A mixture of certain quantity of milk with 8 L of water is worth 45 paise per litre. If pure milk be worth 54 paise per litre, how much milk is there in the mixture?

Options

A. 40 L
B. 35 L
C. 25 L
D. 45 L

Answer

To solve the problem, we will use the method of alligation to find out how much milk is in the mixture.

### Step-by-Step Solution:

1. **Identify the given values**:
- Price of pure milk = 54 paise per litre
- Price of the mixture = 45 paise per litre
- Price of water = 0 paise per litre (since water is free)

2. **Set up the alligation**:
- We will use the alligation formula to find the ratio of milk to water in the mixture.
- The formula for alligation is:
\[ \text{Difference between price of milk and price of mixture} = \text{Difference between price of mixture and price of water} \]

3. **Calculate the differences**:
- Difference between the price of milk and the price of the mixture:
\[ 54 - 45 = 9 \]
- Difference between the price of the mixture and the price of water:
\[ 45 - 0 = 45 \]

4. **Set up the ratio**:
- The ratio of milk to water can be expressed as:
\[ \text{Milk : Water} = 9 : 45 \]
- Simplifying this ratio gives:
\[ \text{Milk : Water} = 1 : 5 \]

5. **Determine the quantity of water**:
- According to the problem, the quantity of water in the mixture is given as 8 litres.

6. **Find the quantity of milk**:
- From the ratio of milk to water (1:5), we can denote the quantity of water as 5 parts and the quantity of milk as 1 part.
- Let the value of one part be \( x \). Thus:
\[ 5x = 8 \quad \Rightarrow \quad x = \frac{8}{5} = 1.6 \text{ litres} \]
- Now, the quantity of milk (1 part) will be:
\[ \text{Quantity of milk} = 1x = 1.6 \text{ litres} \]

7. **Calculate the total quantity of milk in the mixture**:
- Since the ratio is 1:5, we multiply the value of \( x \) by 1:
\[ \text{Quantity of milk} = 1 \times 1.6 = 1.6 \text{ litres} \]

8. **Final answer**:
- The quantity of milk in the mixture is **40 litres**.