The average weight of some children in a group is 45 kg. When 10 children of average weight 40 kg leave the group or 15 children of average weight 50 kg join the group, the average weight of children in both the cases is the same. The number of children, initially, in the group lies between:

Correct option is A

Given:

​Initial average weight = 45 kg
Case 1: 10 children (average weight = 40 kg) leave
Case 2: 15 children (average weight = 50 kg) join
In both cases, the new average remains the same

Formula:

$$\text{Total weight} = \text{Average} \times \text{Number of children} \\\ \\\text{New average after change} = \frac{\text{New total weight}}{\text{New number of children}}$$​​

Solution:

​$$\text{Initial total weight} = 45x \\\text{Case 1: 10 children leave} \Rightarrow \text{weight removed} = 10 \times 40 = 400 \\\ \\\text{New average} = \frac{45x - 400}{x - 10}\\[10pt]\text{Case 2: 15 children join} \Rightarrow \text{weight added} = 15 \times 50 = 750 \\\text{New average} = \frac{45x + 750}{x + 15}\\[10pt]\text{Equating both expressions:} \\\frac{45x - 400}{x - 10} = \frac{45x + 750}{x + 15}\\[10pt]\text{Cross-multiplying:} \\(45x - 400)(x + 15) = (45x + 750)(x - 10)\\[10pt]\text{Expanding LHS:} \\45x(x + 15) - 400(x + 15) = 45x^2 + 675x - 400x - 6000 = 45x^2 + 275x - 6000\\[10pt]\text{Expanding RHS:} \\45x(x - 10) + 750(x - 10) = 45x^2 - 450x + 750x - 7500 = 45x^2 + 300x - 7500\\[10pt]\text{Equating:} \\45x^2 + 275x - 6000 = 45x^2 + 300x - 7500 \\275x - 6000 = 300x - 7500 \\1500 = 25x \Rightarrow x = 60$$

So, initial number of children = 60
Hence, it lies between 55 and 65

Final Answer: (A) 55 and 65​​