In two successive years 100 and 75 students of a school appeared at the final examination. Respectively 75% and 60% of them passed. The average rate of pass is

### Concept & Formula This is a problem based on the **Weighted Average** of percentages. Instead of averaging the percentages directly (which would be incorrect since the base student numbers differ), we must find the total number of passed students and divide it by the total number of students. $$ \text{Overall Pass \%} = \left( \frac{\text{Total Passed Students}}{\text{Total Students}} \right) \times 100 $$ ### Step-by-Step Solution * **Given**: * Year 1: 100 students, 75% passed. * Year 2: 75 students, 60% passed. * **Calculation**: * Passed in Year 1 = $75\%$ of $100 = 75$ * Passed in Year 2 = $60\%$ of $75 = \frac{3}{5} \times 75 = 45$ * Total students = $100 + 75 = 175$ * Total passed = $75 + 45 = 120$ * Overall Average Rate = $\left( \frac{120}{175} \right) \times 100\%$ * **Simplifying**: * Divide numerator and denominator by 5: $\left( \frac{24}{35} \right) \times 100 = \frac{2400}{35}$ * Divide by 5 again: $\frac{480}{7} = 68 \frac{4}{7}\%$ ### Exam Strategy & Shortcut Use the **Ratio Method** for weights to keep numbers small. The ratio of students is $100 : 75 = 4 : 3$. Total parts = 7. Weighted sum = $(4 \times 75\%) + (3 \times 60\%) = 300\% + 180\% = 480\%$. Average pass rate = $\frac{480\%}{7} = 68 \frac{4}{7}\%$. This significantly cuts down calculation time. ### Common Pitfall The most common trap is calculating the simple average of the two percentages: $\frac{75 + 60}{2} = 67.5\%$. This is mathematically invalid because the sample sizes (100 and 75) are different. Always find absolute totals first. ### Final Answer **Therefore, the correct answer is $68 \frac{4}{7}\%$.**