Sharabi chand 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 of 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. Here’s how we can approach it:

### Step 1: Understand the Ratios
We have two mixtures:
- First mixture (M1): Alcohol to Water = 3:4
- Second mixture (M2): Alcohol to Water = 5:6

We need to find the quantity of the first mixture required to make a third mixture (M3) with a total volume of 18 liters and a ratio of alcohol to water = 4:5.

### Step 2: Calculate the Proportions of Alcohol in Each Mixture
1. For the first mixture (M1):
- Total parts = 3 (alcohol) + 4 (water) = 7
- Proportion of alcohol = 3/7

2. For the second mixture (M2):
- Total parts = 5 (alcohol) + 6 (water) = 11
- Proportion of alcohol = 5/11

3. For the third mixture (M3):
- Total parts = 4 (alcohol) + 5 (water) = 9
- Proportion of alcohol = 4/9

### Step 3: Set Up the Alligation
We will set up the alligation to find the ratio in which the two mixtures should be mixed.

- Alcohol proportion in M1 = 3/7
- Alcohol proportion in M2 = 5/11
- Alcohol proportion in M3 = 4/9

Now we will find the differences:
- Difference between M1 and M3:
\[ \frac{3}{7} - \frac{4}{9} \]
To calculate this, we need a common denominator (63):
\[ \frac{3 \times 9}{63} - \frac{4 \times 7}{63} = \frac{27}{63} - \frac{28}{63} = -\frac{1}{63} \]
(We take the absolute value for the ratio, so it becomes \( \frac{1}{63} \))

- Difference between M2 and M3:
\[ \frac{5}{11} - \frac{4}{9} \]
Using a common denominator (99):
\[ \frac{5 \times 9}{99} - \frac{4 \times 11}{99} = \frac{45}{99} - \frac{44}{99} = \frac{1}{99} \]

### Step 4: Form the Ratio
Now we can form the ratio of the two mixtures:
- The ratio of M1 to M2 is given by the differences we calculated:
\[ \text{Ratio} = \frac{1/99}{1/63} = \frac{63}{99} = \frac{7}{11} \]

### Step 5: Calculate the Quantities in the Mixture
Let the quantity of the first mixture (M1) be \( 7x \) and the quantity of the second mixture (M2) be \( 11x \). The total volume of the mixture is 18 liters:
\[ 7x + 11x = 18 \]
\[ 18x = 18 \implies x = 1 \]

### Step 6: Find the Quantity of the First Mixture
Now substituting \( x \) back:
\[ \text{Quantity of M1} = 7x = 7 \times 1 = 7 \text{ liters} \]

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

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