450 litres of a mixture of milk and water contain the milk and water in the ratio 9 : 1. How much water should be added to get a new mixture containing milk and water in the ratio 3 : 1?

Options

A. 54
B. 90
C. 45
D. 63

Answer

To solve the problem, we need to determine how much water should be added to a 450-liter mixture of milk and water, which currently has a ratio of 9:1, in order to change the ratio to 3:1.

### Step-by-Step Solution:

1. **Determine the current quantities of milk and water:**
- The total mixture is 450 liters.
- The ratio of milk to water is 9:1. This means there are 9 parts milk and 1 part water, making a total of 10 parts.
- To find the amount of milk and water, we can calculate:
- Milk = (9/10) * 450 = 405 liters
- Water = (1/10) * 450 = 45 liters

2. **Set up the new ratio after adding water:**
- We want to change the ratio of milk to water to 3:1.
- Let the amount of water to be added be \( x \) liters.
- After adding \( x \) liters of water, the new amount of water will be \( 45 + x \) liters.

3. **Express the new ratio:**
- The new ratio of milk to water can be expressed as:
\[ \text{Milk} : \text{Water} = 405 : (45 + x) \]
- According to the problem, this ratio should equal 3:1. Therefore, we can set up the equation:
\[ \frac{405}{45 + x} = \frac{3}{1} \]

4. **Cross-multiply to solve for \( x \):**
- Cross-multiplying gives us:
\[ 405 = 3(45 + x) \]
- Expanding the right side:
\[ 405 = 135 + 3x \]

5. **Isolate \( x \):**
- Subtract 135 from both sides:
\[ 405 - 135 = 3x \]
\[ 270 = 3x \]
- Divide both sides by 3:
\[ x = 90 \]

6. **Conclusion:**
- Therefore, the amount of water that should be added to the mixture is **90 liters**.