Two alloys P and Q are prepared by mixing tin and lead in the ratios 12 : 5 and 4 : 3 respectively. If equal quantities (by weight) of alloys P and Q are melted together to form a new alloy R, what is the ratio of tin to lead in alloy R?

Introduction / Context:
Alligation and mixture problems involving alloys are standard in aptitude exams. Here, we are given two alloys P and Q made of tin and lead in different ratios. Equal masses of these alloys are melted together to create a third alloy R. The question asks for the resulting ratio of tin to lead in R. This tests understanding of how to convert ratios into fractional compositions and how to combine them correctly when mixing equal quantities.

Given Data / Assumptions:

• Alloy P has tin : lead = 12 : 5.
• Alloy Q has tin : lead = 4 : 3.
• Equal weights of alloy P and alloy Q are mixed.
• Alloys are homogeneous, so the ratio applies uniformly to each unit mass.

Concept / Approach:
The key idea is to express the composition of each alloy in terms of fractional parts of tin and lead. Since equal weights of P and Q are mixed, we can assume 1 unit weight of each alloy for simplicity. We then calculate how much tin and how much lead comes from each alloy, add them, and finally express the combined tin and lead as a simplified ratio. This is a direct application of ratio and fraction manipulation.

Step-by-Step Solution:

Verification / Alternative check:
To check, ensure the ratio is in lowest terms. The greatest common divisor of 76 and 43 is 1 (43 is a prime and does not divide 76), so 76 : 43 is already the simplest form. Also, the tin fraction 152/238 and lead fraction 86/238 simplify to 76/119 and 43/119, reinforcing the same ratio 76 : 43.

Why Other Options Are Wrong:

• 84 : 65, 57 : 35 and 78 : 47 do not match the computed combined tin and lead amounts and do not come from correctly summing the fractional contributions.
• They are simply distractors and do not satisfy the arithmetic derived from the given ratios.

Common Pitfalls:
Students may incorrectly add ratios directly like 12 : 5 with 4 : 3, which is not valid. Another mistake is to mix up numerator and denominator when converting ratios into fractions. Some learners also forget that equal weights are mixed and instead assume equal volumes without standardizing, though here equal quantities by weight are explicitly mentioned, making it straightforward.

Final Answer:
The ratio of tin to lead in alloy R is 76 : 43.