In a milk shoppe there are three varieties of milk, 'Pure,' Cure' and 'Lure'. The 'Pure' milk has `100%` concentration of milk. The ratio of milk is to water in the 'Cure' is 2: 5 and in the Lure it is 3: 8 respectively. Sonali purchased 14 litres od Cure and 22 litres of Lure milk and mixed them. If she wanted to make the concentration of milk in the mixture of purchased milk to `50%`. How many litres of 'Pure' milk she is needed?

Options

A. 6 litres
B. 8 litres
C. 16 litres
D. 18 litres

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the amount of milk in 'Cure' and 'Lure'
- **Cure Milk**: The ratio of milk to water is 2:5. This means for every 2 parts of milk, there are 5 parts of water, making a total of 7 parts.
- Total volume of Cure purchased = 14 liters.
- Milk in Cure = (2/7) * 14 = 4 liters.

- **Lure Milk**: The ratio of milk to water is 3:8. This means for every 3 parts of milk, there are 8 parts of water, making a total of 11 parts.
- Total volume of Lure purchased = 22 liters.
- Milk in Lure = (3/11) * 22 = 6 liters.

### Step 2: Calculate the total amount of milk and water in the mixture
- Total milk from both varieties:
- Total milk = Milk from Cure + Milk from Lure = 4 liters + 6 liters = 10 liters.

- Total volume of the mixture:
- Total mixture = Cure + Lure = 14 liters + 22 liters = 36 liters.

### Step 3: Calculate the amount of water in the mixture
- Total water in the mixture:
- Water from Cure = Total Cure - Milk in Cure = 14 liters - 4 liters = 10 liters.
- Water from Lure = Total Lure - Milk in Lure = 22 liters - 6 liters = 16 liters.
- Total water = Water from Cure + Water from Lure = 10 liters + 16 liters = 26 liters.

### Step 4: Set up the equation for the desired concentration
- Sonali wants the final concentration of milk to be 50%. This means that in the final mixture, the amount of milk should equal the amount of water.
- Let \( x \) be the amount of Pure milk Sonali needs to add.
- The new total amount of milk will be \( 10 + x \) liters.
- The total amount of water remains 26 liters.

### Step 5: Set up the equation for 50% concentration
To achieve a 50% concentration:
\[ 10 + x = 26 \]

### Step 6: Solve for \( x \)
\[ x = 26 - 10 \]
\[ x = 16 \]

### Conclusion
Sonali needs to add **16 liters** of Pure milk to achieve a concentration of 50% in the mixture.

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