In a group of $120$ people, one-fifth are men, one-fourth are women and the rest children. The average age of women is five-sixth of the average age of men. Average age of children is one-fourth of the average age of men. If average age of men is $60$ years, what is the average age of the group?

Concept & Strategy To find the average age of a mixed group, we use the weighted average formula. We need to determine the exact number of people in each subgroup (men, women, children) and their respective average ages, then calculate the total age sum and divide by the total population. $$ \text{Weighted Average} = \frac{n_1A_1 + n_2A_2 + n_3A_3}{n_1 + n_2 + n_3} $$ Step-by-Step Solution * Given: Total people = $120$. Average age of men = $60$ years. * Calculate population sizes: - Men = $1/5$ of $120 = 24$. - Women = $1/4$ of $120 = 30$. - Children = Rest = $120 - (24 + 30) = 120 - 54 = 66$. * Calculate group average ages based on the men's average ($60$): - Women = $(5/6) \times 60 = 50$ years. - Children = $(1/4) \times 60 = 15$ years. * Calculate the total sum of all ages: - Total men's age = $24 \times 60 = 1440$. - Total women's age = $30 \times 50 = 1500$. - Total children's age = $66 \times 15 = 990$. * Total age of the group = $1440 + 1500 + 990 = 3930$. * Calculate group average = $3930 / 120$. Simplify by dropping the zero: $393 / 12 = 131 / 4 = 32.75$. Exam Strategy & Shortcut Instead of large multiplications, you can use the ratio of the population to simplify calculations. The ratio of Men : Women : Children is $24 : 30 : 66 \implies 4 : 5 : 11$. The total parts are $20$. Weighted average = $\frac{(4 \times 60) + (5 \times 50) + (11 \times 15)}{20} = \frac{240 + 250 + 165}{20} = \frac{655}{20} = 32.75$. This keeps numbers smaller and saves time. Common Pitfall A frequent trap is calculating the simple unweighted average of the three ages: $(60 + 50 + 15) / 3 = 125 / 3 \approx 41.6$. This completely ignores the vastly different population sizes (there are nearly three times as many children as men). Always use a weighted average. Final Answer **Therefore, the correct answer is 32.75.**