The average weight of 16 boys in a class is 50.25 kgs and that of the remaining 8 boys is 45.15 kgs. Find the average weight of all the boys in the class.

### Concept & Formula This problem is solved using the concept of **Weighted Average**. When a group is split into two or more subgroups with different averages, the combined average is calculated by finding the total sum of all values and dividing by the total number of individuals. $$Weighted \ Average = \frac{N_1A_1 + N_2A_2}{N_1 + N_2}$$ ### Step-by-Step Solution * **Given:** * Group 1: $N_1 = 16$ boys, Average weight $A_1 = 50.25$ kgs. * Group 2: $N_2 = 8$ boys, Average weight $A_2 = 45.15$ kgs. * **Total Weight Calculation:** * Total weight of the first 16 boys = $16 \times 50.25 = 804$ kgs. * Total weight of the remaining 8 boys = $8 \times 45.15 = 361.2$ kgs. * **Combined Average Calculation:** * Total weight of all boys = $804 + 361.2 = 1165.2$ kgs. * Total number of boys = $16 + 8 = 24$. * Overall average weight = $\frac{1165.2}{24} = 48.55$ kgs. ### Exam Strategy & Shortcut Use the **Ratio (Deviation) Method** to minimize calculation time and avoid multiplying large decimals. 1. Find the ratio of the number of boys: $16 : 8 \rightarrow 2 : 1$. 2. Assume the smaller average ($45.15$) as the base. 3. The larger average ($50.25$) is $+5.1$ more than the base. 4. Calculate the total excess based on the ratio weights: $2 \text{ parts} \times 5.1 = 10.2$. 5. Distribute this excess over the total ratio parts ($2 + 1 = 3$ parts): $\frac{10.2}{3} = 3.4$. 6. Add this to the base average: $45.15 + 3.4 = 48.55$ kgs. ### Common Pitfall Students often waste time executing the full decimal multiplications ($16 \times 50.25$ and $8 \times 45.15$) manually under time pressure, which heavily increases the risk of simple arithmetic errors. Always look for a ratio to simplify the weights first. ### Final Answer **Therefore, the correct answer is 48.55 kgs.**