Vijay purchased two different kinds of alcohol. In the first mixture the ratio of alcohol to water is 3 : 4 and in the second mixture it is 5 : 6. If he mixes the two given mixtures and makes a third mixture of 18 litres in which the ratio of alcohol to water is 4 : 5, the quantity of first mixture (whose ratio is 3 : 4) is required to make the 18 litres of the third kind of mixture is :

Options

A. 6
B. 7
C. 8
D. 9

Answer

To solve the problem step by step, we will use the concept of alligation to find out how much of the first mixture is required to create the third mixture.

### Step 1: Understand the Ratios
- **First Mixture (M1)**: Alcohol to Water ratio is 3:4.
- **Second Mixture (M2)**: Alcohol to Water ratio is 5:6.
- **Third Mixture (M3)**: Alcohol to Water ratio is 4:5.

### Step 2: Calculate the Alcohol Fractions
- For **M1**:
\[ \text{Fraction of Alcohol} = \frac{3}{3+4} = \frac{3}{7} \]

- For **M2**:
\[ \text{Fraction of Alcohol} = \frac{5}{5+6} = \frac{5}{11} \]

- For **M3**:
\[ \text{Fraction of Alcohol} = \frac{4}{4+5} = \frac{4}{9} \]

### Step 3: Set Up the Alligation
Now we will set up the alligation to find the ratio of the quantities of the two mixtures used to create the third mixture.

- **M1**: \( \frac{3}{7} \)
- **M2**: \( \frac{5}{11} \)
- **M3**: \( \frac{4}{9} \)

### Step 4: Calculate the Differences
Using the alligation method, we will subtract the fractions:

1. **Difference between M3 and M1**:
\[ \frac{4}{9} - \frac{3}{7} \]
To perform this subtraction, we need a common denominator (63):
\[ \frac{4}{9} = \frac{28}{63}, \quad \frac{3}{7} = \frac{27}{63} \]
Thus,
\[ \frac{4}{9} - \frac{3}{7} = \frac{28}{63} - \frac{27}{63} = \frac{1}{63} \]

2. **Difference between M3 and M2**:
\[ \frac{5}{11} - \frac{4}{9} \]
Again, using a common denominator (99):
\[ \frac{5}{11} = \frac{45}{99}, \quad \frac{4}{9} = \frac{44}{99} \]
Thus,
\[ \frac{5}{11} - \frac{4}{9} = \frac{45}{99} - \frac{44}{99} = \frac{1}{99} \]

### Step 5: Form the Ratio
Now we have the differences:
- From M1 to M3: \( \frac{1}{63} \)
- From M2 to M3: \( \frac{1}{99} \)

The ratio of the quantities of M1 to M2 is:
\[ \text{Ratio} = \frac{1/63}{1/99} = \frac{99}{63} = \frac{11}{7} \]

### Step 6: Total Quantity in the Mixture
Let the quantities of M1 and M2 be \( 7k \) and \( 11k \) respectively. The total mixture is:
\[ 7k + 11k = 18k \]
Given that the total quantity is 18 liters:
\[ 18k = 18 \implies k = 1 \]

### Step 7: Calculate the Quantity of M1
Now substituting the value of \( k \):
\[ \text{Quantity of M1} = 7k = 7 \times 1 = 7 \text{ liters} \]

### Final Answer
The quantity of the first mixture (whose ratio is 3:4) required to make the 18 liters of the third mixture is **7 liters**.

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