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. If the new average weight becomes 35.7 kg, how many boys are there in the group?

Introduction / Context:
This question is a classic average and replacement problem. Removing a heavier boy and adding a lighter one decreases the average weight slightly. From the size of this decrease we can compute how many boys are in the group.

Given Data / Assumptions:
- Initial average weight of the group = 36 kg. - One boy of weight 42 kg leaves. - A new boy of weight 30 kg joins. - New average weight of the group = 35.7 kg. - Let the initial number of boys be n.

Concept / Approach:
Initially, total weight equals 36n. After the change, the total weight becomes 36n minus 42 plus 30. The number of boys remains n, so the new average is (36n - 12) / n, which is given as 35.7. Solving this equation for n gives the size of the group. This approach directly uses the definition of average as total divided by count.

Step-by-Step Solution:
Step 1: Initial total weight = 36n. Step 2: When the 42 kg boy leaves, total becomes 36n - 42. Step 3: A 30 kg boy joins, so new total weight = 36n - 42 + 30 = 36n - 12. Step 4: New average weight = (36n - 12) / n. Step 5: Given new average = 35.7, so (36n - 12) / n = 35.7. Step 6: Multiply both sides by n: 36n - 12 = 35.7n. Step 7: Rearrange: 36n - 35.7n = 12, so 0.3n = 12 and n = 12 / 0.3 = 40.

Verification / Alternative check:
For n = 40, initial total weight = 36 * 40 = 1440 kg. Removing 42 kg gives 1398 kg. Adding 30 kg yields 1428 kg. New average = 1428 / 40 = 35.7 kg, which matches the question. This confirms that the calculation is correct.

Why Other Options Are Wrong:
- 30, 32, 36 and 56 do not satisfy the equation (36n - 12) / n = 35.7. - Substituting any of these values leads to either a different new average or a non integer group effect.

Common Pitfalls:
Some candidates replace the average directly without forming an equation, or incorrectly subtract both 42 and 30 from the total. Others forget that the number of boys remains unchanged. Always express the new situation in terms of the same variable n and then solve systematically.

Final Answer:
There are 40 boys in the group.