In an examination, $65\%$ of the total examinees passed. If the number of failures is $420$, the total number of examinees is

### Concept & Logic Similar to the previous problem, the sum of passed and failed examinees equals $100\%$ of the total examinees. Find the percentage of failures to match the given absolute number. ### Step-by-Step Solution **Given:** * Percentage of passed examinees = $65\%$ * Number of failures = $420$ **Calculation:** * Determine the percentage of students who failed: $$\% \text{ Failures} = 100\% - 65\% = 35\%$$ * Let the total number of examinees be $x$. $$35\% \text{ of } x = 420$$ $$\frac{35}{100} \times x = 420$$ * Solve for $x$: $$x = \frac{420 \times 100}{35}$$ * Simplify by dividing $420$ by $35$: $$x = 12 \times 100 = 1200$$ ### Exam Strategy & Shortcut Use the unitary method directly. $35\%$ corresponds to $420$. Notice that $35 \times 2 = 70$, and $70 \times 6 = 420$. So, $35 \times 12 = 420$. This means $1\% = 12$. Therefore, $100\% = 1200$. ### Common Pitfall Students sometimes mistake the $420$ failures as the passed students and equate $65\% = 420$. Carefully read whether the absolute number given represents the "passed" or "failed" group. ### Final Answer **Therefore, the correct answer is 1200.**