Two vessels contain mixtures of milk and water in the ratio of 8 : 1 and 1 : 5 respectively. The contents of both of these are mixed in a specific ratio into a third vessel. How much mixture must be drawn from the second vessel to fill the third vessel (capacity 26 gallons) completely in order that the resulting mixture may be half milk and half water?
(a)12 gallons
(b)14 gallons
(c)10 gallons
(d)13 gallons

Options

A. 12 gallons
B. 14 gallons
C. 10 gallons
D. 13 gallons

Answer

To solve the problem step by step, we will use the concept of ratios and the method of alligation.

### Step 1: Understand the Ratios of Milk and Water in Each Vessel
- **First Vessel**: The ratio of milk to water is 8:1.
- Total parts = 8 (milk) + 1 (water) = 9 parts
- Milk in the first vessel = \( \frac{8}{9} \) of the mixture
- Water in the first vessel = \( \frac{1}{9} \) of the mixture

- **Second Vessel**: The ratio of milk to water is 1:5.
- Total parts = 1 (milk) + 5 (water) = 6 parts
- Milk in the second vessel = \( \frac{1}{6} \) of the mixture
- Water in the second vessel = \( \frac{5}{6} \) of the mixture

### Step 2: Determine the Desired Ratio in the Third Vessel
We want the final mixture in the third vessel to be half milk and half water, which means:
- Desired ratio = 1:1
- This means in terms of fractions, we want:
- Milk = \( \frac{1}{2} \)
- Water = \( \frac{1}{2} \)

### Step 3: Apply the Alligation Method
Using the alligation method, we can find the proportion of the mixtures from the two vessels that need to be combined to achieve the desired ratio.

1. **Calculate the difference between the desired ratio and the ratios of the two vessels**:
- For the first vessel (milk):
\[ \text{Difference} = \frac{1}{2} - \frac{8}{9} = \frac{9}{18} - \frac{16}{18} = -\frac{7}{18} \]
- For the second vessel (milk):
\[ \text{Difference} = \frac{1}{6} - \frac{1}{2} = \frac{1}{6} - \frac{3}{6} = -\frac{2}{6} = -\frac{1}{3} \]

2. **Set up the alligation**:
- The positive differences are:
- From the first vessel: \( \frac{7}{18} \)
- From the second vessel: \( \frac{1}{3} \)

3. **Find the ratio**:
- The ratio of the amounts taken from the first vessel to the second vessel is given by the inverses of these differences:
\[ \text{Ratio} = \frac{\frac{1}{3}}{\frac{7}{18}} = \frac{1 \times 18}{3 \times 7} = \frac{18}{21} = \frac{6}{7} \]

### Step 4: Calculate the Amounts Based on the Total Capacity
- Let the amount drawn from the first vessel be \( 6x \) and from the second vessel be \( 7x \).
- The total mixture in the third vessel is 26 gallons:
\[ 6x + 7x = 26 \implies 13x = 26 \implies x = 2 \]

### Step 5: Find the Amount Drawn from the Second Vessel
- Amount drawn from the second vessel:
\[ 7x = 7 \times 2 = 14 \text{ gallons} \]

### Final Answer
Thus, the amount of mixture that must be drawn from the second vessel to fill the third vessel completely is **14 gallons**.