In a school with 600 students, the average age of the boys is 12 years and that of the girls is 11 years. If the average age of the school is 11 years 9 months, then the number of girls in the school is:

### Concept & Logic This problem requires finding the ratio of two subgroups (boys and girls) when their individual averages and the combined average are known. This is a perfect scenario for the **Rule of Alligation**. To avoid messy fractions, it is highly recommended to convert all ages into months first. ### Step-by-Step Solution * **Unit Conversion:** Average age of Boys = $12$ years = $12 \times 12 = 144$ months. Average age of Girls = $11$ years = $11 \times 12 = 132$ months. Combined average age = $11$ years $9$ months = $(11 \times 12) + 9 = 132 + 9 = 141$ months. * **Applying Alligation:** Boys ($144$) ------------- Girls ($132$) \ / Mean ($141$) / \ $(141 - 132)$ ----------- $(144 - 141)$ Ratio $\rightarrow 9 : 3$ Simplifying, the ratio of Boys to Girls is $3 : 1$. * **Calculation:** Total ratio parts = $3 + 1 = 4$ parts. Total students = $600$. Value of 1 part = $600 / 4 = 150$. Number of girls ($1$ part) = $150$. ### Exam Strategy & Shortcut If you prefer decimals over months, convert $9$ months to a fraction of a year: $9/12 = 3/4 = 0.75$ years. Combined average = $11.75$ years. Alligation Ratio = $(11.75 - 11) : (12 - 11.75) = 0.75 : 0.25 = 3 : 1$. The Girls constitute $1/4$ of the total school. $600 / 4 = 150$. Both methods are mathematically identical, but picking the one you are most comfortable calculating mentally saves crucial seconds. ### Common Pitfall The most frequent error is mathematically translating "11 years 9 months" as "$11.9$ years". There are 12 months in a year, not 10. Always use $11 + 9/12$ ($11.75$) or convert entirely to months. ### Final Answer **Therefore, the correct answer is 150.**