In what ratio should sugar costing ₹24 per kg be mixed with sugar costing ₹49 per kg so that by selling the mixture at ₹48.6 per kg, there is a profit of 8%?

1. A

   22:6

2. B

   4:21

3. C

   20:2

4. D

   23:2

Appeared in: DSSSB 13 August 2024

Explanation

• The first step is to calculate the true Cost Price (CP) of the mixture, as the alligation rule applies to costs, not selling prices.
• Using the formula CP = Selling Price / (1 + Profit%/100), the mixture's CP is calculated as 48.6 / 1.08 = ₹45 per kg.
• The rule of alligation is then applied. The ratio is found by cross-subtracting the costs of the individual components from the mean cost price.
• The ratio of cheaper sugar to dearer sugar is (Cost of Dearer - Mean Cost) : (Mean Cost - Cost of Cheaper).
• This gives the ratio (49 - 45) : (45 - 24), which simplifies to 4:21.

Why Other Options Were Wrong

• Option A: This ratio is mathematically incorrect. It does not result from the correct application of the alligation formula using the calculated mean cost price of ₹45.
• Option C: This ratio (which simplifies to 10:1) is incorrect. This error could arise from a significant miscalculation of the mixture's cost price or an error in the subtraction step of the alligation process.
• Option D: This ratio is incorrect and does not satisfy the alligation rule for a mean price of ₹45. Such an error might occur if the selling price was mistakenly used in the alligation formula instead of the cost price.

Related Visual

Clinical Relevance

• Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Application of Mixture and Alligation principles in a Profit and Loss scenario as background academic context rather than a clinical decision trigger.
• While this is a mathematics question, the principle of alligation is directly applicable in healthcare, particularly in pharmacy and nursing.
• Pharmacists and nurses use alligation to calculate the correct amounts of two different concentrations of a solution (e.g., saline or a medication) to mix to achieve a desired intermediate concentration prescribed by a doctor.
• Accurate application of this principle is critical for patient safety, as incorrect medication concentrations can lead to ineffective treatment or adverse effects.

How to Approach the Question

• First, identify all the given values: the cost of the two ingredients, the selling price of the mixture, and the profit percentage.
• Recognize that the rule of alligation requires the cost prices of all components. The selling price of the mixture must be converted to its cost price.
• Use the formula CP = SP / (1 + Profit%/100) to find the mean cost price of the mixture.
• Set up the alligation diagram with the cost of the cheaper ingredient on the left, the dearer on the right, and the calculated mean cost in the center.
• Perform the cross-subtraction: subtract the mean cost from the dearer cost, and subtract the cheaper cost from the mean cost.
• The two results from the subtraction give the required ratio of the cheaper ingredient to the dearer ingredient. Simplify the ratio if necessary and match it with the options.

Concept Tested & Keywords

• Concept Tested: Application of Mixture and Alligation principles in a Profit and Loss scenario.
• Stem keywords: ratio, sugar, costing, mixed, selling price, profit
• Lead-in keywords: In what ratio

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