Shortcut Trick
Weight ratio of Alloy 1 and Alloy 2 = 14 : 21 = 2 : 3.
Sum of internal ratio parts for both alloys is equal (3 + 4 = 7 and 2 + 5 = 7).
Multiply the internal Copper and Gold ratios directly by the weight ratio parts.
Total Copper = (3 × 2) + (2 × 3) = 6 + 6 = 12.
Total Gold = (4 × 2) + (5 × 3) = 8 + 15 = 23.
∴ The correct answer is 12 : 23.
Alternate Method
Alloy 1: 14 kg
Copper (Cu) = 6 kg
Gold (Au) = 8 kg
Alloy 2: 21 kg
Copper (Cu) = 6 kg
Gold (Au) = 15 kg
New Mixture: 35 kg
Total Cu = 6 + 6 = 12 kg
Total Au = 8 + 15 = 23 kg
Given:
Ratio of Copper and Gold in Alloy 1 = 3 : 4
Total weight of Alloy 1 = 14 kg
Ratio of Copper and Gold in Alloy 2 = 2 : 5
Total weight of Alloy 2 = 21 kg
Calculations:
⇒ Total ratio parts in Alloy 1 = 3 + 4 = 7 parts
⇒ Quantity of Copper in Alloy 1 = (3 ÷ 7) × 14 = 6 kg
⇒ Quantity of Gold in Alloy 1 = (4 ÷ 7) × 14 = 8 kg
⇒ Total ratio parts in Alloy 2 = 2 + 5 = 7 parts
⇒ Quantity of Copper in Alloy 2 = (2 ÷ 7) × 21 = 6 kg
⇒ Quantity of Gold in Alloy 2 = (5 ÷ 7) × 21 = 15 kg
⇒ Total Copper in the new mixture = 6 kg + 6 kg = 12 kg
⇒ Total Gold in the new mixture = 8 kg + 15 kg = 23 kg
⇒ Required ratio of Copper to Gold = 12 : 23
∴ The correct answer is 12 : 23.
Additional Information
Equalising Ratios for Mixtures
When mixing two mixtures given in ratios, it is easiest to make the sum of the internal ratio parts equal first (if they are not already) before multiplying with the mixing weights. This avoids complex fraction calculations.
Alligation Method
A foundational rule that enables the calculation of the final ratio in which two or more ingredients of differing concentrations must be mixed to produce a mixture of a desired average concentration.
Compound Ratio
If two or more simple ratios are multiplied together, the resulting expression is called a compound ratio. For ratios a : b and c : d, the final compound ratio is expressed as (a × c) : (b × d).