In one glass, milk and water are mixed in the ratio 3 : 5 and in another glass they are mixed in the ratio 6 : 1. In what ratio should the contents of the two glasses be mixed together so that the new mixture contains milk and water in the ratio 1 : 1 ?

Options

A. `20:7`
B. `8:3`
C. `27:4`
D. `25:9`

Answer

To solve the problem step by step, we need to analyze the mixtures in each glass and determine the ratio in which they should be combined to achieve a final mixture of milk and water in a 1:1 ratio.

### Step 1: Identify the Ratios in Each Glass
- **Glass A**: Milk and Water are mixed in the ratio 3:5.
- **Glass B**: Milk and Water are mixed in the ratio 6:1.

### Step 2: Calculate the Total Parts in Each Glass
- For **Glass A**:
- Total parts = 3 (milk) + 5 (water) = 8 parts.
- For **Glass B**:
- Total parts = 6 (milk) + 1 (water) = 7 parts.

### Step 3: Determine the Amount of Milk and Water in Each Glass
Assuming we take 1 liter from each glass for simplicity:
- **Glass A**:
- Milk = (3/8) liters
- Water = (5/8) liters
- **Glass B**:
- Milk = (6/7) liters
- Water = (1/7) liters

### Step 4: Set Up the Mixture Ratios
We want to mix the contents of Glass A and Glass B to achieve a final ratio of 1:1 (milk to water).

Let the quantities taken from Glass A and Glass B be \( x \) and \( y \) liters respectively.

### Step 5: Express the Total Milk and Water in the New Mixture
- Total Milk from both glasses:
\[ \text{Milk} = \left(\frac{3}{8}x + \frac{6}{7}y\right) \]
- Total Water from both glasses:
\[ \text{Water} = \left(\frac{5}{8}x + \frac{1}{7}y\right) \]

### Step 6: Set Up the Equation for 1:1 Ratio
To achieve a 1:1 ratio, we set the total milk equal to the total water:
\[ \frac{3}{8}x + \frac{6}{7}y = \frac{5}{8}x + \frac{1}{7}y \]

### Step 7: Simplify the Equation
Rearranging the equation gives:
\[ \frac{3}{8}x - \frac{5}{8}x = \frac{1}{7}y - \frac{6}{7}y \]
\[ -\frac{2}{8}x = -\frac{5}{7}y \]
\[ \frac{1}{4}x = \frac{5}{7}y \]

### Step 8: Find the Ratio of x to y
Cross-multiplying gives:
\[ 7x = 20y \]
Thus, the ratio of \( x \) to \( y \) is:
\[ \frac{x}{y} = \frac{20}{7} \]

### Step 9: Conclusion
Therefore, the contents of the two glasses should be mixed in the ratio of **20:7**.

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