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 (CP) = 20%
Concentration of Copper in Alloy Q (CQ) = 60%
Target concentration of Copper in Alloy R (CR): 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.
0.20x+0.60y=0.50(x+y)
0.20x+0.60y=0.50x+0.50y
0.10y=0.30x
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 [(3×20)+(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 [(2×20)+(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 [(3×20)+(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.