In a class there are 32 boys and 28 girls. The average age of the boys in the class is 14 years and the average age of the girls in the class is 13 years. What is the average age of the whole class (rounded to two digits after decimal)?

### Concept & Formula This problem requires calculating a weighted average by combining two distinct groups (boys and girls) of unequal sizes. $$Weighted \ Average = \frac{(N_1 \times A_1) + (N_2 \times A_2)}{N_1 + N_2}$$ where $N$ is the number of individuals and $A$ is their respective average. ### Step-by-Step Solution * **Given:** * Boys: $N_1 = 32$, Average $A_1 = 14$ years. * Girls: $N_2 = 28$, Average $A_2 = 13$ years. * Total students = $32 + 28 = 60$. * **Calculation:** * Total age of boys = $32 \times 14 = 448$ years. * Total age of girls = $28 \times 13 = 364$ years. * Total age of the class = $448 + 364 = 812$ years. * **Average:** * Average age of the class = $\frac{812}{60}$ * Simplifying: $\frac{812}{60} = \frac{406}{30} = \frac{203}{15} \approx 13.5333...$ * **Rounding:** * Rounded to two digits after the decimal, it is 13.53. ### Exam Strategy & Shortcut Use the **Assumed Mean (Deviation) Method** to save time and prevent calculation errors. * Assume the class average is the girls' average: 13 years. * The 32 boys have an average of 14 years, which is $+1$ year higher than the assumed average. * Total surplus age from the boys = $32 \times 1 = +32$ years. * Distribute this 32-year surplus equally among all 60 students: $\frac{+32}{60} = \frac{+16}{30} = \frac{+8}{15} \approx +0.533$ years. * New Average = Assumed Average + Surplus = $13 + 0.533 = 13.53$. This completely avoids the multiplication and addition of large numbers like 448 and 364. ### Common Pitfall A very common mistake is calculating the simple arithmetic mean of the two averages: $\frac{14 + 13}{2} = 13.50$ (Option c). This is incorrect because it ignores the "weight" of the groups. Since there are more boys (32) than girls (28), the true average will pull slightly closer to the boys' average of 14. 13.53 makes logical sense. ### Final Answer **Therefore, the correct answer is 13.53.**