Step 1: Understanding the Question:
The problem asks us to find the ratio in which two liquids of different unit prices should be mixed to achieve a desired intermediate target price.
This is a classic weighted average or alligation problem.
Step 2: Key Formula or Approach:
We can use the Rule of Alligation, which is a visual and quick way to find the mixing ratio.
Ratio = \( \frac{\text{Mean Price} - \text{Price of Cheaper}}{\text{Price of Dearer} - \text{Mean Price}} \).
Alternatively, a simple algebraic equation can be formed: \( C_1 \times Q_1 + C_2 \times Q_2 = C_{\text{mix}} \times (Q_1 + Q_2) \).
Step 3: Detailed Explanation:
Let's use the Rule of Alligation for a quick solution.
The price of the dearer ingredient (Alcohol) is Rs. 3.50 per litre.
The price of the cheaper ingredient (Kerosene) is Rs. 2.50 per litre.
The desired mean price of the resulting mixture is Rs. 2.75 per litre.
By the rule of alligation, we place the prices in a cross pattern.
Difference 1 (Dearer - Mean) = \( 3.50 - 2.75 = 0.75 \).
Difference 2 (Mean - Cheaper) = \( 2.75 - 2.50 = 0.25 \).
The ratio of the quantity of the Dearer ingredient (Alcohol) to the Cheaper ingredient (Kerosene) is equal to Difference 2 divided by Difference 1.
Quantity of Alcohol : Quantity of Kerosene = \( 0.25 : 0.75 \).
To simplify this ratio, divide both sides by 0.25.
\( \frac{0.25}{0.25} : \frac{0.75}{0.25} = 1 : 3 \).
Alternatively, using algebra: Let \( x \) litres of alcohol and \( y \) litres of kerosene be mixed.
Total cost equation: \( 3.50x + 2.50y = 2.75(x + y) \).
Expanding the right side: \( 3.50x + 2.50y = 2.75x + 2.75y \).
Rearranging the terms to group variables: \( 3.50x - 2.75x = 2.75y - 2.50y \).
Simplifying yields \( 0.75x = 0.25y \).
Finding the ratio \( \frac{x}{y} = \frac{0.25}{0.75} = \frac{1}{3} \).
The required proportion is 1:3.
Step 4: Final Answer:
The liquids should be mixed in the proportion 1 : 3.