85 kg of a mixture contains milk and water in the ratio 27: 7. How much more water is to be added to get a new mixture containing milk and water in the ratio 3:1 ?

Options

A. 5 kg
B. 6.5 kg
C. 7.25 kg
D. 8 kg

Answer

To solve the problem, we need to determine how much more water should be added to an existing mixture of milk and water in order to change the ratio of milk to water from 27:7 to 3:1.

### Step-by-Step Solution:

1. **Determine the total parts in the initial ratio**:
The initial ratio of milk to water is 27:7.
Total parts = 27 + 7 = 34.

2. **Calculate the amount of milk in the mixture**:
The total weight of the mixture is 85 kg.
Amount of milk = (27/34) * 85 kg = (27 * 85) / 34 = 67.5 kg.

3. **Calculate the amount of water in the mixture**:
Amount of water = (7/34) * 85 kg = (7 * 85) / 34 = 17.5 kg.

4. **Set up the equation for the new ratio**:
We want the new mixture to have a milk to water ratio of 3:1.
Let x be the amount of water to be added.
The new amount of water will be 17.5 kg + x.

5. **Formulate the equation based on the new ratio**:
According to the new ratio:
\[ \frac{\text{Milk}}{\text{Water}} = \frac{3}{1} \]
This can be rewritten as:
\[ \frac{67.5}{17.5 + x} = \frac{3}{1} \]

6. **Cross-multiply to solve for x**:
Cross-multiplying gives:
\[ 67.5 = 3(17.5 + x) \]
Expanding the right side:
\[ 67.5 = 52.5 + 3x \]

7. **Isolate x**:
Subtract 52.5 from both sides:
\[ 67.5 - 52.5 = 3x \]
\[ 15 = 3x \]
Divide both sides by 3:
\[ x = 5 \]

8. **Conclusion**:
Therefore, the amount of water to be added is **5 kg**.