$37\frac{1}{2}\%$ of the candidates in an examination were girls, $75\%$ of the boys and $62\frac{1}{2}\%$ of the girls passed and $342$ girls failed. The number of boys failed was:

### Concept & Strategy This problem tests standard percentage manipulation, but the key to speed is converting clunky percentages into simple fractions. Memorizing standard percentage-to-fraction conversions allows you to work with ratios instead of decimals, drastically reducing calculation time. ### Step-by-Step Solution * **Given:** Percentage of girls = $37.5\% = \frac{3}{8}$ of the total candidates. Percentage of boys = $100\% - 37.5\% = 62.5\% = \frac{5}{8}$ of the total candidates. Passed girls = $62.5\% = \frac{5}{8}$ of total girls. Passed boys = $75\% = \frac{3}{4}$ of total boys. Failed girls = $342$. * **Calculation / Deduction:** First, find the total number of girls. Since $62.5\%$ of girls passed, the percentage of girls who failed is $100\% - 62.5\% = 37.5\% = \frac{3}{8}$ of all girls. Let the total number of girls be $G$. $\frac{3}{8} \times G = 342$ $G = 342 \times \frac{8}{3} = 114 \times 8 = 912$. So, there are $912$ girls in total. Next, find the total number of boys. The ratio of girls to boys is $37.5\% : 62.5\% = 3 : 5$. Since $3$ ratio units $= 912$, $1$ ratio unit $= 304$. Total boys ($5$ ratio units) = $5 \times 304 = 1520$. Finally, calculate the number of failed boys. Since $75\%$ of the boys passed, $25\%$ ($\frac{1}{4}$) of the boys failed. Failed boys = $\frac{1}{4} \times 1520 = 380$. ### Exam Strategy & Shortcut Immediately convert standard percentages to fractions: $37.5\% \rightarrow \frac{3}{8}$, $62.5\% \rightarrow \frac{5}{8}$, $75\% \rightarrow \frac{3}{4}$. Use the ratio of Girls:Boys directly as $3:5$. Failed girls = $\frac{3}{8}$ of total girls. So, $\frac{3}{8}G = 342 \Rightarrow G = 912$. If $3$ units (girls) $= 912$, then $5$ units (boys) $= 1520$. Failed boys = $\frac{1}{4}$ of $1520 = 380$. ### Common Pitfall A major trap is calculating the total number of candidates first ($x = 2432$). While it gives the correct answer eventually, it introduces larger numbers into your arithmetic, increasing the chance of calculation errors and wasting valuable time. Jump straight to the ratios! ### Final Answer **Therefore, the correct answer is 380.**