Two vessels contain a mixture of spirit and water. In the first vessel the ratio of spirit to water is `8 : 3` and in the second vessel the ratio is `5 : 1`. A 35 litre cask is filled from these vessels so as to contain a mixture of spirit and water in the ratio of `4 : 1`. How many litres are taken from the first vessel?

Options

A. 11 litres
B. 22 litres
C. 16.5 litres
D. 17.5 litres

Answer

To solve the problem step by step, we will use the method of allegations to find out how many liters are taken from the first vessel.

### Step 1: Determine the ratios of spirit and water in both vessels.

- **First Vessel:** The ratio of spirit to water is 8:3.
- Total parts = 8 + 3 = 11
- Water in the first vessel = 3/11

- **Second Vessel:** The ratio of spirit to water is 5:1.
- Total parts = 5 + 1 = 6
- Water in the second vessel = 1/6

- **Desired Mixture:** The desired ratio of spirit to water is 4:1.
- Total parts = 4 + 1 = 5
- Water in the desired mixture = 1/5

### Step 2: Set up the allegation.

We will set up the allegation to find the quantities from each vessel.

1. **Water in the first vessel:** 3/11
2. **Water in the second vessel:** 1/6
3. **Water in the desired mixture:** 1/5

### Step 3: Calculate the differences.

- Difference between water in the first vessel and the desired mixture:
\[ \text{Difference 1} = \frac{3}{11} - \frac{1}{5} \]
To calculate this, find a common denominator (which is 55):
\[ \frac{3}{11} = \frac{15}{55}, \quad \frac{1}{5} = \frac{11}{55} \]
\[ \text{Difference 1} = \frac{15}{55} - \frac{11}{55} = \frac{4}{55} \]

- Difference between water in the second vessel and the desired mixture:
\[ \text{Difference 2} = \frac{1}{6} - \frac{1}{5} \]
Again, find a common denominator (which is 30):
\[ \frac{1}{6} = \frac{5}{30}, \quad \frac{1}{5} = \frac{6}{30} \]
\[ \text{Difference 2} = \frac{5}{30} - \frac{6}{30} = -\frac{1}{30} \]
(Since we are looking for the absolute value, we take it as \(\frac{1}{30}\)).

### Step 4: Set up the ratio of quantities taken from each vessel.

The ratio of the quantities taken from the first vessel to the second vessel is given by the differences calculated:
\[ \text{Ratio} = \text{Difference 2} : \text{Difference 1} = \frac{1}{30} : \frac{4}{55} \]
To simplify this ratio, we can multiply both sides by 30 * 55 to eliminate the fractions:
\[ \text{Ratio} = 55 : 4 \times 30 = 55 : 120 \]
This simplifies to:
\[ 11 : 24 \]

### Step 5: Calculate the total quantity taken from each vessel.

Let the quantity taken from the first vessel be \(11x\) and from the second vessel be \(24x\). The total quantity is given as 35 liters:
\[ 11x + 24x = 35 \]
\[ 35x = 35 \implies x = 1 \]

### Step 6: Find the quantity taken from the first vessel.

Now, substituting \(x\) back:
\[ \text{Quantity from the first vessel} = 11x = 11 \times 1 = 11 \text{ liters} \]

### Final Answer:
The quantity taken from the first vessel is **11 liters**.

---