In a mixture of 45 L, the ratio of milk and water is 3:2. How much water must be added to make the ratio 9 :10 ?

Options

A. 10 L
B. 17 L
C. 15 L
D. 12 L

Answer

To solve the problem step by step, we will follow these calculations:

1. **Understanding the initial ratio**:
The initial ratio of milk to water is given as 3:2. This means that for every 3 parts of milk, there are 2 parts of water.

2. **Setting up the equation**:
Let the quantity of milk be \(3x\) liters and the quantity of water be \(2x\) liters. Since the total mixture is 45 liters, we can write the equation:
\[ 3x + 2x = 45 \]

3. **Solving for \(x\)**:
Combine the terms:
\[ 5x = 45 \]
Now, divide both sides by 5:
\[ x = 9 \]

4. **Calculating the quantities of milk and water**:
Now we can find the actual quantities of milk and water:
- Milk:
\[ 3x = 3 \times 9 = 27 \text{ liters} \]
- Water:
\[ 2x = 2 \times 9 = 18 \text{ liters} \]

5. **Setting up the new ratio**:
We need to add \(y\) liters of water to achieve a new ratio of milk to water of 9:10. The new quantity of water will be \(18 + y\) liters.

6. **Setting up the ratio equation**:
According to the new ratio, we can set up the equation:
\[ \frac{27}{18 + y} = \frac{9}{10} \]

7. **Cross-multiplying to solve for \(y\)**:
Cross-multiply to eliminate the fraction:
\[ 27 \times 10 = 9 \times (18 + y) \]
This simplifies to:
\[ 270 = 162 + 9y \]

8. **Isolating \(y\)**:
Subtract 162 from both sides:
\[ 270 - 162 = 9y \]
\[ 108 = 9y \]
Now, divide both sides by 9:
\[ y = 12 \]

9. **Conclusion**:
Therefore, the amount of water that must be added is **12 liters**.