A group of boys has an average weight of 36 kg. One boy weighing 42 kg leaves the group, and another boy weighing 30 kg joins the group in his place. If the new average weight of the group becomes 35.7 kg, how many boys are there in the group?

Introduction / Context:
This problem is a classic example of how the average of a group changes when one member leaves and another joins. We are told the original average weight of a group of boys, the weight of the boy who leaves, the weight of the boy who joins, and the new average. From this information we must determine the original number of boys in the group. The question tests understanding of average, total, and how small changes affect group statistics.

Given Data / Assumptions:

• Original average weight of the group = 36 kg.
• One boy with weight 42 kg leaves the group.
• A new boy with weight 30 kg joins the group.
• New average weight becomes 35.7 kg.
• The number of boys remains the same, only membership changes.
• We must find the number of boys in the group, call it n.

Concept / Approach:
Average weight is equal to total weight divided by number of boys. Initially, total weight is 36 * n. When the boy of 42 kg leaves, the total decreases by 42. When the boy of 30 kg joins, the total increases by 30. The new total is therefore 36n − 42 + 30. Dividing this new total by n gives the new average 35.7. Setting up this equation and solving for n will give the group size.

Step-by-Step Solution:
Step 1: Let the original number of boys be n. Original total weight = 36n. Step 2: Account for the boy leaving and the boy joining. When a 42 kg boy leaves, new total = 36n − 42. When a 30 kg boy joins, updated total = 36n − 42 + 30 = 36n − 12. Step 3: Use the new average. New average = 35.7 = (36n − 12) / n. Step 4: Multiply both sides by n. 36n − 12 = 35.7n. Step 5: Bring like terms together. 36n − 35.7n = 12. 0.3n = 12. Step 6: Solve for n. n = 12 / 0.3 = 40.
Verification / Alternative check:
Original total weight = 36 * 40 = 1440 kg. After the 42 kg boy leaves, total = 1440 − 42 = 1398 kg. After the 30 kg boy joins, total = 1398 + 30 = 1428 kg. New average = 1428 / 40 = 35.7 kg, matching the given condition.
Why Other Options Are Wrong:
If n were 30, 32, 56 or 28, substituting into the equation would not give a new average of exactly 35.7 kg. Each incorrect n would yield a different total and hence an inconsistent average, so these options cannot be correct.
Common Pitfalls:
A common mistake is to treat the average change as if it directly adds or subtracts a fixed number of kilograms from each boy, without using the total-weight equation. Another typical error is mishandling decimal arithmetic with 35.7, which can lead to incorrect values for 0.3n.
Final Answer:
There are 40 boys in the group.