The correct option is (a) - $1 : 3$ .

[as per provisional answerkey]

Solution

This is a mixture and alligation problem. We need to find the ratio in which Alloy P and Alloy Q should be mixed to achieve a specific concentration in Alloy R.

Step 1: Identify the concentration of one component (e.g., Copper) in each alloy.
Concentration of Copper in Alloy P (C_P) = 20%
Concentration of Copper in Alloy Q (C_Q) = 60%
Target concentration of Copper in Alloy R (C_R): Since R must contain equal amounts of copper and zinc, the concentration of copper must be 50%.

Step 2: Apply the Alligation Rule.
The ratio of the quantities (Quantity of P : Quantity of Q) is given by:
(Concentration of Q - Target Concentration) : (Target Concentration - Concentration of P)
Ratio = $(60 - 50) : (50 - 20)$
Ratio = $10 : 30$

Step 3: Simplify the ratio.
Ratio = $1 : 3$

Alternatively, using the weighted average formula:
Let x be the amount of P and y be the amount of Q.
$0.20 x + 0.60 y = 0.50 (x + y)$
$0.20 x + 0.60 y = 0.50 x + 0.50 y$
$0.10 y = 0.30 x$
$x / y = 0.10 / 0.30 = 1 / 3$

Why the other options are incorrect

Option (b) - 3 : 1: This ratio would result in a copper concentration of $[(3 \times 20) + (1 \times 60)] / 4 = 120 / 4 = 30$ , which is not the required 50%. This is the inverse of the correct ratio.
Option (c) - 2 : 3: This ratio would result in a copper concentration of $[(2 \times 20) + (3 \times 60)] / 5 = (40 + 180) / 5 = 220 / 5 = 44$ , which does not satisfy the "equal amount" condition.
Option (d) - 3 : 2: This ratio would result in a copper concentration of $[(3 \times 20) + (2 \times 60)] / 5 = (60 + 120) / 5 = 180 / 5 = 36$ , which is incorrect.

Key Concept

The Rule of Alligation, which states that the ratio of the weights of two items mixed to achieve a mean price/concentration is inversely proportional to the differences between their individual concentrations and the mean concentration.