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…

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

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 = $(60 - 50) : (50 - 20)$(60−50):(50−20)
Ratio = $10 : 30$10:30

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

Alternatively, using the weighted average formula:
Let x be the amount of P and y be the amount of Q.
$0.20x + 0.60y = 0.50(x + y)$0.20x+0.60y=0.50(x+y)
$0.20x + 0.60y = 0.50x + 0.50y$0.20x+0.60y=0.50x+0.50y
$0.10y = 0.30x$0.10y=0.30x
$x/y = 0.10/0.30 = 1/3$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 $[(3 \times 20) + (1 \times 60)] / 4 = 120 / 4 = 30\%$[(3×20)+(1×60)]/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 $[(2 \times 20) + (3 \times 60)] / 5 = (40 + 180) / 5 = 220 / 5 = 44\%$[(2×20)+(3×60)]/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 $[(3 \times 20) + (2 \times 60)] / 5 = (60 + 120) / 5 = 180 / 5 = 36\%$[(3×20)+(2×60)]/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.