Explanation:

Let's break down the question systematically:

- The first bottle has spirit : water = 1 : 4

? Spirit fraction = 1/5, Water fraction = 4/5

- The second bottle has spirit : water = 4 : 1

? Spirit fraction = 4/5, Water fraction = 1/5

- We want to mix them to get spirit : water = 1 : 3

? Spirit fraction in new mix = 1/4, Water fraction = 3/4

Now, let the two bottles be mixed in x : y ratio.

- Amount of spirit in mix = x(1/5) + y(4/5)

- Amount of water in mix = x(4/5) + y(1/5)

- Ratio required: spirit : water = 1/3

Set up the equation:

$$\frac{x(1/5) + y(4/5)}{x(4/5) + y(1/5)} = \frac{1}{3}$$

Cross-multiplied and simplified:

- numerator: x + 4y

- denominator: 4x + y

So,

$$

\frac{x + 4y}{4x + y} = \frac{1}{3}

$$

Cross multiply:

$$

3(x + 4y) = 4x + y \implies 3x + 12y = 4x + y \implies 12y - y = 4x - 3x \implies 11y = x

$$

So, the bottles should be mixed in a 1 : 11 ratio (x : y = 1 : 11, or y : x = 11 : 1).

Let's check the options:

- Option 1: 5 : 1

- Option 2: 6 : 1

- Option 3: 10 : 1

- Option 4: 11 : 1

The correct answer is Option 4: 11 : 1