Two varieties of sugar cost ₹18 per kg and ₹24 per kg. In what ratio (cheaper sugar : costlier sugar) should they be mixed so that the resulting mixture costs ₹20 per kg (assuming no loss and simple weighted average pricing)?
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A2:1
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B1:2
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C3:1
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D1:3
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E5:4
Answer
Correct Answer: 2:1
Explanation
Introduction / Context:This is a standard alligation problem used for mixing two items with different costs to obtain a target average cost. The method ensures the final mixture cost is a weighted average of the two prices. The ratio depends on how far the target price lies from each original price.
Given Data / Assumptions:
- Cheaper sugar price = ₹18 per kg
- Costlier sugar price = ₹24 per kg
- Target mixture price = ₹20 per kg
- No wastage, so mixture price is a weighted average
Concept / Approach:Alligation rule: cheaper : costlier = (costlier - mean) : (mean - cheaper).
Step-by-Step Solution:
Step 1: Difference between costlier and mean = 24 - 20 = 4 Step 2: Difference between mean and cheaper = 20 - 18 = 2 Step 3: Required ratio (cheaper : costlier) = 4 : 2 Step 4: Simplify 4:2 by dividing by 2 => 2:1Verification / Alternative check:If we take 2 kg at ₹18 and 1 kg at ₹24, total cost = 2*18 + 1*24 = 36 + 24 = 60. Total quantity = 3 kg. Average = 60/3 = ₹20 per kg. Verified.
Why Other Options Are Wrong:
1:2: would overweight ₹24 and raise average above ₹20. 3:1: would overweight ₹18 too much and drop average below ₹20. 1:3: would make average close to ₹24, far above ₹20. 5:4: yields average = (5*18 + 4*24)/9 = (90 + 96)/9 = 20.67, not ₹20.Common Pitfalls:Students sometimes invert the ratio by mistake (using 2:4 instead of 4:2). Another error is taking simple difference of prices (24-18) and guessing the ratio without considering the target mean value.
Final Answer:2:1