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 will follow these steps:

### Step 1: Determine the initial quantities of milk and water in the mixture.

The total volume of the mixture is 450 litres, and the ratio of milk to water is 9:1.

- Let the quantity of milk be \( 9x \) and the quantity of water be \( 1x \).
- Therefore, \( 9x + 1x = 450 \).

Calculating \( x \):
\[ 10x = 450 \\ x = \frac{450}{10} = 45 \]

Now, substituting back to find the quantities:
- Milk = \( 9x = 9 \times 45 = 405 \) litres
- Water = \( 1x = 1 \times 45 = 45 \) litres

### Step 2: Set up the equation for the new ratio.

We want to change the ratio of milk to water to 3:1. Let \( y \) be the amount of water to be added.

After adding \( y \) litres of water, the new quantities will be:
- Milk = 405 litres
- Water = \( 45 + y \) litres

The new ratio of milk to water will be:
\[ \frac{405}{45 + y} = \frac{3}{1} \]

### Step 3: Solve for \( y \).

Cross-multiplying gives:
\[ 405 = 3(45 + y) \\ 405 = 135 + 3y \]

Now, isolate \( y \):
\[ 405 - 135 = 3y \\ 270 = 3y \\ y = \frac{270}{3} = 90 \]

### Conclusion

Thus, the amount of water that should be added is **90 litres**.

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