$76\%$ of the students in a school are boys. If the number of girls is $204$, then the total number of students is

### Concept & Logic In percentage problems involving a binary split (like boys and girls), the percentages of both groups must add up to $100\%$. $$\text{Total Percentage} = \% \text{ Boys} + \% \text{ Girls} = 100\%$$ ### Step-by-Step Solution **Given:** * Percentage of boys = $76\%$ * Number of girls = $204$ **Calculation:** * Find the percentage of girls in the school: $$\% \text{ of Girls} = 100\% - 76\% = 24\%$$ * Let the total number of students be $x$. * We know that $24\%$ of the total students are girls, which equals $204$. $$24\% \text{ of } x = 204$$ $$\frac{24}{100} \times x = 204$$ * Isolate $x$ to find the total number of students: $$x = \frac{204 \times 100}{24}$$ * Simplify the fraction (divide by $12$): $$x = 17 \times 50 = 850$$ ### Exam Strategy & Shortcut If $24\% = 204$, then $1\% = \frac{204}{24} = 8.5$. The total number of students is $100\%$, so multiply by $100$: $$8.5 \times 100 = 850$$ This unitary method bypasses setting up a full algebraic equation and is much faster for mental math. ### Common Pitfall A frequent mistake is calculating $76\%$ of $204$, confusing the given part (girls) with the whole (total students). Always align the percentage with its corresponding absolute value. ### Final Answer **Therefore, the correct answer is 850.**