The average weight of Shivshankar, Gopesh and Reena is 97 kg. The average weight of Shivshankar and Gopesh is 93 kg, and the average weight of Gopesh and Reena is 82 kg. What is the weight (in kilograms) of Gopesh?

Introduction / Context:
This question involves determining an individual's weight when only averages of pairs and the trio are given. It is an example of solving a small system of linear equations that arises naturally from average relationships.

Given Data / Assumptions:

• The average weight of Shivshankar (S), Gopesh (G) and Reena (R) is 97 kg.
• The average weight of S and G is 93 kg.
• The average weight of G and R is 82 kg.
• We need to find the weight of Gopesh, denoted G.

Concept / Approach:
We convert the averages into equations involving sums:

• (S + G + R) / 3 = 97, so S + G + R = 291.
• (S + G) / 2 = 93, so S + G = 186.
• (G + R) / 2 = 82, so G + R = 164.

We now have three linear equations in three variables (S, G, R). Subtracting and combining these equations allows us to isolate G and find its value.

Step-by-Step Solution:
Step 1: Write equations from averages. From trio average: S + G + R = 3 * 97 = 291. From S and G average: S + G = 2 * 93 = 186. From G and R average: G + R = 2 * 82 = 164. Step 2: Use S + G = 186 to express S in terms of G. S = 186 - G. Step 3: Use G + R = 164 to express R in terms of G. R = 164 - G. Step 4: Substitute S and R into S + G + R = 291. (186 - G) + G + (164 - G) = 291. Simplify: 186 - G + G + 164 - G = 291. 350 - G = 291. Step 5: Solve for G. 350 - 291 = G. G = 59 kg.

Verification / Alternative check:
If G = 59 kg, then S + G = 186, so S = 127 kg. Also G + R = 164, so R = 164 - 59 = 105 kg. Check the trio average: (S + G + R) / 3 = (127 + 59 + 105) / 3 = 291 / 3 = 97 kg, which matches the given data. Also, average of S and G is (127 + 59) / 2 = 186 / 2 = 93, and average of G and R is (59 + 105) / 2 = 164 / 2 = 82, so all conditions are satisfied.

Why Other Options Are Wrong:
If G were 72, 56 or 63, the sums S + G and G + R derived from the given averages would not remain consistent with the total S + G + R = 291. Substituting any of these values for G produces contradictions in at least one of the equations, so they cannot be correct. Only G = 59 satisfies all three equations simultaneously.

Common Pitfalls:
A common error is to assume that the weights are symmetric and guess that G equals the overall average of 97, which is not supported by the pairwise averages. Others may misinterpret averages as requiring that each pair has similar weights. The systematic approach is to always convert averages to equations involving sums and then solve the resulting system carefully.

Final Answer:
The weight of Gopesh is 59 kg.