The average weight of $A, B$ and $C$ is 45 kg. If the average weight of $A$ and $B$ be 40 kg and that of $B$ and $C$ be 43 kg, then the weight of $B$ is:

### Concept & Logic When you have the total sum of a complete group, and the sums of two subgroups that overlap on exactly one member, you can find the overlapping member by adding the two subgroups and subtracting the total group. $$\text{Overlapping Member} = (\text{Sum of Group 1} + \text{Sum of Group 2}) - \text{Total Sum}$$ ### Step-by-Step Solution * **Given:** * Average of $A, B, C = 45$ kg * Average of $A, B = 40$ kg * Average of $B, C = 43$ kg * **Calculation:** 1. Find the total weight of all three people ($A, B, C$): $$A + B + C = 3 \times 45 = 135 \text{ kg}$$ 2. Find the total weight of the first subgroup ($A, B$): $$A + B = 2 \times 40 = 80 \text{ kg}$$ 3. Find the total weight of the second subgroup ($B, C$): $$B + C = 2 \times 43 = 86 \text{ kg}$$ 4. Add the two subgroups together. Notice this counts $B$ twice: $$(A + B) + (B + C) = A + 2B + C$$ $$80 + 86 = 166 \text{ kg}$$ 5. Subtract the true total sum from this combined sum to isolate $B$: $$(A + 2B + C) - (A + B + C) = 166 - 135 = 31 \text{ kg}$$ ### Exam Strategy & Shortcut **Net Deviation Method:** Compare the subgroup averages to the main group average of 45 kg. Subgroup 1 ($A, B$): Deviation is $-5$ kg per person. Total deviation = $2 \times (-5) = -10$ kg. Subgroup 2 ($B, C$): Deviation is $-2$ kg per person. Total deviation = $2 \times (-2) = -4$ kg. Total combined deviation = $-10 + (-4) = -14$ kg. Because $B$ is the overlapping element tying these averages together, $B$ must be exactly this total deviation away from the main average to balance the scales. Weight of $B = 45 - 14 = 31$ kg. ### Common Pitfall A frequent mistake is attempting to solve for $A$ and $C$ individually before finding $B$. While possible (e.g., finding $C$ by subtracting $A+B$ from the total), it adds unnecessary steps. Combining the subgroups and subtracting the total is the most direct mathematical route. ### Final Answer **Therefore, the correct answer is 31 kg.**