The average weight of P, Q and R together is 71 kg. The average weight of P and Q is 66 kg, while the average weight of Q and R is 76.5 kg. What is the weight (in kg) of Q?

Introduction / Context:
This question uses averages of overlapping groups to determine individual values. Three people P, Q and R have certain average weights in different pairs and as a whole. From these overlapping averages, we need to deduce the exact weight of one person, Q. This type of problem relies on forming and solving simple linear equations.

Given Data / Assumptions:

• Average weight of P, Q and R together = 71 kg.
• Average weight of P and Q = 66 kg.
• Average weight of Q and R = 76.5 kg.
• We are asked to find the weight of Q.

Concept / Approach:
If the average of some people is known, we can multiply by the number of people to get the total weight. Let the weights of P, Q and R be p, q and r respectively. From the given averages, we form three equations: p + q + r, p + q and q + r. These equations can be combined to eliminate p and r, leaving us with q. This is a classic system of linear equations problem.

Step-by-Step Solution:
Let weights be: P = p, Q = q, R = r. Average of P, Q, R = 71 kg. So, p + q + r = 71 * 3 = 213. (1) Average of P and Q = 66 kg. So, p + q = 66 * 2 = 132. (2) Average of Q and R = 76.5 kg. So, q + r = 76.5 * 2 = 153. (3) Add equations (2) and (3): (p + q) + (q + r) = 132 + 153. p + 2q + r = 285. (4) From (1), p + q + r = 213. Subtract (1) from (4): (p + 2q + r) - (p + q + r) = 285 - 213. q = 72.
Verification / Alternative check:
If q = 72, then from (2): p + 72 = 132, so p = 60. From (3): 72 + r = 153, so r = 81. Check total: p + q + r = 60 + 72 + 81 = 213. Average = 213 / 3 = 71, which matches the given value.
Why Other Options Are Wrong:
Option A (60): This is actually the weight of P, not Q. Option C (81): This is the weight of R after solving, not Q. Option D (75): Does not satisfy all three average conditions. Option B (72): Correct, as shown by the equations and verification.
Common Pitfalls:
Students may confuse which person corresponds to which equation and misassign values. Some try to average the given averages directly rather than forming proper equations. Arithmetic mistakes in multiplying averages by the number of persons can also lead to wrong totals and thus wrong answers.
Final Answer:
The weight of Q is 72 kg.