frac{155 \div 5}{55 \div 5} = \frac{31}{11} \]

### Final Answer
The resulting ratio of water to syrup in the mixture is **31 : 11**.

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