The ratio of spirit and water in the two vessels is 5 : 1 and 3 : 7 respectively. In what ratio the liquid of both the vessels should be mixed such that a new mixture containing half spirit and half water is obtained?

Options

A. `3 : 5`
B. `5 : 7`
C. `1 : 1`
D. `4 : 7`

Answer

To solve the problem of mixing liquids from two vessels with different ratios of spirit and water to achieve a new mixture that is half spirit and half water, we can follow these steps:

### Step 1: Understand the Ratios
- The first vessel has a spirit to water ratio of 5:1.
- The second vessel has a spirit to water ratio of 3:7.

### Step 2: Calculate the Fraction of Spirit and Water in Each Vessel
- For the first vessel:
- Total parts = 5 (spirit) + 1 (water) = 6 parts.
- Fraction of spirit = 5/6.
- Fraction of water = 1/6.

- For the second vessel:
- Total parts = 3 (spirit) + 7 (water) = 10 parts.
- Fraction of spirit = 3/10.
- Fraction of water = 7/10.

### Step 3: Set Up the Equation for Mixing
Let the quantities of liquid taken from the first and second vessels be \( x \) and \( y \) respectively. We want the final mixture to have equal parts of spirit and water, i.e., a ratio of 1:1.

The total spirit in the mixture can be expressed as:
\[ \text{Spirit from first vessel} + \text{Spirit from second vessel} = \frac{5}{6}x + \frac{3}{10}y \]

The total water in the mixture can be expressed as:
\[ \text{Water from first vessel} + \text{Water from second vessel} = \frac{1}{6}x + \frac{7}{10}y \]

### Step 4: Set Up the Equality for Half Spirit and Half Water
Since we want the final mixture to have equal amounts of spirit and water, we can set up the equation:
\[ \frac{5}{6}x + \frac{3}{10}y = \frac{1}{6}x + \frac{7}{10}y \]

### Step 5: Simplify the Equation
To eliminate the fractions, we can multiply through by the least common multiple of the denominators, which is 30:
\[ 30\left(\frac{5}{6}x + \frac{3}{10}y\right) = 30\left(\frac{1}{6}x + \frac{7}{10}y\right) \]
This simplifies to:
\[ 25x + 9y = 5x + 21y \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ 25x - 5x = 21y - 9y \]
\[ 20x = 12y \]

### Step 7: Find the Ratio of x to y
Dividing both sides by 4 gives:
\[ 5x = 3y \]
Thus, the ratio of \( x \) to \( y \) is:
\[ \frac{x}{y} = \frac{3}{5} \]

### Conclusion
The ratio in which the liquids from both vessels should be mixed to obtain a new mixture containing half spirit and half water is **3:5**.

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