An alloy P contains 20% copper and 80% zinc by weight. Another alloy Q contains 60% copper and 40% zinc by weight. A third alloy R is to be prepared from P and Q so that it contains equal amount of copper and zinc. In what ratio, amounts of P and Q be mixed in order to get R?

The correct option is A (1 : 3).

Why the correct option is correct: This problem falls under the Quantitative Aptitude section of the UPSC CSAT (General Studies Paper-II) and is best solved using the Rule of Alligation. The rule states that when two ingredients are mixed, the ratio of their quantities is inversely proportional to the differences between their individual concentrations and the desired mean concentration.

Let us calculate based on the concentration of copper:

• Alloy P contains 20% copper.
• Alloy Q contains 60% copper.
• Alloy R must contain an equal amount of copper and zinc, meaning its target copper concentration is 50%.

Applying the Rule of Alligation formula: Ratio (P : Q) = (Concentration in Q - Mean) : (Mean - Concentration in P) Ratio (P : Q) = (60% - 50%) : (50% - 20%) Ratio (P : Q) = 10 : 30 = 1 : 3. Thus, 1 part of P mixed with 3 parts of Q will correctly yield Alloy R.

Why the incorrect options are wrong:

• Option B (3 : 1) is incorrect: Mixing 3 parts of P and 1 part of Q yields a weighted copper concentration of (3 × 20 + 1 × 60) / 4 = 120 / 4 = 30%, which is well below the target 50%.
• Option C (2 : 3) is incorrect: Mixing 2 parts of P and 3 parts of Q yields a copper concentration of (2 × 20 + 3 × 60) / 5 = 220 / 5 = 44%.
• Option D (3 : 2) is incorrect: Mixing 3 parts of P and 2 parts of Q yields a copper concentration of (3 × 20 + 2 × 60) / 5 = 180 / 5 = 36%.

Takeaway: For rapid CSAT calculations, visualize the Alligation Cross Method: Subtract the mean value from the higher concentration and assign the result to the lower ingredient's ratio. Subtract the lower concentration from the mean and assign the result to the higher ingredient's ratio.