Ram prepares solutions of alcohol in water according to customers' needs. This morning Ram has prepared 27 litres of a 12% alcohol solution and kept it ready in a 27 litre delivery container to be shipped to the customer. Just before delivery, he finds out that the customer had asked for 27 litres of 21% alcohol solution. To prepare what the customer u wants, Ram replaces a portion of 12% solution by 39% solution. How many litres of 12% solution are replaced?

Options

A. 5
B. 9
C. 10
D. 12

Answer

To solve the problem, we need to determine how many liters of the 12% alcohol solution Ram needs to replace with the 39% alcohol solution in order to create a final solution that is 27 liters of 21% alcohol.

### Step-by-Step Solution:

1. **Define the Variables**:
Let \( x \) be the amount (in liters) of the 12% solution that Ram replaces with the 39% solution.

2. **Calculate the Amount of Alcohol in the Original Solution**:
The original 27 liters of 12% solution contains:
\[ \text{Alcohol from 12% solution} = 27 \times \frac{12}{100} = 3.24 \text{ liters} \]

3. **Calculate the Amount of Alcohol in the New Solution**:
After replacing \( x \) liters of the 12% solution with \( x \) liters of the 39% solution, the amount of alcohol in the new solution will be:
- Alcohol from the remaining 12% solution:
\[ \text{Remaining 12% alcohol} = (27 - x) \times \frac{12}{100} = \frac{12(27 - x)}{100} \]
- Alcohol from the 39% solution:
\[ \text{Alcohol from 39% solution} = x \times \frac{39}{100} = \frac{39x}{100} \]

4. **Set Up the Equation for the Final Solution**:
The total amount of alcohol in the final solution must equal the amount of alcohol in 27 liters of 21% solution:
\[ \text{Total alcohol in final solution} = \frac{12(27 - x)}{100} + \frac{39x}{100} \]
The amount of alcohol in 27 liters of 21% solution is:
\[ 27 \times \frac{21}{100} = 5.67 \text{ liters} \]

5. **Equate and Solve the Equation**:
Set the total alcohol from the new solution equal to the alcohol in the 21% solution:
\[ \frac{12(27 - x)}{100} + \frac{39x}{100} = 5.67 \]
Multiply through by 100 to eliminate the fraction:
\[ 12(27 - x) + 39x = 567 \]
Expanding gives:
\[ 324 - 12x + 39x = 567 \]
Combine like terms:
\[ 27x + 324 = 567 \]
Subtract 324 from both sides:
\[ 27x = 243 \]
Divide by 27:
\[ x = 9 \]

6. **Conclusion**:
Ram needs to replace **9 liters** of the 12% solution with the 39% solution.