The average weight of P, Q and R is 45 kg. The average weight of P and Q is 36.5 kg and the average weight of Q and R is 52 kg. What is the weight (in kg) of Q?

Introduction / Context:
This problem deals with averages of overlapping groups. You are given the average weight of three persons together and the averages of two different pairs. The goal is to find the weight of the common person in both pairs, which requires forming and solving linear equations.

Given Data / Assumptions:
• Persons are P, Q, and R.• Average weight of P, Q, and R together = 45 kg.• Average weight of P and Q together = 36.5 kg.• Average weight of Q and R together = 52 kg.• All weights are in kilograms.

Concept / Approach:
The average of a group multiplied by the number of members gives the total weight. Using this property, we can write equations for P + Q + R, P + Q, and Q + R. From these, we can isolate individual weights. The main idea is to subtract equations to find R and then back substitute to find Q.

Step-by-Step Solution:
Step 1: Let the weights be P, Q, and R.Step 2: From the given average of P, Q, R: (P + Q + R) / 3 = 45, so P + Q + R = 3 * 45 = 135.Step 3: Given average of P and Q: (P + Q) / 2 = 36.5, so P + Q = 2 * 36.5 = 73.Step 4: Given average of Q and R: (Q + R) / 2 = 52, so Q + R = 2 * 52 = 104.Step 5: From P + Q + R = 135 and P + Q = 73, subtract to find R: R = 135 - 73 = 62.Step 6: Use Q + R = 104 to find Q: Q + 62 = 104.Step 7: Therefore, Q = 104 - 62 = 42.

Verification / Alternative check:
Check the computed values. With Q = 42 and R = 62, then Q + R = 42 + 62 = 104, so the pair average is 104 / 2 = 52 as given. Also, P + Q = 73 gives P = 73 - 42 = 31. Now P + Q + R = 31 + 42 + 62 = 135, and 135 / 3 = 45, which matches the given average for all three. This confirms that Q = 42 kg is correct.

Why Other Options Are Wrong:
44, 46, 48: Substituting any of these values for Q will break at least one of the given average relationships. The sums P + Q, Q + R, or P + Q + R will not match the required totals.

Common Pitfalls:
Students may treat the averages independently without converting them to sums, or they may overlook that Q appears in both pair averages. Another mistake is incorrect arithmetic when subtracting totals. Writing the equations clearly and checking each step prevents such errors.

Final Answer:
The weight of Q is 42 kg.